Advanced Numerical Characterization of Metal with …eprints.qut.edu.au/54621/1/Qiang_Fu_Thesis.pdf · Parametric studies including specimen size and loading rate, ... Advanced Numerical

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


Jan 2012

Advanced Numerical Characterization of Silicon with Defects by Nanoindentation


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.



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.



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.



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.



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



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



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



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



4.3.3 Phase Transformation and Atomic Configuration .......................................................... 74

4.3.4 Discussion ..................................................................................................................... 77

4.4 Effect of multiple cavities .................................................................................................... 80





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



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



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



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



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

http://en.wikipedia.org/wiki/Microprocessor

http://en.wikipedia.org/wiki/Microcontroller

http://en.wikipedia.org/wiki/Static_Random_Access_Memory

http://en.wikipedia.org/wiki/Digital_logic



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



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



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



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.



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.



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



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]



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)



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)



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)



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)



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)



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:



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)



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.



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



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].



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]



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



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.



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



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)



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)



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)



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)



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)



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.



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



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



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)



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.



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).



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



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,



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)



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.



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



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



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.



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



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.



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


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)



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.



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



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



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.



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.



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.



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.



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



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



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


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.



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



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



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.



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



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.



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



67

(a)

(b)



68

(c)

(d)



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.



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.


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)



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.


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.



72

(a)

(b)

Figure 19 .Load-displacement curves of Group f: (a) Offset in –y direction; (b) Offset in –z and +z directions.



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



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



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.



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



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



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



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)



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.



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.


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



81


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.



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



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.



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





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.



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.



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



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.



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



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.



90 6 Bibliography

1. Sinano, I., Advanced Nanoelectronics Technology. 2009.

2. Sinano, I., Sinano Institute Vision, 2009.

3. Oda, S. and D.K. Ferry, Silicon nanoelectronics2006: CRC.

4. KAWAMOTO, H. and K. OKUWADA, Development Trend for High Purity Silicon Raw Material Technologies.

5. Pharr, G., W. Oliver, and D. Harding, New evidence for a pressure-induced phase transformation during the indentation of silicon. Journal of Materials Research, 1991. 6(06): p. 1129-1130.

6. Zarudi, I. and L. Zhang, Structure changes in mono-crystalline silicon subjected to indentation--yexperimental findings. Tribology international, 1999. 32(12): p. 701-712.

7. Domnich, V., Y. Gogotsi, and S. Dub, Effect of phase transformations on the shape of the unloading curve in the nanoindentation of silicon. Applied Physics Letters, 2000. 76: p. 2214.

8. Kim, D. and S. Oh, Atomistic simulation of structural phase transformations in monocrystalline silicon induced by nanoindentation. Nanotechnology, 2006. 17: p. 2259.

9. Sanz-Navarro, C., S. Kenny, and R. Smith, Atomistic simulations of structural transformations of silicon surfaces under nanoindentation. Nanotechnology, 2004. 15: p. 692.

10. ZHANG, L.C. and H. Tanaka, On the mechanics and physics in the nano-indentation of silicon monocrystals. JSME international journal. Series A, Solid mechanics and material engineering, 1999. 42(4): p. 546-559.

11. Cheong, W.C.D. and L. Zhang, Effect of repeated nano-indentations on the deformation in monocrystalline silicon. Journal of materials science letters, 2000. 19(5): p. 439-442.

12. Vodenitcharova, T. and L. Zhang, A new constitutive model for the phase transformations in mono-crystalline silicon. International Journal of Solids and Structures, 2004. 41(18-19): p. 5411-5424.

13. Zarudi, I., et al., The difference of phase distributions in silicon after indentation with Berkovich and spherical indenters. Acta Materialia, 2005. 53(18): p. 4795-4800.

14. Tan, T. and U. Gösele, Point defects, diffusion processes, and swirl defect formation in silicon. Applied Physics A: Materials Science & Processing, 1985. 37(1): p. 1-17.



91 15. Fahey, P.M., P. Griffin, and J. Plummer, Point defects and dopant diffusion in silicon. Reviews of

modern physics, 1989. 61(2): p. 289.

16. Bullis, W.M., Current trends in silicon defect technology. Materials Science and Engineering B, 2000. 72(2-3): p. 93-98.

17. Ciszek, T. and T. Wang, Silicon defect and impurity studies using float-zone crystal growth as a tool. Journal of Crystal Growth, 2002. 237: p. 1685-1691.

18. Malvido, J. and J. Whitten, Ab initio treatment of silicon defect clusters. The unrelaxed, neutral monovacancy. Physical Review B, 1982. 26(8): p. 4458.

19. Sinno, T., Z.K. Jiang, and R.A. Brown, Atomistic simulation of point defects in silicon at high temperature. Applied Physics Letters, 1996. 68(21): p. 3028-3030.

20. Medyanik, S.N. and W.K. Liu, Multiple time scale method for atomistic simulations. Computational Mechanics, 2008. 42(4): p. 569-577.

21. Fischer-Cripps, A., A review of analysis methods for sub-micron indentation testing* 1. Vacuum, 2000. 58(4): p. 569-585.

22. Oliver, W. and G. Pharr, Improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research, 1992. 7(6): p. 1564-1583.

23. Weppelmann, E., J. Field, and M. Swain, Observation, analysis, and simulation of the hysteresis of silicon using ultra-micro-indentation with spherical indenters. Journal of Materials Research, 1993. 8(04): p. 830-840.

24. Fischer-Cripps, A.C., Critical review of analysis and interpretation of nanoindentation test data. Surface and Coatings Technology, 2006. 200(14-15): p. 4153-4165.

25. Yu, N., A.A. Polycarpou, and T.F. Conry, Tip-radius effect in finite element modeling of sub-50 nm shallow nanoindentation. Thin Solid Films, 2004. 450(2): p. 295-303.

26. 'Finite Element Method', http://en.wikipedia.org/wiki/Finite_element_method (Last updated on 21 June 2012).

27. Fischer-Cripps, A.C., Nanoindentation. 2004 Springer-Verlag. New York

28. Fischer-Cripps, A.C., The Handbook of Nanoindentation. 2009 Fischer-Cripps Laboratories. Forestville NSW Australia

29. Schwaiger, R., et al., Some critical experiments on the strain-rate sensitivity of nanocrystalline nickel. Acta Materialia, 2003. 51(17): p. 5159-5172.

http://en.wikipedia.org/wiki/Finite_element_method



92 30. Chen, J., L. Lu, and K. Lu, Hardness and strain rate sensitivity of nanocrystalline Cu. Scripta

Materialia, 2006. 54(11): p. 1913-1918.

31. Oliver, W.C. and G.M. Pharr, Improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research, 1992. 7(6): p. 1564-1583.

32. Fischer-Cripps, A.C., Introduction to contact mechanics2000: Springer Verlag.

33. Zhan, H., Y. Gu, and P. Yarlagadda, Advanced numerical characterization of mono-crystalline copper with defects. Advanced Science Letters, 2011. 4(5): p. 1293-1301.

34. Smith, G., E. Tadmor, and E. Kaxiras, Multiscale simulation of loading and electrical resistance in silicon nanoindentation. Physical Review Letters, 2000. 84(6): p. 1260-1263.

35. Rubinsky, B., Irreversible electroporation2010: Springer Verlag.

36. Jordan, C., Calculus of finite differences1965: Chelsea Pub Co.

37. Jameson, A., W. Schmidt, and E. Turkel, Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA paper, 1981. 81: p. 1259.

38. Liu, B., et al., The atomic-scale finite element method. Computer methods in applied mechanics and engineering, 2004. 193(17-20): p. 1849-1864.

39. Li, M., et al., 3D finite element simulation of the nanoindentation process. Acta Mechanica Sinica, 2003. 35(3): p. 257-64.

40. Bressan, J.D., A. Tramontin, and C. Rosa, Modeling of nanoindentation of bulk and thin film by finite element method. Wear, 2005. 258(1-4 SPEC ISS): p. 115-122.

41. Rakesh, L., et al., Computer-aided applications of nanoscale smart materials for biomedical applications. Nanomedicine, 2008. 3(5): p. 719-739.

42. Ghoniem, N.M., et al., Multiscale modelling of nanomechanics and micromechanics: an overview. Philosophical Magazine, 2003. 83(31-34): p. 3475-3528.

43. Zhang, L. and H. Tanaka, Towards a deeper understanding of wear and friction on the atomic scale--a molecular dynamics analysis. Wear, 1997. 211(1): p. 44-53.

44. Zhang, L., K. Johnson, and W. Cheong, A molecular dynamics study of scale effects on the friction of single-asperity contacts. Tribology Letters, 2001. 10(1): p. 23-28.

45. Zhang, T., et al., Measurement of mechanical properties of MEMS materials. Advances in Mechanics, 2002. 32(4): p. 545-562.



93 46. Cheong, W. and L. Zhang, Molecular dynamics simulation of phase transformations in silicon

monocrystals due to nano-indentation. Nanotechnology, 2000. 11(3): p. 173-180.

47. Schiotz, J. and K. Jacobsen, A maximum in the strength of nanocrystalline copper. Science, 2003. 301(5638): p. 1357.

48. Xiao, K. and L. Zhang, The stress transfer efficiency of a single-walled carbon nanotube in epoxy matrix. Journal of Materials Science, 2004. 39(14): p. 4481-4486.

49. Lin, Z.-C., J.-C. Huang, and Y.-R. Jeng, 3D nano-scale cutting model for nickel material. Journal of Materials Processing Technology, 2007. 192-193: p. 27-36.

50. Wang, H., et al., Quasicontinuum simulation of indentation on FCC metals. Transactions of Nonferrous Metals Society of China, 2008. 18(5): p. 1164-1171.

51. Allen, M.P., Introduction to Molecular Dynamics Simulation. Computational Soft Matter: From Synthetic Polymers to Proteins,Lecture Notes,, 2004.

52. Alder, B.J. and T.E. Wainwright, Phase Transition for a Hard Sphere System. The Journal of Chemical Physics, 1957. 27: p. 1208-1209.

53. Wang, C., Q. Meng, and Y. Wang, MOLECULAR DYNAMICS SIMULATION OF THE INTER-ACTION BETWEEN 30◦ PARTIAL DISLOCATION AND MONOVACANCY IN Si. ACTA METALLURGICA SINICA, 2009: p. 400-404.

54. Cavaliere, P., Mechanical properties of nanocrystalline metals and alloys studied via multi-step nanoindentation and finite element calculations. Materials Science and Engineering: A, 2009. 512(1-2): p. 1-9.

55. Lin, Y., et al., Atomic-level simulations of nanoindentation-induced phase transformation in mono-crystalline silicon. Applied Surface Science, 2007. 254(5): p. 1415-1422.

56. Wen, Y.H., R.Z. Zhu, and F.X. Zhou, An overview on molecular dynamics simulation. Advances in Mechanics, 2003. 33(1): p. 65-73.

57. Nosé, S., A unified formulation of the constant temperature molecular dynamics methods. The Journal of Chemical Physics, 1984. 81: p. 511.

58. Hoover, W.G., Canonical dynamics: Equilibrium phase-space distributions. Physical Review A, 1985. 31(3): p. 1695-1697.

59. HINCHLIFFE, A., Molecular modelling for beginners. 2008. Wiley England

60. Haile, J.M., Molecular dynamics simulation. 1992: Wiley New York.



94 61. Lennard-Jones, J., On the forces between atoms and ions. Proceedings of the Royal Society of

London. Series A, Containing Papers of a Mathematical and Physical Character, 1925: p. 584-597.

62. Brown, I., The Chemical Bond in Inorganic Chemistry-The Bond Valence Model. IUCr monographs on Crystallography 12, 2002, Oxford University Press.

63. Tersoff, J., Modeling solid-state chemistry: Interatomic potentials for multicomponent systems. Physical Review B, 1989. 39(8): p. 5566.

64. Daw, M.S. and M.I. Baskes, Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Physical Review B, 1984. 29(12): p. 6443-6453.

65. Stillinger, F.H. and T.A. Weber, Computer simulation of local order in condensed phase of silicon. Physical Review B, 1985. 31: p. 5262-71.

66. Lehmann, V., Electrochemistry of silicon. Vol. 97. 2002: Wiley Online Library.

67. Vodenitcharova, T. and L. Zhang, A mechanics prediction of the behaviour of mono-crystalline silicon under nano-indentation. International Journal of Solids and Structures, 2003. 40(12): p. 2989-2998.

68. Hu, J.Z., et al., Crystal data for high-pressure phases of silicon. Physical Review B, 1986. 34(7): p. 4679.

69. Nelmes, R., et al., Crystal structure of ZnTe III at 16 GPa. Physical Review Letters, 1994. 73(13): p. 1805-1808.

70. Hanfland, M., et al., Crystal structure of the high-pressure phase silicon VI. Physical Review Letters, 1999. 82(6): p. 1197-1200.

71. Duclos, S.J., Y.K. Vohra, and A.L. Ruoff, Experimental study of the crystal stability and equation of state of Si to 248 GPa. Physical Review B, 1990. 41(17): p. 12021.

72. Cheong, W.C.D. and L.C. Zhang, A stress criterion for the β-Sn transformation in silicon under indentation and uniaxial compression. Key Engineering Materials, 2002. 233: p. 603-608.

73. Plimpton, S., Fast parallel algorithms for short-range molecular dynamics. Journal of Computational Physics, 1995. 117(1): p. 1-19.

74. Plimpton, S., P. Crozier, and A. Thompson. LAMMPS Molecular Dynamics Simulator. 2007.

75. Cheong, W.C.D., L.C. Zhang, and H. Tanaka, Some essentials of simulating nano-surfacing processes using the molecular dynamics method. Key Engineering Materials, 2001. 196: p. 31-42.



95 76. Mylvaganam, K. and L.C. Zhang, Atomistic Deformation of Silicon under Triaxial Stresses. Key

Engineering Materials, 2003. 233(236): p. 615-620.

77. Zarudi, I., et al., Atomistic structure of monocrystalline silicon in surface nano-modification. Nanotechnology, 2004. 15: p. 104.

78. Chen, S. and F. Ke, MD simulation of the effect of contact area and tip radius on nanoindentation. Science in China Series G: Physics Mechanics and Astronomy, 2004. 47(1): p. 101-112.

79. Liu, C.-L., T.-H. Fang, and J.-F. Lin, Atomistic simulations of hard and soft films under nanoindentation. Materials Science and Engineering: A, 2007. 452-453: p. 135-141.

80. Dolbow, J. and M. Gosz, Effect of out-of-plane properties of a polyimide film on the stress fields in microelectronic structures. Mechanics of Materials, 1996. 23(4): p. 311-321.

81. Bhushan, B. and X. Li, Micromechanical and tribological characterization of doped single-crystal silicon and polysilicon films for microelectromechanical systems devices. Journal of Materials Research, 1997. 12(01): p. 54-63.

82. Sjöström, H., et al., Reactive magnetron sputter deposition of CNx films on Si (001) substrates: film growth, microstructure and mechanical properties. Thin Solid Films, 1994. 246(1-2): p. 103-109.

83. Bolesta, A.V. and V.M. Fomin, Molecular dynamics simulation of sphere indentation in a thin copper film. Physical Mesomechanics, 2009. 12(3-4): p. 117-123.

84. Hu, J. and I. Spain, Phases of silicon at high pressure. Solid State Communications, 1984. 51(5): p. 263-266.

85. Pfrommer, B.G., et al., Relaxation of Crystals with the Quasi-Newton Method* 1. Journal of Computational Physics, 1997. 131(1): p. 233-240.

86. Piltz, R., et al., Structure and properties of silicon XII: A complex tetrahedrally bonded phase. Physical Review B, 1995. 52(6): p. 4072.

87. Crain, J., et al., Reversible pressure-induced structural transitions between metastable phases of silicon. Physical Review B, 1994. 50(17): p. 13043.

88. Shimojo, F., et al., Molecular dynamics simulation of structural transformation in silicon carbide under pressure. Physical Review Letters, 2000. 84(15): p. 3338-3341.

Advanced Numerical Characterization of Metal with …eprints.qut.edu.au/54621/1/Qiang_Fu_Thesis.pdf · Parametric studies including specimen size and loading rate, ... Advanced Numerical

Documents