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Mechanical behavior of nano-copper interconnects
subjected to thermal loading
Ana Mónica Carvalho Fidalgo
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisors: Prof. Paulo Jorge Matos Fernandes Martins Ferreira
Prof. Augusto Manuel Moura Moita de Deus
Examination Committee
Chairperson: Prof. Fernando José Parracho Lau
Supervisor: Prof. Augusto Manuel Moura Moita de Deus
Member of the Committee: Prof. Nuno Miguel Rosa Pereira Silvestre
May 2019
ii
I declare that this document is an original work of my own authorship and that it
fulfils all the requirements of the Code of Conduct and Good Practices of the
Universidade de Lisboa.
iii
Dedicated to my mother and father for their endless love and spirit of sacrifice.
iv
Acknowledgements
I would like to express my deepest gratitude to Professor Augusto Moita de Deus and Professor
Paulo J. Ferreira, my supervisors, for entrusting a very Materials-oriented thesis to an Aerospace
student, for their unceasing enthusiasm, patient guidance, active encouragement and for their rich,
rigorous and constructive critiques of my work.
A very special thanks to the staff at INL – International Iberian Nanotechnology Laboratory, for the
warmest welcome during my Summer Internship, in which I was able to work on my thesis while
collecting experience in a research environment. I am particularly grateful to Professor Paulo J. Ferreira,
Dr. Cristiana Alves and Dr. Sebastian Calderon Velasco for their hospitality and assistance during my
stay at INL. I would like to thank them for letting me observe their work and answering all my questions
regarding electron microscopy techniques and equipment.
I must acknowledge CeFEMA - Center of Physics and Engineering of Advanced Materials, for the
financial support provided with regards to my participation and oral presentation at “Materiais 2019 –
XIX Congresso da Sociedade Portuguesa de Materiais and X International Symposium on Materials” in
Lisbon, from 14-17 April 2019.
I would also like to extend my gratitude to Luís Castelo Branco, who is developing a similar work
under the same supervisors, for our continuously shared information and ideas and for always offering
me a car ride to the train station on rainy days.
Since it is my belief that a student’s success is also his/her teachers’, I would like to give a very
special thanks to the Math teachers that carried me through my senior high school years, Mr. António
Teixeira and Mrs. Ana Maria Espregueira and to my English teacher, Mrs. Ana Tavares, for giving me
the tool with which I was able to write this work, and for her inexhaustible encouragement throughout
the years.
Finally, I wish to thank my family, friends and colleagues for making life’s most challenging moments
bearable. I cannot thank my parents enough for their continuous emotional and financial support and
for always prioritizing their children’s education above all else.
To my best friend and soulmate: thank you for your endless love, friendship and support.
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Resumo
À medida que os componentes electrónicos vêem o seu tamanho reduzido, o aquecimento por
efeito Joule desempenha um papel mais determinante na degradação ou mesmo falha de componentes
electrónicos. As altas temperaturas observadas em tais dispositivos são um obstáculo ao desempenho
de circuitos à escala nano. Nesse contexto, é crucial entender a forma como a arquitectura dos circuitos
integrados baseados em silício sujeitos a carregamento térmico influencia as tensões geradas nas
linhas interconectoras de cobre.
O objectivo deste trabalho é produzir um modelo de Elementos Finitos capaz de prever as
deformações e tensões térmicas em linhas interconectoras. Este modelo foi utilizado, em particular,
para o estudo da influência de diferentes parâmetros geométricos nas tensões e deformações
observadas nas linhas de cobre.
O software de Elementos Finitos AbaqusTM foi utilizado para simular as condições de trabalho do
circuito integrado. Uma abordagem inicial bidimensional foi usada para uma configuração de referência,
a partir da qual foram estabelecidas tendências entre a arquitectura do modelo e as tensões térmicas.
Finalmente, um modelo tridimensional foi desenvolvido e os resultados foram comparados com os
estudos bidimensionais.
No presente trabalho, confirmou-se que o modelo bidimensional apresentou resultados
semelhantes ao modelo 3D para espessuras acima de 40µm. As tensões nas linhas de cobre tendem
a estabilizar para razões de largura/altura superiores a 1.5 e distâncias entre linhas superiores à largura
das mesmas. Configurações com larguras de linha menores são submetidas a maiores tensões de
compressão. Os resultados obtidos fornecem orientações úteis para a optimização e design de
circuitos.
Palavras-chave
Interconectores de cobre, carregamento térmico, método dos elementos finitos, tensões residuais
vi
Abstract
As electronic components are continuously downscaled, Joule heating plays an increasingly critical
role in the degradation or even failure of electric components. The high temperatures observed in such
devices challenge the creation of smaller circuits and compromise their performance. In this context, it
is crucial to understand the way thermal loading and silicon integrated circuit architectures influence
stresses in nano-copper interconnect lines.
The goal of this work is to produce a Finite Element (FE) model capable of predicting the thermal
stresses at interconnect lines and use it to study the influence of different geometric parameters on the
thermal stresses and deformation observed in copper lines.
The FE analysis software AbaqusTM was used to simulate the integrated circuit’s working conditions
and an initial bidimensional approach was used for a reference configuration from which several
geometric features were studied (line width, aspect ratio and line distance) and trends between the
model architecture and stresses were established. Finally, a three-dimensional model was developed,
and the results were compared with the bidimensional studies.
In the present work, the bidimensional model was confirmed to provide matching results, as
compared with 3D, for architectures with thicknesses above 40µm. Stresses at the copper lines were
found to stabilize for aspect ratios higher than 1.5 and line distances higher than their respective width.
Configurations with smaller line widths are subjected to higher compressive stresses. The results
obtained from these models can provide useful guidelines for the optimal design of circuits.
Keywords
Copper interconnects, thermal loading, finite element method, residual stresses
vii
Table of Contents Acknowledgements ........................................................................................................................... iv
Resumo ..............................................................................................................................................v
Abstract ............................................................................................................................................. vi
Table of Contents ............................................................................................................................. vii
List of Figures .................................................................................................................................... ix
List of Tables .................................................................................................................................... xii
List of Symbols and Abbreviations .................................................................................................. xiii
1 Introduction .............................................................................................................................. 1
1.1 Overview ........................................................................................................................... 2
1.2 Outline of the Thesis ........................................................................................................ 3
2 State of the art .......................................................................................................................... 4
2.1 The finite element method ................................................................................................ 5
2.1.1 Introduction ............................................................................................................ 5
2.1.2 Mesh refinement .................................................................................................... 6
2.2 Thermal expansion and thermal stresses ........................................................................ 8
2.2.1 Introduction ............................................................................................................ 8
2.2.2 Thermal stresses in design .................................................................................... 9
2.3 Copper interconnects in the semiconductor industry ..................................................... 10
2.3.1 Introduction .......................................................................................................... 10
2.3.2 The Damascene process ..................................................................................... 10
2.3.3 Physical vapor deposition (PVD) ......................................................................... 12
2.3.4 Ionized physical vapor deposition (iPVD) ............................................................ 12
2.3.5 Chemical vapor deposition (CVD) ....................................................................... 12
2.4 Thermal stresses in copper interconnects ..................................................................... 13
2.4.1 Finite element modelling of stresses in metal interconnects ............................... 13
2.4.2 Electromigration ................................................................................................... 14
2.4.3 Microstructure ...................................................................................................... 15
2.5 Analytical solutions ......................................................................................................... 17
3 Methodology ........................................................................................................................... 19
3.1 Modeling ......................................................................................................................... 20
3.1.1 Copper interconnect architecture ........................................................................ 20
3.1.2 Dimensional analysis ........................................................................................... 21
3.1.3 Material properties ............................................................................................... 22
3.1.4 Thermal loading and boundary conditions .......................................................... 28
3.1.5 Simplifications and assumptions ......................................................................... 29
3.2 Domain size studies ....................................................................................................... 29
3.2.1 Bidimensional model ........................................................................................... 29
viii
3.2.2 Three-dimensional model .................................................................................... 32
3.2.3 Study of the influence of the diffusion barrier ...................................................... 34
3.3 Convergence studies ...................................................................................................... 35
3.3.1 Bidimensional model ........................................................................................... 35
3.3.2 Three-dimensional model .................................................................................... 40
4 Results and discussion .......................................................................................................... 42
4.1 Plasticity studies ............................................................................................................. 43
4.2 Geometry studies ........................................................................................................... 48
4.2.1 Overview .............................................................................................................. 48
4.2.2 Aspect ratio .......................................................................................................... 48
4.2.3 Line distance ........................................................................................................ 52
4.2.4 Line width ............................................................................................................. 55
4.3 Three-dimensional studies ............................................................................................. 60
5 Conclusions ............................................................................................................................ 63
5.1 Conclusions .................................................................................................................... 64
5.2 Future work ..................................................................................................................... 65
References .................................................................................................................................... 67
Appendix A. Dimensional analysis ................................................................................................... 70
Appendix B. Wafer length study (i) .................................................................................................. 72
B.1 Discussion of results ...................................................................................................... 74
Appendix C. Wafer length study (ii) ................................................................................................. 75
C.1 Discussion of results ...................................................................................................... 77
Appendix D. Convergence studies (2D) ........................................................................................... 78
Appendix E. Aspect ratio studies ..................................................................................................... 81
Appendix F. Line distance studies ................................................................................................... 83
Appendix G.Line width studies ........................................................................................................ 85
ix
List of Figures
Figure 2.1 Finite element analysis of an aircraft’s bearing bracket carried out in the web
browser with SimScaleTM [5] ................................................................................ 6
Figure 2.2 Mesh refinement through the decrease in element size [6] ................................. 7
Figure 2.3 Mesh refinement through the increase in element order [6] ................................ 7
Figure 2.4 Mesh refinement through global adaptative meshing [6] ..................................... 7
Figure 2.5 Mesh refinement through local adaptative meshing [6] ....................................... 7
Figure 2.6 Manual mesh refinement [6] ................................................................................ 8
Figure 2.7 Thermal expansion joints in the Auckland Harbour Bridge in New Zealand [9] .. 9
Figure 2.8 Damascene process steps [10] ......................................................................... 10
Figure 2.9 Antique Japanese damascene Amita bug pin [16] ............................................. 11
Figure 2.10 (a) Schematic of the model interconnect structure and coordinate system adopted
by Shen. (b) Predicted average longitudinal stress in periodic Al lines as a
function of line aspect ratio for various line distances. [20] ............................... 13
Figure 2.11 TEM planar view of 180nm interconnect lines showing the diffusion barrier (d.b.)
and the interlayer dielectrics (ILD) [1] ................................................................ 14
Figure 2.12 FIB and TEM images of electromigration damaged samples for (a) SiCxHyNz
capped line and (b) Ta/TaN coated line [24]. .................................................... 15
Figure 2.13 (a) Bright-field STEM image on which the stress analysis was performed. (b)
Color-coded replica of image (a) containing stiffness and orientation information
for each grain. (c) Application of a fine mesh on image (b). (d) Finite mesh and
FEM stress solution for every grain. (e) FEM stress solution without the mesh [27]
........................................................................................................................... 16
Figure 2.14 Geometry used by Hsueh when formulating an approximate general solution for
the stress distributions in passivated interconnects. The unit cell has infinite
length in the z-direction and the interconnect is bonded to the substrate at x=0
[30] ..................................................................................................................... 17
Figure 2.15 Volume-averaged thermal stresses in the interconnect calculated by Hsueh; (a)
𝜎𝑥𝑖 , (b) 𝜎𝑦
𝑖 and (c) 𝜎𝑧𝑖 as a function of the aspect ratio, 𝑡/2𝑙. The asymptotic
stresses are also shown [30] ............................................................................. 18
Figure 3.1 Cross-section of the problem for n=4 ................................................................ 20
Figure 3.2. (a) Unit cell. (b) Half unit cell. ............................................................................. 21
Figure 3.3. Main differences between all constitutive models ............................................. 23
Figure 3.4. True stress-strain curves of copper at different temperatures ........................... 25
Figure 3.5. True stress-strain curves of tantalum at different temperatures ........................ 25
Figure 3.6. True stress-strain curve of silicon at 20ºC ......................................................... 26
Figure 3.7. True stress-strain curve of silicon dioxide at 20ºC ............................................ 26
x
Figure 3.8. True stress-strain curve of silicon nitride at 20ºC ............................................. 26
Figure 3.9. Compressive stress-strain curves of the unalloyed tantalum under dynamic and
quasi-static deformation at various temperatures [34]....................................... 27
Figure 3.10. Thermal loading condition used in the simulation .............................................. 28
Figure 3.11. Boundary conditions used in the model ............................................................. 28
Figure 3.12. Control points used for the bidimensional wafer size study .............................. 30
Figure 3.13. Influence of the wafer length on the stresses observed at control point B ........ 30
Figure 3.14. Influence of the wafer length on the Von Mises stresses observed at various
control points inside the copper lines................................................................. 31
Figure 3.15. Location of CPedge .............................................................................................. 33
Figure 3.16. Plot demonstrating the behaviour of the Von Mises stress along CPedge for a wafer
size of w=3μm and model thickness Z=4μm ..................................................... 33
Figure 3.17. Results of the wafer size study for a 3D model of thickness Z=4μm ................. 34
Figure 3.18. Results of the diffusion barrier influence study for Model 1 ............................... 35
Figure 3.19. Results of the diffusion barrier influence study for Model 2 ............................... 35
Figure 3.20. Control points for the bidimensional convergence study ................................... 36
Figure 3.21. Convergence study for control point A............................................................... 36
Figure 3.22. Convergence study for control point B............................................................... 37
Figure 3.23. Biased mesh convergence study for control point A ......................................... 38
Figure 3.24. Uniform mesh convergence study at various control points – Von Mises stress
........................................................................................................................... 39
Figure 3.25. Von Mises stress distribution (in MPa) for the refined biased mesh ................. 39
Figure 3.26. Von Mises stress distribution (in MPa) for the uniform 0.005µm mesh ............. 40
Figure 3.27. Convergence study for a 3D model of thickness Z=4µm ................................... 41
Figure 3.28. Location of control point B for the 3D model ..................................................... 41
Figure 3.29. Mesh used for the 3D model studies ................................................................. 41
Figure 4.1. Von Mises and hydrostatic stress values (MPa) at control point B for all
constitutive models, at t=1s ............................................................................... 44
Figure 4.2. Von Mises and hydrostatic stress values (MPa) at control point B for all plastic
constitutive models, at t=2s ............................................................................... 44
Figure 4.3. Von Mises stress distribution (MPa) for all constitutive models at t=1s ............ 46
Figure 4.4. Distribution of plastic strain for all plastic constitutive models at t=1s ............... 47
Figure 4.5. Results of the aspect ratio study for all constitutive models, at t=1s ................. 48
Figure 4.6. Results of the aspect ratio study for all plastic constitutive models, at t=2s ...... 49
Figure 4.7. Results of the aspect ratio study for constitutive model Elast2, at t=1s ............ 50
Figure 4.8. Results of the aspect ratio study for constitutive model Plast, at t=1s .............. 50
Figure 4.9. Results of the aspect ratio study for constitutive model Plast, at t=2s .............. 51
Figure 4.10. Results of the line distance study for all constitutive models, at t=1s ............... 52
Figure 4.11. Results of the line distance study for all plastic constitutive models, at t=2s .... 52
Figure 4.12. Results of the line distance study for constitutive model Elast2, at t=1s ........... 53
xi
Figure 4.13. Results of the line distance study for constitutive model Plast, at t=1s ............. 54
Figure 4.14. Results of the line distance study for constitutive model Plast, at t=2s ............. 54
Figure 4.15. Results of the convergence study for the adjusted base model ........................ 56
Figure 4.16. Results of the line width study for all constitutive models, at t=1s ..................... 57
Figure 4.17. Results of the line width study for all plastic constitutive models, at t=2s ......... 57
Figure 4.18. Results of the line width study for constitutive model Elast2, at t=1s ................ 58
Figure 4.19. Results of the line width study for constitutive model Plast, at t=1s .................. 58
Figure 4.20. Results of the line width study for constitutive model Plast, at t=2s .................. 59
Figure 4.21. Comparison between the three-dimensional and bidimensional results, at t=1s,
for constitutive model Elast1 .............................................................................. 60
Figure 4.22. Comparison between the three-dimensional and bidimensional results, at t=1s,
for constitutive model Plast ................................................................................ 61
Figure 4.23. Stress (MPa) distributions at CPedge along thickness z (µm) for a 3D model of total
thickness 40µm (at t=1s, for constitutive model Elast1) .................................... 62
Figure B.1. Influence of the wafer CTE on stresses at control points B and Si. .................. 73
Figure B.2. Influence of the wafer CTE on displacements at control points B and Si. ......... 73
Figure C.1. Influence of the wafer length on stresses at control points B and Si. ................ 76
Figure C.2. Influence of the wafer length on displacements at control points B and Si ....... 76
Figure D.1. Convergence study for control point C. ............................................................. 79
Figure D.2. Convergence study for control point D. ............................................................. 79
Figure D.3. Convergence study for control point Si .............................................................. 79
Figure D.4. Uniform mesh convergence study – S11. .......................................................... 80
Figure D.5. Uniform mesh convergence study – S22 ........................................................... 80
Figure D.6. Uniform mesh convergence study – S33. .......................................................... 80
Figure E.1. Highlight of the “half” copper line shown in the following results ....................... 82
Figure E.2. Stress distribution (MPa) in a “half” copper line, for the line aspect ratio study of
model Elast2, at t=1s (H), for AR=0.25 (*) and AR=2.5 (**).. ............................ 82
Figure E.3. Stress distribution (MPa) in a “half” copper line, for the line aspect ratio study of
model Plast, at t=1s (H) and t=2s (C) for AR=0.25 (*) and AR=2.5 (**).. .......... 82
Figure F.1. Stress distribution (MPa) in a “half” copper line, for the line distance study of
model Elast2, at t=1s (H), for Dist=0.01µm (*) and Dist=1.08µm (**). ............... 84
Figure F.2. Stress distribution (MPa) in a “half” copper line, for the line distance study of
model Plast, at t=1s (H) and t=2s (C) for Dist=0.01µm (*) and Dist=1.08µm (**)
........................................................................................................................... 84
Figure G.1. Stress distribution (MPa) in a “half” copper line, for the line width study of model
Elast2, at t=1s (H), for Width=10nm (*) and Width=180nm (**).. ....................... 86
Figure G.2. Stress distribution (MPa) in a “half” copper line, for the line width study of model
Plast, at t=1s (H) and t=2s (C), for Width=10nm (*) and Width=180nm (**) ...... 86
xii
List of Tables
Table 3.1. Materials and measurements used, in µm. ........................................................ 20
Table 3.2. Material properties used for copper in Elast1 and for all other materials in the
remaining models............................................................................................... 23
Table 3.3. Coefficient of thermal expansion used for copper in all models except for Elast1
........................................................................................................................... 23
Table 3.4. Values used in the reconstruction of the true stress-strain curves of copper and
tantalum (for various temperatures) ................................................................... 24
Table 3.5. Values used in the reconstruction of the true stress-strain curves of all materials
(for 20ºC)............................................................................................................ 24
Table 3.6. Von Mises stress at CPB, in MPa, for different types of finite elements ............ 37
Table 4.1. Main architectural differences between the models used in the linewidth study.
........................................................................................................................... 55
Table A.1. Dimensional analysis. ........................................................................................ 71
xiii
List of Symbols and
Abbreviations
𝑎, 𝑏, 𝑏′ Distances (exclusive to section 2.5)
𝑑 Distance between the passivation layer and
the wafer (µm)
𝑐𝑗, 𝑄𝑗, 𝑃𝑗 Coefficients (miscellaneous)
𝑢𝑅,𝑖 𝑖 rotation
𝜀𝑖𝑗 Strain tensor
𝜀𝑛 Engineering strain
𝜀𝑇 True strain
𝜀𝑦 Yield strain
𝜀𝑝𝑙 Plastic strain
𝜀𝑈𝑇𝑆 Maximum uniform strain
𝜀̇ Strain rate
𝜙𝑗 Approximation function
AR Aspect ratio
CI Copper interconnect(s)
CAD Computer aided design
CMP Chemical mechanical polishing
CPedge Control edge
CPi, CP_i Control point 𝑖
CTE, 𝛼 Coefficient of thermal expansion
CVD Chemical vapor deposition
Dist Line distance
E, 𝐸𝑖 Young’s Modulus
ECD Electrochemical deposition
EM Electromigration
FE Finite element
FEA Finite element analysis
FEM Finite element method
FIB Focused ion beam
IC Integrated circuit(s)
iPVD Ionized physical vapor deposition
ℎ, 𝑡, 𝑝, 𝑤 Distances (exclusive to section 2.4.1)
xiv
n Number of interconnect lines
PVD Physical vapor deposition
RAM Random access memory
RC Resistance-capacitance
RIE Reactive ion etching
𝑆𝑖𝑗, 𝜎𝑖𝑗 Stress tensor
𝜎𝑛 Engineering stress
𝜎𝑇 True stress
𝜎𝑦 Yield stress
SLA Scanned laser annealing
STEM Scanned transmission electron microscope
T Temperature, ºC
t Time, s
TEM Transmission electron microscope
Ui, 𝑢𝑖 Displacement in the 𝑖 direction
Umag Displacement magnitude
w Wafer length
Z 3D model thickness
Δ𝑇 Temperature step
𝑙𝑤 Line width (nm)
𝑥,1 Horizontal axis
𝑦, 2 Vertical axis
𝑧, 3 Longitudinal axis
𝜈 Poisson’s coefficient
1
Chapter 1
Introduction
This chapter provides a brief overview of the research work. The motivation behind the work is
presented, as well as its original contributions. The current State-of-the-Art concerning the scope of the
work is also addressed. At the end of the chapter, an outline of the work is provided.
2
1.1 Overview
In the last 60 years, integrated circuits (IC) have successfully made their way into our daily lives in
applications, such as computers, TV/radio/video, cell phones, digital clocks, robotics,
telecommunications, automotive and aerospace industries, medical equipment and military.
As the demand for higher computational power has been increasing, electronic components have
been continuously downscaled. In the context of these electronic components, interconnects (IC) are
wires (stacked in multi-levels and with various line widths), responsible for transmitting signal, power
and connecting various components within integrated circuits (IC). As of 1997, copper had become the
metal of choice for IC. Due to the narrow line widths (currently at the nanometer scale), Joule heating
plays an increasingly critical role in the degradation or even failure of components, such as
microprocessors. The high temperatures observed in those devices are a challenge for creating smaller
circuits without compromising their performance. In particular, when circuits are subjected to Joule
heating under normal working conditions, the mismatch between the thermal expansion coefficient of
copper and all other materials present in the IC promotes the development of thermal stresses, which
pose a serious threat to IC performance.
In this context, it is paramount to understand the way thermal loading and silicon integrated circuit
architectures influence stresses in nano-copper interconnect lines. Many approaches can be taken to
address this problem, such as testing, characterization or modelling. In this study, a modelling approach
by means of the Finite Element (FE) method is featured and AbaqusTM was the Finite Element Analysis
(FEA) software chosen.
To the best of the author’s knowledge, some FE studies exist for interconnects, but most were
performed in the mid 90’s for aluminum lines. Such studies focused mainly on the annealing conditions
(simply cooling from a temperature of around 350ºC to room temperature) instead of working conditions
(heating to a working/critical temperature followed by cooling to room temperature).
Bidimensional models based on plane stress or generalized plane strain conditions (later validated
by experimental results) are also more commonly found than three-dimensional models/studies and, in
recent years, more effort has been put into studying stresses at copper thin films instead of copper
interconnect lines.
The main goal of this work is to produce a FE model capable of predicting the thermal stresses
occurring in copper interconnect lines and subsequently use it to study the influence of geometric
parameters, such as line width, aspect ratio and distance between lines on thermal stresses and
deformation observed in copper lines. This study’s original contributions are centered around the
simulation of the ICs critical working conditions (instead of annealing conditions) and a detailed
examination and discussion of different geometries, material modelling options and their implications.
Thermal stresses are examined both at maximum (critical) temperature and at room temperature after
cooling (residual stresses).
An initial bidimensional approach with plane strain configuration was used for a base-model with a
line width 180nm and aspect ratio 1.5. This base model’s architecture was based on a test structure
from Freescale Semiconductor, Inc.TM found on a 2007 paper from Met et al. [1]. Based on this reference
3
configuration, several geometric features were studied, such as different combinations of line widths,
aspect ratios and line distances, so that trends between the architecture of the lines and stresses could
be established. Different constitutive models concerning material behavior were tested against each
other (pure elasticity, ideal plasticity, work-hardening). The multidimensionality of the problem and its
consequences are also discussed in detail. Finally, a three-dimensional model was developed, and the
results were compared with the bidimensional studies to validate the latter, justifying its application.
1.2 Outline of the Thesis
This thesis is composed of 5 chapters.
Chapter 1 – Introduction: this chapter provides a brief overview of the research work. The
motivation behind the work is presented, as well as its original contributions. The current State-
of-the-Art concerning the scope of the work is also addressed. At the end of the chapter, an
outline of the work is provided.
Chapter 2 – State of the art
2.1 – The finite element method
2.2 – Thermal expansion and thermal stresses
2.3 – Copper interconnects in the semiconductor industry
2.4 – Thermal stresses in copper interconnects
2.5 – Analytical solutions
This chapter introduces the finite element method and the thermal stresses associated with the
copper interconnects (CI). In addition, it provides an overview of the use of CIs in the
semiconductor industry, as well as interconnect microstructure, fabrication, and thermal loading.
Chapter 3 – Methodology
3.1 – Problem modelling
3.2 – Domain size studies
3.3 – Convergence studies
This chapter details the methods, constitutive models and assumptions used throughout the
work. Convergence and domain size studies are presented and discussed for both the
bidimensional and three-dimensional models.
Chapter 4 – Results
4.1 – Plasticity studies
4.2 – Geometry studies
4.3 – Three dimensional studies
This chapter presents the main results. A comparison between all constitutive models is
featured. The influence of geometric parameters on the stress distributions at copper
interconnect lines is studied and the bidimensional studies are validated by a three-dimensional
analysis.
Chapter 5 – Conclusions: this chapter summarizes the work, provides the conclusions and
points out work to be developed in the future.
4
Chapter 2
State of the Art
This chapter introduces the finite element method and the thermal stresses associated with the
copper interconnects (CI). In addition, it provides an overview of the use of CIs in the semiconductor
industry, as well as interconnect microstructure, fabrication, and thermal loading.
5
2.1 The finite element method
2.1.1 Introduction
When modeling a scientific problem, knowledge in mathematics and physics can be used to
transform an assembly of raw data into a set of mathematical relations. Yet, it is critical to use
assumptions about the way physical processes work. This may result in the establishment of a set of
differential equations, whose solution provided by exact methods is often a challenging task. Therefore,
approximate methods can be used to find solutions for these problems, such as the finite difference
method and variational methods, such as the Rayleigh-Ritz and Galerkin methods [2]. To determine the solution of a differential equation by variational methods, the equation is
transformed into an equivalent weighted-integral form and the approximate solution is a linear
combination, Σ𝑗c𝑗ϕ𝑗, of properly chosen approximation functions, ϕ𝑗, and undetermined coefficients, c𝑗
[2]. The major difference between variational methods lies in the distinct integral forms, weight functions
and/or approximation functions chosen. However, approximation functions for problems with arbitrary
domains are very difficult to construct [2]. The finite element method overcomes this disadvantage
because it provides a systematic procedure to compute the approximation function over subregions of
a domain, in particular:
- A domain that is geometrically complex can be characterized as a collection of
geometrically simple subdomains called finite elements;
- The approximation functions are computed over each finite element;
- Algebraic relations between the undetermined coefficients (nodal values) are computed
while satisfying the problem’s governing equations over each element.
This idea of breaking down an arbitrary domain into a collection of discrete subregions is neither
new nor unique to the finite element method. The first estimation of π was done by Archimedes of
Syracuse (287-212 BC) by approximating the area of a circle using the Pythagorean Theorem to
calculate the areas of two regular polygons: one inscribed within the circle and one within which the
circle was circumscribed [3]. The real area of the circle would have to be a value between the calculated
areas of both polygons. By continuously increasing the number of sides of the polygons, an increasingly
more accurate number interval for pi is estimated. Archimedes showed that pi is a value between 31
7
and 310
71 [3].
The finite element method offers great freedom of discretization both in the types of elements used
and in the basis functions. The usefulness of this method and its applicability to a wide range of problems
was first recognized at the start of the 1940s by the German-American mathematician Richard Courant.
It took, however, several decades before this approach was generalized for fields outside of structural
mechanics [4].
Presently, the finite element method is used across all fields of physical sciences and engineering,
not only for structural analysis but also heat transfer, electromagnetism and fluid mechanics, etc.
6
Scientists and engineers of the twenty first century have an extensive catalog of finite element analysis
(FEA) software packages to choose from, among which AbaqusTM, ANSYSTM, COMSOL MultiphysicsTM,
NastranTM, FemapTM and many others.
A FEA usually begins with a computer-aided design (CAD) model of the domain of the problem.
Aside from the geometry, knowledge of the materials (and their properties) is needed, as well as a good
understanding of the applied loads and constraints present in the physical process that is being
simulated. With this information, a prediction of the real-world behavior can be achieved with satisfying
accuracy. Figure 2.1 exemplifies the finite element analysis of an aircraft bearing bracket with the
software SimScaleTM [5].
2.1.2 Mesh refinement
The accuracy of a FEA model is directly linked to the finite element mesh used for the simulation. A
mesh is what subdivides the CAD model into the smaller subregions called finite elements. When a
mesh is refined, its finite elements are made increasingly smaller and it is expected that the computed
solution will approach the real solution. This process of mesh refinement is a very important step towards
validating the FEA model and producing reliable results.
Initially, when a model is being tested for bugs or inconsistencies in its formulation (material
properties, loads or constraints) it is reasonable to start with a coarse mesh. A mesh with very large
elements will require smaller computational resources and although the simulation might generate
inaccurate results, they can still be used to roughly verify if the model is behaving as desired.
Next, a mesh refinement process is applied, during which the simulation is successively resolved
for increasingly finer meshes. Then, results for different meshes are compared by either analyzing the
fields at control points (appropriately chosen by the analyst) in the model or by evaluating the integral
of a field over certain subdomains or boundaries. The convergence of the solution with respect to the
mesh refinement is then judged by the analyst by comparing these scalar quantities. If the formulation
of the model is correct, the changes between solutions will become increasingly smaller until the model
Figure 2.1 Finite element analysis of an aircraft’s bearing bracket carried out in the web browser with SimScaleTM [5]
7
is considered to have converged. This consideration often depends on the sensibility and experience of
the analyst, acceptable uncertainty of the model and computational power available.
Ideally, a mesh can be infinitely refined for increasingly more precise results and a solution given by
a very fine mesh can be taken as an approximation to the real solution. The art of mesh refinement has
various techniques, illustrated in figures 2.2-2.6 [6], namely:
- Increasingly reducing the element size: simple method, but no preferential mesh
refinement in regions where a finer mesh could be more adequate.
- Increasing the element order: instead of remeshing the domain, the order of the finite
elements is changed; this technique is useful for 3D geometries (where remeshing becomes
very time consuming) but requires more computational power than other techniques.
- Global adaptative mesh refinement: the FEA software estimates the point in the model
where the local error is largest and generates a new mesh accordingly; this process is
automatically done by the software so the user has no control over the mesh and, which may
results in excessive meshing of areas of less interest (where a larger error is acceptable);
- Local adaptative mesh refinement: similar to the aforementioned, but the error is
evaluated only over a subregion of the domain; the software still creates a new mesh for the
entire model, but with the objective of reducing the error in a specific chosen region;
- Manual mesh adjustment: the analyst manually manipulates the mesh according to the
physics of the problem; this approach requires experience and greater understanding and
sensibility but, if done right, it can significantly lessen computational time and resources.
Figure 2.2 Mesh refinement through the decrease in
element size [6]
Figure 2.3 Mesh refinement through the increase in
element order [6]
Figure 2.4 Mesh refinement through global adaptative
meshing [6]
Figure 2.5 Mesh refinement through local adaptative
meshing [6]
8
2.2 Thermal expansion and thermal stresses
2.2.1 Introduction
Most solid materials expand when heated and contract when cooled. Thermal expansion is a result
of heat’s ability to increase a material’s kinetic energy: as temperature rises, molecules that would
otherwise be closely located start vibrating at increasingly faster speeds and push away from each other,
enlarging the average distance between individual atoms and augmenting the volume of the structure
[7].
The change in length with temperature for a solid material is directly proportional to its linear
coefficient of thermal expansion and to the applied temperature change [8]. The linear coefficient of
thermal expansion (CTE) has reciprocal temperature units [℃−1 𝑜𝑟 ℉−1] and represents the extent to
which a material expands upon heating; its magnitude increases with rising temperature [8].
Thermal stresses are consequences of the temperature-driven changes of dimensions in a body.
These stresses can potentially be the cause of fractures or undesirable plastic deformations in a
structure, and thus have a serious effect on its strength and stability.
If a homogeneous and isotropic rod experiencing a uniform temperature change is free to expand
(or contract), no thermal stresses will develop. However, if the axial motion of the rod is rigidly restrained,
the magnitude of the thermal stress caused by a temperature change (Δ𝑇 = 𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙 − 𝑇𝑓𝑖𝑛𝑎𝑙) would be
σ = 𝐸𝛼 Δ𝑇 (2.1)
Where E is the modulus of elasticity and 𝛼 is the CTE of the rod’s material.
If the rod is heated, Δ𝑇 < 0 the stress is compressive 𝜎 < 0 because the material’s expansion is
constrained. Inversely, when the rod is cooled Δ𝑇 > 0 tensile stresses 𝜎 > 0 develop. The same stress
would be required to elastically compress (or elongate) the rod back to its original length after it had
been allowed to freely expand (or contract) with the same temperature change.
Figure 2.6 Manual mesh refinement [6]
9
2.2.2 Thermal stresses in design
Thermal stresses are usually unwelcomed in design. In welding, for example, undesirable stresses
develop when an assembly of different materials (with different CTE) is joined together and cooled down.
Since some areas of the welding contract more than others (due to the difference in CTEs), residual
stresses occur within the region of the weld, which tends to compromise the durability of the structure.
In some cases, thermal expansion and stresses can be embraced in design. “Shrink-fitting” is a
process in which an internal component is mated with its (previously heated to the point of expansion)
external counterpart to form a joint that becomes stronger as the two components reach the same
temperature [7]. Another example of this are the thermal expansion joints frequently used in bridges to
allow a length change without buckling. Thermal expansion joints are used in other applications as well,
such as ducted air systems and railways [9].
When electric current flows through a solid or liquid with finite conductivity, its energy is converted
into heat through resistive losses in the material. This process is known as Joule heating (i.e. resistive
or ohmic heating) and its effect is especially undesirable in electrical systems and components because
heating can cause degeneration or even melting of the microstructures.
Figure 2.7 Thermal expansion joints in the Auckland Harbour Bridge in New Zealand [9]
10
2.3 Copper interconnects in the semiconductor industry
2.3.1 Introduction
Interconnects are becoming the dominant factor for system performance and power dissipation in
integrated circuits (IC). They are responsible for distributing clock and other signals and providing
power/ground to the various IC [10]. The interconnection of over 1100 million transistors requires almost
35 km of interconnects at the 65nm technology node (achieved in 2007) [11].
The first working microprocessor using copper was made by IBM® in 1997 [10]. Historically,
aluminum had been the dominant material used in interconnects. Copper emerged at the 220nm
technology node in microprocessors [11] as the next heir to the interconnect throne due to its lower
resistivity (40% less, reducing of the resistance-capacitance (RC) delay, which in turn increases the IC
speed) [10]. As another advantage, copper also has twice the thermal conductivity and 100 times more
resistance to electromigration failures than aluminum [10]. Electromigration is the transport of conductor
materials as a result of high current densities, which may lead to void formation in the conductor [12].
The use of copper also reduces the power consumption by 30% at a specific frequency [10]. and
allows the simplification of the interconnect routing, which reduces the number of process steps, saving
costs and favoring a higher device yield [10].
2.3.2 The Damascene process
The replacement of aluminum by copper was a hurdle for semiconductor process engineers: while
aluminum was deposited over the entire wafer surface and then patterned by reactive ion etching (RIE),
copper could not be patterned [10]. To overcome this predicament, a new process had to be developed.
The now industry-standard Damascene Process [10]. (which borrowed its name from the city of
Damascus, in Syria) was originally developed for jewelry manufacturing. The Damascene art had been
known for centuries in countries like Egypt, Greece and Japan. In the XV century, and it also became
popular in Europe, with the city of Toledo (in Spain) becoming known as the European capital of the
damascene industry [13].
The Damascene Process consists of three main steps: deposition of a seed layer, electroplating and
chemical mechanical polishing. They can be described as follows:
Figure 2.8 Damascene process steps [10]
11
(1) A barrier layer such as Ta (tantalum) is deposited by ionized physical vapor deposition
(iPVD) over the patterned dielectric. This prevents interaction between the metal and the
dielectric [12]. The diffusion of copper into silicon would cause deep level impurity, degrading
device performance. A copper seed layer is then applied over the barrier layer by PVD, iPVD or
CVD [12]. The seed layer must be uniform (to avoid voiding) and its surface must be free of
oxides for efficient charge transfer during plating [10] [14].
(2) The pattern is filled by electroplating (i.e. electrochemical deposition, ECD). In ECD, a
conductive surface is immersed in a solution with metal ions and linked to a power supply. The
electrical current passing from the surface to the solution causes the metal to be deposited on
the surface [10].
(3) The excess material is removed by chemical mechanical polishing (CMP). A slurry
containing abrasive particles (such as silica) is used to physically grind the excess material.
Since oxidizing agents (such as hydrogen peroxide) are also present in the slurry [15], chemical
action is also promoted to planarize the wafer surface.
A variation of this technique called Dual Damascene Process makes it possible to deposit copper in
interconnect and via, reducing the number of process steps required.
In figure 2.9, a piece of antique Japanese damascene jewelry is shown [16].
Figure 2.9 Antique Japanese damascene amita bug pin [16]
12
2.3.3 Physical vapor deposition (PVD)
Physical vapor deposition (PVD) is a vacuum coating technique where a solid metal is vaporized to
a plasma of atoms or molecules and deposited on electrically conductive substrates as a pure metal or
alloy coating. The process usually takes place in a high vacuum chamber at 10−2 to 10−4 millibar and
between 150° and 500℃ [17].
PVD coatings can be optimized for different applications depending on the type of material used:
from microchips and semiconductors to surgical and medical implants, aerospace and automotive parts
(graphite and titanium), cutting tools, etc. These coatings also tend to be highly resistant to tarnishing
and corrosion, making them the perfect choice for decorative finishes [17].
2.3.4 Ionized physical vapor deposition (iPVD)
Ionized physical vapor deposition (iPVD) distinguishes itself from PVD because, unlike neutral
atoms, ions can have their motion controlled by an electric field. These ions are accelerated towards the
substrate and their bombardment energy is be controlled by application of a bias voltage to the
substrate. By guiding of the deposition material to the desired areas of the substrate, the bottom of deep
narrow trenches or vias of high aspect ratio IC structures can be filled. A neutral flux would tend to
deposit on the upper part of sidewalls, resulting in very little film coverage and possible void formation
at the bottom [18].
2.3.5 Chemical vapor deposition (CVD)
Chemical vapor deposition (CVD) is a materials-processing technology where a precursor gas (or
gases) flows into a chamber containing one or more heated objects to be coated. The chemical reactions
that occur on (and near) the hot surfaces promote the deposition of a thin film [19].
CVD films are generally conformal, and thus can be applied to elaborately shaped pieces, insides
and undersides of features and high-aspect ratio holes (unlike classical PVD techniques). A wide variety
of materials can be deposited with very high purity and deposition rates. This process also doesn’t
require as high a vacuum as PVD processes [19].
The main disadvantages of CVD are the cost, volatility, explosivity, corrosivity and high toxicity of
the precursors. The byproducts can also be hazardous [19]. By contrast, PVD coatings can be applied
with no toxic or environmentally unfriendly residues or by-products [17].
13
2.4 Thermal stresses in copper interconnects
2.4.1 Finite element modeling of stresses in metal interconnects
The finite element method has been a tool for modelling stresses in metal interconnects before
the introduction of copper in the semiconductor industry. Studies about the effects of line distance and
aspect ratio in periodic aluminum interconnect lines of very simple geometry were performed by Y. -L.
Shen in 1997. Shen realized that the effects of line aspect ratio on stresses are not independent of the
spacing between lines [20]. Figure 2.10 illustrates this behavior, where p/w is a measure of the distance
between lines (p/w=2 corresponds to a scenario in which the line distance equals the line width). These
studies were carried out for annealing conditions, from an initial stress-free temperature of 400ºC
(passivation deposition temperature) to room temperature, 24ºC.
Shen demonstrated the importance of considering line periodicity in FE analyses through the
insertion of symmetry boundary conditions, as large errors are observed when a single-line approach is
used to model thermal stresses in periodically aligned interconnects. He also highlighted that the study
of thermal stresses in metal interconnects is of the outmost importance to the understanding of the
voiding phenomenon, since the hydrostatic stress magnitude is a strong indicator of the propensity of
voiding damage in metal interconnects [20].
By the same author, a more recent study regarding copper interconnects concluded that the stiff
barrier layers surrounding the copper lines promote stress accumulation at the corners, which is likely
to stimulate electromigration and voiding [21]. This study was also performed for annealing conditions
and used experimental data from copper thin films to model plastic strain hardening in encapsulated
copper films. The experimental results suggested that treating copper as an elastic-perfectly plastic solid
with temperature-dependent yield strength (as was usually the practice for Aluminum) may come at the
cost of significant errors in stresses generated inside the copper lines.
Figure 2.10 (a) Schematic of the model interconnect structure and coordinate system adopted by Shen. (b)
Predicted average longitudinal stress in periodic Al lines as a function of line aspect ratio for various line
distances. [20]
(a)
(b)
14
In a 2007 study by H. Mei et al., a FE model was implemented to examine the way residual stresses
in copper interconnect lines are altered due to transmission electron microscopy (TEM) sample
preparation. TEM is a technique that has been used to study the correlations between residual stresses
in copper interconnects and their microstructure. Since it requires electron transparent specimens, one
or more film layers must be sliced off the sample, which ultimately alters the state of stresses that initially
characterized the specimen [1].
Figure 2.11 displays a planar view of the 180nm copper interconnect lines on which the base model
for this thesis’s study is based.
2.4.2 Electromigration
Electromigration (EM), as first mentioned in section 2.3.1, is a major failure mechanism in metal
interconnectors in which various atomic migration processes (promoted by temperature gradients,
thermomechanical stress gradients and electron wind force) are involved. Due to its current relevance,
various studies have been made about electromigration in recent years.
A study of the effects of grain structures on the rate of electromigration-induced failure of copper
interconnects was performed by C. S. Hau-Riege and C. V. Thompson, using scanned laser annealing
(SLA) to produce interconnects with different grain structures [22]. Differences in grain structure were
previously found to lead to orders of magnitude changes in lifetimes for Al-based interconnects.
However, electromigration experiments carried out on copper interconnects with very different grain
structures (from long-grained bamboo structures to polygranular structures in which the average grain
size is less than the linewidth) showed no significant differences in failure rates. These results suggested
that EM in copper interconnects with standard liners and interlevel diffusion-barrier layers occurs by
mechanisms which are faster than grain boundary self-diffusion.
Wei Li et al. studied dynamic physical processes of EM such as void nucleation and void growth
[23]. The intersection of the barrier layer and the copper line was again confirmed to play a vital role in
void formation. Vacancies moving through the grain boundaries become immobilized at that intersection
due to its higher strain energy (higher stresses at this intersection are a consequence of the thermal
mismatch of the different materials present). Since the movement and nucleation of vacancies occurs
to reduce the strain energy, the barrier layer-grain boundary intersections are the weakest points for the
void formation in metal interconnects.
Figure 2.11 TEM planar view of 180nm interconnect lines showing the diffusion barrier (d.b.) and the interlayer
dielectrics (ILD) [1]
15
Many techniques and solutions to mitigate EM have been investigated. C.-K. Hu et al. compared the
effects of EM in copper interconnects with different capping layers. According to the study’s conclusions,
a thin Ta/TaN cap on top of the copper line surface improves electromigration lifetime more effectively
than a SiNx or SiCxNyHz cap. The activation energy for electromigration was shown to have increased
from 0.87eV for cap-less lines to 1.0-1.1eV for SiNx or SiCxNyHz caps and 1.4eV for 1.4 eV for Ta/TaN
capped samples [24]. In figure 2.12, retrieved from that same paper, samples with different caps
damaged by electromigration can be observed. These images were obtained via FIB (focused ion beam)
and TEM (transmission electron microscopy).
2.4.3 Microstructure
A complete study of the behavior of copper interconnects subjected to thermal loads is incomplete
without looking at the texture of copper lines. While macro scale studies may provide insight on where
high stress concentrations can be found in different copper interconnect geometries, the influence that
different crystallographic orientations have on the concentration of stresses at the interconnect lines,
together with copper’s anisotropic behavior cannot be neglected.
Depending on the orientation, dimensions, surroundings, and location of the grains, thermally
stressed copper lines can be subjected to very significant variations (x10) in normal stresses along the
grain boundaries [25]. Normal stresses observed in copper interconnect lines can thus become higher
depending on the orientation of the grains, introducing weak points inside the metal lines where
delamination and accumulation of vacancies is most likely to occur. Annealing and polishing treatments
can significantly alter the texture and strain observed in copper damascene lines [26].
An expeditious method to analyze local texture, local stress and stress gradients in copper
interconnect lines with small width-to-thickness ratios is exemplified by K. J. Ganesh et al [27]. An
automated scanning transmission electron microscope (STEM) diffraction technique was used to map
the local grain orientation of 120nm wide copper interconnect lines, after which an Objected Oriented
Figure 2.12 FIB and TEM images of electromigration damaged samples for (a) SiCxHyNz capped line and (b)
Ta/TaN coated line [24]
16
Finite Element Method software for microstructure analysis (OOF2) is used to analyze the local stresses.
The step by step methodology for this combined technique is illustrated in figure 2.13.
OOF2 is a finite element method software especially designed by NIST [28] for microstructure
analysis. After assigning material properties to an experimental or simulated micrograph, the user
generates a finite element mesh which is then used to compute and visualize the microscopic response
of the microstructure to requested conditions, thus aiding the design of material microstructures of
tailored performance. A more detailed explanation of the inner workings of this software can be read in
“Modelling Microstructures with OOF2” [29].
Figure 2.13 (a) Bright-field STEM image on which the stress analysis was performed. (b) Color-coded replica of
image (a) containing stiffness and orientation information for each grain. (c) Application of a fine mesh on image
(b). (d) Finite mesh and FEM stress solution for every grain. (e) FEM stress solution without the mesh [27]
17
2.5 Analytical Solutions
Various analytical models have been developed to calculate thermal stresses in interconnects. Older
models have had their applicability improved by newer ones, adding more features and considering
more effects and variables upon each iteration. In 2002, Chun-Hway Hsueh developed an analytical
model capable of obtaining approximate general solutions for the stress distributions in passivated
interconnect systems that agreed with the existing finite element calculations corresponding to the same
geometry [30]. The geometry used in these calculations was that of a periodic unit cell cross-section
consisting of a substrate and a planar passivated interconnect, shown in figure 2.14.
The author first derived the asymptotic solutions for the two limiting cases when the aspect ratio of
the interconnect line approaches its limiting values (zero and infinite). After obtaining the closed-form
solutions, the derivation of the general solution followed.
The general solutions obtained by Hsueh [30], for the stress distributions in the x, y and z directions
are shown below as equations 2.1, 2.2 and 2.3, respectively.
𝜎𝑥
𝑖 = 𝜎𝑟𝑒𝑓 + 𝑄1 exp(𝑄3𝑥) + 𝑄2 exp(−𝑄3𝑥) (𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 𝑎)
(2.1)
𝜎𝑦
𝑖 = [−𝑄4(𝛼𝑖 − 𝛼𝑠) − 𝑄5(𝛼𝑖 − 𝛼𝑝)]Δ𝑇/𝑄6 + 𝑄8exp (𝑄7𝑦) + 𝑄9 exp(−𝑄7𝑦) (𝑓𝑜𝑟 |𝑦| ≤ 𝑏)
(2.2)
𝜎𝑧
𝑖 =1
1 − 𝜈𝑖𝜈𝑠
[(𝜈𝑖 − 𝜈𝑠)𝜎𝑦𝑖 − (1 + 𝜈𝑠)𝐸𝑖(𝛼𝑖 − 𝛼𝑠)Δ𝑇]
(2.3)
Where 𝑖, 𝑠, 𝑝 stand for interconnect, substrate and passivation, respectively. 𝐸 is the Young Modulus,
𝜈 is the Poisson’s coefficient and 𝛼 is the coefficient of thermal expansion. Δ𝑇 is the temperature step
Figure 2.14 Geometry used by Hsueh when formulating an approximate general solution for the stress
distributions in passivated interconnects. The unit cell has infinite length in the z-direction and the interconnect is
bonded to the substrate at x=0. [30]
𝒂
𝒃
𝒃′
18
and 𝜎𝑟𝑒𝑓, 𝑄𝑗 are material and geometry dependent coefficients, as found in “Modeling of thermal stresses
in passivated interconnects” [30].
After deriving the general solutions, specific results were obtained for the volume averaged stresses,
for a system of aluminum interconnect lines on silicon substrate with silicon dioxide passivation and
plotted as a function of the aspect ratio in figure 2.15.
Figure 2.15 Volume-averaged thermal stresses in the interconnect calculated by Hsueh; (a) �̅�𝑥𝑖 , (b) �̅�𝑦
𝑖 and (c) �̅�𝑧𝑖
as a function of the aspect ratio, 𝑎/2𝑏. The asymptotic stresses are also shown. [30]
(a) (b)
(c)
19
Chapter 3
Methodology
This chapter details the methods, constitutive models and assumptions used throughout the work.
Convergence and domain size studies are presented and discussed for both the bidimensional and
three-dimensional models.
20
3.1 Modeling
3.1.1 Copper interconnect architecture
There is a wide range of different architectural choices for interconnect lines and different levels of
interconnectors can have increasingly more complex three-dimensional geometries. In the interest of
this thesis’ study, however, a specific architecture was chosen as base-model from which slightly
different variations are performed.
The architecture chosen is mostly based on the multilayer copper interconnect test structure
provided by Freescale Semiconductors in a paper by Mei et al [1], except for the aspect ratio value
which was set to 1.5, as opposed to 1.67, for greater simplicity. The cross-section of the problem is
represented in figure 3.1 (not up to scale).
The measurements and materials used for the model are shown in table 3.1, with total model width
depending on the number of copper lines, 𝑛 ∈ 𝑁.
Table 3.1 Materials and measurements used, in µm.
Component Material Length, Y Width, X Thickness Obs.
Capping Layer Silicon Dioxide 0.25 - -
Cu Line Copper 0.27 0.18 - AR=1.5
Distance between Cu
lines
Silicon Dioxide - 0.18 -
Passivation Layer Silicon Nitride 0.05 - -
Diffusion Barrier Tantalum - - 0.01
Si Wafer Silicon 749 - -
The total length of the cross-section is 750 µm.
0.7 µm
Figure 3.1 Cross-section of the problem for n=4.
𝒙 ≡ 𝟏
𝒚 ≡ 𝟐 749 µm
0.25 µm
0.05 µm
0.2
7 µ
m
0.18 µm 0.01 µm
0.18 µm
21
Upon brief inspection of figure 3.1, it becomes evident that the total cross-section of the problem
can be represented by the repetition (n times) of a smaller geometry that is symmetric to the vertical
axis and infinite in the z direction, which will be called “unit cell”. This unit cell also possesses vertical
symmetry. This is true for both bidimensional and three-dimensional studies. As a result, the full domain
of the problem can be understood through the analysis of a small slice of the geometry with double
vertical symmetry (a “half unit cell”) if a proper symmetry boundary condition is applied to the finite
element model.
This simplification, while greatly decreasing the time needed for the computer simulation to run, still
allows the visualization of a bigger slice of the domain with any number of copper lines desired. This is
achieved through a post-result visualization feature present in the FEA software Abaqus, which admits
the reflection and patterning of the half unit cell analysis results. A visual representation of both the unit
cell and half unit cell can be found in figure 3.2.
3.1.2 Dimensional analysis
The finite element software used, AbaqusTM, is dimensionless. The output units obtained from the
simulations will solely depend on the input unit system chosen by the user. However, if the unit system
used is not coherent, the result output will have no straightforward interpretation. Abaqus also has a
minimum allowed input of 10−5 (geometry-wise) i.e. if the input is given in SI units (meter), the lowest
dimension allowed in the model is 10µm.
Given that our architecture’s lowest dimension is 10 nm, a micrometer-based unit system must be
used. For this purpose, a dimensional analysis was performed, the results of which may be found in
Appendix A.
Figure 3.2 (a) Unit cell. (b) Half unit cell.
(a) (b)
22
3.1.3 Material properties
Material properties were modelled in distinct ways according to necessity. A total of six constitutive
models are used throughout this work (and their abbreviations will be consistently referenced in
upcoming sections).
For pure elastic regime:
Elast1: All materials are linear elastic solids whose properties are shown in table 3.2. Linear
elasticity is an assumption that implies there is a linear relationship between stress and engineering
strain, and it is useful in the case of small deformations/loads.
Elast2: Same as “Elast1”, but copper’s coefficient of thermal expansion (CTE) is treated as
temperature-dependent in accordance with table 3.3.
For plastic regime:
IdPlast: Same as “Elast2” but copper’s yield stress is treated as temperature-dependent in
accordance with table 3.4. In other words, copper is treated as an isotropic elastic-perfectly plastic
solid, meaning that it experiences no work-hardening during plastic deformation, while the rest of
the model undergoes no plastic deformation.
WHardCu: Same as “Elast2” but copper undergoes non-temperature dependent work-
hardening while the rest of the model undergoes no plastic deformation. The true stress-strain curve
used for copper (at 20ºC) is shown in figure 3.4. Work hardening (or strain hardening) is the
strengthening of a material by plastic deformation.
WHard: Same as “Elast2” but the whole model undergoes non-temperature dependent work-
hardening and the true stress-strain curves used for each material (at room temperature, 20ºC) are
shown in figures 3.4-3.8.
Plast: Same as “Elast2” but copper and tantalum undergo temperature dependent work
hardening and the true stress-strain curves of these two materials are shown in figures 3.4 and 3.5.
The rest of the model undergoes non-temperature dependent work-hardening and the true stress-
strain curves used (at room temperature, 20ºC) are shown in figures 3.6-3.8. To the author’s
knowledge, reliable sources for stress-strain curves (at different temperatures) of all other materials
present in the model are scarce, not free of charge and, in many cases, not directly applicable to
the present scenario; so although this constitutive model is the most complete out of all presented,
it is still an intermediate step of a more comprehensive representation of reality. Silicon, however, is
known to have brittle behavior for temperatures under 500ºC [31]. Consequently, it is safe to assume
this material is not being completely mischaracterized by a temperature independent plastic model.
Figure 3.3 is a comprehensive diagram detailing the main differences between each constitutive
model.
23
Table 3.2 Material properties used for copper in Elast1 and for all other materials in the remaining models
Material CTE
(× 𝟏𝟎−𝟔 ℃−𝟏)
Young’s Modulus
(MPa)
Poisson
Ratio
Silicon [1] 4.2 165 000 0.22
Silicon Dioxide [32] 0.55 68 000 0.15
Silicon Nitride [1] 2.9 290 000 0.27
Tantalum [32] 6.6 183 000 0.35
Copper [1] 17 128 000 0.36
Table 3.3 Coefficient of thermal expansion used for copper in all models except for Elast1
Material Temperature
(ºC)
CTE
(× 𝟏𝟎−𝟔 ℃−𝟏)
Copper [33]
20 15.4
27 15.4
77 15.8
100 15.9
127 16.2
177 16.5
200 16.6
227 16.9
277 17.3
Constitutive models
Elastic
𝐶𝑇𝐸𝐶𝑢= 𝑐𝑜𝑛𝑠𝑡.
Elast1
𝐶𝑇𝐸𝐶𝑢= 𝑓(𝑇)
Elast2
Plastic
Only copper
Ideally plastic
IdPlast
Temperature independent
work-hardening
WHardCu
Entire model
Temperature independent
work-hardening
WHard
Copper and tantalum -
temperature dependent work-
hardening. Rest of the model -temperature
independent work-hardening.
Plast
Figure 3.3 Main differences between all constitutive models
24
When using plastic assumptions, knowledge of true stress-strain curves of all materials is needed.
Such detailed information regarding material behavior is usually either unavailable or not free of charge.
Besides, the same material can have considerably different stress-strain curves depending on its purity
and/or fabrication methods.
As a result, a reconstruction of what could be the stress-strain curves of every material was
elaborated based on information that is publicly available. Since this reconstruction uses only 2 points
(corresponding to the yield and ultimate tensile stresses, this constitutive model is a linear hardening
model.
Although the values chosen were based on acceptable value intervals, these curves are tentative.
For silicon and silicon dioxide, an effort was made for the reconstructed stress-strain curve to reflect the
brittle nature of these materials. Tables 3.4 and 3.5 show the material properties used in the
reconstruction of the true stress-strain curves of all materials in the model.
Table 3.4 Values used in the reconstruction of the true stress-strain curves of copper and tantalum (for various
temperatures)
Material Temperature
(ºC)
Yield Stress
(MPa)
Ultimate Tensile Stress
(MPa)
Copper [33]
20 211 250
100 205 230
200 195 200
250 170 175
Tantalum [34]
20 204 404
100 166 366
200 148 349
250 145 345
Table 3.5 Values used in the reconstruction of the true stress-strain curves of all materials (for 20ºC)
Material Yield Stress
(MPa)
Ultimate Tensile
Stress (MPa) Elongation
Silicon [32] 180 180 0.0015
Silicon Dioxide [32] 50 50 0.00075
Silicon Nitride [32] 250 250 0.09
Tantalum [34] 204 404 0.18
Copper [33] 211 250 0.3
The stress values shown are engineering values which must be converted to true values. The
elongation values presented correspond to the post-fracture strains and not the ultimate tensile stress
abscissae. In the absence of an experimental value, the maximum uniform strain can be taken as
approximately equal to the elongation at break (𝜀𝑈𝑇𝑆 = 𝑒𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛).
Considering that Hooke’s Law is valid until the yield stress;
𝜀𝑦 =𝜎𝑦
𝐸 (3.1)
25
To obtain the true stress and strain values;
𝜎𝑇 = 𝜎𝑁(1 + 𝜀𝑁) (3.2)
𝜀𝑇 = ln(1 + 𝜀𝑁) (3.3)
And finally, to calculate the true plastic strain (necessary input in Abaqus - please note that the true
plastic strain at the yield point is zero);
𝜀𝑝𝑙 = 𝜀𝑇 −
𝜎𝑇
𝐸
(3.4)
In figures 3.4—3.8 the reconstructed stress-strain curves of each material are shown. The
coarseness of this approximation is evident and to be expected since appropriate values for the stresses
and strains could only be tentatively guessed at two points (corresponding to the yield and ultimate
tensile stresses).
Figure 3.4 True stress-strain curves of copper at different temperatures
0
50
100
150
200
250
300
350
0 0.05 0.1 0.15 0.2 0.25 0.3
Tru
e s
tre
ss (
MP
a)
True strain
20ºC
100ºC
200ºC
250ºC
0
50
100
150
200
250
300
350
400
450
500
0 0.05 0.1 0.15 0.2
Tru
e s
tre
ss (
MP
a)
True strain
20ºC
100ºC
200ºC
250ºC
Figure 3.5 True stress-strain curves of tantalum at different temperatures
26
The properties shown for tantalum in tables 3.4 and 3.5 were based on a constitutive model featured
in article by Chen and Gray III [34]. The values of the yield and ultimate tensile stress of tantalum, for
the temperatures required in this study, were estimated as follows: in the article, the constitutive relations
for Ta and Ta-W alloys were derived for different models. For the purpose of this study, the Zerilli-
Armstrong (ZA) model and its constitutive relations are used:
𝜎 = 𝑃1 + 𝑃2 ∙ exp(−𝑃3 ∙ 𝑇 + 𝑃4 ∙ 𝑇 ∙ ln 𝜀̇) + 𝑃5 ∙ 𝜀𝑝𝑙𝑃6 (3.5)
Where 𝑃1−6, are constants for this model and, according to table II of the article, for tantalum, these
fitting parameters are 140MPa, 1750MPa, 0.00975/K, 0.000675/Ks, 650MPa and 0.650, respectively.
The strain rate, 𝜀,̇ used in the following calculations was 0.1/s. Using equation 3.5, for 𝜀𝑝𝑙 = 0 and 𝑇 =
0
50
100
150
200
0 0.0025 0.005 0.0075 0.01
Tru
e S
tre
ss (
MP
a)
True Strain
Figure 3.6 True stress-strain curve of silicon at 20ºC
0
10
20
30
40
50
60
0 0.00025 0.0005 0.00075 0.001
Tru
e S
tre
ss (
MP
a)
True Strain
Figure 3.7 True stress-strain curve of silicon dioxide at
20ºC
0
100
200
300
0 0.05 0.1 0.15 0.2
Tru
e S
tre
ss (
MP
a)
True Strain
Figure 3.8 True stress-strain curve of silicon nitride at 20ºC
27
293, 373, 473, 523𝐾 the values of the nominal Yield Stress for 20, 100, 200 and 250ºC were obtained,
as shown in table 3.4.
For the ultimate tensile stress, figure 3.9 is used. Curve 6 of this figure corresponds to a strain rate
equal to the one chosen previously at a temperature of 25ºC. For the purpose of this study, this curve
was assumed to be comparable to a stress-strain curve for the same strain rate at 20ºC. The maximum
elongation, for this case, would be around 0.18 according to figure 3.9 and will be assumed to remain
constant regardless of temperature (the same was previously assumed for the maximum elongation of
copper as well) and to be approximately equal to the maximum uniform strain (𝜀𝑈𝑇𝑆 = 𝑒𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛).
The (nominal) plastic strain can be obtained from equation 3.6:
𝜀𝑝𝑙 = ln(1 + ε) − (
𝜎
𝐸)
(3.6)
Solving equations 3.5 and 3.6 simultaneously, 𝜀𝑈𝑇𝑆 = 0.18, will result in the nominal values of
ultimate tensile stresses registered in table 3.4 and in the plastic strain values later used to reconstruct
the stress-strain curves of figure 3.5.
Figure 3.9 Compressive stress-strain curves of the unalloyed tantalum under dynamic and quasi-static
deformation at various temperatures [34]
28
3.1.4 Thermal loading and boundary conditions
The thermal load is simulated through a global (isothermal) application of a 1 second heating step
(Δ𝑇 = 230°𝐶) from room temperature, 20°𝐶, to a maximum temperature of 250ºC; followed by a 1
second cooling back to room temperature (figure 3.10). The finite element model used in this study is
time independent because viscoplasticity and stress relaxation effects, although relevant, are not
considered in the present study. The maximum temperature chosen is referred to as “critical” because
it is the temperature at which the effects of stress meet the effects of diffusion and voiding ensues.
The boundary conditions used in the model are schematized in figure 3.11. The symmetry condition,
as discussed in section 3.1.1, is meant to simulate the periodicity of the copper lines. The top surface is
free to move during deformation and the bottom surface is pinned to prevent rigid body motion.
These loading and boundary conditions are unchanged for every study elaborated in the scope of
this work and results will always be observed at either t=1s (maximum temperature) or t=2s (after
cooling).
Figure 3.11 Boundary conditions used in the model
Vertical symmetry condition, 𝑢𝑥 = 𝑢𝑅,𝑦 = 𝑢𝑅,𝑧 = 0
Pinned condition, 𝑢𝑥 = 𝑢𝑦 = 𝑢𝑧 = 0
𝑥 ≡ 1
𝑦 ≡ 2
0
50
100
150
200
250
0 1 2
Te
mp
era
ture
, ºC
Time, s
Figure 3.10 Thermal loading condition used in the simulation
29
3.1.5 Simplifications and assumptions
The following simplifications are common to all studies elaborated in this work:
• The initial stress distribution present in the model is zero. In a real-life scenario the copper lines
are subjected to residual stresses introduced during fabrication. A way to simulate these
residual stresses in a finite element simulation would be, for example, through the application
of an initial state of stress/strain whose values would be dependent on the fabrication method
used for the copper lines. Results are still meaningful, however, as it is a limit case with practical
interest.
• All materials are assumed isotropic. In other words, their physical properties are considered
constant in all directions/orientations.
• Slip is not considered at the materials interfaces, where materials share the same nodes.
• For all bidimensional studies, plane strain finite elements are used:
𝜀𝑥𝑧 = 𝜀𝑦𝑧 = 𝜀𝑧𝑧 = 0 (3.5)
3.2 Domain size studies
3.2.1 Bidimensional model
Examining figure 3.1 the domain size, in particular the length of the Si wafer, is found to be a relevant
issue. Its large size (compared with the other sub-regions of the model) prevents a clear visualization of
the entire domain. Moreover, it leads to higher computational times, therefore demanding more
processing power.
Consequently, a study of the influence of the silicon wafer size (depth, y) on the thermal stresses at
the copper lines is needed to understand if the finite element model can be further simplified.
To study the influence of the silicon wafer size on the thermal stresses at the copper lines, multiple
models with different wafer lengths, w, were created. Note that {𝑤 ∈ 𝑅| 0 ≤ 𝑤 ≤ 749} and all lengths
are represented in µm. A coarse mesh with an approximate global size of 0.037µm (minimum allowed
size by the FEA software for the 𝑤 = 749µ𝑚 model) was used for all models, with plane strain linear
quadrangular elements.
Given that all models share the same loading and boundary conditions and the same mesh size,
the difference in results will depend only on each model’s geometry, more specifically, on the wafer size.
For this kind of coarse analysis, the shape of the element (be it triangular or quadrangular) is not very
relevant and would not bias the results. The material properties used in this study correspond to
constitutive model Elast1 detailed in section 3.1.3 (chosen due to its simplicity).
Various control point located inside the copper lines were chosen. An analysis was performed for
each model and the values of the different stresses registered for the same control points. In figure 3.12
the locations of the chosen control points are represented. In figures 3.13 and 3.14, the influence of the
30
wafer length in the stress values registered at different points, for each model, is shown. Note that these
stress values were taken at t=1s, or in other words, at maximum temperature.
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
0 100 200 300 400 500 600 700 800
Str
esse
s a
t C
P_B
in
MP
a
Wafer size in µm
Mises S11 S22 S33
Figure 3.13 Influence of the wafer length on the stresses observed at control point B.
𝑥 ≡ 1
𝑦 ≡ 2
0.5
µm
µm
Figure 3.12 Control points used for the bidimensional wafer size study.
CPB w
µm
CPB1 CPB2
CPB3
CPSi
31
From figures 3.13 and 3.14, it is reasonable to conclude that the length of the wafer used for the
finite element model does not significantly impact the stresses observed at the copper lines. Since the
focus of this study is to investigate the thermal loads at the copper lines, it is plausible to assume that
the choice of the wafer length used in the model is arbitrary for this specific type of analysis.
That said, a decision still needs to be made and a compromise must be achieved. While the model
should be visually appealing and intuitive (the wafer should be visibly larger than the rest of the
component), the global size of the geometry should be small enough to allow a smooth mesh
manipulation and fast computing. All things considered, a wafer size of 3µm seems both reasonable
and convenient.
To understand exactly how the wafer length influences the finite element analysis, however, two
other studies were performed for further investigation:
i) The model with wafer length 𝑤 = 3𝜇𝑚 was utilised to run four simulations, each with a different
silicon CTE value. The objective was to understand how much the order of magnitude of the
wafer CTE influenced the analysis results.
ii) Seven models with different wafer lengths, 𝑤 ∈ {1,5,10,50,100,500,749} were chosen for the
second study, where the CTE of the wafer was kept at its real value (4.2 × 10−6). The purpose
was to take models with different sized wafers for a more detailed analysis, where not only the
Von Mises stress but also other outputs are examined.
Only two of the previously defined control points (CPB and CPSi) were used in these studies. Instead
of only considering stresses present in the copper lines, the values of the normal stresses S11 (𝑥-
direction) and S22 (𝑦-direction), as well as the shear stress S12, the spatial displacements U1, U2 and
the spatial displacement magnitude (Umag) were examined at both control points (nodes).
Both studies used, again, the material properties corresponding to constitutive model Elast1 detailed
in section 3.1.3 and results were taken at the end of the heating step. The mesh used throughout all
0
100
200
300
400
500
600
0 100 200 300 400 500 600 700 800
Vo
n M
ise
s s
tre
ss in
MP
a
Wafer size in µm
CP_B CP_B1 CP_B2 CP_B3
Figure 3.14 Influence of the wafer length on the Von Mises stresses observed at various control points inside
the copper lines.
32
studies belonging to this section remains unchanged. The results of both studies (i) and (ii) are shown
and discussed in Appendix B and C, respectively.
The plots presented in Appendix B and C, except for all x-displacements (U1) of CPSi, have no other
null values registered, only very small ones (~10−4). Because the values registered in both control points
are, for the most part, very different, it might seem like the values for S22 and S12 of CPSi are zero when
in fact they are not.
Although the stresses at the copper lines are shown to be unaffected by the wafer length, the reason
for which the real model has a 749 µm-sized wafer can be thermal and/or structural. Because in a real-
life scenario the part is subjected to other types of loadings while inside the electrical circuit, a bigger
substrate could greatly increase the structural stability of the whole component.
3.2.2 Three-dimensional model
It is necessary to confirm if, for the 3D model, the stress values at the copper lines are still
independent of the wafer length, since it might be incorrect to assume that the 3D model can have the
same wafer size, 𝑤 = 3µ𝑚 , as the 2D model.
The stress values at surface control point B, for the 3D model, are lower than intuitively expected
because it is located on a free surface, where stresses tend to be lower. A “control edge” CPedge was
created along which stresses were plotted against the model’s thickness (z-dimension). This control
edge is a line segment that begins at control point B and is perpendicular to the front surface (parallel
to the z-axis) so that it passes through the middle of the (half unit cell’s) copper line, i.e. it is the geometric
centerline of the (half unit cell’s) interconnect line. The location of this control edge is exemplified in
figure 3.15 (the dashed line indicates that the edge is “inside” the model).
The results were as expected: along the thickness z (CPedge), the Von Mises stress distribution is
symmetrical and tends to a maximum at around 𝑍 =𝑧
2𝜇𝑚 (in this case, for 𝑍 = 4µ𝑚, the maximum Von
Mises stress is registered at 𝑍 = 2µ𝑚). Figure 3.16 illustrates this behavior for a model with 𝑍 = 4µ𝑚
and 𝑤 = 3µ𝑚.
In the wafer size study, models of different wafer sizes 𝑤 ∈ {2,3,4,5,9,15} (in µm) with constant Z
thickness equal to 4µm were created and stress values were taken at 𝑍 = 2µ𝑚, where they are highest.
A biased hex linear 0.05/0.02µm mesh was used (this choice will be explained in section 3.3.2 when
discussing 3D model convergence). The results of the wafer size study for the 3D model are displayed
in figure 3.17. The material properties used in this study correspond to constitutive model Elast1 detailed
in section 3.1.3 (chosen due to its greater simplicity).
From figure 3.17, the stress dependence on the wafer size of the model became more evident for
the 3D model. Nevertheless, the percentual errors measured between a wafer size of 15µm and 3µm
were 3.24%, 0.55%, 0.125% and 1.71% for the Von Mises stress, S11, S22 and S33, respectively. For
engineering purposes, these errors can be considered small and/or even negligible.
Since the main purpose of the 3D model is to draw a comparison with the 2D studies and given that
the percentual error is so small, the wafer size for the 3D model was kept at w=3μm to stay in
concordance with the 2D model studies as much as possible.
33
Figure 3.16 Plot demonstrating the behavior of the Von Mises stress along CPedge for a wafer size of 𝑤 = 3𝜇𝑚
and model thickness 𝑍 = 4𝜇𝑚.
Figure 3.15 Location of CPedge.
CPB
CPedge
𝑥 ≡ 1
𝑦 ≡ 2
𝑧 ≡ 3
Str
ess (
Von M
ises,
MP
a)
Z distance along path (µm)
34
3.2.3 Study of the influence of the diffusion barrier
In figure 3.16, a disturbance is shown at the start and end of the stress curve. To try to understand
the reason behind such behavior, a small study was performed to verify if the diffusion barrier section
of the geometry could be the cause. The diffusion barrier section was the first study candidate due to its
unique geometry, setting it apart from the rest of the model: its L-shape contrasts with the squared shape
of most of the model sections.
Taking the 𝑤 = 3𝜇𝑚 and 𝑧 = 4𝜇𝑚 3D model from the previous section and using a biased hex linear
0.05/0.02µm mesh (discussed ahead, in section 3.3.2), two hypotheses were tested:
(1) The strange behavior is due to the diffusion barrier properties: the fact that a L-shaped section
of the domain has distinct properties creates a discontinuity in the stress curve along CPedge;
(2) The origin of the discontinuity is purely geometrical (due to the L-shape) and has nothing to do
with the material properties of the diffusion barrier section;
To test these two hypotheses, two slightly different models were created and tested against the
model used in the previous section (3.2.2) and whose stress curve was represented in figure 3.16. Model
1 kept the initial geometry but had the properties of tantalum replaced by those of copper at the diffusion
barrier section. In Model 2 the diffusion barrier section was eliminated and replaced by an enlargement
of the copper line’s domain. All material properties used in this study correspond to constitutive model
Elast1 detailed in section 3.1.3 (chosen due to its greater simplicity).
For both cases, the Von Mises stress curve along CPedge was obtained and is displayed in figures
3.18 and 3.19, respectively. The results unquestionably corroborate hypothesis 2. Please note that, in
Model 2, CPedge no longer passes through the geometric centre of the halved copper line and is instead
kept in the exact same location as in Model 1, in other to allow for a direct comparison between models.
-600
-500
-400
-300
-200
-100
0
100
200
300
400
0 2 4 6 8 10 12 14 16
Str
esse
s a
lon
g C
P_
ed
ge
at
Z=
2µ
m
Wafer size in µm
Von Mises S11 S22 S33
Figure 3.17 Results of the wafer size study for a 3D model of thickness 𝑍 = 4𝜇𝑚.
35
3.3 Convergence studies
3.3.1 Bidimensional model
One of the most important steps towards validating the finite element model is assuring its
convergence. As the number of finite elements used in the model increases, the simulation results are
expected to tend to and stabilize around a given value. The following convergence study was performed
for a model of wafer size 3µm, as discussed previously in sections 3.2.1 and 3.2.2.
The material properties used in this study correspond, again, to constitutive model Elast1 detailed
in section 3.1.3 (chosen due to its greater simplicity) and more control points (figure 3.20) were added
to the ones used in previous sections.
A mesh with linear quadrangular plane strain elements of initial approximate global size 0.03 µm
was increasingly and uniformly refined. For each new mesh size iteration, the output values of the
averaged Von Mises stress and displacement magnitude (Umag) were registered at each control point
at the end of the heating step (i.e. at maximum temperature). The element shape was chosen to
Figure 3.18 Results of the diffusion barrier influence study for Model 1
Figure 3.19 Results of the diffusion barrier influence study for Model 2
Str
ess (
Vo
n M
ises,
MP
a)
Z distance along path (µm)
Str
ess (
Vo
n M
ises,
MP
a)
Z distance along path (µm)
36
elegantly match the geometry of the problem. The results for control points A and B can be found in
figures 3.21 and 3.22, respectively. Results for control points C, D and Si are compiled in Appendix D.
Figure 3.20 Control points for the bidimensional convergence study
CPD
CPC
CPA
CPB
CPSi
3 µ
m
0.5
µm
CPB1
CPB3
CPB2
CPB4
𝑥 ≡ 1
𝑦 ≡ 2
0
200
400
600
800
1000
1200
1400
1600
Vo
n M
ise
s s
tre
ss a
t C
PA
, M
Pa
Number of finite elements
0.00583080
0.00583100
0.00583120
0.00583140
0.00583160
0.00583180
0.00583200
0.00583220
0.00583240
0.00583260
Um
ag
at C
PA
, µ
m
Number of finite elements
Figure 3.21 Convergence study for control point A
37
Except for control point A, all other results suggest that convergence is reached at around 30 400
finite elements, or in other words, when the mesh has an approximate global size of 0.005µm.
This conclusion is true regardless of the shape and geometric order chosen for the finite elements.
To support this statement, four simulations were run with the same global mesh size of 0.005µm but
different types of finite elements. The results obtained in each simulation for the Von Mises stress (MPa)
at control point B (geometric center of the half-unit cell copper line) are shown in table 3.6.
Table 3.6 Von Mises stress at CPB, in MPa, for different types of finite elements
Geometric order
Linear Quadratic
Ele
men
t S
hap
e
Triangular 417.301 417.298
Quadrangular 417.308 417.300
As previously mentioned, control point A was yet to converge for a mesh of 2 113 439 finite elements.
The simulation time and computing power needed started being so considerable that a different
approach had to be taken.
Point A poses a challenge to the convergence of the model, for two reasons: it is the interface
between 3 materials with different thermal expansion coefficients and it is also a 90-degree corner,
where stresses usually tend to concentrate.
To make sure that convergence is achieved even for point A, a biased mesh had to be created.
While the approximate global element size is kept at 0.005µm (the value for which the rest of the model
converged), the mesh is increasingly tightened around point A with each iteration.
417.2
417.3
417.4
417.5
417.6
417.7
417.8
417.9
418
418.1
418.2
418.3
Vo
n M
ise
s s
tre
ss a
t C
PB
Number of finite elements
0.00519800
0.00519900
0.00520000
0.00520100
0.00520200
0.00520300
0.00520400
0.00520500
0.00520600
0.00520700
Um
ag
at C
PB
, µ
m
Number of finite elements
Figure 3.22 Convergence study for control point B
38
In addition, and since control point E converged sooner than any other (Appendix D), a biased mesh
was also implemented on the silicon wafer section: elements were created at the bottom surface of the
wafer with an initial size of 0.02µm (bigger than the approximate global size) and made to increasingly
shrink along the vertical direction until a size of 0.005µm is reached at the silicon-silicon dioxide
interface. This approach greatly economizes the simulation in terms of time consumption through a
drastic reduction in the number of finite elements used. The results for the biased mesh convergence
study of control point A can be found in figure 3.23.
Due to the increasingly more significant element distortion observed in smaller meshes (which, if
too big, can trigger warnings and/or bias simulation results), control point A was considered to have
converged at 38 000 elements, with a mesh of approximate global element size 0.005µm and minimum
element size of 0.00006 µm. Although it was not assumed to have converged with the same fineness
as the other control points, it is still worth noticing that the percentual error between the 61636-element
mesh (the finest one shown in figure 3.23) and the mesh chosen is 0.115% so, for practical purposes,
convergence can be said to have been achieved.
This biased mesh, however, is not always the most convenient because:
(a) Bigger versions of the base model will greatly increase simulation time and, sometimes,
require a bigger amount of computer memory;
(b) For geometry studies, the base model needs to be reshaped. If the same biased mesh is
applied to certain reshaped geometry variants, distorted or misshaped finite elements arise,
which can badly compromise results.
One solution would be to apply a uniform linear quadrangular mesh of global seed size small enough
to provide significant results and big enough not to compromise time or computer efficiency. Uniform
meshes of different sizes were tested and the values of stresses S11, S22 and S33 taken at control
points A, B, B1, B2, B3 and B4 (all located inside the copper lines) with the intent of understanding whether
a biased mesh is strictly necessary. The results for the Von Mises stress are shown in figure 3.24 and
in Appendix D for S11, S22 and S33.
1040
1045
1050
1055
1060
1065
1070
1075
1080
1085
1090
0 20000 40000 60000 80000
Vo
n M
ise
s s
tre
ss a
t C
PA
, M
Pa
Number of finite elements
0.005
0.0051
0.0052
0.0053
0.0054
0.0055
0.0056
0.0057
0.0058
0.0059
0 20000 40000 60000 80000
Um
ag
at C
PA
, µ
m
Number of finite elements
Figure 3.23 Biased mesh convergence study for control point A
39
Except for control points A and B4 (which, as discussed previously, are corner points and therefore
“problematic”), the rest seem to have reached convergence at around 30400 finite elements, or in other
words, for a uniform mesh of 0.005µm. For a point at the geometric centre of the (half) copper line (point
B), the difference between stresses observed when a uniform mesh of 0.005µm is used (as opposed to
the biased mesh for which the corner nodes converge) results in an relative error of only 0.003%.
These results suggests that the aforementioned “problematic” behavior is restricted to the “corner”
nodes and so a uniform mesh of 0.005µm will not gravely mischaracterize any results and/or
conclusions, at the very least for not the geometry studies (section 4.2), where the tendencies in stress
(between models) are the main object of study. As further reinforcement to this conclusion, figures 3.25
and 3.27 show the Von Mises stress distribution (in MPa) of the same base-model, for both the refined
biased mesh at which point A converged and the uniform 0.005µm mesh, respectively.
0
200
400
600
800
1000
1200
Vo
n M
ise
s s
tre
ss, M
Pa
Number of finite elements
CP_A CP_B CP_B1 CP_B2 CP_B3 CP_B4
Figure 3.24 Uniform mesh convergence study at various control points – Von Mises stress
Figure 3.25 Von Mises stress distribution (in MPa) for the refined biased mesh
40
3.3.2 Three- dimensional model
Further validation of the 2D model and its studies could be obtained through the study of an
equivalent 3D model: the plane strain assumption is validated if, for a certain thickness Z, the stress
values obtained in the 3D simulation align with the ones obtained for an equivalent 2D model.
However, the addition of a new dimension increases the computational power and memory needed
to run the finite element simulation. Besides, a study of the stresses at the copper lines as a function of
the new dimension thickness Z, is necessary to validate the 2D studies. This means that the mesh
chosen needs to stay constant for all thicknesses studied, so it must be coarse enough that the increase
in model volume does not compromise the computer’s ability to run the simulation.
Given that the computer used for these studies has 8GB of RAM, the usage of the previous 0.005µm
mesh is unsuited for the 3D model (and so convergence for the “corner” nodes had to be adapted). The
best compromise, in this case, is to have a linear hex biased mesh of 0.02µm size around the copper
lines and 0.05µm in the rest of the model (corresponding to 78320 finite elements for the model used in
the study). Please note that, for this convergence study, an arbitrary thickness of 𝑍 = 4µ𝑚 and wafer
size 𝑤 = 3µ𝑚 was used as the third-dimension value. Detailed studies of the z-dimension and wafer
sizes will be elaborated in section 4.3.
The convergence study that led to this conclusion is presented below in figure 3.27. The control
point used for this study (still called CPB) is located on the frontal surface of the 3D model, where control
point B was located for the 2D studies (on the geometric center of the halved copper line). Figure 3.28
illustrates the exact location of CPB, while figure 3.29 shows the final mesh chosen along with the 3D
model used for this convergence study.
Figure 3.26 Von Mises stress distribution (in MPa) for the uniform 0.005µm mesh
41
Figure 3.29 Mesh used for the 3D model studies. Figure 3.28 Location of control point B for the 3D model.
CPB
𝑥 ≡ 1
𝑦 ≡ 2
𝑧 ≡ 3
0
20
40
60
80
100
120
140
0 50000 100000 150000 200000
Vo
n M
ise
s s
tre
ss @
CP
_B
; M
Pa
Number of finite elements
Figure 3.27 Convergence study for a 3D model of thickness Z=4µm.
42
Chapter 4
Results and discussion
This chapter presents the main results. A comparison between all constitutive models is featured.
The influence of geometric parameters on the stress distributions at copper interconnect lines is studied
and the bidimensional studies are validated by a three-dimensional analysis.
43
4.1 Plasticity studies
In chapter 3.1.3, six constitutive models were presented. In this section, they are studied to
understand which one(s) should be given greatest emphasis on the upcoming geometry (and 3D)
studies.
In previous studies, constitutive model Elast1 was chosen due to its simplicity. It is not uncommon,
when using finite elements, to start with a simpler model and increasingly build on top of it. For that
reason, constitutive model Elast1 was the obvious choice for all initial studies (convergence, wafer size,
etc.). As the need for a more faithful representation of a real-life scenario arrives, however, other
methods of simulating material behavior must be employed to assure that the results obtained are as
meaningful as possible.
Because the copper lines are the most critically important region in the model, their characterization
should be made with an added level of detail. Constitutive model Elast2 is a consequence of that need
grouped with the fact that, to the author’s knowledge, reliable sources for detailed properties of all other
materials present in the model are scarce. The Young’s modulus of copper is still considered constant
because, although it is known to change with temperature, the change is very small and therefore
negligible in the small temperature interval considered.
To model plasticity, a different number of approaches can be taken. As touched upon briefly in
section 2.4.1, experimental measurements have suggested that ignoring work hardening in copper lines
can induce significant errors in stresses generated inside the copper lines [18]. However, modelling
work hardening with little to none access to precise experimental data and detailed material properties
results in the use of the coarsely constructed stress-strain curves, which nonetheless are closer to the
real-life scenario.
Even the best constitutive model used in this study, Plast, can be viewed as an approximation to a
real-life scenario due to the lack of temperature dependent data on silicon, silicon dioxide and silicon
nitride. Nonetheless and given that each new model was created just slightly more complex than the
previous one, a comparison between the six models could prove useful in understanding the way each
assumption affects the stresses observed at the copper lines.
Constitutive models IdPlast and WHardCu represent the two cases where purely temperature-
dependent plasticity (yield strength) is tested against temperature independent work-hardening. In both
models, only copper experiences plastic deformation while the rest of the model is in pure elastic regime.
A comparison between these two models can help understand the individual effects of temperature
dependency and work-hardening on stresses and, when compared to a model where all materials
undergo plastic deformation (WHard, Plast), the differences and consequences of considering plastic
deformation on copper versus the entirety of the model can also be observed.
Model WHard considers temperature independent work hardening for the whole model and, when
compared to model WHardCu, is used to understand whether focusing solely on copper’s properties is
a plausible approach. Finally, model Plast is the most complete model featured in this study and
presents temperature dependent work-hardening in copper and tantalum and temperature independent
work-hardening for the rest of the materials.
44
A study was made to test all six constitutive models against each other: for the refined biased mesh
for with control point A achieved convergence in section 3.3.1, the values of the Von Mises stress and
hydrostatic stress (1
3(𝑆11 + 𝑆22 + 𝑆33)) were taken at control point B (figure 3.20) for each of the six
constitutive models.
In the case of plastic constitutive models, results were taken at maximum temperature (t=1s) and
after cooling (t=2s). For pure elastic regime results are taken only at t=1s, as no residual stresses occur
in elastic simulations. The outcome of this study is shown in figures 4.1 and 4.2.
Figure 4.1 Von Mises and hydrostatic stress values (MPa) at control point B for all constitutive models, at t=1s
-500 -400 -300 -200 -100 0 100 200 300 400 500
Elast1
Elast2
IdPlast
WHardCu
WHard
Plast
Hydrostatic Von Mises
0 50 100 150 200 250
IdPlast
WHardCu
WHard
Plast
Hydrostatic Von Mises
Figure 4.2 Von Mises and hydrostatic stress values (MPa) at control point B for all plastic constitutive models, at
t=2s
45
The results for models Elast1 and Elast2 are very similar for both Von Mises and hydrostatic
stresses, so considering the CTE of copper constant or temperature dependent does not produce large
differences in results.
In the models where only copper is considered plastic (IdPlast and WHardCu), the hydrostatic stress
values observed at t=1s are lower than in the case of pure elasticity and higher than in globally plastic
models (WHard and Plast). The residual hydrostatic stresses (t=2s), however, are higher for the models
where all materials can deform plastically.
For plastic constitutive models, the Von Mises stress values at t=1s are similar for temperature
independent plastic models (WHard and WHardCu) and higher than for temperature dependent plastic
models (IdPlast, Plast). The residual Von Mises stresses seem independent of the plastic model
considered, which may be an indication that, for the upcoming geometry studies, evaluating hydrostatic
stresses can prove more useful than Von Mises stresses when comparing constitutive models.
Figures 4.3 and 4.4 show how different constitutive models translate to different distributions of
stress and plastic strain when the temperature is highest. A tendency for stress and/or plastic strain
accumulation at the interface between the diffusion barrier and the interconnect lines is observable and
had been previously mentioned in the literature [21].
Results are gravely influenced by the constitutive model used and, when modelling material
behavior, assumptions and simplifications chosen must be given great thought. In the upcoming
geometry studies (section 4.2), results will first be shown and compared for all constitutive models (in
terms of hydrostatic stresses) and later in detail (S11, S22, S33 and Von Mises) for models Elast2 and
Plast only. For the three-dimensional studies, models Elast1 and Plast were chosen to demonstrate how
a three-dimensional model of appropriate thickness tends to the bidimensional solutions.
46
Figure 4.3 Von Mises stress distribution (MPa) for all constitutive models at t=1s
WHardCu
Elast2
IdPlast
Elast1
WHard Plast
47
Figure 4.4 Distribution of plastic strain for all plastic constitutive models at t=1s
IdPlast
WHard Plast
WHardCu
48
4.2 Geometry studies
4.2.1 Overview
It is most relevant to understand how stresses at the interconnect lines change as a function of the
model’s geometry. Three different geometry studies were performed in this section: one based on aspect
ratio, another one based on the distance between lines and lastly one regarding the influence of line
width.
The mesh used for all these studies was the 0.005µm uniform linear quadrangular mesh discussed
previously in section 3.3.1. All stress values plotted are for control point B which, regardless of each
model’s architecture, is always defined as being located at the geometric center of the (half) copper line
(represented in both figures 3.12 and 3.20). Results are taken, for plastic regime simulations, both at
maximum temperature (t=1s) and after cooling (t=2s). For pure elastic regime simulations, results are
taken only at t=1s since no residual stresses would be present by the end of cooling.
4.2.2 Aspect ratio
From the base model architecture presented in section 3.1.1, the width (x-dimension) of the
interconnect lines was kept at 180nm, while the length (y-dimension) was changed to create different
aspect ratios, 𝐴𝑅 ∈ {0.25, 0.4, 0.5, 0.6, 0.75, 1.0, 1.5 (𝑏𝑎𝑠𝑒 𝑚𝑜𝑑𝑒𝑙), 2.0, 2.5 } with 𝐴𝑅 =𝑑𝑒𝑝𝑡ℎ
𝑤𝑖𝑑𝑡ℎ. All
other dimensions remained unchanged.
The hydrostatic stresses present at control point B for different aspect ratios are plotted and
compared in figures 4.5 (maximum temperature) and 4.6 (at the end of cooling) for all constitutive
models.
-600
-500
-400
-300
-200
-100
0
0 0.5 1 1.5 2 2.5 3
Hyd
rosta
tic s
tre
sse
s a
t C
P_B
(M
Pa
)
Line aspect ratio
Elast1 Elast2 IdPlast WHardCu WHard Plast
Figure 4.5 Results of the aspect ratio study for all constitutive models, at t=1s
49
Although the effects of line aspect ratio on stresses are not independent of the spacing between
lines [20], that does not invalidate this study’s results. It should be noted, however, that this analysis
corresponds to the specific case for which the spacing the between lines is equal to the linewidth.
There is a clear tendency for stresses to stabilize for aspect ratios greater than 1.5 (original model)
independently of the constitutive model used. For constitutive models Elast1, Elast2, WHard and Plast;
smaller aspect ratios translate to higher (compressive) hydrostatic stresses at t=1s (figure 4.5). For
models IdPlast and WHardCu, however, smaller aspect ratios tend to decrease hydrostatic stresses.
This distinct behavior comes from the fact that, in models IdPlast and WHardCu, only copper is
considered plastic while all other materials were assumed perfectly elastic.
The residual stresses (figure 4.6) show approximately the same tendency for stabilization after
aspect ratios higher than 1.5. Stresses behave likewise for all plastic models: at smaller aspect ratios,
higher (positive) hydrostatic stresses are observed.
Detailed results for constitutive models Elast2 and Plast are shown in figures 4.7-4.9 at maximum
temperature (Elast2 and Plast) and at the end of cooling (Plast only). Images of the stress (MPa)
distributions for the 0.25 and 2.5 aspect ratio geometries are compiled in Appendix E, for the same two
models, as a confirmation of the representativity of control point B.
0
50
100
150
200
0 0.5 1 1.5 2 2.5 3Hyd
rosta
tic s
tre
sse
s a
t C
P_
B (
MP
a)
Line aspect ratio
IdPlast WHardCu WHard Plast
Figure 4.6 Results of the aspect ratio study for all plastic constitutive models, at t=2s
50
For constitutive model Elast2 (figure 4.7), smaller aspect ratios tend to increase normal stresses
S11 and S33 (only slightly) and lower normal stress S22. For model Plast (figure 4.8), smaller aspect
ratios tend to increase the normal stresses S11 and S33, but S22 is approximately constant. Overall the
most notable difference between the elastic and plastic models (at t=1s) is that, although the stresses
show similar tendencies (with geometry), plasticity attenuates stresses observed at the copper lines, as
is to be expected.
Figure 4.7 Results of the aspect ratio study for constitutive model Elast2, at t=1s
-900
-600
-300
0
300
600
900
0 0.5 1 1.5 2 2.5 3
Str
esse
s a
t C
P_
B (
MP
a)
Line aspect ratio
Von Mises S11 S22 S33 Hydrostatic
-300
-200
-100
0
100
200
0 0.5 1 1.5 2 2.5 3
Str
esse
s a
t C
P_B
(M
Pa
)
Line aspect ratio
Von Mises S11 S22 S33 Hydrostatic
Figure 4.8 Results of the aspect ratio study for constitutive model Plast, at t=1s
51
From figure 4.9, the residual (normal) stresses S11 greatly increase for smaller aspect ratios while
the residual (normal) stresses S22 decrease and S33 has a maximum for an aspect ratio of 0.5.
In all cases, the normal (longitudinal) stress S33 seems to have the greatest influence over the Von
Mises and hydrostatic stress values. The residual Von Mises stresses are constant for all geometries in
this study. This was also observed in all plastic constitutive models. This parameter is thus unable of
reflecting the influence of geometry on stresses at the copper lines (as previously discussed in section
4.1). This is the reason why, in figures 4.5 and 4.6, as well as in upcoming sections 4.2.3 and 4.2.4, the
parameter used to compare all constitutive models is the hydrostatic stress.
0
100
200
300
400
0 0.5 1 1.5 2 2.5 3
Str
esse
s a
t C
P_
B (
MP
a)
Line aspect ratio
Von Mises S11 S22 S33 Hydrostatic
Figure 4.9 Results of the aspect ratio study for constitutive model Plast, at t=2s
52
4.2.3 Line distance
For this study, all dimensions of the base-model (section 3.1.1), except for the distance between
copper lines, are unchanged. The smallest distance between lines studied corresponds to the limit
situation at which the distance separating the interconnect lines is equal to the thickness of the diffusion
barrier and the upper limit studied corresponds to a distance six times greater than that of the base
model. All distances studied were 𝐷𝑖𝑠𝑡 ∈ {0.01, 0.03, 0.06, 0.09, 0.18 (base model), 0.36, 0.54, 0.72,
0.9, 1.08 } (units in µm). The original model has a line distance of 0.18µm, which is equal to its line width.
The hydrostatic stresses present at the copper lines for different line distances are plotted and
compared in figures 4.10 (maximum temperature) and 4.11 (at the end of cooling) for all constitutive
models.
Figure 4.10 Results of the line distance study for all constitutive models, at t=1s
-600
-500
-400
-300
-200
-100
0
0 0.2 0.4 0.6 0.8 1 1.2
Hyd
rosta
tic s
tre
sse
s a
t C
P_
B (
MP
a)
Distance between lines (µm)
Elast1 Elast2 IdPlast WHardCu WHard Plast
0
50
100
150
200
0 0.2 0.4 0.6 0.8 1 1.2
Hyd
rosta
tic s
tre
sse
s a
t C
P_B
(M
Pa
)
Distance between lines (µm)
IdPlast WHardCu WHard Plast
Figure 4.11 Results of the line distance study for all plastic constitutive models, at t=2s
53
There is a tendency for stresses to stabilize for line distances greater than 0.18 µm (which is the line
distance and line width of the original model) for models Elast1 and 2, IdPlast and WHardCu. The stress
curves for models WHard and Plast show a maximum (minimum absolute value) for a line distance of
0.09µm and stabilize for line distances greater than 0.54µm. For model Elast1 and 2, smaller line
distances increase (compressive) hydrostatic stresses at t=1s (figure 4.10). For models IdPlast and
WHardCu, however, smaller line distances decrease hydrostatic stresses at t=1s (as discussed in the
previous section).
The residual stresses (figure 4.11) show the same tendency for stabilization for line distances of
0.54µm and beyond. Stresses behave likewise for models IdPlast and WHardCu, and a minimum stress
is reached for a line distance of 0.18µm (original model). In models WHard and Plast, a similar tendency
can be observed, but with overall higher residual stresses (when compared to models IdPlast and
WHardCu) and hydrostatic stresses minimize at a line distance of 0.09µm.
Detailed results for models Elast2 and Plast are shown in figures 4.12-4.14 at maximum temperature
(Elast2 and Plast) and at the end of cooling (Plast only).
Figure 4.12 Results of the line distance study for constitutive model Elast2, at t=1s
-900
-600
-300
0
300
600
900
0 0.2 0.4 0.6 0.8 1 1.2
Str
esse
s a
t C
P_B
(M
Pa
)
Distance between lines (µm)
Von Mises S11 S22 S33 Hydrostatic
54
For constitutive model Elast2 (figure 4.12), smaller line distances increase compressive normal
stresses S11 and S33 and decrease S22. For model Plast (figure 4.13), smaller line distances promote
a decrease in the absolute value of normal stresses S22 and S33 and substantially increase S11.
The residual stresses (figure 4.14) show a tendency for stabilization for line distances higher than
0.54µm. Smaller line distances increase normal stresses S11 and decrease S22 and S33, which
nonetheless has considerably higher values than the other two.
Images of the stress (MPa) distributions for the 0.01µm and 1.08µm line distance models are
compiled in Appendix F as a confirmation of the representativity of control point B.
Figure 4.13 Results of the line distance study for constitutive model Plast, at t=1s
-300
-250
-200
-150
-100
-50
0
50
100
150
200
0 0.2 0.4 0.6 0.8 1 1.2
Str
esse
s a
t C
P_
B (
MP
a)
Distance between lines (µm)
Von Mises S11 S22 S33 Hydrostatic
-100
0
100
200
300
400
0 0.2 0.4 0.6 0.8 1 1.2
Str
esse
s a
t C
P_B
(M
Pa
)
Distance between lines (µm)
Von Mises S11 S22 S33 Hydrostatic
Figure 4.14 Results of the line distance study for constitutive model Plast, at t=2s
55
4.2.4 Line width
There is interest in knowing how stresses at the copper lines behave for increasingly smaller
geometries and line widths, given that interconnect line sizes are now reaching beyond the 10nm mark
and below. For that reason, a study involving models of different sizes is advised.
Usually, 10nm width lines used in the industry tend to have an aspect ratio of 2.0 and a diffusion
barrier thickness of only 1nm. In order to keep the changes made between models used for this study
as minimal as possible, the base-model (180nm line width, see section 3.1.1) was slightly tweaked from
an aspect ratio of 1.5 to 2.0 and the thickness of its tantalum diffusion barrier decreased from 10nm to
1nm. This way, the same aspect ratio and diffusion barrier thickness can be kept constant throughout
all models with different line widths.
The distance between lines should not be kept constant throughout all models, however. Not only
does it influence the stresses observed at the interconnect lines (section 4.2.3) but keeping a constant
distance of 180nm would completely mischaracterize smaller geometries. The best compromise is to
keep the distance between lines always equal to the respective line’s width. It is desirable, in this study,
that the highest possible degree of familiarity is kept between models, so that a meaningful comparison
between them can be made between them. Table 4.1 shows the differences between all models used
for this study.
Table 4.1 Main architectural differences between the models used in the linewidth study
Linewidth, 𝒍𝒘 (nm) 180 120 80 40 20 10
Distance between lines, 𝑫𝒊𝒔𝒕 (nm) 180 120 80 40 20 10
Aspect Ratio, 𝑨𝑹 2.0 2.0 2.0 2.0 2.0 2.0
Diffusion barrier thickness (nm) 1 1 1 1 1 1
For smaller linewidth models, keeping the base-model’s length (vertical dimension) is unsustainable
from a computational point of view if the same relative mesh size is to be kept. Having a y-dimension
almost twenty times bigger than the x-dimension creates, again, a multidimensionality issue that must
be solved. A solution would be to dimension each model’s vertical measurements in direct proportion to
the base-model, the constant of proportionality being:
𝑙𝑤
180, 𝑙𝑤 ∈ {120, 80, 40, 20, 10}
(4.1)
For the adjusted 180nm linewidth model, a brief convergence study was made to verify if the
0.005µm uniform mesh discussed in section 3.3.1 can still be used with some degree of reliability even
after the diffusion barrier thickness is downsized.
As a rule of thumb, the number of finite elements inside the diffusion barrier should be such that at
least 5 elements can be found in every direction. However, since the diffusion barrier region is so small
compared to the rest of the model, trying to achieve full convergence inside that subdomain usually
results in very fine meshes that do not directly translate to a finer convergence for elements inside the
56
copper lines (with the exception of the 2 “corner nodes”), which is the subdomain relevant to the study
(see section 3.3.1).
A very fine and/or biased mesh can be difficult to keep constant or even comparable between models
with very different dimensions, so a quick convergence study with exclusively uniform meshes was
elaborated for the tweaked 180nm model. Control points B, B1, B2 and A were used, and their location
(relative to the copper lines) is, as always, represented by figure 3.20 regardless of each model’s
architecture. The results, presented in Figure 4.15, are analogous to those of the previous convergence
studies from section 3.3.1. The uniform linear quadrangular 0.005µm mesh (~30 000 finite elements)
seems to be, again, good enough to provide meaningful results inside the relevant domain of the study.
Control point A (and consequently, B4) having not yet converged for this mesh, as was to be expected
but, as discussed in section 3.3.1, this is not drastic enough to significantly bias results in the region of
interest.
Although a 0.005µm uniform mesh is appropriate for the tweaked 180nm line model, the same
cannot be said for the smaller line width models because, for increasingly smaller models, the same
mesh will produce results increasingly less precise (the mesh becomes increasingly coarser). To avoid
doing a convergence study for each different width model, another solution is to choose a uniform mesh
size such that the number of finite elements (approximately 30 000) is kept constant throughout the six
model architectures to be studied. This way, all meshes used should provide results of similar accuracy.
This approach is not the most rigorous, but it is the most pragmatical. Given that the focus of this
study is to understand tendencies (between the geometry of the models and the stresses observed at
the copper lines) and not absolute values, these meshes should be more than enough to fulfil that
purpose. Keeping the number of elements constant throughout the study also guarantees that different
results are not a consequence of the use of meshes of different fineness.
0
200
400
600
800
1000
1200
1400
1600
Vo
n M
ise
s s
tre
ss (
MP
a)
Number of finite elements
CP_A CP_B CP_B1 CP_B2
Figure 4.15 Results of the convergence study for the adjusted base model
57
The hydrostatic stresses present at the copper lines for different line widths are plotted and
compared in figures 4.16 (maximum temperature) and 4.17 (at the end of cooling) for all constitutive
models.
There is a tendency for smaller line widths to increase (compressive) hydrostatic stresses
independently of the constitutive model used, for t=1s (figure 4.16), i.e., models with smaller copper line
width are subjected to higher compressive stresses when compared to architectures with larger widths.
The residual stresses are seemingly independent of line width for models IdPlast and WHardCu
(figure 4.17). For models WHard and Plast, the hydrostatic stresses are highest for smaller line widths.
This distinct behaviour between plastic models is a direct consequence of assuming, in models IdPlast
Figure 4.16 Results of the line width study for all constitutive models, at t=1s
-500
-400
-300
-200
-100
0
0 20 40 60 80 100 120 140 160 180 200
Hyd
rosta
tic s
tre
sse
s a
t C
P_
B (
MP
a)
Line width (nm)
Elast1 Elast2 IdPlast WHardCu WHard Plast
0
50
100
150
0 20 40 60 80 100 120 140 160 180 200
Hyd
rosta
tic s
tre
sse
s a
t C
P_B
(M
Pa
)
Line width (nm)
IdPlast WHardCu WHard Plast
Figure 4.17 Results of the line width study for all plastic constitutive models, at t=2s
58
and WHardCu, that only copper behaves plastically while the rest of the materials are in pure elastic
regime.
Detailed results for models Elast2 and Plast are shown in figures 4.18-4.20 at maximum temperature
(Elast2 and Plast) and at the end of cooling (Plast only). Images of the stress (MPa) distributions for the
10nm and 180nm linewidth models are compiled in Appendix G as a confirmation of the representativity
value of control point B.
-900
-750
-600
-450
-300
-150
0
150
300
450
600
0 20 40 60 80 100 120 140 160 180 200
Str
esse
s a
t C
P_
B (
MP
a)
Line width (nm)
Von Mises S11 S22 S33 Hydrostatic
Figure 4.18 Results of the line width study for constitutive model Elast2, at t=1s
-300
-200
-100
0
100
200
0 20 40 60 80 100 120 140 160 180 200
Str
esse
s a
t C
P_B
(M
Pa
)
Line width (nm)
Von Mises S11 S22 S33 Hydrostatic
Figure 4.19 Results of the line width study for constitutive model Plast, at t=1s
59
Smaller line widths slightly increase all normal stresses (S11, S22 and S33) regardless of the
constitutive model used, for t=1s (figures 4.18 and 4.19). The largest stress increment is observed from
the 20nm to the 10nm node.
For constitutive model Plast, the residual normal stresses S22 and S33 are highest for smaller line
widths, while S11 is almost unchanged (figure 4.20).
0
100
200
300
0 20 40 60 80 100 120 140 160 180 200
Str
esse
s a
t C
P_
B (
MP
a)
Line width (nm)
Von Mises S11 S22 S33 Hydrostatic
Figure 4.20 Results of the line width study for constitutive model Plast, at t=2s
60
4.3 Three dimensional studies
In the bidimensional studies a plane strain working hypothesis was used. In other words,
𝜀𝑥𝑧 = 𝜀𝑦𝑧 = 𝜀𝑧𝑧 = 𝜎𝑥𝑧 = 𝜎𝑦𝑧 = 0 (4.2)
This simplification is frequently used for very thick models, while its counterpart (plane stress) is the
alternative for very thin models. The true model thickness (in the z-direction) is an unknown but it would
be reasonable to assume that the model would be at least as thick as it is lengthy (y-direction). Since
the true wafer length (y-direction) is 749µm (see section 3.1.1), a true model thickness of ~750µm is not
a senseless estimate.
If the bidimensional model is to be verified by the three-dimensional model, then as the size of the
third dimension of the model is increased, the maximum stresses observed along CPedge are expected
to tend towards values like the ones obtained for the bidimensional analysis at CPB. Visual
representations of the locations of both CPedge and CPB can be found in section 3.2.2 (figure 3.15) and
section 3.3.2 (figure 3.28), respectively.
Keeping the wafer size of the base model (section 3.1.1) at 3µm (section 3.2.2) and using the biased
hex linear mesh from section 3.3.2, models with the same base geometry (same cross-section, figures
3.1 and 3.2(b)) and different thicknesses 𝑍 ∈ {1, 4, 8, 12, 16, 20, 32, 40, 50} were created.
The values of the Von Mises, S11, S22 and S33 stresses, taken at the mid cross section, 𝑍 =𝑧
2
(where stresses are higher) are represented in figures 4.21 and 4.22 (for constitutive models Elast2 and
Plast, respectively) next to their bidimensional (plane strain) counterparts obtained for the uniform
0.005µm mesh (section 3.3.1). All stress values were taken at t=1s (when the temperature is highest).
Figure 4.21 Comparison between the three-dimensional and bidimensional results, at t=1s, for constitutive
model Elast1
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
0 10 20 30 40 50
Ma
xim
um
str
esse
s a
lon
g C
P_e
dg
e (
MP
a)
Model thickness (µm)
Von Mises, 2D Von Mises, 3D S11, 2D S11, 3D
S22, 2D S22, 3D S33, 2D S33, 3D
61
The elastic constitutive model used in this study was Elast1 (constant and temperature independent
material properties), instead of Elast2, due to its greater simplicity. This choice was made to minimize
the computational time and processing power needed as much as possible. Even so, for thicknesses of
20µm and above, the finite element simulations took several hours to complete.
In figure 4.21, the reliability of the elastic bidimensional model becomes clear for models of thickness
equal or higher than 40µm. The same study performed for plastic conditions (model Plast) is shown in
figure 4.22, in which the 3D model tends to the bidimensional results faster, at Z=8µm. The study shown
in figure 4.21 is still, however, an upper limit after which the bidimensional model is certain to provide
meaningful results. Overall these results confirm the reliability of the studies performed for the
bidimensional models for 3D models of at least 40µm thickness.
The Von Mises, S11, S22 and S33 stress distributions along CPedge (at t=1s) are shown in figure
4.23 for a three-dimensional model of thickness Z=40µm (Elast1 assumption was used).
As previously mentioned, stresses observed at the free (front/back) surfaces (Z=0µm and Z=40µm,
respectively) are the lowest and the model displays a symmetric stress distribution with the highest
stress values registered at 𝑍 =𝑧
2.
Figure 4.22 Comparison between the three-dimensional and bidimensional results, at t=1s, for constitutive
model Plast
-300
-200
-100
0
100
200
0 2 4 6 8 10
Ma
xim
um
str
esse
s a
lon
g C
P_
ed
ge
(M
Pa
)
Model thickness (µm)
Von Mises, 2D Von Mises, 3D S11, 2D S11, 3D
S22, 2D S22, 3D S33, 2D S33, 3D
62
Str
es
s (
Vo
n M
ise
s,
MP
a)
Str
es
s (
S1
1,
MP
a)
Str
es
s (
S2
2,
MP
a)
Str
es
s (
S3
3,
MP
a)
(X1000)
(X1000) (X1000)
Figure 4.23 Stress (MPa) distributions at CPedge along thickness z (µm) for a 3D model of total thickness 40µm (at t=1s,
for constitutive model Elast1)
Z distance along path (µm) Z distance along path (µm)
Z distance along path (µm) Z distance along path (µm)
63
Chapter 5
Conclusions
This chapter summarizes the work, provides the conclusions and points out work to be developed
in the future.
64
5.1 Conclusions
The aim of this work was to understand the behavior of nanosized copper interconnect lines in the
presence of thermal loads. Therefore, working conditions were simulated through the application of an
isothermal temperature of 230ºC, corresponding to a critical operation temperature of 250ºC, after which
the system is cooled back to room temperature, 20ºC. Stresses were analyzed both at maximum
temperature (t=1s) and at the end of cooling (t=2s).
In section 4.1, different constitutive models (described in section 3.1.3) were tested against each
other with the objective of understanding how the simulation results are influenced by material properties
and working assumptions chosen. The stress distributions observed in the interconnect lines are
severely influenced by the constitutive model chosen. For very small thermal loads an elastic model
can be considered, but in general a plastic model with temperature dependent strain hardening
should be considered.
Von Mises stresses were much higher at the copper lines (≈420MPa) in the pure elastic regime
(constitutive models Elast 1 and 2). For plastic models, Von Mises stresses at the copper lines were
lower (≈170MPa) when the temperature dependency is considered (IdPlast and Plast) and higher
(≈210MPa) when the model undergoes temperature independent work-hardening (model WHardCu and
WHard).
On the assumption that the initial (t=0s) state of stresses is null, the entirety of the interconnect lines
are subjected to negative (compressive) stresses when the temperature is at its highest. This is because
copper possesses a significantly bigger CTE than all other materials present in the component, and thus
when the copper lines try to expand due to the increase in temperature, their dilation is partially restricted
by the surrounding materials, resulting in compression forces.
At peak thermal load the highest stress and plastic strains were registered at the interface between
the diffusion barrier and the interconnect lines. This result supports E. Ege and Y.-L. Shen’s
observations [21] that the stress accumulation at the corners is likely to promote electromigration and
voiding since, according to Shen, the stress magnitude can be a strong indicator of the propensity of
voiding damage in metal interconnects [20].
In section 4.2, several geometric features were studied to establish trends between the architecture
of the lines and the stress distribution. In the line aspect ratio and distance studies, regardless of their
initial behavior and independently of the constitutive model used, stresses were found to stabilize for
aspect ratios and line distances greater than 1.5 and 0.18µm (0.54µm for models WHard and Plast),
respectively. These results suggest that commercial circuits use aspect ratios and line distances
that stabilize (and lower) thermal stresses in the interconnect lines.
For the line width study, when the temperature is highest, there is a clear tendency for smaller line
widths to increase stresses. In other words, models with smaller line widths are subjected to higher
compressive stresses, at working temperatures, when compared to architectures with larger widths.
In all geometry studies, the residual normal stress S33 (longitudinal stress) registered the highest
(absolute) values. This stress could be the most important in residual stress analysis in metal
interconnects (at the very least it is the most influential value for the calculation of the hydrostatic stress).
65
Finally, in section 4.3, a three-dimensional model was created to test the bidimensional plane strain
hypothesis. The highest stress values observed along the geometric centerline of the “half” copper lines
(in other words, along CPedge) for a 3D model of enough thickness (depends on the constitutive model)
very closely match the stress values observed at the geometric center of the “half” copper lines (control
point B) in the bidimensional model. For the plastic regime (constitutive model Plast), the 3D stress
values tend to the bidimensional stress values faster than for elastic regime (constitutive model Elast1).
The behaviors observed in bidimensional studies have a direct correlation to the ones potentially
observed in their three-dimensional counterparts if the thickness of the third dimension is 40µm (upper
limit) or higher. This is advantageous in circuit design studies because CPU time for a 2D model is
typically less than 5% as compared with 3D. Subsequently, the results obtained from these studies
can provide useful guidelines for the optimal design of circuits.
5.2 Future work
Since modeling can typically complex, most studies are ultimately constrained by the simplifications
and assumptions atop which they are built. This study is no exception and thus there are unexplored
and/or underdeveloped topics. For instance, access to more detailed and accurate experimental data
of the properties of all materials present in the model would eliminate (or at least mitigate) most of the
uncertainty and simplifications involved in the creation of all constitutive models. Ideally, a viscoplastic
constitutive model considering both temperature dependency and work-hardening in all materials
present in the model should be featured.
The decision to keep a null initial state of stress/strain was made as an initial simplification because
the initial state of stresses is dependent on the fabrication methods used for the interconnect lines. In
order to get as close as possible to reality, the number of simplifications made in the constitutive models
was kept to a minimum so that the model would not be constructed on top of a lot of uncertainties.
Experimental values of the initial stresses could be obtained through stress field testing. A detailed finite
element simulation attempting to recreate the annealing conditions of the interconnect lines could also
provide useful guidelines for the modelling of more realistic initial conditions.
The studies elaborated in the scope of this work are, however, still meaningful. After annealing,
copper slightly shrinks in size when it cools down to room temperature and the initial state of stress at
the copper lines would be an initial state of tension. Since the lines will be compressed after heating
during working conditions, there is an initial stress relaxation after which the lines are be exposed to
increasingly higher compressive stresses. This would produce a final state of stress after heating (t=1s)
lower (less negative) than in the case of zero initial stress, so the case studied in this work corresponds
to an upper limit for thermal stresses.
Another feature that was considered in an approximate fashion was the application of the thermal
load. This is because, in operating conditions, the circuit temperature increases due to Joule heating,
not due to an isothermal temperature step imposition. In future work, the next step would be to model
the heating condition through the application of a current density (or volume) passing through the
interconnect lines.
66
In this work, the thermal load was only applied once (in each given simulation), so the residual stress
fields corresponded to only one thermal cycle. However, in working conditions, circuits are subjected to
multiple cycles. The recyclability and thermal fatigue of the copper lines could be studied through the
application of multiple temperature loading cycles.
The results obtained in the geometry studies could also be used, in the future, as basis for the
creation of an optimization program (with the joint use of PythonTM and AbaqusTM) that would, for any
given feature, calculate an optimized architecture which minimizes stresses at the interconnect lines.
Lastly, the microstructure of the copper lines was not considered in this work, as this research was
focused on a macroscopic approach to the problem. However, as the dimension of the interconnect
lines becomes increasingly smaller, the effects of texture become increasingly more relevant. The
anisotropy of copper was also unaddressed, which is something that should not be neglected. While
macroscopic anisotropy may be modeled with AbaqusTM, microstructure modeling would have to be
done with the OOF2 software or equivalent (mentioned previously in section 2.4.3), after a surface
characterization is made using electron microscopy. Especially at working temperatures, the dynamics
of crystal effects become increasingly important. Dislocation pile-up, vacancy migration and void
formation can be a consequence of stress but can in turn affect the mechanical response of the system.
These effects should also be taken into consideration in a more complete model.
The modelling of copper interconnect lines’ behavior and the study of their failing mechanisms is,
undoubtfully, very nuanced and sophisticated and requires understanding from different fields of science
and engineering, as well as access to state-of-the-art technology (such as innovative electron
microscopy techniques). The joint effort and interest built around this topic will surely lead to a satisfying
conclusion in which the future of computing is no longer compromised.
67
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70
Appendix A
Dimensional analysis
Dimensional analysis performed to calculate the multipliers needed to change from a meter-based
system to a micrometer base system.
71
Table A.1 Dimensional analysis
Parameter Units(m) Dimensions Units(µm) Dimensions Multiplier m-˃µm
Length 𝑚 - µ𝑚 - 106
*Current 𝐴 - 𝑝𝐴 - 1012
Force 𝑁 [𝑘𝑔. 𝑚
𝑠2] µ𝑁 [
𝑘𝑔. 𝜇𝑚
𝑠2] 106
Mass 𝑘𝑔 - 𝑘𝑔 - -
Time 𝑠 - 𝑠 - -
Temperature ℃ - ℃ - -
Thermal Expansion
1
℃ -
1
℃ - -
Stress 𝑃𝑎 [𝑘𝑔
𝑚. 𝑠2] 𝑀𝑃𝑎 [
𝑘𝑔
𝜇𝑚. 𝑠2] 10−6
Density - [𝑘𝑔
𝑚3] - [
𝑘𝑔
𝜇𝑚3] 10−18
Power 𝑊 [𝑘𝑔. 𝑚2
𝑠3] 𝑝𝑊 [
𝑘𝑔. 𝜇𝑚2
𝑠3] 1012
Energy 𝐽 [𝑘𝑔. 𝑚2
𝑠2] 𝑝𝐽 [
𝑘𝑔. 𝜇𝑚2
𝑠2] 1012
Thermal Conductivity
[𝑊
𝑚℃] [
𝑘𝑔. 𝑚
𝑠3℃] [
𝑝𝑊
𝜇𝑚℃] [
𝑘𝑔. 𝜇𝑚
𝑠3℃] 106
Heat Flux [𝑊
𝑚2] [
𝑘𝑔
𝑠3] [
𝑝𝑊
𝜇𝑚2] [
𝑘𝑔
𝑠3] 1
Specific Heat [𝐽
𝑘𝑔. ℃] [
𝑚2
℃. 𝑠2] [
𝑝𝐽
𝑘𝑔. ℃] [
𝜇𝑚2
℃. 𝑠2] 1012
Convection Coefficient
[𝑊
𝑚2℃] [
𝑘𝑔
𝑠3℃] [
𝑝𝑊
𝜇𝑚2℃] [
𝑘𝑔
𝑠3℃] 1
Voltage 𝑉 [𝑘𝑔. 𝑚2
𝐴. 𝑠3] 𝑉 [
𝑘𝑔. 𝜇𝑚2
𝑝𝐴. 𝑠3] 1
**Electrical Conductivity
[𝑆
𝑚] [
𝐴2. 𝑠3
𝑘𝑔. 𝑚3] [
𝑝𝑆
𝜇𝑚] [
𝑝𝐴2. 𝑠3
𝑘𝑔. 𝜇𝑚3] 106
Electrical Resistivity
[Ω𝑚] [𝑘𝑔. 𝑚3
𝐴2 . 𝑠3] [TΩμ𝑚] [
𝑘𝑔. 𝜇𝑚3
𝑝𝐴2. 𝑠3] 10−6
*For simplicity when working with Volts
**[𝑆] = [𝐴2.𝑠3
𝑘𝑔.𝑚2]
72
Appendix B
Wafer length study (i)
Study of the influence of the silicon wafer coefficient of thermal expansion (CTE) on stresses and
displacements at control points B and Si.
73
-2400-2200-2000-1800-1600-1400-1200-1000
-800-600-400-200
0200400600800
10001200140016001800200022002400
-10 -5 0 5 10 15 20 25 30 35 40 45
Str
esse
s (
MP
a)
Wafer CTE (x1 000 000)
Von Mises, CP_B Von Mises, CP_Si S11, CP_B S11, CP_Si S12, CP_B
S12, CP_Si S22, CP_B S22, CP_Si S33, CP_B S33, CP_Si
Figure B.1 Influence of the wafer CTE on stresses at control points B and Si
-0.0075-0.005
-0.00250
0.00250.005
0.00750.01
0.01250.015
0.01750.02
0.02250.025
0.02750.03
0.03250.035
0.03750.04
0.04250.045
0.04750.05
-10 -5 0 5 10 15 20 25 30 35 40 45
Dis
pla
ce
me
nts
(µ
m)
Wafer CTE (x 1000 000)
Umag,CP_B Umag, CP_Si U1, CP_B U1, CP_Si U2, CP_B U2, CP_Si
Figure B.2 Influence of the wafer CTE on displacements at control points B and Si
74
B.1 Discussion of results
The influence of the silicon thermal expansion coefficient on the stresses and displacements
observed both in the copper lines (CPB) and the silicon wafer (CPSi) is presented. Because of the
symmetry condition applied to the model, 𝑢𝑥 = 𝑢𝑅,𝑦 = 𝑢𝑅,𝑧 = 0, nodes located along the side surfaces
of the model are only free to move in the y direction.
When the wafer attempts to expand uniformly in all directions due to an increase in temperature,
both the geometric and physical symmetries present in the wafer enforce U1≈0 for CPSi. The negative
increase in S11 (and thus the increase in Von Mises stress) for CPSi corroborates this statement because
the bigger the CTE, the bigger the compressive stresses needed in the x direction to keep U1≈0.
Since all nodes in the model are free to move in the y direction, U2 and Umag for both control points
increase with the increase in CTE while all other unmentioned variables such as S22, remain
independent (constant).
75
Appendix C
Wafer length study (ii)
Study of the influence of the wafer length on the stresses and displacements at control points B
and Si.
76
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800
Str
esse
s (
MP
a)
Wafer length in µm
Von Mises, CP_B Von Mises, CP_Si S11, CP_B S11, CP_Si S12, CP_B
S12, CP_Si S22, CP_B S22, CP_Si S33, CP_B S33, CP_Si
-0.1-0.05
00.050.1
0.150.2
0.250.3
0.350.4
0.450.5
0.550.6
0.650.7
0.750.8
0.850.9
0.951
1.051.1
1.151.2
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800
Dis
pla
ce
me
nts
(µ
m)
Wafer length in µm
Umag, CP_B Umag, CP_Si U1, CP_B U1, CP_Si U2, CP_B U2, CP_Si
Figure C.1 Influence of the wafer length on stresses at control points B and Si
Figure C.2 Influence of the wafer length on displacements at control points B and Si
77
C.1 Discussion of results
The study in Appendix C focuses on the influence of the wafer length on the stresses and
displacements perceived in the same control points mentioned previously for Appendix B. In this case,
the CTE of silicon was kept constant while the wafer length is increased.
This means that the wafer is expanding at the same rate across all simulations, which explains why
S11 is now constant in the silicon wafer, CPSi.
The loading conditions are the same for all different models in the study (𝛥𝑇 = 𝐶𝑇𝐸 = 𝑐𝑜𝑛𝑠𝑡.), so
𝜀 = 𝐶𝑇𝐸. 𝛥𝑇 = 𝑐𝑜𝑛𝑠𝑡. Because the stresses are given by [𝜎] = [𝐸][𝜀] and since [𝐸] and [𝜀] are constant,
then S11, S22, S12 and the Von Mises stresses are also constant (independent of the wafer size).
The displacement 𝑢 = 𝛥𝐿 = 𝐿 − 𝐿0, where 𝐿 is the final length after deformation and 𝐿0 the initial
length. Because 𝜀 = 𝛥𝐿/𝐿0 = 𝑐𝑜𝑛𝑠𝑡., as the initial vertical length of the wafer (and thus the whole model)
increases, 𝑢2 must also increase proportionally for both CPB and CPSi, as is verified.
78
Appendix D
Convergence studies (2D)
Results of the convergence studies performed in section 3.3.1 for a bidimensional 3µm wafer base-
model.
79
0.00523600
0.00523650
0.00523700
0.00523750
0.00523800
0.00523850
0.00523900
0.00523950
0.00524000
0.00524050
0.00524100
0.00524150
Um
ag
at C
P_
C,
µm
Number of finite elements
442
443
444
445
446
447
448
449
Vo
n M
ise
s s
tre
ss a
t C
P_
C,
MP
a
Number of finite elements
Figure D.1 Convergence study for control point C
10.75
10.8
10.85
10.9
10.95
11
11.05
11.1
Vo
n M
ise
s s
tre
ss a
t C
P_
D,
MP
a
Number of finite elements
0.00456820
0.00456840
0.00456860
0.00456880
0.00456900
0.00456920
0.00456940
0.00456960
0.00456980
Um
ag
at C
P_D
, µ
m
Number of finite elements
Figure D.2 Convergence study for control point D
0
50
100
150
200
250
Vo
n M
ise
s s
tre
ss a
t C
P_E
, M
Pa
Number of finite elements
0.00377729
0.00377729
0.00377730
0.00377730
0.00377731
0.00377731
0.00377732
Um
ag
at C
P_E
, µ
m
Number of finite elements
Figure D.3 Convergence study for control point Si
80
-400
-300
-200
-100
0
S1
1, M
Pa
Number of finite elements
CP_B1 CP_B2 CP_B4 CP_A CP_B3 CP_B
Figure D.4. Uniform mesh convergence study – S11
-400
-300
-200
-100
0
100
200
S2
2, M
Pa
Number of finite elements
CP_B1 CP_B2 CP_B4 CP_A CP_B3 CP_B
Figure D.5. Uniform mesh convergence study – S22
-800
-700
-600
-500
-400
-300
-200
-100
0
S3
3, M
Pa
Number of finite elements
CP_B1 CP_B2 CP_B4 CP_A CP_B3 CP_B
Figure D.6. Uniform mesh convergence study – S33
81
Appendix E
Aspect ratio studies
Stress distributions observed at the copper lines in models with different line aspect ratios.
82
Figure E.1 Highlight of the “half” copper line shown in the following results.
Figure E.2 Stress distribution (MPa) in a “half” copper line, for the line aspect ratio study of model Elast2, at
t=1s (H), for AR=0.25 (*) and AR=2.5 (**).
H**
H*
H**
H*
H**
H*
Figure E.3 Stress distribution (MPa) in a “half” copper line, for the line aspect ratio study of model Plast, at t=1s (H) and
t=2s (C) for AR=0.25 (*) and AR=2.5 (**).
H** C**
C*
H*
H** C**
C*
H*
H** C**
C*
H*
83
Appendix F
Line distance studies
Stress distributions observed at the copper lines in models with different line distances.
84
Figure F.1 Stress distribution (MPa) in a “half”
copper line, for the line distance study of model
Elast2, at t=1s (H), for Dist=0.01µm (*) and
Dist=1.08µm (**).
H* H**
H**
H**
H*
H*
Figure F.2 Stress distribution (MPa) in a “half” copper line,
for the line distance study of model Plast, at t=1s (H) and
t=2s (C) for Dist=0.01µm (*) and Dist=1.08µm (**).
C* C** H** H*
H* C* C** H**
H* C* C** H**
85
Appendix G
Line width studies
Stress distributions observed at the copper lines in models with different line widths.
86
Figure G.2 Stress distribution (MPa) in a “half” copper line, for
the line width study of model Plast, at t=1s (H) and t=2s (C),
for Width=10nm (*) and Width=180nm (**).
Figure G.1 Stress distribution (MPa) in a “half”
copper line, for the line width study of model
Elast2, at t=1s (H), for Width=10nm (*) and
Width=180nm (**).
H* H**
H* H**
H* H**
H* H** C** C*
H* H** C** C*
H* H** C** C*