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Queensland University of Technology
Brisbane Australia
Advanced Numerical Characterization of Silicon with
Defect by Nanoindentation
Qiang Fu
Principal Supervisor: Associate Professor Yuantong Gu
Associate Supervisor: Associate Professor Cheng Yan
A thesis submitted in fulfilment of the requirements for the degree of master of engineering
Faculty of Science and Engineering
Queensland University of Technology
Jan 2012
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
I Acknowledgement
The author of this thesis would like to take this opportunity to acknowledge those who
have offered their assistance and support during the research.
Firstly, the author would sincerely express his gratitude to his principal and associate
supervisor, Professors Yuantong Gu and Cheng Yan, for the guidance, advice, patience and
encouragement. Without their knowledge, vision and support, this work would not have
been possible.
Secondly, the author would express his appreciation to the QUT High Performance
Computing & Research Support Team. With their help, the massive computational
simulations have been completed efficiently. Special thanks extended to Mr. Haifei Zhan,
for the knowledge of the MD simulation field.
At last but not least, the author would thanks to his beloved family for their always support
and encouragement throughout the completion of this work and his life.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
II Publication
During the course of this project, one journal paper has been accepted. It is listed below
for reference.
Fu. Q, Zhan. HF and Gu. YT Atomistic investigations of single-crystal silicon with pre-
existing defect. Accepted by Advanced Science Letters in 2011.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
III Abstract
Nano silicon is widely used as the essential element of complementary metal–oxide–
semiconductor (CMOS) and solar cells. It is recognized that today, large portion of world
economy is built on electronics products and related services. Due to the accessible fossil
fuel running out quickly, there are increasing numbers of researches on the nano silicon
solar cells.
The further improvement of higher performance nano silicon components requires
characterizing the material properties of nano silicon. Specially, when the manufacturing
process scales down to the nano level, the advanced components become more and more
sensitive to the various defects induced by the manufacturing process.
It is known that defects in mono-crystalline silicon have significant influence on its
properties under nanoindentation. However, the cost involved in the practical
nanoindentation as well as the complexity of preparing the specimen with controlled
defects slow down the further research on mechanical characterization of defected silicon
by experiment. Therefore, in current study, the molecular dynamics (MD) simulations are
employed to investigate the mono-crystalline silicon properties with different pre-existing
defects, especially cavities, under nanoindentation.
Parametric studies including specimen size and loading rate, are firstly conducted to
optimize computational efficiency. The optimized testing parameters are utilized for all
simulation in defects study. Based on the validated model, different pre-existing defects
are introduced to the silicon substrate, and then a group of nanoindentation simulations of
these defected substrates are carried out. The simulation results are carefully investigated
and compared with the perfect Silicon substrate which used as benchmark.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
IV It is found that pre-existing cavities in the silicon substrate obviously influence the
mechanical properties. Furthermore, pre-existing cavities can absorb part of the strain
energy during loading, and then release during unloading, which possibly causes less
plastic deformation to the substrate. However, when the pre-existing cavities is close
enough to the deformation zone or big enough to exceed the bearable stress of the crystal
structure around the spherical cavity, the larger plastic deformation occurs which leads the
collapse of the structure. Meanwhile, the influence exerted on the mechanical properties
of silicon substrate depends on the location and size of the cavity. Substrate with larger
cavity size or closer cavity position to the top surface, usually exhibits larger reduction on
Young’s modulus and hardness.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
V Certification of Thesis
I hereby declare that no part of this work has previously been accepted for the award of
any other person in any university or institute. This thesis was completed during my
enrolment for degree of master by research at Queensland University of Technology, and
to the best of my knowledge the material presented is original except where due reference
is made in the text of this thesis.
Qiang Fu
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
1 Table of Contents
1 Chapter 1 Introduction ....................................................................................................................... 8
1.1 Background ........................................................................................................................... 8
1.2 Current Research of Nano Silicon .......................................................................................... 9
1.3 Objective............................................................................................................................. 10
1.4 Scope .................................................................................................................................. 12
1.5 Structure of Thesis .............................................................................................................. 13
2 Chapter 2 Literature Review ............................................................................................................. 14
2.1 Nanoindentation ................................................................................................................. 14
2.1.1 Young’s Modulus .......................................................................................................... 16
2.1.2 Hardness ...................................................................................................................... 16
2.1.3 Other Mechanical Properties ........................................................................................ 17
2.1.3.1 Strain-Rate Sensitivity ........................................................................................... 17
2.1.3.2 Activation Volume ................................................................................................. 18
2.2 Contact Mechanics .............................................................................................................. 18
2.2.1 Hertz Contact Theory .................................................................................................... 18
2.2.2 Oliver and Pharr method .............................................................................................. 21
2.2.3 Comments .................................................................................................................... 23
2.3 Methodology Review .......................................................................................................... 23
2.3.1 FEM Models ................................................................................................................. 24
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
2 2.3.2 Molecular Dynamics ..................................................................................................... 24
2.3.3 Multi-scale Method ...................................................................................................... 26
2.3.4 Discussion .................................................................................................................... 27
2.4 Review of Molecular Dynamic ............................................................................................. 27
2.4.1 Initial Condition ............................................................................................................ 28
2.4.2 Interatomic Potentials .................................................................................................. 29
2.4.2.1 Pair Potential ........................................................................................................ 29
2.4.2.1.1 Lennard-Jones potential (L-J) ............................................................................. 29
2.4.2.1.2 Born –Lande potential........................................................................................ 30
2.4.2.1.3 Morse potential and Johnson potential .............................................................. 30
2.4.2.1.4 Tersoff potential ................................................................................................ 31
2.4.2.2 Multi-body Potential ............................................................................................. 31
2.4.2.2.1 Embedded Atom Method (EAM) ........................................................................ 32
2.4.2.2.2 Stillinger-Weber (SW) Multiple-Body Potential ................................................... 33
2.4.3 Integration Algorithms.................................................................................................. 33
2.4.4 Molecular Dynamics in Different Ensembles / Temperature conversion ........................ 34
2.5 Phase Transformation of Silicon .......................................................................................... 34
3 Chapter 3 Characterization of Mono-crystalline silicon and Parametric Study .................................. 36
3.1 Numerical Implementation ................................................................................................. 36
3.2 Interatomic potentials ......................................................................................................... 37
3.3 Loading-Displacement Curve ............................................................................................... 40
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
3 3.4 Results of indentation of the perfect substrate.................................................................... 43
3.5 Parametric Studies of Specimen Size and Loading Rate........................................................ 44
3.5.1 The influence of substrate lateral size ........................................................................... 45
3.5.2 The influence of Substrate Thickness ............................................................................ 49
3.5.3 The Influence of Loading Rate....................................................................................... 52
3.6 Conclusion .......................................................................................................................... 54
4 Chapter 4 Characterization of Mono-crystalline Silicon with Defects ................................................ 57
4.1 Computational Model and Defects Description ................................................................... 58
4.2 Effect of the Cavity Size ....................................................................................................... 59
4.2.1 Description of Defect Cases ........................................................................................... 59
4.2.2 Load-Displacement Curve and Test Results ................................................................... 59
4.2.3 Phase Transformation and Atomic Configuration .......................................................... 62
4.2.4 Discussion ..................................................................................................................... 65
4.3 Effect of cavities’ positions .................................................................................................. 70
4.3.1 Description of Defect Cases ........................................................................................... 70
4.3.2 Load-Displacement Curve and Test Results ................................................................... 71
4.3.3 Phase Transformation and Atomic Configuration .......................................................... 74
4.3.4 Discussion ..................................................................................................................... 77
4.4 Effect of multiple cavities .................................................................................................... 80
4.4.1 Description of Defect Cases ........................................................................................... 80
4.4.2 Load-Displacement Curve and Test Results ................................................................... 81
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
4 4.4.3 Phase Transformation and Atomic Configuration .......................................................... 82
4.4.4 Discussion ..................................................................................................................... 83
5 Chapter 5 Conclusion and Future Work ............................................................................................ 86
5.1 Conclusions ......................................................................................................................... 86
5.2 Recommended Future Work ............................................................................................... 89
6 Bibliography ..................................................................................................................................... 90
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
5 Table List
Table 1 Equations to determine the load P for different indenters [28] ................................................. 20
Table 2 Parameters in Tersoff Potential for silicon .............................................................................. 39
Table 3 Parameter in the Morse potential for Interaction of C-Si ......................................................... 40
Table 4 Loading force at maximum indentation depth for different lateral size .................................... 46
Table 5 Young’s modulus and hardness for different lateral size substrate........................................... 46
Table 6 Loading force at maximum indentation depth for different thickness ...................................... 50
Table 7 Young’s Modulus and Hardness for different lateral size substrate ......................................... 50
Table 8 Group d Defect Cases ............................................................................................................ 59
Table 9 Loading force at maximum penetration depth for Group d ...................................................... 60
Table 10 Estimated Young’s modulus and hardness for Group d ........................................................ 62
Table 11 Coordination Number for All Single Cavity Cases................................................................ 66
Table 12 Group f Defect Cases ........................................................................................................... 70
Table 13 Estimated Young’s modulus and hardness for cases f1- f4 in Group f ................................... 73
Table 14 Estimated Young’s modulus and hardness in Group f ........................................................... 74
Table 15 Estimated Young’s modulus and hardness for Group e ......................................................... 82
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
6 Figure list
Figure 1 A representation of the h-P (displacement vs. force) diagram [26] ......................................... 15
Figure 2 (a) Geometry of loading a preformed impression of radius Rr with a rigid indenter radius Ri.(b)
Compliance curve (load vs. displacement) for an elastic-plastic specimen loaded with a spherical
indenter showing both loading and unloading responses. Reprinted from [21] .................................... 21
Figure 3 Time and space scale of modern numerical methods and their applications [42]. ................... 25
Figure 4 Phase I silicon gradually transformed into Phase II silicon under indentation stress [55]. ....... 35
Figure 5 (a) Nanoindentation simulation model; (b) Schematic of cavities’ positions. ......................... 37
Figure 6 Loading –Displacement curve for prefect case ...................................................................... 41
Figure 7 Potential- Distance Curve plot in according the Morse potential used in the simulation model
........................................................................................................................................................... 42
Figure 8 Loading –Displacement curve for five lateral sizes................................................................ 45
Figure 9 (a) Hardness and (b) Young’s modulus – lateral size curves for five lateral sizes ................... 48
Figure 10 Loading –Displacement curve for four thicknesses 22a, 18a, 14a and 10a. .......................... 49
Figure 11 Trend curve for hardness of substrates with four different thicknesses. ................................ 51
Figure 12 Load – Displacement curve for five different loading rates ................................................. 53
Figure 13 Young’s modulus quickly converges with the decrease of the speed .................................... 54
Figure 14 Coordinate system for defining the location of defects ........................................................ 58
Figure 15 Load-displacement curves of Group d. ................................................................................ 61
Figure 16 Atomic configurations of d0 case at four different stages: (a) - (d): substrate with1.5a radius
defect at the indentation depth of 0.657 nm, 1.857 nm, unloading to depth 0.657nm and full unloaded;
Atoms with the CN value between 0 and 13 are visualised. ................................................................. 63
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
7 Figure 17 Atomic configurations of case d3 at four different stages. (a) - (d): substrate with1.5a radius
defect at the indentation depth of 0.657 nm, 1.857 nm, unloading to depth 0.657nm and full unloaded;
Atoms with the CN value between 0 and 13 are visualised. ................................................................. 64
Figure 18 Number of atoms with specified CNs (6, 7 and 8) versus time: (a) case d0; (b) case d3; (c)
case d4; (d) case d5; and (e) case d6. .................................................................................................. 69
Figure 19 .Load-displacement curves of Group f: (a) Offset in –y direction; (b) Offset in –z and +z
directions............................................................................................................................................ 72
Figure 20 Atomic configurations of cases f3 and f6 at three different stages: (a1)-(d1): case f3 at the
indentation depth of 0.657 nm, 1.857 nm, 0.657 nm unloading and full unloaded; (a2)-(d2): case f6 at
the indentation depth of 0.657 nm, 1.857 nm, 0.657 nm unloading and full unloaded. Atoms with the
CN value between 0 and 13 are visualised. ......................................................................................... 75
Figure 21 Atomic configurations of cases f1 and f4 at three different stages: (a1)-(d1): case f1 at the
indentation depth of 0.657 nm, 1.857 nm, 0.657 nm unloading and full unloaded; (a2)-(d2): case f4 at
the indentation depth of 0.657 nm, 1.857 nm and full unloaded. Atoms with the CN value between 0
and 13 are visualised. ......................................................................................................................... 77
Figure 22 Number of atoms with specified CNs (6, 7 and 8) versus time: (a) case f3; (b) case f6. ........ 79
Figure 23 Group e: Four cavities cases (a) Top view (b) Isometric view .............................................. 80
Figure 24 Load-displacement curves of Group e, 4 cavities with the radii of 0.5a, 1a, 1.5a and 2a,
respectively. ....................................................................................................................................... 81
Figure 25 Atomic configurations of cases e1 and e4 at two different stages: (a1)-(a4) at the indentation
depth of 1.857 nm; (b1)-(b4) full unloaded; Atoms with the CN value between 0 and 13 are visualised.
........................................................................................................................................................... 83
Figure 26 Number of atoms with specified CNs (6,7 and 8) versus time for case e1. ........................... 84
Figure 27 Number of atoms with specified CNs (6, 7 and 8) versus time for case e4 ........................... 85
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
8 1 Chapter 1 Introduction
1.1 Background
In the field of nano-material research, silicon (Si) is one of the most widely researched
substrates. It is recognized that today, 16% of the world economy is built on electronics
products and related services [1, 2]: communication, computing, consumer electronics,
health, transport, security, and environment, etc. This percentage is growing every year. As
the essential element of complementary metal oxide semiconductor(CMOS), silicon is used
for microprocessors, microcontrollers, static RAM, and other digital logic circuits. Within
next 15 years, the critical feature size of the elementary nano-electronic devices will drop
from 25nm in 2007 to 4.5nm in 2022 [2, 3]. Within the foreseeable further, silicon will
maintain its position as the main semiconductor material. The further improvement for
higher performance and low/ultra low power application generates the needs on the new
material, technologies and device architectures. According to the vision of SINANO
institute (a nano-electronic research net work based on Europe), several future CMOS
developments are expected [1, 2]:
Advancing core technology for CMOS ;
Adding functionality to CMOS;
Characterisation at the nano-scale; and etc.
In the coming decades, the core technology for CMOS will continuously scale down the size
of silicon based device to nano level. The lifetime of the devices will likely be extended as
the maximum benefit is derived from a particular set of device dimensions. Smaller
devices enable the possibility of parallel computing in multi-core processors, rather than by
increasing absolute clock frequency, which will be limited to about 4 GHz constrained by
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
9 cooling limits. The smaller scale of CMOS is the prerequisite to allow adding the
functionality to CMOS. The additional architectures need to integrate nanostructures in
electronic substrates, e.g. nanopores, nanogaps, nanorods, nanotubes, nanowire,
nanocrystals and etc [1, 2].
Another important application of mono-crystalline silicon is for solar cells. Solar energy has
the potential to become the primary energy supplying global need. The production of solar
cell system increased 30% to 35% in past 10 years. The power generated by solar cell
systems reached about 1.7GW in 2007 [3, 4]. The thin-film technologies have significant
contribution to the improvement of solar cells. The nanostructured silicon thin-film solar
cell lower manufacturing costs by reducing the amount of materials required in creating
the active material for solar cells. From technical perspective, the thin-film structure
strongly enhances the light trapping of the solar cells. Furthermore, the plan to produce
more efficient solar cells with nanowire silicon has been introduced. The potential
advantage of the nanowire silicon solar cells over the thin-film solar cells include reduced
reflection, enhanced light trapping, band gap tuning, and silicon compatible. Due to the
accessible fossil fuel running out quickly, there are increasing numbers of researches on
the property of nano silicon by experiment and simulation[4].
It is clear that silicon will be a significant material for the further technology development.
However the size of silicon components is scaled down to nano level, it is crucial to have a
comprehensive understanding of nano silicon properties.
1.2 Current Research of Nano Silicon
In order to bridge the gap between the theoretical concepts and the design of nano-scale
silicon components, the study focus is on characterizing the material properties, i.e. the
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
10 strength of components needs to be investigated and designed to boost the carrier mobility.
As the manufacturing process scales down to the nano level, the advanced devices will be
more and more sensitive to the various defects induced by the manufacturing process.
Therefore, a fundamental understanding of silicon properties and deformation processes is
crucial for its application and manufacturing purpose. Various experimental approaches
such as nanoscratch and nanoindentation are traditionally employed to exploit the
characteristics of nano-scale silicon. As one of the mature techniques to investigate
mechanical properties of a material, nanoindentation has been extensively employed to
explore mechanical properties of single-crystal silicon for more than a decade [5, 6]. For
instance, Domnich et al. [7] found that β-Si (Si-II) phase is formed in a high-stress region
and the structure becomes amorphous after unloading. Besides of the experimental
studies, affluent atomistic investigations have also been carried out to interpret the
deformation mechanisms of single-crystal Si under nanoindentation [8, 9]. Zhang and his
group members [10-13] have conducted a serial of molecular dynamics (MD) studies of
single-crystal silicon under nanoindentation. Several important results have been
concluded, e.g., Si-I phase may transform into metastable β-Si (Si-II) phase when loading is
large enough to cause plastic deformation, and may further convert into an amorphous
phase during unloading.
1.3 Objective
It is known that materials used in production always contain defects, such as point defects,
cavities (vacancies), impurities et, al. [14, 15]. Such nano-scale defects are one of the most
important factors that affect the material’s properties. For example, the crystal originated
pits (COPs) on the polished surface of the silicon wafer are reported degrading gate oxide
quality[16]. However, the experimental research on the nano material is constrained by
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
11 time and funding. Nano-scale experiment has to be in the controlled environment, factors
such as temperature, humanity, vibration as well as the preparation of the sample could
have significant influence on the results of experiments. Particularly, as the research is
based on such a small scale, the nano defects become one of important factor which could
affect the results. It is no doubt that physical preparation of the specimen with controlled
defects is extremely hard. Therefore, to experimentally characterize the influence from
defect is even more difficult. Only a few studies are found in regard to defects studies, e.g.,
Ciszek and Wang [17] employed the float-zone (FZ) method to study the defect and
impurity effect on the material properties of minority charge carrier lifetime and
photovoltaic efficiency. Under such circumstances, researchers refocused to the numerical
methods, including ab initio calculations [18] and molecular dynamics (MD) simulations.
Particularly, the molecular dynamics (MD) simulation, not only has the ability to precisely
control and alter the defects within specimen, but also has the ability to provide in-time
deformation insights. It becomes an effective and popular tool to study the defect’s effect.
For instance, Sinno et al. [19] used MD to estimate the equilibrium and transport
properties of self-interstitials and vacancies in crystalline silicon at high temperatures. It is
seen that, although the perfect single-crystal silicon is well studied by previous researchers,
the study of defect’s effect on silicon mechanical properties is still rare.
Therefore, the aim of this project is to use numerical tool to investigate and explore the
defects influences on silicon material properties. Therefore, a set of specific objectives are
achieved in this research. Firstly, a numerical nanoindentation model (molecular dynamics)
is developed. Then the reliability and effectiveness of the simulation techniques is verified.
In this thesis, parametric studies of the geometric size of the specimen (lateral size and
thickness) and the loading rate are conducted to optimize the simulation model. Basing on
the validated model, different pre-existing defects are introduced to the silicon substrate,
and then a group of nanoindentation simulations of these defected substrates are carried
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
12 out. The simulation results are carefully investigated and compared with the results from
the benchmarking model of perfect silicon substrate.
1.4 Scope
In this project, molecular dynamics (MD) simulations are carried out to reproduce the
mono-crystalline silicon nanoindentation experiment with a rigid hemispherical indenter.
The simulations produce load-displacement curves and hence mechanical properties, e.g.
Young’s modulus and hardness can be determined. The model is validated by comparing
the calculated Young’s modulus and hardness with the results reported in referred
literatures.
In addition, the simulations are further performed by varying main parameters to optimize
the computational efficiency and simulation accuracy. The influences of parameters to the
calculated results are analyzed and then the acceptable test parameters are selected for
further studies. The tendencies of the influence are checked against the similar cases
investigated by other researchers to provide supportive evidence of model validation.
The cases of mono-crystalline silicon with single cavity and multiple cavities are both
investigated. In order to systematically investigate the influence from single to multiple
cavities, the defect cases are divided into three groups. For the first group, the centers of
spherical cavities are coincident with the substrate lateral center, while the radius of the
cavities varied from case to case. There are two parts of research in second group. For one
part of research, the positions of cavities with the same size remain unchanged on
thickness direction, but gradually moved away from the substrate lateral center. In another
part of research, the cavities positions are always on substrate lateral center, and moved in
the thickness direction. In the third group, simple multi-cavities cases are investigated.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
13 The load-displacement curves and calculated mechanical properties are compared and
analyzed. The preliminary conclusions are drawn based on the tendency of loading force,
Young’s modulus and hardness. Furthermore, the number of silicon atoms with coordinate
number (CN) of six, seven and eight are monitored during loading-unloading process. The
influences of the pre-existing cavity defects to the mono-crystalline silicon phase
transformation are discussed and the visualized atomic configurations are presented to
assist with explaining the phenomena.
1.5 Structure of Thesis
The thesis is organized as following. Chapter 1 introduces the current research of the nano
Silicon, and outlines the objective and the scope of this thesis. In the chapter 2, the
technical components involved in the nanoindentation are reviewed. Additionally, an
overview of numerical simulation methods as well as the MD simulation is included in this
chapter is included.
In chapter 3, we developed and validated the MD simulation model for the
nanoindentation. The parametric studies were carried out to optimize the MD simulation
model.
In chapter 4, we utilized the optimized model to characterize the Si with defects. The cases
with single cavity of different sizes and locations are investigated. Cases with certain
multiple cavities are also considered.
Finally, the conclusion and the further work are discussed in chapter 5.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
14 2 Chapter 2 Literature Review
In this chapter, the technical components including the nanoindentation, contact
mechanics and silicon mechanical properties are reviewed. Additionally, an overview of
numerical simulation methods is included, and particular attentions are paid to the MD
simulation, which is believed to be the most suitable method to investigate the silicon
properties with available resources.
2.1 Nanoindentation
Nanoindentation is one of mature techniques to investigate nano-scale material’s
properties. It uses the recorded depth of the indenter into the specimen along with the
measured applied load to determine the area of contact and hence the material
mechanical properties. The load applied to the indenter increased from zero to the
maximum and then reduced from maximum to zero, which is usually termed as loading
and unloading. During this process, usually both elastic and plastic deformations are
involved. Due to the plastic deformation, a residual impression is left on the surface of the
testing substrate. The results of nanoindentation rely on the accurate determination of the
initial contact of the indenter with the specimen surface, corrections for any penetration
that arises during this initial contact, corrections for compliance of the loading column,
corrections for the departure of ideal shape of the indenter, and corrections for materials-
related issues such as piling-up and indentation size-effect, residual stress, etc [20].
However, the nano-scale residual impression cannot be accurately measured by
conventional method. Using elastic equation of Hertz contact, the area of contact can be
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
15 estimated from the depth of penetration and the geometry of the indenter, and obtain the
Young’s modulus and hardness from the unloading curve [21]. Oliver and Pharr [22] and
Field and Swain [23] presented their procedures to extract the Young’s modulus and
hardness. The procedure they presented also considers the indentified systematic errors
for particular type of tests [24].
Usually, the primary properties extracted from nanoindentation are hardness and Young’s
modulus, however, many other mechanical properties can also be obtained from the
experimental load–displacement curves(Figure 1), these properties such as the strain-
hardening index, fracture toughness, yield strength and residual stress can also be
obtained in certain circumstances [20, 25].
Figure 1 A representation of the h-P (displacement vs. force) diagram [26]
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
16 The methods of estimating Young’s modulus and hardness as well as other mechanical
properties, such as the strain-rate sensitivity and the activation volume are introduced in
the following sections.
2.1.1 Young’s Modulus
The most straight forward parameter that can be obtained from load–displacement curve
is the Young’s modulus. In the practical experiments, the result always contains
deformation from the test specimen and indenter itself [20, 24, 27], the reduced Young’s
modulus rE can be calculated as:
)(2
1
c
rhA
SE
where, contact stiffness S is the slope of unloading curve dhdP / (shown in Figure 1). A is the
area of contact at the depth of indentation Ch , and is the constant related to the geometry
of the indenter. Equation (1) actually derived according to Hertz contact theory. The more
specific introduction of contact mechanicals will be included in section 2.2.
2.1.2 Hardness
There are different methods to measure hardness, Include but not limited to Meyer,
Martens, Brinell, Vickers and Knoop [28]. The equation expression for these measures of
hardness as below:
Meyer: APH /
Martens: 243.26 h
FHM
(1)
(14)
(2)
(3)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
17 where, h is the measured depth from specimen free surface.
Brinell: 22(
2
dDDD
PBHN
where, D is the diameter of the indenter and d is the diameter of residual impression
Vickers: 2
136sin
22
d
PHV
where, P is force and d is the length of diagonal of the indenter
Knoop:
2
130tan
2
5.172cot
2
2d
PKHN
where, d is the length of long diagonal of residual impression.
In the Martens, Brinell and Vickers’ methods, the indented area is the actual measured
area, but for the Knoop and Meyer’s methods, the area of indented is projected area. The
Meyer hardness is widely used for material being indented [20, 24, 27].
2.1.3 Other Mechanical Properties
2.1.3.1 Strain-Rate Sensitivity
The strain-rate sensitivity of the flow stress m is defined as [29, 30]
.
ln
ln
m
(7)
(16)
(4)
(5)
(6)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
18 where is the flow stress and
.
is the strain rate produced under the indenter.
m can be determined from
hndmdHd lnlnln.
2.1.3.2 Activation Volume
Activation volume is the estimated volume swept out by dislocations during thermal
activation, the activation volume *V can be expressed [30]:
HTkV B
.
ln9*
where T is the temperature and Bk is Boltzmann's constant.
2.2 Contact Mechanics
As mentioned earlier, the elastic modulus and hardness for nanoindentation can be
estimated by using the Hertz contact theory. The following provides an overview of using
Hertz’s contact theory and the Oliver-Pharr method [31] to determine the elastic modulus
and hardness of the substrate.
2.2.1 Hertz Contact Theory
Hertz firstly studied the spherical indenters press against the flat specimen in late 19
century [21, 27, 32]. Hertz determined radius of contact circle a in relation to the load,
combined radius, and Young’s modulus by equation:
(8)
(17)
(9)
(18)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
19 *
3
4
3
E
PRa
*E and R are combined elastic modulus and radius of the spherical indenter and the
testing specimen respectively. *E and R can be expressed by equation (11) and (12)
respectively:
'
2'2
*
)1()1(1
E
v
E
v
E
where E and 'E are the Young’s modulus of the indenter and the substrate, respectively.
v and 'v are the Poisson ratio of the indenter and the substrate, respectively. For the rigid
indenter used in the simulations, the elastic modulus of the indenter equals infinity,
therefore the indenter contribution to the combined elastic modulus equals zero.
21
111
RRR
where 1R and 2R are the radius of the indenter and the substrate respectively. For the flat
specimen the radius equals infinity, therefore the contribution of specimen radius item
equals zero and the indenter’s radius is the combined radius.
The maximum depth of the indentation can be determined as:
a
P
Eh
4
3*
Combining equations (10) and (13) together, we can obtain equations (14) and (15).
R
P
Eh
22
*
3
4
3
(10)
(11)
(12)
(13)
(14)
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Queensland University of Technology
20 R
ah
2
Rearrange the equation to express the load P,
2
3
2
1
*
3
4hREP
Similar equations are developed for other different types of indenters, as they are different
in the contact areas and the depth of contact circle. The equations to determine the load
P for different indenters are listed in Table 1 [28].
Table 1 Equations to determine the load P for different indenters [28]
Indenter Load Area of contact
Spherical 2
3
2
1
*
3
4hREP
cc RhRhA 2)tan2( 2
Conical 2
* tan2h
EP
22tanchA
Cylindrical haEP *2 )( 2A
All the equations of contact presented above are applied to the fully elastic contact only
[28]. Hertz method does not consider the systematic error contains in the raw data.
However, usually the plastic deformation participants in the process, and the systematic
error from the raw data is difficult to be eliminated [21]. As shown in Figure 2, after the
load is removed, the indentation depth th of the indenter at the maximum loading
(15)
(16)
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21 recovers to residual impression depth rh . The residual impression depth reflects the plastic
deformation of the indentation. Therefore, it is reasonable to assume rte hhh as the
elastic part of indentation deformation. As the deformation from rh to th is elastic, the
depth of the contact circle is half eh from the original free surface of the substrate. The ph
is the remaining depth from 2/eh to th .
Figure 2 (a) Geometry of loading a preformed impression of radius Rr with a rigid indenter radius Ri.(b) Compliance curve (load vs. displacement) for an elastic-plastic specimen loaded with a spherical indenter showing both loading and unloading responses. Reprinted from [21]
2.2.2 Oliver and Pharr method
From the stipulation of the Hertz contact theory, the determination of the correct contact
area for the different indenter tip shapes is critical [21] [28].
Oliver and Pharr method [31] provides the solution to determine the distance from the
total depth to the depth of contact circle area ph . Derivate the equation (16) in respective
to depth h . The slope of the unloading curve can be expressed as:
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
22 2
12
1
*2 ehREdh
dP
Substituting with the load expression for spherical indenter (Table 1), we get
pdP
dhhe
2
3
As the depth of the contact circle sh is half of eh from the original free surface of the
substrate,
pdP
dhhs
4
3
and
2
etp
hhh
Thus the radius a of the contact circle can be determined from
pippi hRhhRa 22 2
The hardness of material can be calculated through APH / , A is the contact area
computed from radius a .The elastic displacement has been defined by the above equation,
the equation can be then rewritten:
aE
R
aRE
dh
dP *
2
12
1
* 22
By rearranging the equation, the Young’s modulus can be expressed as
adh
dPE
2
1*
(17)
(18)
(19)
(20)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
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23 2.2.3 Comments
The Hertz contact theory incorporate with the Oliver-Pharr method is frequently utilized
for the calculation of the Young’s modulus of the specimen during nanoindentation with
the spherical indenter, e.g. Zhan et al. [33] used this method to investigate the elastic
modulus of copper with defects; and Smith et al. [34] also employed the same method to
determine the Young’s modulus for silicon under nanoindentation simulation. Since there
are numerous of successful cases, the same technique is adopted in this project.
2.3 Methodology Review
There are some inevitable difficulties involved in the nanoindentation experiments. Firstly,
as mentioned before, the condition of the sample may be not ideal: the surface roughness;
surface contamination; defects or micro crack beneath the surface or the accuracy of the
indenter profile may significantly affect the test results. Secondly, the nanoindentation
experiments are not able to reveal the detailed crystal structure change during the
indentation. Particularly, in this research, the defects in the material need to be controlled
in size and position. However, it’s extremely difficult to physically prepare samples with
controlled defects. On the contrary, these drawbacks could be perfectly overcome by using
the numerical simulation.
This section firstly reviews the numerical simulation methods commonly used in material
research and then the most suitable approach to characterize silicon on nano-scale is
selected by comparing benefits and drawbacks of each method.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
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24 2.3.1 FEM Models
The Finite Element Method (FEM) is the numerical method used to determine the
approximate solutions of partial differential equations (PDE) and integral equations. The
solution approach is based either on eliminating the differential equation completely , or
rendering the PDE into an approximating system of ordinary differential equations [35],
which are then solved using standard techniques such as finite difference [36], Runge-
Kutta, etc [37].
Although the FEM is originated from the needs for solving complex elasticity, structural
analysis problems in civil engineering and aeronautical engineering, it is also long-term
used in studies of nanoindentation experiments [38]. Yu et al. [25] used finite element
simulations to investigate the tip-radius effect during shallow nanoindentation whose
indentation depth is lower than 20nm. Li et al. [39] used 3D FEM method to study the
influence of friction, slide and sample size. While Bressan et al. [40] investigated the
nanoindentation of bulk and thin film by FEM method.
The macro scale material behaviors are connected to the conventional continuum FEM
usually via empirical and macro scale experiments determined model [38]. The
conventional continuum methods such as the FEM are not applicable to the investigation
of the nano-scale atomic behaviors, as they are unable to accurately predict the interaction
between atoms.
2.3.2 Molecular Dynamics
Molecular Dynamics (MD) is a numerical simulation method wherein molecules are
interacted for a period under physics law [41]. It provides a view of the motion of the
molecules. However, MD systems generally consist of a vast number of particles; the time
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
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25 of the processing could be unbearable [38]. Thus, the common application of MD is to
solve the micro/nano-scale problem. It represents an interface between laboratory
experiments and theory.
FEM and MD both have limitations on length and time scale. Figure 3 shows different
numerical methods and applications in accordance with different length and time scale.
Continuum mechanics method, such as FEM, is located in area where time length is up to
seconds and geometry scale at meters. Classical MD, on the other hand, is located at nm-
ps area. In this project, the study of mechanical properties on material with defects is
based on the atomic scale, in which, FEM appears deficient in providing satisfied results.
Figure 3 Time and space scale of modern numerical methods and their applications [42].
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
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26 In fact, the MD simulation for nanoindentation of covalent material has been attempted,
and the results shows MD simulation has the potential to be a powerful tool in the nano-
scale material research [10, 43, 44], [11, 45, 46]. In this thesis, since it is almost impossible
to obtain the specimens with controlled defects, the advantage of MD simulation becomes
more obvious: it can accurately define the defects by rearranging the atoms positions.
Although the MD method can perfectly solve such problem, as the involved scale exceeds
10μm, the computational process usually becomes unacceptably complicated [38].
Therefore, in this project, only simple defects are in the consideration due to the
computational capacity of MD. The complex dimensional defects problem extends the
investigation to the macro scale level; and thus the multi-scale method should be
employed in this case, which is out of the scope of this thesis
2.3.3 Multi-scale Method
The atomic modeling can reveal nano-scale mechanisms such as discrete dislocation [47],
but it is impossible to handle large length and time scales. In contrast, continuum
modeling, e.g., FEM, predicts deformation by averaging atomic scale dynamics and defect
evolution, which is valid only for large systems that include a substantial number of
atoms/defects/damages [48]; however it is not possible to handle the atomic dimensions
because the scale is too coarse to capture the fine details.
Multi-scale method, coupling the micro and macro scale simulation together, to simulate
the engineering problem. This method has been employed by many researchers to explore
various nano-scale problems, e.g. Lin et al. [49] used multi-scale method to investigate the
stress and strain of single crystal nickel material during nano-scale cutting; Wang et al. [50]
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
27 applied multi-scale simulation to study the process of incipient plastic deformation on FCC
metals.. However, in this thesis, the multi-scale method is not attempted.
2.3.4 Discussion
In the case of characterizing the silicon with defects, due to the needs of having insight of
atomic deformation and configuration during and after the experiment, FEM will not be
applied.
By coupling FEM (or other continuum simulation method) with MD, the multi-scale
method is able to capture both atomic scale details and macro scale deformations.
However, this thesis is focusing on the simple defect influence to the material mechanical
property, the number of defect and severity of the defect is based on atomic level,
therefore, to use multi-scale simulation is not necessary. In addition, another significant
benefit of using MD simulation is that defects can be defined by easily altering the atoms
arrangements during the numerical implementation stage. Therefore, MD is chosen for the
purpose of characterizing the silicon with defects.
2.4 Review of Molecular Dynamics
MD simulation, one of the popular numerical simulation approaches, has been widely used
in the research of nano silicon material. As a complement to conventional experiments,
MD provides the possibility of understanding the property of the molecular assembly
structure and the interaction between molecules. MD represents an interface between
laboratory experiments and theory [51]. As the results of the discussion in section 2.3, MD
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
28 is selected as the most suitable numerical method for this thesis, thus, the technical
components of MD are reviewed in the following paragraphs.
Alder and Wainwright [52] utilized the method of molecular dynamics to investigate gas
and liquid phase transition with hard-sphere system in 1957, which can be considered as
the first application of MD Simulation. In decades, with the development of computer
performance, the MD simulation technology has been improved significantly.
MD simulations not only provide the details of molecule motion, but also provide a chance
to observe the process of the “virtual experiment”. The advantage makes MD simulation to
become very attractive in many fields, such as physics, chemistry, material science, and
biology etc.
2.4.1 Initial Condition
The initial condition of MD simulation mainly refers to the initial position and velocity of
molecules. The initial conditions can be obtained from experiments results, assigned by the
theoretical model, or the combination of the two. Typically, the initial positions of
molecules are sited on a regular lattice (FCC, BCC etc.) [10, 43-46, 53-56]. The initial
velocity can be assigned by Maxwell-Boltzmann distribution at certain temperatures. When
the initial positions are set up, the Nose-Hoover thermostat [57, 58] is typically employed
to keep the simulation system at certain required temperature. After a period of time, the
position and velocity is equilibrated, and the model is ready for further simulation. This
process is also known as relaxation.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
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29 2.4.2 Interatomic Potentials
The potential function is critical for the reliability of simulation results. The potential
function is usually the combination of experience equation/coefficient and experimental
results, therefore, choices of the potential function for the particular situation are very
important [56].
Potential function considers the effect contributed by the electron cloud. The potential
function has been developed from pair potential to multiple-body potential. Pair potential
only considers the interaction between two molecules, neglecting the interaction of other
molecules in the system. The pair potential is a simplified function in order to represent
the basic intermolecular potentials. Actually, in the multiple-body system, the electron
clouds are interfered from each others, the density of electron cloud is changed upon the
positions of the molecules in a certain range. General speaking, the multiple-body
potential is able to give more reliable description of the interaction between molecules [56,
59, 60].
2.4.2.1 Pair Potential
Alder and Wainwright [52] used pair potential in MD simulation in 1957, with the decades
of research, many pair potential functions have been developed with the purpose of
describing the interaction between different molecules. Some typical potential are
discussed as follow:
2.4.2.1.1 Lennard-Jones potential (L-J)
Lennard-Jones potential is initially developed to describe the interaction between the
molecules of inert gases. L-J potential describes the relevant weak interaction, it usually
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
30 used in simulation of gas or liquid. It also can be used to describe the BCC transition metal,
such as chromium, molybdenum, tungsten, etc [56, 59, 60]. The equation expression of L-J
potential as below:
])()[(4)( 612
ijij
ijrr
rV
where is the depth of the potential well, is the distance at which the inter-particle
potential is zero, and ijr is the distance between particles[61].
2.4.2.1.2 Born –Lande potential
Born–Lande potential is a derivation of Born–Lande equation, which is proposed by Max
Born and Alfred Lande[56]. The Born–Lande equation divides the lattice energy into
electrostatic potential and repulsive potential energy. It’s commonly used to describe the
potential between ion atoms in MD [56, 62].
2.4.2.1.3 Morse potential and Johnson potential
Both Morse potential and Johnson potential are used to describe the solid metal. These
potential functions are commonly used in the solid metal simulation [56, 59]. Morse
potential is also employed to describe the interaction between the carbon indenter and
silicon [46]. The Morse potential can be expressed as follow:
)}}(exp{2)}(2{exp{)( 00 rrrrDrUjiijij Rcrij (22)
The detailed introduction of Morse potential is included in chapter 3.
(21)
(1)
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31 2.4.2.1.4 Tersoff potential
Tersoff potential is a pair-like potential where the bond order of the atoms is affected by
its environment. Tersoff potential can be expressed as follow [63]:
Cheong & Zhang [46] employed Tersoff potential to simulate the indentation of mono-
crystalline silicon, they successfully reproduced the nanoindentation experiment and
observed the transformation from the diamond cubic structure into a body centered
tetragonal form during the simulation.
2.4.2.2 Multi-body Potential
Multi-body potential appeared in the mid of 1980s. In the multiple-atoms systems, the
electron clouds are interfered from each others, the density of electron cloud changes with
ji
ij
i
i WEE2
1
)]()()[( ijAijijRijCij rfbrfrfW
)exp()( ijijijijR rArf
)exp()( ijijijijA rBrf
0
)(
)(cos
1
)(21
21
ijij
ijij
ijCRS
Rrrf
ijij
ijijij
ijij
Rr
SrR
Rr
(23)
(1)
(24)
(2) (25)
(3) (26)
(4)
(27)
(5)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
32 the positions of the atoms in the system, thus the potential energy of the atoms changes
correspondingly. In the MD simulation, it is assumed that only the electron field in a
certain size around of the atom contributes to the interference.
2.4.2.2.1 Embedded Atom Method (EAM)
Daw & Baskes [64] developed the EAM in 1984. The main idea of the EAM is to divide the
total potential into the pair potential and the electron cloudy potential [54, 56, 64]. EAM
can be expressed as:
ij
ijij
i
ii rFU )(2
1)(
where, the first term i
iiF )( is the electron cloudy potential; the send term ij
ijij r )(2
1
is the pair potential; and i is the total electron density constructed by all the atoms.
For metallic material:
2
1
3 ()()(k
ijkijkkijj rRHrRAr )
6
1
3 ()()(k
ijkijkkijij rrHrrar )
where kA , kR , ka and kr are the coefficient depending on different metal.
For the covalence materials, the fdps ,,, level of electron is considered, thus, there are
four different i . The total i can be expressed as:
(28)
(6)
(29)
(7)
(30)
(8)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
33 2
3
0
)(2 )()( h
i
h
h
ii t
hence, the electron cloudy potential in (28) becomes:
ln)( AEF
where, A and E are the coefficients depending on the different material.
2.4.2.2.2 Stillinger-Weber (SW) Multiple-Body Potential
The Stillinger-Weber (SW) multiple-body potential explicitly involves two and three bodies’
potential. SW multiple-body potential uses an experience potential function and a simple
body potential function to describe defects in the material [53, 65].
2.4.3 Integration Algorithms
In order to obtain the details of the atoms motion, there are various of integration
algorithms available, such as Leap-frog algorithm, Beeman algorithm, Gear algorithm,
Rahman algorithm and Verlet algorithm, etc [51, 56, 59, 60].
In the MD Simulation, the most commonly used time integration algorithm is known as the
Verlet algorithm. The basic idea is to write two third-order Taylor expansions for the
positions )(tr , one is at t forward step, and at t backward step.
)()()6/1()()2/1()()()( 432 tOttbttattvtrttr
)()()6/1()()2/1()()()( 432 tOttbttattvtrttr
(31)
(9)
(32)
(10)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
34 Add above two equations up,
)()()()(2)( 42 tOttattrtrttr
2.4.4 Molecular Dynamics in Different Ensembles / Temperature conversion
The Molecular Dynamics Simulation can be carried out in different ensembles. In this thesis,
the NVT ensemble is preferred in most cases. In the NVT ensemble, the atoms number N ,
volume V and temperature T remain constant. Because the kinetic energy is directly
related to the temperature, the kinetic energy is a constant as well [51, 56, 59].
There are a few methods for temperature conversion. The most direct method is to limit
the velocity by multiplying a coefficient in every step. Alternatively, the virtual thermal
bath can help the temperature remain in the range of requirement. The most commonly
used thermal bath includes Berendsen thermal bath, Gaussian thermal bath, and Nose-
Hoover thermal bath, etc [56-58].
Another method used in the MD simulation for nanoindentation is surrounding the atoms
system with thermostat atoms [46]. During the process, the heat is transferred out of the
system through thermostat atoms, and the thermostat atoms remain at a setting
temperature, therefore the controlled volume temperature remains stable.
2.5 Phase Transformation of Silicon
Mono-crystalline silicon is a brittle material with a gray metallic appearance. The typical
Young’s modulus is from 130 GPa to 190 GPa which varies slightly with crystal orientation.
(33)
(13)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
35 The atomic weight of silicon is 4.6638x10-23g per atom. Mono-crystalline silicon has the
same crystal structure as diamond with a lattice constant of 0.543095 nm [66].
It has been confirmed that mono-crystalline silicon experiences the brittle-ductile solid
phase transformation under certain loading conditions. Vodenitcharova and Zhang
summarize some major transformation phase [12, 67]: Si II (β phase Si) and Si VII (a
hexagonal close-packed structure) is reported by Hu et al.[68] in 1986; Si XI, Imma silicon is
observed at 13.2-15.6 by McMahon et al.[69]; Si V and Si VI, is found at 14 GPa and 38 GPa
[70]; and Si X a face-centered cubic, claimed to be discovered at 248 GPa [71].
Figure 4 Phase I silicon gradually transformed into Phase II silicon under indentation stress [55].
In this work, Si II (β phase Si) is the primary phase to be investigated. The β phase Si have
intensively used [11, 46, 55, 72] as an indication of deformation during the
nanoindentation process.
Figure 4 shows the crystal structure gradually deforms from Phase I Si to Phase II Si. The
coordination number for the phase I Si is four and the phase with a coordination number
of six, is the b-Si structure (Si-II). The transformation is usually induced by stress.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
36 3 Chapter 3 Characterization of Mono-crystalline silicon
and Parametric Study
In this chapter, the MD simulation model for the nanoindentation is developed and
implemented, and the load-displacement curves are recorded for analysis. From the curves,
the elastic modulus and hardness are extracted and compared with reported results to
validate the simulation model. The influence of key simulation parameters including lateral
size, thickness of the substrate and loading rate, are further investigated to optimize the
simulation model. Meanwhile, the atomic configuration and the coordination number of
the perfect silicon are obtained and used for the comparison with defected cases.
3.1 Numerical Implementation
MD simulations were carried out to reproduce the nanoindentation on mono-crystalline
silicon by a hemispherical indenter tip. Figure 5 shows the initial model developed to
simulate the nanoindentation experiment. The specimen is a mono-crystalline silicon
substrate contains 189,216 atoms with the size of 36ax36ax18a. The diamond structured
carbon hemispherical indenter tip with a radius of 5a is assumed to be a rigid body. Herein,
a is the silicon lattice constant. The x , y and z axes represent the lattice direction [100],
[010] and [001], respectively. Five atomic layers at the bottom of the substrate are fixed to
provide structural stability and prevent the substrate from moving. The remaining layers
are thermal control layers used to impose the substrate temperature. Periodic boundary
condition is imposed in the two lateral directions ),( yx and free surfaces along the
thickness direction )(z . Different potentials are selected to describe the interaction of
different atoms. The Tersoff potential is adopted to simulate the behavior of silicon
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
37 substrate. The Morse potential is used to describe the atomic interaction between carbon
indenter and silicon substrate. During the simulation, the substrate is firstly relaxed to a
minimum energy state using conjugate gradient energy minimization method and then the
Nose-Hoover thermostat [57, 58] is employed to equilibrate the substrate at 0.01K. For the
initial model, a constant speed of 0.02 nm/Ps is adopted during both loading and unloading
processes. At the maximum indentation depth, the simulation model is relaxed for 120 ps.
The equations of motion are integrated with time using a velocity Verlet algorithm [73]. All
simulations were carried out by the open source code LAMMPS [74].
Figure 5 (a) Nanoindentation simulation model; (b) Schematic of cavities’ positions.
3.2 Interatomic potentials
The selection of interatomic potentials and relevant parameters are crucial for the
reliability of the MD simulation results. It has been proven that Tersoff potential is capable
to predict stable phases of diamond cubic silicon and body centered tetragonal β phase
silicon. Tersoff potential is widely employed by researchers, e.g. Zhang [46, 75-77] and his
team to investigate the phase transformation of silicon induced by nanoindentation.
(a) (b)
z
Cavity Y
x
z
y
x
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
38 Therefore, in this thesis, Tersoff potential is adopted to describe the interaction between
of silicon atoms.
The Tersoff potential is expressed by Equations (23)-(26) in Chapter 2(Literature Review,
Page 31). For the convenience of reading, equation (23) and (24) are repeated at below:
Equation (23) is the equation expression of total potential energy E , which is the sum of
atomic potential energy ijW . Within the atomic potential energy defined by equation (24),
a Repulsive pair potential is expressed by function Rf , which includes the
orthogonalization energy, and the attractive pair potential associated with the bonding
force is expressed by Af . The function Cf is the smooth cut-off function, which defines the
range of potential energy influenced. The equation expression of Rf , Af and cf have been
introduced in the equation (25), (26) and (27), respectively. The parameter ijb , which is a
monotonically decreasing function of the atoms i and j coordination, describes the bond
order [63]. Tersoff, who firstly developed this method, determined a satisfactory form for
ijb in late 80s [63],
nn
ij
n
ijb 2
1
)1(
jik
ikijijkikcij rrgrf,
33
3 ])(exp[)()(
])cos(/[/1)( 22222 hdcdcg
ji
ij
i
i WEE2
1
)]()()[( ijAijijRijCij rfbrfrfW
(23)
(24)
(34)
(24
(36)
(35)
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
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39 Tersoff [63] also determined the parameter values for silicon, and now widely used by
researchers [46, 55, 63]. The parameters adopted in the modeling are listed in Table 2.
Table 2 Parameters in Tersoff Potential for silicon
Parameters in Tersoff Potential for silicon
Parameter Si-Si value
A (eV) 1830.8
B (eV) 471.18
(Å-1
) 2.4799
(Å-1
) 1.7322
1.1× 10-6
n 0.78734
c 1.0039 × 105
d 16.217
h -0.59825
R (Å) 2.7
S (Å) 3.0
sisi 1
The two bodies Morse potential is imposed to describe the interaction between silicon and
carbon atoms, as the accuracy and feasibility of using Morse potential for the Si-C
interaction is verified by researchers [46, 55] through good agreements between
experiment and simulation results.
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
Queensland University of Technology
40 For the convenience of reading, the equation expression of Morse potential is repeated as
below:
)}}(exp{2)}(2{exp{)( 00 rrrrDrUjiijij Rcrij (22)
The Parameter of Morse potential is listed in Table 3:
Table 3 Parameter in the Morse potential for Interaction of C-Si
Parameter in the Morse potential for Interaction of C-Si
Parameter Value Notes
D (eV) 0.435 Cohesion Energy
(Å-1
) 4.6487 Elastic Modulus
0r (Å) 1.9475 Equilibrium distance between atoms
cR 1.5r0 Cut-off Distance
3.3 Loading-Displacement Curve
During nanoindentation, a typical load-displacement curve is obtained, which describes the
corresponding force during both loading and unloading process. Based on this curve, the
mechanical properties such as Young’s modulus and hardness can be determined.
The load-displacement curve in Figure 6 records simulation results of the initial model
described in the Section 3.1. The curve is divided into four parts for the purpose of analysis,
which represent four different statuses including approaching, (AB) Loading (BC),
Relaxation (CD), and Unloading (CD).
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
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41 At the beginning of the simulation, the indenter keeps static, and the substrate is under a
relaxation process for 40000 steps (80ps). After that, a constant Speed is applied to the
indenter. Before the indenter and substrate is making contact, the indenter needs to travel
the pre-set distance between them. As shown in Figure 6, the loading curve presents zero
loads until the start point of approaching stage A.
The AB section of the loading curve represents the approaching stage of the indentation.
After relaxation, the indenter moves towards to substrate free surface with the given
speed. When the indenter is close enough to the surface of silicon substrate, the negative
loading is detected as shown in Figure 6.
Figure 6 Loading –Displacement curve for prefect case
This phenomenon can be explained by the force-distance curve of the Morse potential for
silicon and carbon atoms. As indicated in Figure 7, when the distance is further than 0r , the
atoms present attractive force, and the attractive force increases when the distance is
C
D
E
A
B
D
C
E
E
Advanced Numerical Characterization of Silicon with Defects by Nanoindentation
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42 closer to 0r . When the indenter moves even closer to substrate, the distance between bulk
silicon and carbon atoms is shorten to less than 0r , the force quickly turns into repulsive
force. It is worth to mention that even the indenter has not made physical contact with the
substrate at this moment, which means the load equal zero, the point is still considered as
the contact point of the indentation because of the repulsive force presenting between
carbon and silicon atoms [78].
Figure 7 Potential- Distance Curve plot in according the Morse potential used in the simulation model
At point B, the repulsive force resumes back to zero and grows with increasing of the
indentation depth until the indenter reach the maximum depth at point C. It is discussed
by A.C Fischer-Cripps [24, 27, 32], that the loading usually involved both elastic and plastic
deformation. Because the plastic deformation has completed at the maximum indentation
depth, the unloading curve reflects the recovery of elastic deformation only. Therefore,
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43 unloading curve (section DE) is usually selected for analysis of the material mechanical
properties.
The indenter stops at maximum depth of 2.4nm, and then hold for 500 steps before it
starts to retract upwards at the same speed as loading. Another phenomenon revealed in
the load-displacement curve is the slight descending of the load during the holding period
at the maximum depth (section CD), which is widely observed during nanoindentation
simulations [55, 79]. This phenomenon is due to the relaxation of the system, similar to the
relaxation at the beginning of the simulation, the atoms are interacting with each other to
achieve the minimum energy state during this period.
3.4 Results of indentation of the perfect substrate
As aforementioned, using the Hertz theory incorporated with the Oliver-Pharr method, the
reduced modulus *E and hardness H can be estimated. The reduced modulus is defined
in equation (11), for the convenience of discussion, the equation is repeated here:
'
2'2
*
)1()1(1
E
v
E
v
E
Where E and v , and 'E and 'v , are the Young’s modulus and Poisson’s ratio of the
substrate and indenter, respectively.
For a perfectly rigid indenter applied in this work, the indenter will not contribute to the
reduced modulus *E , as a result, the reduced *E can fully reflect deformation of the
substrate. The substrate Young’s modulus is simply given by: )1( 2* vEE , and the
Poisson’s ratio of the silicon substrate is 0.22, from reference [80]. The value of unloading
curve slope dhdP / and the radius of the circle of contact a can be extracted from the load-
(11)
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44 displacement curve (Figure 6) for initial indentation modeling. By employing the
aforementioned method, the Young’s modulus and hardness are calculated as 17.123* E
GPa and 6.38H GPa, respectively, which agree with those from the previous research by
others. e.g., Bhushan and Li [81] reported the single crystal silicon with a Young’s modulus
of 179 GPa, Sjöström et al. [82] reported a hardness around 36.3 GPa for uncoated silicon
substrate.
3.5 Parametric Studies of Specimen Size and Loading Rate
During the modeling, it is found that the specimen size and the indenter speed could have
significant influence to simulation results. These phenomena are also observed by other
researchers [46], [79], and [83]. Therefore, in the following section, parametric studies of
specimen size (substrate lateral size and thickness) and indenter speed are conducted.
Three groups of simulations are designed to investigate the influence of these parameters.
The first group studies the influence of substrates lateral size. The second group is used for
studying influence of substrate thickness. The third group investigates the influence of
different loading rates.
To be noted, as this parametric study is qualitative investigation on the trends of
mechanical properties under the changing parameters, the accuracy of results is not the
priority. Therefore, some modification is made on the initial model to improve the
computational efficiency. The modifications include changing the indenter speed to 0.04
nm/Ps and changing the maximum indentation depth to 1.457 nm.
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45 3.5.1 The influence of substrate lateral size
The size of the substrate could affect the results of nanoindentation. Cheong and Zhang
[46] discussed that the effect of the test specimen size by analysis of the displacement field
of the atoms at maximum indentation depth. They believe that as long as the substrate
size is larger than displacement region, no influence should be made to the testing results.
Within the first group of parametric study, a series of simulations with different substrate
lateral sizes are performed. Different Silicon substrates sizes including 18ax18ax18a,
24ax24ax18a, 30ax30ax18a, 36ax36ax18a and 42ax42ax18a are considered.
Figure 8 Loading –Displacement curve for five lateral sizes
Figure 8 shows five load-displacement curves for the silicon substrates with different
lateral sizes. From the loading portions of these curves, it is found that smaller substrate
leads to steeper curve. At the maximum depth of indentation, the loading values on the
substrates with smaller lateral sizes are greater than those on the substrates with larger
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46 lateral sizes. During the relaxation period (at the maximum depth), the loading values on
the smaller substrates subjects to larger reduction than those on larger substrates. The
loading values before and after relaxation at maximum indentation depth for different
lateral sizes are listed in Table 4.
Table 4 Loading force at maximum indentation depth for different lateral size
Lateral Size 42ax42ax18a 36ax36ax18a 30ax30ax18a 24ax24ax18a 18ax18ax18a
Before
Relaxation
498.5 502.1 523.5 550 575.6
After
Relaxation
386.5 379.8 371.6 361.9 359.2
*Loading unit is in nN
Table 5 Young’s modulus and hardness for different lateral size substrate
Lateral Size 42ax42ax18a 36ax36ax18a 30ax30ax18a 24ax24ax18a 18ax18ax18a
Young’s
modulus
176.92 201.2 295.5 373.1 514.0
hardness 26.19 24.8 22.6 21.5 20.8
*Value unit is in GPa
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47 From above two phenomena, an assumption can be made that strain energy transfer rate
is lower in the smaller substrate. During the loading process, the energy is more
concentrated for the smaller substrate, which requires more efforts to achieve strain
energy equilibrium. For the same reason, during unloading, the steeper curve is observed
for the substrate with smaller lateral size. In accordance with the Hertz’s theory, the
steeper unloading curve leads to higher Young’s modulus. The estimated mechanical
properties from these curves are listed in Table 5.
The estimated hardness presents an opposite trend of Young’s modulus in this group: as
shown in Figure 9(a), the larger lateral size leads to higher hardness. The finding is
consistent with our previous assumption. As shown in Figure 8, the strain energy transfer
rate is lower in the smaller substrate which makes the system harder to achieve
equilibrium, thus the smaller force is observed at the same depth during unloading. It leads
to the smaller estimated value of hardness because the estimated contact area A is
unchanged for the same indentation depth.
The influence of lateral size to the testing results is convergent while the lateral size is
increasing. From Figure 9(b), we find that Young’s modulus for the 18ax18ax18a substrate
peak up to 514 GPa, the value drop to 373.1 GPa for the 24ax24ax18a substrate. However,
for the larger lateral size cases 36ax36ax18a and 42ax42ax18a, the reduction of Young’s
modulus is only 24.28 GPa. Based on above observation, the larger lateral size substrate is
less susceptible to un-equilibrium strain energy. However, while taking both accuracy and
computational cost into account, lateral size 36ax36a is selected for the further research.
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48
(a)
(b)
Figure 9 (a) Hardness and (b) Young’s modulus – lateral size curves for five lateral sizes
18ax18a
24ax24a
30ax30a
36ax36a
42ax42a
18ax18a 24ax24a
30ax30a
36ax36a
42ax42a
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49 3.5.2 The influence of Substrate Thickness
The second group of simulations is designed to investigate the influence of the substrate
thickness. Within this group, a series of simulation on the substrates with different
thickness were performed. Simulation setting same as adopted in the first group are
applied.
Figure 10 shows the load-displacement curves of silicon substrates with same lateral size
but different thicknesses. The silicon substrates with size of 36ax36ax10a, 36ax36ax14a,
36ax36ax18a and 36ax36ax22a were considered in this group. The loading curves show
the corresponding force at maximum depth is increasing while the thickness is reducing.
During the relaxation, the force of the thinner substrates subjects to larger reduction than
those on thicker substrates.
Figure 10 Loading –Displacement curve for four thicknesses 22a, 18a, 14a and 10a.
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50
Table 6 Loading force at maximum indentation depth for different thickness
Thickness 36ax36ax22a 36ax36ax18a 36ax36ax14a 36ax36ax10a
Before Relaxation 476.5 502.1 541.4 568.4
After Relaxation 369.5 380.2 394.4 420.4
*Loading unit is in nN
Table 7 Young’s Modulus and Hardness for different lateral size substrate
Thickness 36ax36ax22a 36ax36ax18a 36ax36ax14a 36ax36ax10a
Young’s
modulus
151.6 201.2 237.9 324.8
Hardness 26 24.8 25 25.7
*Value unit is in GPa
Table 6 shows that the greatest reduction is 148 nN on the 10a thickness and the least is
107 nN on 22a thickness. This trend is similar to what we found in the first group.
Therefore, it is believed that the transfer rate of strain energy in the thinner thickness is
lower, which is similar with case of smaller lateral size. For the same reason, it is expected
that Young’s modulus is descending when the substrate becomes thicker (Table 7)
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51 In terms of hardness, however, we find it is not decreasing monotonically along with the
decrease of thickness. The hardness under the influence of the lower layer material, which
is rigid in this case, rises up after lowest point at thickness 18a. As shown in Figure 11, the
turning point of the trend of hardness is between thickness 18a and 14a. Therefore, it is
reasonable to use 18a as the minimum thickness of specimen without major influence
from the lower layer material. The similar phenomenon is reported by Bolesta and Fomin
[83] that the mechanical properties of the thin film intend to have more influence from the
base material’s properties when the film becomes thinner. For the simulations in future
studies, thickness 18a is selected with a comprehensive consideration of computational
efficiency and accuracy.
Figure 11 Trend curve for hardness of substrates with four different thicknesses.
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52 3.5.3 The Influence of Loading Rate
Liu and his team [79] discussed the significant influence of indenter’s speed to the MD
simulation results. In accordance with their conclusions, the higher indentation speed of
indenter leads to the higher strength, and the value of the strength quickly converge with
the decrease of the speed. The purpose of this section is to verify their conclusion and
select the suitable loading rate for future simulations.
Based on the conclusion of previous section 3.5.1 and 3.5.2, the substrate size
36ax36ax18a was employed. All other simulation settings are same as the first group and
second group simulations, except giving different loading rates to the indenter.
Figure 12 displays the load-displacement curve for the cases with different loading rates.
The simulations are performed with indenter speed at 0.01, 0.02, 0.03, 0.05, and 0.08
nm/Ps. The trend of estimated Young’s modulus for this group is presented in Figure 13.
Young’s modulus rockets up to 1293.3 GPa at the 0.08 nm/Ps loading rate, then reduces
down to 124.26 GPa at 0.01 nm/Ps. Convergence presented in Figure 13 is consistent with
Liu and his team’s finding [79].
The un-equilibrium strain energy can also be used to explain the trend of Young’s modulus in
this group. For the substrates with same size, the faster loading rate leads to less time for
atoms to interact and recover to the strain energy equilibrium condition. Therefore, higher
Young’s modulus is induced at faster loading speed. When the loading rate reduces, the
atoms have enough time to settle down, and the Young’s modulus is much lower.
Another phenomenon, which reflects the un-equilibrium strain energy, is the loading
reduction during the relaxation process. From Figure 12, it is observed that the greatest
reduction of loading happens when the loading rate is 0.08 nm/Ps. The relaxation effect
becomes less significant when the loading rate slows down, and finally disappears at 0.01
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53 nm/Ps. That means less effort needs to be made to achieve the strain energy equilibrium
at lower loading rate, and particularly, at 0.01 nm/Ps loading rate, the atoms have enough
time to fully interact with each other, and the Young’s modulus estimated from that curve
is free from the influence of un-equilibrium strain energy.
Liu et al. [79] believed that the ratio of the elastic energy to total energy decreases while
the loading rate increases. As shown in Figure 12, the loading values after relaxation are
390.9, 389.3, 379.8, 369.6 and 337.7 nN for loading rate 0.01, 0.02, 0.04, 0.05, and 0.08
nm/Ps, respectively.
Figure 12 Load – Displacement curve for five different loading rates
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54
-
Figure 13 Young’s modulus quickly converges with the decrease of the speed
It is stipulated in the previous section that only elastic deformation contribute to the
unloading curve. Therefore, it is reasonable to postulate that the substrate with smaller
force after relaxation receives more plastic deformation which absorbs more energy.
To sum up, the load rate 0.02 nm/Ps is adopted for simulations of future studies with
consideration of the balance between the efficiency and accuracy.
3.6 Conclusion
In this chapter, the MD simulation model was built for the nanoindentation on mono-
crystalline silicon with hemispherical rigid indenter. Through comparison with reported
results, the MD model has been verified.
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55 The parametric studies of the substrate lateral size, substrate thickness and loading rate
indicate that these simulation parameters have significant influences on the
nanoindentation results. Preliminary, substrate with smaller lateral size and thinner
thickness intends to induct higher Young’s modulus. However, the trend of estimated
hardness under the influences of substrate lateral size is different from the one of
substrate thickness. The hardness is monotonically decreasing with the lateral size. But,
when the substrate thickness decreases, the curve of hardness represent the decreasing
trend, until the influence from hardness of lower layer material (which is rigid in this case)
rises the hardness up. In accordance with our simulation results, the minimum substrate
thickness without having significant influence from lower layer is between 18a and 14a.
The loading rate influence to the nanoindentation results is also examined. The higher
indentation speed of indenter will lead to the higher strength, and the value of the
strength quickly converges with the decrease of the speed. The observation is consistent
with Liu’s finding [79].
The substrates with either smaller lateral size or thinner thickness, intend to have lower
strain energy transfer rate. During loading, the strain within the smaller lateral size or
thinner substrate is more concentrated thus leads to higher force. And then, during the
relaxation period, the more reduction in the loading value will be expected since the more
un-equilibrium strain energy has been released. The opposite circumstance is observed
during unloading. The strain energy transfer rate of the substrate with smaller lateral size
or thinner thickness is lower, which leads the slower elastic deformation recovery. Because
the elastic recovery is slower in the substrate with smaller lateral size or thinner thickness,
unloading curve becomes steeper, thus the Young’s modulus is larger.
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56 The faster loading rate leads to less time for atoms to interact and recover to the strain
energy equilibrium condition. Therefore, the substrate presents higher estimated Young’s
modulus at faster loading speed.
In summary, according to the simulation results, the selected substrate size used in future
studies is 36ax36ax18a and the selected loading rate is 0.02 nm/Ps.
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57 4 Chapter 4 Characterization of Mono-crystalline
Silicon with Defects
This chapter concentrates on the investigation of the mono-crystalline silicon with
different defects by nanoindentation simulation. The controlled single cavities with
different sizes, locations, as well as multiple cavities are introduced into the model by
removing certain atoms from the perfect silicon. Three groups of simulation models
(groups d, e, and f) with different pre-existing internal cavities have been devised.
Specifically, in the group d, cases with single spherical cavity located at the lateral centre of
the substrate with six different radii are considered. Group f includes eight cases with the
same size cavities at different locations. The first four cases are used to investigate cavities
located at 1a, 2a, 3a and 4a on y positive direction (a is the silicon lattice constant) and
then the other four cases are employed to investigate cavities located ±1a and ±2a on z
direction. Multiple defects are considered in the group f. In detail, four same size defects
are distributed 90 degrees evenly around the central point of the surface. The defects with
sizes of 0.5a, 1a, 1.5a and 2a radius are considered.
The following investigation focuses on the influences from cavities to the mechanical
properties including Young’s modulus and hardness. The phenomenon of phase
transformation from diamond structured crystalline to a body-centered tetragonal form (β-
silicon) upon loading of the indenter are observed for every defected case.
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58 4.1 Computational Model and Defects Description
The computational model used to characterize the silicon with defects is clarified in this
section. The model described in section 3.1 was used as the basic model with clarification
as below:
Substrate size 36ax36ax18a is chosen.
Loading rate of 0.02 nm/Ps is adopted.
The maximum indentation depth is 1.857 nm.
In the following paragraphs, a unique code is given to each case to simplify the discussion.
Each code is given with the description of the defects in terms of size and location. In order
to describe the location of defect, a coordinate system with the original point at the centre
of the substrate was established, which is shown in
Figure 14. For the convenience of discussion, we signed a unique code for each case with
the description of defects in the tables at the beginning of each group.
Figure 14 Coordinate system for defining the location of defects
y
x
y
z
18a
18a
9a
9a
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59 4.2 Effect of the Cavity Size
In group d, we considered seven cases to investigate the influence of cavity size. For
comparison purpose, the non-defect case for benchmarking is also included. The trends of
the estimated mechanical properties are summarized, and the cavity size’s influence to
silicon phase transformation is discussed by analyzing the atomic coordinate number (CN)
and atomic configuration.
4.2.1 Description of Defect Cases
The cavities are located at the center of lateral face and 8a deep from the surface of the
substrate. The radius of the cavity was varied from 0 to 3a with 0.5a increment. The
following table defines the codes for the Group d cases, and the location is in expression of
a ),,( zyx coordination.
Table 8 Group d Defect Cases
Case code d0 d1 d2 d3 d4 d5 d6
Size (Radius) 0a 0.5a 1a 1.5a 2a 2.5a 3a
`Location (0,0,1) (0,0,1) (0,0,1) (0,0,1) (0,0,1) (0,0,1) (0,0,1)
4.2.2 Load-Displacement Curve and Test Results
Figure 15 shows the load-displacement curves of Group d. To note that, each load-
displacement curve contains three regions: loading region, relaxation region and unloading
region, as pointed in Figure 15. Basically, it is found that, with the increase of the cavity’s
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60 size, the force required to achieve the same indentation depth decreases. This trend
becomes more obvious with the increase of the indentation depth. For instance, at the
maximum indentation depth, the loading after relaxation for the case with 0.5a radius
cavity is 523.9 nN, which is greater than 470.1 nN for the case with 3a radius cavity. Table 9
shows the loading force before and after relaxation at maximum indentation depth.
Table 9 Loading force at maximum penetration depth for Group d
Case Code d0 d1 d2 d3 d4 d5 d6
Cavity radius 0 0.5a 1.0a 1.5a 2.0a 2.5a 3a
Before Relaxation 596.9 595.7 591.7 580.9 578.3 558.3 543.4
After Relaxation 524.8 523.9 524.8 518.3 509.4 497.8 470.1
*Loading unit is in nN
It is worth to mention that small defect sizes, in our cases, with radius 0.5a and 1.0a,
shows negligible influence to the loading. The evidence can be observed from load-
displacement curves (Figure 15), the loading and unloading curves for the silicon substrate
with 0.5a and 1a radius cavity are overlapped with the curve of prefect substrate.
Furthermore, the calculated Young’s modulus and hardness are all determined by the least
squares fitting method based on the unloading data, and the fitting method may have
influence on the values. Therefore the Young’s modulus and hardness for the case with
0.5a and 1a radius cavity should not be taken for consideration in this section.
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61
Figure 15 Load-displacement curves of Group d.
Young’s modulus and hardness were estimated as listed in Table 10. For the perfect silicon
substrate case, Young’s modulus and hardness were found around 127.17 GPa and 33.36
GPa, respectively. Generally, in accordance with Table 10, the presentation of cavity
induces obvious influence to Young’s modulus and hardness. Both calculated Young’s
modulus and hardness reduce while the radius of the defect increases. Comparing with the
perfect substrate, Young’s modulus appears 1.60%, 3.79%, 5.73%, and 9.06% reduction for
the 1.5a, 2a, 2.5a, and 3a radius defect case respectively. The hardness decreases 1.05%,
2.46%, 4.86% and 11.12% for the aforementioned defected case respectively.
Loading
Relaxation
Unloading
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62 Table 10 Estimated Young’s modulus and hardness for Group d
Case d0 d1 d2 d3 d4 d5 d6
Radius of
Defects 0 0.5a 1a 1.5a 2a 2.5a 3a
Young’s
Modulus 123.17 123.10 125.51 121.20 118.50 116.11 112.01
Hardness 33.36 33.29 33.04 33.01 32.54 31.74 29.65
*Young’s modulus and hardness unit in GPa.
4.2.3 Phase Transformation and Atomic Configuration
According to Hu et al. [68], five phase transformations (Si-XI, Si-V, Si-VI, Si-VII and Si-X)
appear during loading and unloading process. Researchers found that, for compression
about 10-12.5 GPa, silicon transforms from the original diamond structure (Si-I) to the
metallic β-Si phase (Si-II) [84, 85]. More specifically, phase transformation from Si-II to Si-III
or Si-XII takes place when the indentation pressure is gradually released. Under a high
releasing rate, phase transforms from Si-II to a mixture of Si-VII and Si-IX or amorphous
phase [86, 87]. To illustrate the phase transformation process during nanoindentation,
several sectional views of the atomic configurations of cases d0, d3 are presented in Figure
16 and Figure 17, respectively. As it is known, Si-I has an atom coordination number (CN) of
four, and β-Si (Si-II) has a CN of six, which is gradually formed due to relative sliding
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63 between atoms along the compressive direction [88]. Further, it is accepted that Si-I
transforms to metallic Si-II during loading [55].
(a) 0.657 nm (Loading) (b) 1.857 nm
(c) 0.657 nm (Unloading) (d) Unloaded
Figure 16 Atomic configurations of d0 case at four different stages: (a) - (d): substrate with1.5a radius defect at the indentation depth of 0.657 nm, 1.857 nm, unloading to depth 0.657nm and full unloaded; Atoms with the CN value between 0 and 13 are visualised.
For the perfect substrate, from Figure 16(a) and (b), we found atoms with the CN of six (Si-
II), seven or eight (metastable phases) are formed from Si-I. Similar as reported by Lin et al.
[55], the β-Si phase is surrounded by the metastable phases and the phase transformation
occurs and propagates anisotropically [8]. According to Figure 16(c), the deformation
under the indenter starts to recover, and as shown in the Figure 16(d), after being fully
unloaded, a large part of the deformed region underneath the indenter is found recovered
to Si-I phase, with some region still in the mixture of Si-II and amorphous phases, which is
believed consequently formed due to plastic deformations [11] that occur during loading.
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64 As demonstrated by Zarudi et al. [13], only a qualitative prediction of structural phase
transformation from Si-II to the amorphous phase is available because of the high loading
and unloading rates in MD simulations.
(a) 0.657 nm (Loading) (b) 1.857 nm
(c) 0.657 nm (Unloading) (d) Unloaded
Figure 17 Atomic configurations of case d3 at four different stages. (a) - (d): substrate with1.5a radius defect at the indentation depth of 0.657 nm, 1.857 nm, unloading to depth 0.657nm and full unloaded; Atoms with the CN value between 0 and 13 are visualised.
According to Figure 17, a similar deformation process is found in case d3 with a pre-
existing cavity located at the middle of the substrate. Obviously, we observed the
deformed region in case d3, as shown in Figure 17(d), is relatively smaller than that in case
d0, as shown in Figure 16(d), after being fully unloaded. This phenomenon can be
explained by comparing Figure 16(b) and Figure 17(b). From Figure 17(b), the atomic
structures around spherical cavity absorb part of the strain energy during loading, and it
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65 induces the deformation of the cavity. This strain energy is then released during unloading
process, which consequently benefits the recovery of the deformed region to Si-I phase. As
seen in Figure 17(d), the cavity recovers to the original spherical shape. Meanwhile, fewer
atoms are found in the deformed zone comparing with case d0. Similar deformation
processes have also been observed in cases d4, d5, and d6.
4.2.4 Discussion
To further explain the pre-existing cavities’ effect, the changes of crystal structures are
tracked by different atom coordination numbers (CNs), including CN=6, 7 and 8 in all cases.
It is consistent with the results from previous researchers [55], that the phases with CN
more than six are gradually disappeared during unloading, according to Figure 18(a). While,
for β-Si phase (Si-II) with CN equals six, is still existed and is distributed in the permanent
amorphous phase. This observation was also reported by Cheong and Zhang [11], who
proposed that, only β-Si structure is the absolute formed phase during indentation. For
case d5 with defected substrate, we find similar changing trends of CNs. A summary table
of CNs for all single cavity cases is presented in Table 11.
The number of atoms with the CN of six in case d5 (about 701) is observed to be smaller
than that in case d0 (about 718) at the maximum indentation depth before relaxation, and
they are almost the same when fully unloaded (around 386). This observation has proved
our assumption that, the cavity has absorbed certain strain energy during loading; thus, it
leads to less plastic deformation during loading. Comparing with the other cases with
defects, we also found although the atoms with CN of six after relaxation have similar
amount for the defect cases with 1.5a, 2a, 2.5a, when the cavity size increases to 3a in
radius(case d6), the number of atoms with the CN of six has a considerable decrease to
around 332.
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66
Table 11 Coordination Number for All Single Cavity Cases
CN Period
Cases
d0 d3 d4 d5 d6
CN=6 Loading 718 721 694 701 691
Relaxation 719 694 684 692 685
Unloading 393 371 392 386 332
CN=7 Loading 455 476 443 449 499
Relaxation 405 455 444 437 489
Unloading 202 222 181 163 164
CN=8 Loading 324 289 303 299 254
Relaxation 323 286 288 282 235
Unloading 82 86 71 57 29
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67
(a)
(b)
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68
(c)
(d)
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69
(e)
Figure 18 Number of atoms with specified CNs (6, 7 and 8) versus time: (a) case d0; (b) case d3; (c) case d4; (d) case d5; and (e) case d6.
Another interesting phenomenon found in the Figure 16 is that, in the cases d3, d4 d5 and
d6, the number of atoms with CN=6 peaks up before relaxation, and then as soon as the
relaxation starts, the number quickly reduces to a stable value. This phenomenon is not
observed on the non-defected case. The phenomena may be another evidence to support
aforementioned assumption i.e. the atomic structures around spherical cavity absorbs
some energy, some elastic deformation can be recovered during the relaxation period, and
thus the number of atoms β-Si phase (Si-II) reduced. With the cavity size increases, atomic
structures around the spherical cavity intend to become weaker, therefore, the ability to
absorb elastic deformation is reduced, and thus the number of β-Si phase (Si-II) has less
significant reduction.
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70 4.3 Effect of cavities’ positions
Eight cases are considered in Group f to investigate the effect of cavities’ position. In the
first four cases we offset the cavities along the lateral direction. For the other four cases,
the cavity locations were offset in the thickness direction. The trends of the estimated
mechanical properties are summarized, and the silicon phase transformation is analyzed
for each case according to the atomic coordinate number (CN) and atomic configuration.
4.3.1 Description of Defect Cases
In this section, we adopt the defect case d5 as the benchmarking model. The case d5 has
the with a cavity radius of 2.5a, which is located at the lateral centre of the substrate. For
the first four cases, the 2.5a radius cavity was offset 1a, 2a, 3a and 4a in y direction,
respectively. Then in the second four cases the cavity locations were offset on the
thickness direction ),( zz . Cases f5, f6, f7, and f8 represent the cavity offset 1a, 2a, -1a
and -2a in the thickness direction, respectively.
Table 12 Group f Defect Cases
Case code d5
(benchmarking)
f1 f2 f3 f4
Size (radius) 2.5 2.5 2.5 2.5 2.5
Location (0,0,1) (0,1,1) (0,2,1) (0,3,1) (0,4,1)
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71 Case code f5 f6 f7 f8
Size (radius) 2.5 2.5 2.5 2.5
Location (0,0,2) (0,0,3) (0,0,0) (0,0,-1)
Table 12 defines the case codes and defects for simulations in group f, the location is
expressed by a coordinate ),,( zyx , and for the comparison purpose, the description of
case d5 is also included.
4.3.2 Load-Displacement Curve and Test Results
Figure 19 shows the load-displacement curves of cases f1-f4 and f5-f8, respectively. As
illustrated in Figure 19(a), we found that, during the loading process, all the load-
displacements curves are almost consistent with each other until the indentation depth
reaches around 1.2 nm. After that, for a larger offset in the lateral direction )( y , a larger
force is usually observed at the same indentation depth, which is more obvious at the
maximum indentation depth of 1.857 nm. For cases f5-f7 with the cavities offset in the
thickness direction, we found obvious differences from the load-displacement curves in
Figure 19(b). Basically, during the loading process, a greater force is normally observed
when the cavity is nearer to the top surface of the substrate. For instance, at the
indentation depth of 1.857 nm after relaxation, the force is around 439.1 nN in case f6 (2a
offset in +z direction), while in case f8 (2a offset in -z direction), the force is about 515.1 nN.
In all, it is found that cases f5-f8 show more variations on load-displacement curves, which
suggests the testing results appears more sensitive to the vertical distance between the
cavity and the substrate’s top surface.
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72
(a)
(b)
Figure 19 .Load-displacement curves of Group f: (a) Offset in –y direction; (b) Offset in –z and +z directions.
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73 Young’s modulus and hardness were also estimated according to the unload-displacement
curves, as listed in Table 13. Comparing with case d5 in Group d, we found Young’s
modulus and hardness in cases f1-f4 is generally greater than that in case d5. When the
cavity is at 4a offset of the center, the calculated Young’s modulus is almost the same as
that of the perfect case d0.
Table 13 Estimated Young’s modulus and hardness for cases f1- f4 in Group f
Offset by 1a, 2a, 3a and 4a in the Lateral Direction
Cases d5 f1 f2 f3 f4 d0**
Positions (0,0,1) (0,1,1) (0,2,1) (0,3,1) (0,4,1) N/A
Young’s modulus 116.11 118.92 117.87 118.20 123.45 123.17
Hardness 31.74 30.89 31.25 31.77 31.60 33.36
*Young’s modulus and hardness unit in GPa **Case d0 contains no defect
For the hardness, slight differences were found. It is worth to note that the force after
relaxation is used to calculate the hardness. From Figure 19(a), it is obvious that if we apply
the force at the maximum indentation before relaxation to calculate the hardness, a
generally greater hardness in cases f1-f2 than in case d5 can be found. Therefore, the
greater offset of the cavity in the lateral direction, the less influence to Young’s modulus
and hardness can be introduced.
For cases f5-f8, more uniform changing trends of Young’s modulus and hardness were
found. In general, the nearer position the cavity to the substrate’s top surface has, the
larger decrease to Young’s modulus and hardness appears. For instance, in case f6 with the
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74 cavity offset 2a in z direction, Young’s modulus and hardness were calculated about
104.31 GPa and 27.75 GPa, respectively. Meanwhile, in case f7 with the cavity offset 1a in
z direction, Young’s modulus and hardness were around 122.80 GPa and 31.88 GPa,
respectively.
Table 14 Estimated Young’s modulus and hardness in Group f
Offset by 1a and 2a in +x, +z and –z direction
Cases d5 f5 f6 f7 f8
Positions (0,0,1) (0,0,2) (0,0,3) (0,0,0) (0,0,-1)
Young’s modulus 116.11 114.85 104.31 122.80 121.50
Hardness 31.74 28.9 27.75 31.88 32.65
*Young’s modulus and hardness unit in GPa.
4.3.3 Phase Transformation and Atomic Configuration
Figure 20 presents the sectional views of the atomic configurations of cases f3 and f6 at
three different stages. As seen in Figure 20(a2), the cavity is eccentrically compressed
during loading. After unloading, the strain energy stored in the deformed cavity is released,
which recovers the cavity back to spherical structure. As shown in Figure 20 (a4), this
releasing process changes the left part of deformation zone underneath the indenter back
to Si-I phase. However, the strain energy cannot always be fully released, as shown in the
case of f6. Due to the cavity in f6 is much nearer to the substrate’s top surface than f3,
which makes it receiving larger structure deformation.
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75
Figure 20 Atomic configurations of cases f3 and f6 at three different stages: (a1)-(d1): case f3 at the indentation depth of 0.657 nm, 1.857 nm, 0.657 nm unloading and full unloaded; (a2)-(d2): case f6 at the indentation depth of 0.657 nm, 1.857 nm, 0.657 nm unloading and full unloaded. Atoms with the CN value between 0 and 13 are visualised.
Si-I phase Recovered
(a1) 0.657nm (a2) 0.657nm
(b1) 1.857nm (b2) 1.857nm
(c1) 0.657nm unloading (c2) 0.657nm unloading
(d1) Unloaded (d2) Unloaded
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76 According to Figure 20 (b2), the cavity is found seriously compressed or nearly collapsed
during loading. Such severe deformation has caused the cavity fails to recover to original
spherical structure, as show in Figure 20 (d2). The middle part of the deformed zone
underneath the indenter is almost fully changed back to Si-I phase after unloading.
The above discussions are also applicable to explain the trend of Young’s modulus for cases
with the cavity lateral offset. As depicted in Figure 21 (d1 and d2), the strain energy that
stored in the deformed cavity is released, this releasing process has changed the left
deformed zone underneath the indenter back to Si-I phase. In both case f1 and f4, the
recovery are observed, and it also can be easily found that case f4 have more atoms
recovered during the unloading process. The calculated values of Young’s moduli listed in
Table 13 and Table 14 show the same trend in case f3 and case f6, which is 118.20 and
104.31 GPa, respectively.
(a1) 0.657nm (a2) 0.657nm
(b1) 1.857nm (b2) 1.857nm
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77
Figure 21 Atomic configurations of cases f1 and f4 at three different stages: (a1)-(d1): case f1 at the indentation depth of 0.657 nm, 1.857 nm, 0.657 nm unloading and full unloaded; (a2)-(d2): case f4 at the indentation depth of 0.657 nm, 1.857 nm and full unloaded. Atoms with the CN value between 0 and 13 are visualised.
4.3.4 Discussion
The CN pattern provides the insight of the cavities position effects on the substrate
deformation. The number of atoms with different CNs (6, 7 and 8) in cases f3 and f6 were
recorded against the time step, as illustrated in Figure 22. Similar as in group d, the phases
with CN of seven and eight gradually decrease during unloading. At the end of unloading,
the number of atoms with CN of seven and eight reduce to a negligible level. The β-Si
phase (Si-II) with CN of six still exists and is distributed in the permanent amorphous phase.
Comparing with case d5, less reduction on number of atoms with CN of six (Figure 22) is
observed in case f3 (300 compare to 306 in case d5) at the maximum indentation depth
during unloading. In other words, there are fewer atoms with CN of six involved in the
elastic recovery during unloading. For case f6, the cavity moves 2a upward, and from
Recovered Recovered
(c1) 0.657nm unloading (c2) 0.657nm unloading
(d1) Unloaded (d2) Unloaded
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78 Figure 22, the number of recovered atoms with CN of six is 227, which is much less than
case d5. In accordance with the pervious findings, it is postulated that the cavity located at
shallow thickness and cavity located closer to the top surface leads to more plastic
deformation, i.e., the number of atoms with CN=6 after unloading are normally larger in
these cases (Figure 22).
The number of atoms with the CN of six is a good indication of the deformation situation.
However, the deformation profile of Si substrate does not solely depend on the number of
atoms with the CN of six.
(a)
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79
(b)
Figure 22 Number of atoms with specified CNs (6, 7 and 8) versus time: (a) case f3; (b) case f6.
Preliminarily, when a cavity moves away from the region beneath of the indenter, the
effects brought by the cavity became smaller, and eventually the influence will be
negligible at the far enough distance. It is found that defects with a radius of 2.5a at a
distance of 4a from the lateral centre of the substrate have negligible effect on the Young’s
modulus and hardness. For the case of a cavity moving in the thickness direction, we found
that when the cavity is closer to the surface, the measured Young’s modulus and hardness
intend to be smaller. With the cavities moving to a deeper position of the substrate, the
effects brought by cavities become negligible. Furthermore, for the cavity with a radius of
2.5a, a depth of 9a is deep enough to eliminate the effect to the Young’s modulus and
hardness. Conclusively, the position of cavity is playing an important role on the cavities’
effects on Young’s modulus and hardness, as well as the phase transformation.
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80 4.4 Effect of multiple cavities
Multiple cavity cases are considered in Group e. Due to the complexities of the multiple
cavities and the limitation of available resources, only simple cases of multiple cavities,
were designed and studied in this group to verify the conclusions we have drawn in last
two sections.
4.4.1 Description of Defect Cases
In this section, a group of cases with four cavities cases is considered. As shown in Figure
23, four cavities are located at 8a deep a depth of 8a according to from the substrate
surface, and each cavity is offset with a distance of 6a from the lateral center on both yx,
directions. Four cases with the cavity radii of 0.5a, 1a, 1.5a and 2a were investigated. The
four cases are named e1 to e4, respectively.
(a) (b)
Figure 23 Group e: Four cavities cases (a) Top view (b) Isometric view
36a
36a
12a
12a
8a
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81
4.4.2 Load-Displacement Curve and Test Results
From the load-displacement curves of cases e1-e4, as illustrated in Figure 24, during the
loading process all load-displacements curves are very close to each other. However, the
detailed analysis reveals that the unloading curves become steeper while the size of the
cavity becomes smaller. Interestingly, when the cavity is excluded in the model, the curve
becomes less steep than those cases with cavities in this group. This trend can also be
observed in the calculated Young’s modulus presented in Table 15. Young’s modulus
increases from 122.79 GPa at case e4 to 130.85 GPa in case e1 then jump back to 123.17
GPa for the perfect substrate.
Figure 24 Load-displacement curves of Group e, 4 cavities with the radii of 0.5a, 1a, 1.5a and 2a, respectively.
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82
Table 15 Estimated Young’s modulus and hardness for Group e
Case d0 e1 e2 e3 e4
Cavity Radius 0 0.5a 1a 1.5a 2a
Young’s modulus 123.17 130.85 126.53 125.21 122.79
Hardness 33.36 32.41 33.64 32.85 32.70
*Young’s modulus and hardness unit in GPa.
4.4.3 Phase Transformation and Atomic Configuration
Similar to the single cavity cases discussed in previous section, the deformation of spherical
structured cavity absorbs energy. Figure 25 shows the atomic configuration for the cases
e1 and e4. The location of the cavity is relatively far away from the deformation zone, and
in both cases, the atomic structures around cavities are elastically compressed during
loading. After unloading, the strain energy that stored in the deformed cavity is released,
and the atomic structure around the cavity recovers back to Si-I phase. Although there is
some plastic deformation can be observed on the surface of the silicon substrate, there is
nothing other than Si-I structure is found around the cavities. Therefore, it can be
postulated that the cavities are not directly involved in the effect to the phase
transformation, the increase of the Young’s modulus and hardness is the result of the
energy absorption due to cavity structural elastic deformation.
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83
Figure 25 Atomic configurations of cases e1 and e4 at two different stages: (a1)-(a4) at the indentation depth of 1.857 nm; (b1)-(b4) full unloaded; Atoms with the CN value between 0 and 13 are visualised.
4.4.4 Discussion
Figure 26 and Figure 27 show the number of atoms with CNs of six, seven, and eight for
cases e1 and e4. In case e1, after the relaxation process, the number of atoms with the CN
of six is 712. After being fully unloaded, this number becomes 380, with a reduction of 332.
(a1) Case e1; 1.857nm
(a2) Case e2; 1.857nm
(a3) Case e3; 1.857nm
(a4) Case e4; 1.857nm (b4) Case e4; Unloaded
(b3) Case e3; Unloaded
(b2) Case e2; Unloaded
(b1) Case e1; Unloaded
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84 The number of atoms with CN of six for case e4 decreases more, i.e., 731 at the end of
relaxation and then the number of atoms with CN of six reduces from 397 to 334 after
being fully unloaded. For the prefect case d0 the number of atoms with CN of six recovered
during the unloading process is 325. Comparing to case e1, the effect of elastic
deformation is obvious.
Figure 26 Number of atoms with specified CNs (6,7 and 8) versus time for case e1.
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85
Figure 27 Number of atoms with specified CNs (6, 7 and 8) versus time for case e4
It is observed that in group e, the existence of some small cavities can enhance the
mechanical performance of silicon substrates. From the results of simulations, it is found
that the Young’s moduli of cases e1, e2, and e3 are larger than the prefect case d0. The
investigation of the atomic configurations for cases e1 and e4 shows that atomic structures
around the cavities do not transform permanently. Those atomic structures recover back
to Si-I phase during unloading. The recovery of those atomic structures has contributions
to the Young’s modulus of silicon substrate. When the cavities are small enough, the
mechanical properties are enhanced because the recovery of those atomic structures
around the cavities overcomes the weakening effect brought by the cavities, so the
Young’s moduli of those cases with small cavities exceed that of the perfect case. It is
interesting to conclude that the existing small cavity does not lead to the weakness of the
atomic structure, such as the substrate contains the cavities with a radius of 0.5a.
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86 5 Chapter 5 Conclusion and Future Work
5.1 Conclusions
The MD simulation is performed to reproduce the nanoindentation on the mono-
crystalline silicon. By comparing the simulation results with other successful simulation
cases, the MD model is validated at first. Utilizing this simulation model, influences from
some key simulation parameters are investigated. Those parameters, including the lateral
size and thickness of silicon substrate and the loading rate applied on the indenter, have
significant influences to the results. According to MD results, the following conclusions can
be made:
The smaller lateral size of the silicon substrate leads to higher corresponding force
at the maximum indentation depth. For the silicon substrate with smaller lateral
size, more strain energy is expected to be relieved, during relaxation, larger force
reduction is induced. Both loading and unloading curves for the substrate with
smaller lateral size are steeper. Thus, the Young’s modulus calculated from the
unloading curve is greater than those substrates with larger lateral size. The
Young’s modulus converges to the value of substrate with infinity lateral size. The
hardness calculated from the unloading force and indentation depth shows
opposite trend to the Young’s modulus.
The influence from substrate thickness is observed and analysed. Young’s modulus
is larger for the thinner substrates, but the hardness does not increase
monotonically along with the increase of the thickness. The hardness under the
influence of rigid atoms layer rises up at thinner thickness. The similar phenomena
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87 is also reported by A.V.Bolesta and V.M.Fomin[83], who investigated mechanical
properties of Cu thin film.
The loading rate has significant influence on the estimated mechanical properties
of silicon substrate. The higher loading speed leads to steeper loading curve and
higher loading value at maximum indentation depth. During the relaxation process,
the corresponding force with higher loading speed has larger reduction. Higher
indentation speed of the indenter results in the higher Young’s modulus, and
Young’s modulus quickly converges with the decrease of the speed. This conclusion
consist with Liu and his team’s [79] finding. It is also noted that when the loading
speed reduces down to 0.01 nm/Ps, there is no reduction of loading due to the
relaxation effect.
Based on the parametric study preformed, the optimized parameters are adopted to
ensure the accuracy and computational efficiency. The validated MD model is employed to
investigate the mono-crystalline silicon properties with different pre-existing cavities under
nanoindentation. Cavities with different radii and positions are considered. The factors
including Young’s modulus, hardness and numbers of atoms with the coordination
numbers of six, seven and eight have been obtained to quantify the cavities’ effect. Main
conclusions can be drawn as follows:
Pre-existing cavities in the silicon substrate have obvious influences on the
mechanical properties of silicon under nanoindentation;
Pre-existing cavities can absorb part of the strain energy during loading and then
release during unloading. It possibly causes less plastic deformation to the
substrate.
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88
The larger offset of the cavity in the lateral direction, the less influence we found,
and the higher Young’s modulus and hardness has been found. When the cavity
moves nearer to the substrate’s top surface, the larger influence is induced. In our
cases, we found that, for a cavity with a radius of 2.5a, when it is located at a deep
of 9a beneath the surface, or at a depth of 8a and a distance of 3a from the lateral
centre. It will eliminate the influence of cavity brought to the estimated mechanical
results.
The combination of the location closer to indenter and larger size of cavity may
introduce more plastic deformation around the cavities. When the pre-existing
cavities are close enough to the deformation zone or big enough to exceed the
bearable stress for the spherical cavity, larger deformation occurs, which results in
the ‘ collapse’ of the cavity, and the transformation of the silicon due to stress will
not able to recover. Furthermore, some cavity cases do not have visible plastic
deformation, but they still cut into the “effect zone” and have direct impact to the
silicon phase transformation.
When the cavity is far enough from the ‘effect zone’ or the cavity is small enough,
even there is visible elastic deformation, the cavity is considered as no significant
influence to the plastic deformation, minor increase of Young’s modulus and
hardness is observed.
When substrate contains multi-cavities with small radius, the mechanical properties
of the substrate can be enhanced, because the elastic recovery of compressed
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89 atomic structures around the cavities overcomes the weakening effect brought by
the cavities.
5.2 Recommended Future Work
Due to the limitations of available resource and timeframe, there is a couple of possible
extended topics have not been included in this thesis. Therefore, we list them in this
section for recommended future work.
In this thesis, we only examined a simple multiple defect case. The main parameters to
define a multiple defects only include the location, size, and cavitations density. In future
study, the same MD simulation can be employed, and the different multiple cavities can
further be designed by removing atoms out of the substrate. The expected outcome is to
have quantitative results of the influence on the mechanical properties with respect of
location, size and cavitations density.
Another topic is to investigate different types of defects. In this thesis, only cavity defect
cases are considered. In the future project, the more types of defects can be included, such
as grain boundary (GB), impurity embed, dislocation or even different sharp of cavitations.
MD modeling is able to satisfy all the needs to investigate the existing defects in the silicon
substrates.
More complex defects were not considered in the present project. It is feasible to upgrade
the MD model to multi-scale model, and extend this project to investigate the mechanism
of crack propagation on the atomic level. The benefit of multi-scale simulation is able to
couple the continuum modeling and atomic modeling together, in order to unify the
theory of atomic scale and macro scale.
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