STRUCTURES
by
for the degree of
ii
ACKNOWLEDGEMENT
First of all, I wish to express my gratitude to my supervisor, Dr. Yoon Tiem
Leong, for their professional guidance and suggestions throughout the whole period of
my project and thesis writing. The same also goes to Dr. Lim Thong Leng from
Multimedia University, Melaka, who has provided a lot of inspiring guidance and
academic advice to make the work in this thesis a success. I would like to acknowledge
the collaboration with the Complex Liquid Lab, National Central University. Taiwan,
led by Prof. Lai San Kiong, along with his group members, for their academic support.
Ideas, motivation, comments as well as the tireless commitment from individuals
mentioned above are essential contributing factors in leading my learning path to
become a researcher. For their help and concerns in my studies, I am greatly indebted
to all of them.
I would like to thank the Ministry of Higher Education for financial support in
term of Fundamental Research Grant Scheme (FRGS) (Project number:
203/PFIZIK/6711348) as well as MyMaster scholarship in covering the tuition fees.
For the immediate colleagues from the theoretical and computational group, I
would like to thank them for helping me in my research and pleasantly accommodate
my presence. We shared many fruitful discussions that involved a lot of general
knowledge, essentially a wide coverage on the latest world news that brings insights
to each of us. Special thanks to Mr. Ng Wei Chun, my junior who has offered all kinds
of support. Next in line are juniors Ms. Soon Yee Yeen and Ms. Ong Yee Pin for their
help in organizing various meetings and sharing of paperwork.
iii
Last but not least, I am grateful for my family who supports me in all aspects.
They understand and respect my decisions during the completion of my project, and
thesis. The hard works and sacrifices they have made encourage me even more to
succeed both in life and in academic.
iv
1.2 Problem statements 3
CHAPTER 2:
BACKGROUND THEORY
2.3 Silicene 13
Chapter 3:
Research Methodology
3.1.1 Empirical Potential 20
3.1.3 Simulation Details 31
3.2.1 Empirical Potential 32
3.2.3 Simulation Details 34
3.3 Annealing of ZnO surfaces 39
3.3.1 Structure of ZnO 39
3.3.2 Simulation Details 40
3.3.3 Empirical Potential 42
4.1.2 One-layer Graphene 45
4.3 Annealing of ZnO surfaces 69
4.3.1 Sublimation of O atoms at and beyond a
threshold temperature 69
function of annealing temperature 69
4.3.3 Sublimation of surface O atoms in pairs 70
4.3.4 Comparison with results from experiment
measurements 71
4.3.6 Partial charge of the sublimated O atoms 81
4.3.7 The Zn-terminated surfaces, (0 0 0 1) 82
CHAPTER 5:
Page
Table 3.1 LAMMPS input requires for the data files which provides
information required for constructing a rhombus shape 6H-SiC
substrate. The information is extracted from http://cst-
www.nrl.navy.mil/lattice/struk/6h.html.
23
Table 3.2 Crystal structure of wurtzite ZnO, as obtained from [54]. 39
viii
Figure 2.1 Types of carbon nanotubes and its chirality. 7
Figure 2.2 Low-dimensional carbon allotropes: fullerene (0-D), carbon
nanotube (1-D) and graphene (2-D).
7
Figure 2.3 The nanotubes and its chiral angle. 9
Figure 2.4 Single wall nanotube (left) and multiwall nanotube (right). 10
Figure 2.5 A graphite structure. 10
Figure 2.6 (a) The structure of a free-standing silicene, (b) the bond
length and bond angle of silicene and (c) the buckling
parameter of a silicene.
particle not only interacts with every other particle in the
system but also with all other particles in the copies of the
system. The arrows from the particles point to nearest copy of
other particles in the system [46].
19
Figure 3.1 Visualization of the SiC unit cell. Si atom is in light blue. The
type of atom can be read off from column 3 in the inset. The
-coordinate of each atom, which are labeled No. 1 to No. 12
in the first column of the inset, are clearly shown in the last
column of the inset.
25
Figure 3.2 The unit cell as shown in Fig. 3.1, when repeated along the x-
direction and y-direction via periodic boundary condition will
form an infinite substrate along these two directions as shown,
in which the structure is viewed from the sideway (i.e., from
the -direction).
26
Figure 3.3 Substrate of Si-terminated 6H-SiC (0001) taken as the initial
configuration input in LAMMPS software for performing
simulated annealing. Standard periodic conditions are applied
along the x- and y-directions.
27
ix
Figure 3.4 Figure 3.4: Modified 6H-SiC substrate after removing the Si
atom (that labelled No. 2 in Fig. 3.1) and replacing it by the C
atom (that labelled No. 1 in Fig. 3.1).
28
Figure 3.5 Preparation of a two C-rich bilayers substrate for growing a
two-layer graphene. This is done by systematically relocating
the atoms in the original unit cell of Fig. 3.1. The essential
seperations between the atom layers are labelled. The system
is then energy-minimized. The values in black and red are
those before and after energy minimization. The value =
1.35 is found by manual tunning (see text).
30
Figure 3.6 A silicene sheet created from diamond structure of silicon in
Si (1 1 1) orientation It is a 2D honeycomb shape similar to
graphene (left) with dimensions of 130.563 Å × 150.761 Å
and buckling parameter of 0.44 Å (right). The average bond
length of Si-Si is 2.4 Å.
34
Figure 3.7 The 15×15×1 supercell of wurtzite ZnO slab used as initial
structure in this MD simulation. Left: Direct surface view
from the direction +; Right: Edge-on view. The (0 0 0 1)
surface terminates with oxygen atoms while (0 0 0 1) Zn
atoms. In these figures, the (0 0 0 1) surface is in the direction
pointing along + , while the (0 0 0 1) surface in the -
direction
40
Figure 3.8 A typical temperature vs. step profile in the simulation. 41
Figure 4.1 The binding energy (per atom) for an infinite free
graphene layer calculated with the Tersoff (solid circles or
dashed line) and TEA (open circles or full line) potentials at
different values of lattice constant 0.
45
(second column) and (b) TEA (third column) potentials. In the
second and third columns at the bottom corner on the right,
the integer is the hexagon number. The average distance of
separation between the graphene buffer layer and surface is
about 2.43 Å for TEA potential.
47
Figure 4.3 The variation of the binding energy (per atom) plotted
against the equilibrium annealing time steps (Δ = 0.5 fs) at
(a) 1200 K for Tersoff potential, (b) 1100, and (c) 1200 K for
TEA potential.
Figure 4.4 Comparison of the average bond-length (Å) versus
annealing temperature (in units Kelvin) between results
calculated using TEA (open circle) and Tersoff (solid circle)
potentials.
50
Figure 4.5 Same as Fig. 4.4 except for the binding energy (per atom) . 50
Figure 4.6 (a) Pair correlation function () of carbon atoms obtained
using TEA potential at different annealing temperature (in
units of Kelvin) for the one-layer graphene which emerges for
≥ 1200 K. At < 1200 K, it displays typical
crystalline structure. (b) Same as Fig. 8(a) except that the MD
simulations were done using Tersoff potential. The one-layer
graphene emerges at ≥ 1500 K. At < 1300 K, it
displays typical crystalline structure. Only few hexagons are
seen at = 1400 K.
52
Figure 4.7 The result of binding energy against temperature when the
substrate used is with thickness 15.11 Å (or = 1), it is hard
to pinpoint the formation temperature of graphene when the
substrate is too thin. The overall structure has very poor
thermal stability.
54
Figure 4.8 A 6H-SiC unit cell with a thickness = 2 substrate. 55
Figure 4.9 Two-layer graphene overlaid on 6H-SiC (0001) obtained by
simulated annealing method with TEA potential. In the
second and third columns at the bottom corner on the right,
the integer is the hexagon number.
57
temperature T (in units of Kelvin) obtained by the simulated
annealing using TEA potential for two-layer graphene.
Notations used are: first-layer graphene, open circle; second-
layer, solid circle. (b) The binding energy (per atom) (in
units of eV) versus annealing temperature (in units of
Kelvin) obtained by simulated annealing using TEA potential
for two-layer graphene.
Figure 4.11 Three-layer graphene overlaid on 6H-SiC (0001) obtained by
simulated annealing method with TEA potential. The first
graphene “buffer” layer refers to one closest to the top surface
of substrate and has an average distance of separation about
2.6 Å, and the third layer corresponds to one next to the
second graphene layer and these graphene layers are separated
by an average distance about 3.2 Å. And the seperation
59
xi
between the second layer and the first layer (the top most
layer) is 2.8 Å.
Figure 4.12 (a) The binding energy (per atom) (in units of eV) versus
annealing temperature T (in units of Kelvin) obtained by
simulated annealing using TEA potential for three-layer
graphene. (b) The average bond-length (Å) versus annealing
temperature T (in units of Kelvin) obtained by the simulated
annealing using TEA potential for three-layer graphene. The
result above shown is only applied for the third-layer
graphene.
60
Figure 4.13 Simulated annealing of silicene by using optimized Stillinger-
Weber potential. Initially the silicene has equilibrium bond
length of ~2.5 Å. An abrupt change can be viewed at 1500 K
in which the sheet is tearing apart and the melting process is
thus begins. After the melting point, the Si particles settle
down to form four smaller “islands” which has equilibrium
bond length of ~2.8 Å.
65
Figure 4.15 Pair correlation function of the system at various
temperatures. At room temperature a sharp peak occurs at
~2.5 Å. The distribution curve begins to widen up and the
peak at ~2.5 Å is lowered as target temperature T increases.
67
Figure 4.16 Potential energy plot (caloric curve) of the system ( eV )
against temperature (K) measured for target temperature, = 2000 K. The sharp and abrupt drop of potential energy at
1500 K indicates occurrence of melting at that temperature.
67
Figure 4.17 Global similarity index plot against temperature. The silicene
structure is compared with its 300 K state.
68
Figure 4.18 Amorphous silicon “cylinder” after quenching of the melted
silicene. The structure is unable to revert to a sheet form.
68
Figure 4.19 Ratio of O atoms sublimated (normalized to original number
of O atoms on the surface) for T = 300 to 1300 K
70
Figure 4.20 Snapshots at various stages of the partial charge distribution
in the ZnO slab when undergoing annealing at = 300 K.
(a) At the beginning, the slab of ZnO has only gone through
energy minimization at 0.1 K but not any thermal treatment.
(b) and (c) are snapshots during which the slab is being
77
xii
annealed at the temperature plateau T. (d) Slab at the end of
thermal history as depicted by Fig. 3.8. The vertical axis is in
units of e. The z-axis is in units of
Figure 4.21 Snapshots at various stages of the partial charge distribution
in the ZnO slab when undergoing annealing at = 1300 K.
78
Figure 4.22 Partial charge distributions at the end of an MD run for eight
annealing temperatures ranging from 400 K to 1100 K.
79
Figure 4.23 Charge density () as a function of depth from the (0 0 0 1)
surface, , for annealing temperature = 500 K, = 700
K, = 800 K and = 1000 K. The vertical axis is in units
of /3 . The -axis is in units of . Note the qualitative
change of the density profile (especially the region close to
the (0 0 0 1) end) when crossing from = 700 K to = 800 K
80
Figure 4.24 Average partial charge per sublimated atom as a function of
annealing temperature.
TEA Tersoff-Erhart-Albe
SW Stillinger-Weber
temperature (T)
energy (E)
PL Photoluminescence
STRUKTUR NANO YANG TERPILIH
Premis utama dalam tesis ini adalah menggunakan kaedah dinamik molekul
(MD) untuk menyimulasi dan mengukur tiga sistem nano yang berbeza, termasuklah
(i) pertumbuhan grafen secara epitaksial pada permukaan 6H-SiC (0001) yang
didorongkan oleh pemanasan simulasi, (ii) silicene yang tergantung bebas tertakluk
kepada pemanasan yang ekstensif, dan (iii) kepingan ZnO berbentuk wurtzite yang
tertakluk kepada pemanasan simulasi. Pertumbuhan grafen secara epitaksial pada
permukaan (0001) daripada substrat 6H-SiC disimulasikan melalui kaedah dinamik
molekul dengan meggunakan kod LAMMPS. Pembentukan grafen secara epitaksial di
permukaan substrat disimulasikan melalui satu protocol yang direka khas untuk
mencapai pembinaan semula permukaan. Dua keupayaan empirik, iaitu keupayaan
Tersoff dan keupayaan TEA digunakan dalam simulasi MD supaya mekanisme
pertumbuhan yang dipaparkan oleh mereka dapat diselidiki dan dibandingkan.
Keputusan yang diperolehi daripada simulasi MD dalam tesis ini menunjukkan
bahawa keupayaan TEA lebih tepat dalam menggambarkan proses pertumbuhan untuk
membentukkan grafen, di mana keputusannya adalah lebih fizikal dan realistik secara
umumnya. Dalam simulasi MD dengan menggunakan keupayaan TEA, grafen muncul
secara tepat pada suhu pemanasan ~1200 K, setanding dengan yang diperhatikan
dalam eksperimen yang dilaporkan di mana grafen ternukleat pada suhu pembentukan
lubang 1298 K. Penilaian secara berangka ke atas panjang ikatan purata, tenaga ikatan
serta fungsi korelasi pasangan dalam eksperimen MD membenarkan pengukuran dan
kuantifikasi dilakukan ke atas grafen yang terbetuk. Grafen yang berlapisan dua dan
tiga boleh ditumbuh berdasarkan subtrat yang sama selepas lapisan grafen pertama
xv
dibentukkan. Teknik untuk menumbuh grafen berlapisan dua dan tiga di atas grafen
lapisan tunggal yang sedia terbentuk menyerupai prosedur untuk menumbuh grafen
lapisan pertama dengan sedikit pemubahsuian. Selain daripada pertumbuhan grafen
secara epitaksial, tesis ini juga melakukan simulasi MD untuk mengukur takat lebur
silicene yang tergantung bebas dengan menggunakan keupayaan Stillinger-Weber
(SW) yang dioptimumkan oleh Zhang et al.. Data ini dianalisis secara sistematik
dengan mengunakan beberapa petunjuk yang berbeza secara kualitatif, termasuk
fungsi lengkung kalori, fungsi taburan jejarian dan petunjuk berangka yang dikenali
sebagai indeks kesamaan global. Keupayaan SW yang dioptimumkan menghasilkan
takat lebur secara konsistennya pada 1500 K untuk simulasi silicene yang tergantung
bebas serta tak-terhingga. Sistem berskala nano yang ketiga yang disiasat dalam tesis
ini melalui MD adalah kepingan ZnO yang tebal berbentuk wurtzite yang ditamatkan
pada dua permukaan, iaitu (0001) (yang ditamatkan oleh oksigen) dan (0001) (yang
ditamatkan oleh Zn). Eksperimen MD dilakukan untuk mengukur kesan pemanasan
haba ke atas kepingan ZnO. Untuk tujuan ini, medan daya reaktif (ReaxFF) digunakan.
Sebagai akibat pemanasan, untuk julat suhu ambang 700 K < ≤ 800 K,
permukaan oksigen mula memejalwap dari permukaan (0001), sementara tiada atom
meninggalkan permukaan (0001). Nisbah oksigen yang meninggalkan permukaan
meningkat dengan peningkatan suhu (untuk ≥ ). Keamatan kependarkilauan
relatif pada puncak sekunder dalam spektrum foto-kependarkilauan (PL), ditafsirkan
sebagai ukuran jumlah kekosongan pada permukaan sampel, bersetuju dengan
simulasi MD secara kualitaif. Simulasi MD juga mendedahkan pembentukan dimer
oksigen di permukaan serta evolusi pengagihan caj separa semasa proses pemanasan.
Keputusan daripada simulasi MD berdasarkan ReaxFF adalah konsisten dengan
pemerhatian eksperimen.
structures
ABSTRACT
The core premise of this thesis is the adoption of molecular dynamics (MD) in
simulating and measuring three different nanoscale systems. namely (i) epitaxial
graphene growth on 6H-SiC (0001) surface induced by simulated annealing, (ii) free-
standing silicene subjected to extensive thermal heating, and (iii) wurtzite ZnO slab
which is subjected to simulated annealing. Epitaxial growth of graphene on the (0001)
surface of 6H-SiC substrate is simulated via molecular dynamics using LAMMPS
code. A specially designed protocol to reconstruct the surface via a simulated
annealing procedure, is prescribed to simulate the epitaxial graphene formation on the
substrate surface. Two empirical potentials, the Tersoff potential and the TEA
potential are used in the MD simulations to investigate and compare the growth
mechanisms resulted. Results obtained from MD simulated in this thesis show that
TEA potential is more accurately in describing the growth process of graphene
formation, in which the result is generally more physical and realistic. Graphene is
shown in the MD simulation using TEA potential to be accurate at an annealing
temperature of ≈ 1200 K, comparable to that observed in a reported experiment in
which graphene nucleates at a pit-forming temperature of 1298 K. The numerical
evaluation of the average bond-length, binding energy as well as pair correlation
function in the MD experiments allows for the measurement and quantification of the
graphene formed. Double and triple layer graphene can also be grown from the same
substrate after the first layer of graphene is formed. The technique to grow double and
triple layer graphene on top of the already-formed single layer graphene follows a
similar but slightly modified procedure used in growing the first layer graphene. In
xvii
addition to epitaxial graphene growth, MD simulations are also performed in this thesis
to measure the melting temperature of free-standing silicene by using optimized
Stillinger-Weber (SW) potential by Zhang et al.. The data are systematically analysed
using a few qualitatively different indicators, including caloric curve, radial
distribution function and a numerical indicator known as global similarity index. The
optimized SW potential consistently yields a melting temperature of 1500 K for the
simulated free-standing, infinite silicene. The third nanoscale system investigated in
this thesis via MD is a thick wurtzite ZnO slab terminated in two surfaces, namely,
(0001) (which is oxygen terminated) and (0001) (which is Zn-terminated). The MD
experiment is performed to measure the effect of thermal annealing on the ZnO slab.
To this end, reactive force field (ReaxFF) is used. Is it observed that annealing results
in the sublimation of surface oxygen atoms from the (0001) surface at a threshold
temperature range of 700 K < ≤ 800 K, while no atoms leave the (0001) surface.
The ratio of oxygen leaving the surface increases with temperature (for ≥ ).
The relative luminescence intensity of the secondary peak in the photoluminescence
(PL) spectra, interpreted as a measurement of amount of vacancies on the sample
surfaces, qualitatively agrees with the threshold behaviour as found in the MD
simulations. The formation of oxygen dimers on the surface and evolution of partial
charge distribution during the annealing process has also been depicted in the MD
simulations. The MD simulations have also revealed the formation of oxygen dimers
on the surface and evolution of partial charge distribution during the annealing process.
The results from the MD simulations based on the ReaxFF are consistent with
experimental observations.
1.1 Motivation of Study
The discovery of graphene has opened a doorway to endless possibility in material
science that no one can imagine. It sparks a field of debate and controversial (especially
germanene and stanene which is still hypothetical [1] among material scientists. The
experimental discovery of graphene in particular, and other 2D nanomaterials in
general, have since driven researchers to intensify research effort to investigate their
respective properties which has proven tremendous applications such as a substitution
to our current conventional devices. Since the discovery of the graphene in 2010, it
has revolutionized the field of material science. Many researchers around the world
have since delved into the field of low dimensional structure in the hope to utilize
graphene for wide range of application especially in nanoelectronics and N/MEMS.
But now, they even look for the alternative for graphene (i.e. silicene, germanene,
stanene and heterostructure of 2D materials) [1] to further improve the performance of
various devices.
One of the interests in studying low dimensional nanostructures is their
enhanced properties due to scaling effect as compared to their respective bulk
properties. By understanding the properties of the nanostructures (i.e.
thermodynamical properties, electronic properties, optical properties etc.) equips us
with the necessary knowledge to turn them into applications. For example, a
nanostructure with high thermal and electrical conductivity is suitable for fabrication
of computer nanochip which is small, high in operating efficiency, saving electricity
2
and generating less heat. In order to turn nanomaterials into real applications, it is
necessary to understand the technique to produce high quality nanostructures with
minimum defect. Growing a large surface area of 2D nanostructures with minimum
defect is a desirable achievement among material scientists working in the field.
However, experimental investigation on materials at nanoscale requires high
precision technologies and can often be difficult to carry out in practice. Detailed
dynamics occurring at atomistic level in these nanomaterials demands expensive and
ultraprecision technique if it is to be revealed experimentally. However, there are
alternative approaches to physically measuring these nanosystems for atomistic
information, e.g., computational approach, of which molecular dynamics (MD) is an
excellent representative. Nanomaterials, which are made up of atoms and molecules
that interact among themselves via potential fields (a. k. a force fields) at classical level,
can be simulated by building atomistic models that mimic their realistic behavior.
Time evolution of the dynamical details in the simulated systems can be followed
atom-by-atom. In this way, many physical properties, such as thermodynamical and
mechanical properties, can be derived from ensembles of atoms mimicking these
nanomaterials by applying classical physics and standard statistical mechanics
techniques on the molecular dynamics data. Despite not able to capture physical
properties of nanomaterials that are driven by quantum mechanical effect (generally
known as the electronic structures), molecular dynamics is still a powerful, convenient
and relatively cheap way to simulate nanomaterials at atomistic scale. The study of
this thesis resolves around the theme of simulating thermal properties of low
dimensional nanostructures of three distinct systems via molecular dynamics
simulation.
3
1.2 Problem Statements
To reveal the temperature-driven dynamics of the atoms making up materials at
nanoscale via experimentation techniques requires expensive and ultra-precision
equipment. It is not possible to do so in local settings due to many pragmatic
constraints. As an alternative approach to gain physical insight into three distinctive
nanosystems considered in this thesis, the detailed dynamics of thermally-induced
effects at atomistic level are computationally ‘measured’ through MD instead. The
following problems are the core concerns to be addressed in this thesis:
1. What is the dynamical mechanism that drives the formation of graphene islands
on the (0001) surface of a 6H-SiC substrate?
2. When heated up from room temperature, at what temperature graphene begins
to form on the (0001) surface of a 6H-SiC substrate?
3. How to use MD to simulate the epitaxial growth of multilayered graphene on
the (0001) surface of a 6H-SiC substrate?
4. When heated up from room temperature, at what temperature a free-standing
graphene begins melt?
5. What happens to the surface atoms of a nano size ZnO slab upon heating
beyond 1000 K?
6. When heated up from room temperature, will oxygen be released from the
surfaces of a ZnO slab of nano size? If they do, willl they be released in the
form of monoatom or molecular? At which temperature oxygen begins to
sublimate?
4
1.3 Objectives
The aim of the research presented is to perform MD simulation on epitaxial growth
of graphene on 6H-SiC (0001) substrate [2]. A prediction regarding the temperature of
the formation of graphene on 6H-SiC (0001) substrate is being made. The quality
(numbers of hexagonal rings formed) of graphene is determined. The aim of this
project is to come up with an effective strategy and method such as binding energy,
average bond length and pair correlation function to qualitatively and quantitatively to
determine the formation of graphene. The objective is to come up with the optimal
condition of annealing of high quality graphene.
Next, the melting point of silicene is determined through MD simulations. MD
simulations is used to quantify the melting points of graphene through some physical
quantities such as caloric curve, pair correlation function and global similarity index.
A novel indicator known as global similarity index was used to predict the melting
point of the silicene and compare it with the existing conventional methods.
Finally, the surfaces of oxygen-terminated and zinc-terminated ZnO slab is
characterized using MD simulations via annealing. The results thus obtained are
compared with those experimental results obtained in Sharom et al. [3]. experiment
(which will be detailed in Chapter 3 and Chapter 4). The vacancies formed at the
surfaces of annealed ZnO slab is evaluated for various temperatures and charges
distribution is quantified.
All the simulations above will be compared with the existing experimental results
and to predict the thermal behavior of the above selected nanostructures, the reliability
of MD simulations is then determined.
5
1.4 Overview of the thesis
An introduction to the research presented in the theses, motivations, problem
statements and objectives ware provided in Chapter 1 of the thesis. Chapter 2 is
generally a literature review and background theory of some selected nanostructures
such as carbon nanostructures (graphene especially will be made as the priority
subject), history of the epitaxial growth of graphene, silicene and some background of
zinc oxide. Chapter 3 will focus on methodology such as the implementation of
empirical potential, construction of the nanostructures and simulation details. Chapter
4 presents the findings and results of the MD simulations on all three nano systems,
namely, 6H-SiC (0001), free-standing silicene and ZnO nano slab. Chapter 5 will be
the conclusion of this thesis.
6
Carbon nanostructures comprising fullerene (0-D), carbon nanotubes (1-D),
graphene (2-D) and graphite (3-D) are all but derived from carbon and has a
characteristic dimension of a few or tens of nanometer in size. Most of these carbon
nanostructures are sp2-hybridized. The bond length between carbon chains is
approximately 1.42 Å. Carbon nanostructures as shown in Fig. 2.1 have attracted
researchers around the world due to their superior and unique proprieties as compared
to their bulk materials, especially in optical, semiconducting and mechanical properties
[4]. However, these nanostructures are rarely produced as free-standing entities but are
often grown on a substrate by using a suitable catalyst. The first graphene sheet was
synthesized through “scotch tape cleaving” method of on three-dimensional graphite.
The synthesis is halted when it reaches a single layer of carbon atoms [5].
7
Figure 2.2: Low-dimensional carbon allotropes: (a) fullerene (0-D), (b) carbon
nanotube (1-D) and (c) graphene (2-D).
8
To fully understand the nature of graphene, the attention is first turn to carbon
nanotubes. If one cuts along the wall parallel to the axis running through the cylinder,
and roll out, a two-dimensional graphene is formed. Fig. 2.1 shows carbon nanotubes
with three different chirality, namely armchair, zig-zag and chiral. The chirality of the
nanotubes hinges on the orientation of the tube and the rolling angle. The tube chirality
of chiral vector defines the characteristics of a carbon nanotube. The chiral vector, ,
1 and 2 are as shown in Fig. 2.3. Mathematically, these three parameters can be
written as the combination of lattice basis vector,
= 1 + 2 (2.1)
The integers (, ) denote the number of steps along the zig-zag carbon bonds of the
hexagonal lattice while 1 and 2 are the unit vectors. Zig-zag and armchair
configurations of carbon nanotubes can only be observed under limited circumstances
when the chiral angle, , is at 0° and 30° respectively due to the geometry of the carbon
bonds around the circumference of the nanotube.
9
Figure 2.3: The nanotubes and its chiral angle.
The length of chiral vector is defined as the circumference of the carbon
nanotube. The diameter, , of the nanotube is thus
=
= √2 + + 2 (2.2)
Lattice constant of 2.49 Angstrom of the carbon honeycomb is also the lattice
parameter for carbon nanotube.
Figure 2.4: Single wall nanotube (left) and multiwall nanotube (right).
Figure 2.5: A graphite structure.
11
The carbon nanostructures are generally greyish-black in color, opaque and
have a lustrous black sheen. Both metal and non-metal properties can be observed in
the carbon nanostructures. It is hard but brittle. It has excellent thermal and electrical
conductivity and is chemically inert. The stacking of graphene sheets will form
graphite as shown in Fig. 2.5. The interlayer spacing between the carbon layers is 3.35
Å. A three-million-layer graphene will form bulk graphite with an aggregate layer
thickness of 1 mm [6].
2.2 Epitaxial growth of graphene
The discovery of graphene has revolutionized our fundamental understanding
of material science. This unique two-dimensional nanostructure comprising pure
monolayer carbon atoms formed sp, sp2 and sp3 hybridization, allowing more stable
formation as compare to other carbon allotrope. Notable electronic properties are high
electrical conductivity (typically ~2 mΩ−1 ) [7] or high carrier mobility [8]
(typically ~(2 − 5) × 103cm2V−1 ) (value as high as 5 × 103cm2V−1 has been
reported also [9]) and superior thermal conductivity ( ~ 3 − 5 × 103 Wm−1K−1) [10].
Single-layer graphene also presents unusual mechanical properties such as high-in-
plane stiffness (single-layer graphene with an effective thickness ~ 6Å) and extremely
hard [11], i.e. intrinsic strength around 130 Gpa or Young’s modulus value around 1
Tpa).
A number of studies have been conducted for pre-graphene formation on the
SiC surface by using scanning tunneling and atomic force microscopy, providing
detailed view on the surface reconstruction of SiC surface [12]. Experimentally,
several methods reported producing high quality graphene layers. One popular method
12
is the epitaxial graphene technique, where 4H- or 6H-SiC surfaces are heated up to
high temperature. Epitaxial growth refers to the deposition of a crystalline overlayer
on a crystalline substrate. This strategy involves graphitization of SiC whereby Si
sublimation occurred during high temperature annealing in vacuum. To gain insight
into the growth of epitaxial graphene, Hannon and Tromp [13] studied the formation
of graphene using the low-energy electron microscopy. It is worth noting that Hannon
and Tromp observed the formation of smooth steps and the step height was measured
through atomic-force microscopy under prolonged high temperature annealing at 1298
K in vacuum. It is believed that this terracing feature would give rise to pit formation
which hinders the formation of flat graphene layers at temperature T < 1300 K. In a
separate study, Borysiuk et al. [14] independently observed similar carpet-like
corrugation panorama using transmission electron microscopy. Recent works of Tang
et al. [15], Lampin et al. [16], Jakse et al. [17] using computer simulation have also
provided important insights on the formation of epitaxial graphene.
The occurrence of epitaxial graphene is not only limited to SiC substrate but is
also extended to various transition elements. Li et al. successfully grew large area of
graphene on Cu (1 1 1) surface [18] and Sutter et al. surprisingly fabricated a large
graphene domain with uniform thickness across Ru metal surface [19]. Computer
simulation has also been conducted by Enstone et al. by using Monte Carlo model on
graphene/Cu (1 1 1) [20]. To differentiate graphene from the substrate, it is to be noted
that the graphene that grew epitaxially has the advantage of having higher electronic
carrier mobility relative to the SiC substrate. It is, however, important to note also that
removal of graphene from its respective metal substrate might damage the graphene
layer.
13
2.3 Silicene
Silicene, a two-dimensional nanosheet made up of silicon atoms arranged in
the form of honey comb lattice, has been predicted theoretically by Takeda and
Shiraishi [21] in year 1994. Subsequent DFT calculations by Guzman-Verri and Voon
[22] revived the interest on silicene by showing that silicene was indeed energetically
stable, and of feasible possibility to being experimentally produced. Silicene, unlike
graphene which prefers sp2 hybridization, is not flat. Rather, due to the preference of
sp3 hybridization, the silicene sheet has a buckled configuration, where the out-of-
plane buckle parameter is predicted to be 0.44 Å according to DFT calculations.
Having a close resemblance to graphene, silicene offers many possibilities as a
functional material of advanced applications, such as photovoltaic, optoelectronic
devices [23], thin-film solar cell absorbers beyond bulk Si [24] and hydrogen storage
[25]. One advantage of silicene over other 2D materials is that it is, in principle, easier
to get integrated into nano devices which are mainly silicon-based.
Silicene is a rather new form of 2D material and was synthesized on supported
substrates in a series of discovery since 2007 [26]. Following the successful synthesis
of silicene on supported substrate, many theoretical studies and simulations on the
structural, mechanical, electronic and thermal properties of silicene on supported
substrate have been published [27]. The structural properties of a free-standing
silicene sheet, as was originally investigated by Jose et al. [25], however, being
modified when grown on a substrate. Thus, the silicene experimentally synthesized so
far is not free-standing but sitting on a substrate. One has yet to see any report of
experimentally synthesized free-standing silicene. Having said that, investigation of
free-standing silicene serves the purpose of understanding the pristine system in the
absence of interactions with surfaces. The understanding on the basic properties of
14
silicene without the interference from substrate shall provide useful insight for higher
level manipulation of silicene. One of the envision is the ‘van der Waals'
heterostructures envisaged by Geim [27], which it deals with heterostructures and
devices made by stacking different 2D crystals on top of each other. Strong covalent
bonds provide in-plane stability of 2D crystals, whereas relatively weak, van der
Waals-like forces are sufficient to keep the stack together.
Figure 2.6: (a) The structure of a free-standing silicene, (b) the bond length and bond
angle of silicene and (c) the buckling parameter of a silicene.
In this thesis, there are very limited work on the melting behavior and thermal
stability of free-standing silicene is reported in the literature. Bocchetti et al. simulated
the melting behavior of free-standing silicene via Monte Carlo method with original
and modified version of Tersoff potential parameter set (known as ARK) for silicon
atom [28]. According to Bocchetti et al., original Tersoff parameters for silicon atom
results in a melting of the free-standing silicene at 3600 K, meanwhile the melting
temperature obtained using ARK parameter set is only ~ 1750 K. Berdiyorov et al.
simulated the influence of defect on the thermal stability of free-standing silicene via
MD using Reactive force-field (ReaxFF) [29], where it is found that pristine silicene
15
is stable up to 1500 K. As a general observation, melting properties and thermal
stability of free-standing silicene obtained in MD simulations varies from cases to
cases depending on the details of the simulation procedure. Furthermore, the
simulation results are strongly force-field dependent. Apart from simulating thermal
stability, MD simulation has also been applied to investigate or predict thermal
conductivity of free-standing silicene. A wide range of potentials is employed in these
simulations, and the potentials are of semi-empirical type. For example, Zhang et al.
developed a set of Stillinger-Weber potential parameters specifically for a single-layer
Si sheet to simulate the thermal conductivity [30]. Most researchers would often use
Tersoff potential with original parameter sets though.
2.4 Zinc Oxide
Zinc oxide (ZnO) has been extensively studied, both theoretically and
experimentally, due to its many promising applications in piezoelectric devices,
transistors, photodiodes, photocatalysis and antibacterial function [31-33]. The
physical properties of ZnO, especially its surface properties, can be experimentally
modified at the atomic level to engineer the material for desired functionality. Since
ZnO contacts with its external environment through its surfaces, knowing how the
surface properties respond to external perturbation (e.g. thermal treatment) would
provide valuable information on how to manipulate ZnO for application purposes in
future. And one of the simplest way researchers known is to heat ZnO to high
temperature (below its melting point). Heating ZnO can be easily carried out in
practice, and many works had been reported along this line [34-36].
16
ZnO crystals are dominated by four surfaces with low Miller indices: the non-
polar (1 0 1 0) and (1 1 2 0) surfaces and the polar surfaces which are the zinc-
terminated surface (0 0 0 1) and the oxygen terminated surface (0 0 0 1). Surface
energy of polar surfaces in an ionic model diverges with sample size due to the
generation of macroscopic electrostatic field across the crystal [37]. This kind of
behavior was well investigated by Tasker [38]. Accordingly, wurzite ZnO is also
labeled as Tusker-type surfaces, and these surfaces are formed by alternating layers of
oppositely charged ions.
It is interesting to investigate what will happen to the atomic configuration of
the surface when a ZnO slab with finite thickness is heated without melting.
Sublimation of atoms from the polar surfaces due to temperature effect will be studied
using molecular dynamics (MD) simulation, where the trajectories of all atoms at a
given temperature are followed quantitatively. As will be reported in Chapter 4 when
the result of the MD simulation of ZnO is represented, in which the reactive force field
(ReaxFF) for ZnO is used, sublimation of O atoms from ZnO polar surface is observed.
ReaxFF for ZnO allows bond formation and charge transfer among the selected atoms.
When sublimation of atoms occurs, point vacancies are created on the surface.
Quantitative information of the amount and type of atoms sublimated, as well as point
vacancies created on the surface at different annealing temperatures can thus be
obtained.
Experimentally, if a ZnO wurtzite surface is heated to an elevated temperature
and investigate the resultant surface using photoluminescence (PL) measurement, the
spectrum should reflect the amount of point vacancies created. It is expected that an
increase of annealing temperature will create more point vacancies. In this thesis,
predictions from MD simulation are compared with PL data.
17
2.5 Molecular Dynamics Simulation
Molecular dynamics (MD) simulation is used to solve equation of motion of
particles in different phases [39]. It could reliably predict the physical properties of
material even in non-ground state. Particles interact with each other at finite
temperature for an extended period in a MD simulation. The atoms or molecules
evolve in the system made possible by the interactive forces or so-called empirical
inter-atomic potential. The forces that govern the motion of atom are in accordance
with Newton’s Second law. Using equation of motions of all particles in the system,
the evolution of the system is solved as,

(2.3)

() = = −∇ = −∇[ΣV2 (r , r) + Σ,V3(r , r , r) + ]
(2.4)
where denotes the particle in the system, is the interactive potential between
particles, r is the position of the particles while refers to the velocity of the system.
The initial conditions to commence an MD simulation include positional
coordinates, initial random seed of the velocity and appropriate empirical potential
(which will be elaborated in detail in Chapter 3) so as to derive the forces between
particles. Regardless of the merits of the other algorithms in the simulation code
(integrators, pressure and thermostat etc.), whether or not the simulation produces
realistic results depends ultimately on the empirical potential. Empirical potentials are
also the computationally most intensive parts of a molecular dynamics simulation code,
consuming up to 95% of the total simulation time. The simulated particles are placed
in the simulation box with a defined boundary condition. There are two types of
18
boundary condition: periodic boundary condition and fixed boundary condition.
Periodic boundary condition eliminates the edge effect on the simulation box. Periodic
boundary condition artificially creates the simulation box with infinite volume
appropriate for simulating bulk or periodic crystal. This is achieved by replicating the
simulation box in such a way that the particles within the simulation box would interact
with their neighboring particles. As for fixed boundary condition, the simulation box
is enclosed by “wall” or “edge” with a defined volume. The particles would be
reflected to the simulation box when the interacting particles reach the boundary of the
simulation box. This fixed boundary is suitable for the simulation of finite size
particles such as clusters, surfaces and nanoparticles.
It is essential that thermalization process is performed onto the system to enable
the system to achieve thermal equilibrium and minimum energy. Microcanonical
ensemble (NVT) with Nose-Hoover [40] thermostat is best used to equilibrate a system
to its local minima. The molarity, volume and temperature of the system are conserved.
The ensemble performs time integration on Nose-Hoover style non-Hamiltonian
equations of motion to generate positions and velocities. When used correctly, the
time-averaged temperature of the particles will match the target values specified [41].
Sometimes, canonical ensemble (NVE) is also used, during which the system molarity,
volume and energy are conserved. Thus, the equilibrating of the system can also be
ascertained using NVE. The summary of concept of periodic boundary condition is
shown in Figure 2.7.
19
Figure 2.7: Periodic boundary condition of molecular dynamics. Each particle not only
interacts with every other particle in the system but also with all other particles in the
copies of the system.
In practice, MD simulation is performed by using existing computational
packages. There exist many full-fledged, multi-functional software packages
implementing MD. The code Large-scale Atomic/Molecular Massively Parallel
Simulator (LAMMPS) [42] is among the best known. It will be used exclusively in
this thesis for simulating the thermal behavior of the three chosen nanosystems.
20
3.1.1 Empirical Potential
Simulation of epitaxial graphene growth involve Si and C atoms. In the context
of MD simulation, the so-called force field, which refers to the interaction among these
atoms has to be determined. The force field used in a MD simulation plays a vital role
as the correctness of the simulated results is directly determined by it. For the case of
carbon and silicon atoms, due to their wide applications in current materials science
and semiconducting technology, many high-quality force fields have been historically
developed. Considered as the most widely used in MD simulation of materials science
involving carbon and silicon atoms is the prototype force field by Tersoff [43] and its
more refined form, or the so-called TEA potential (Tersoff-Erhart-Albe) [44]. Both are
empirical force fields developed based on experimental input and rigorous physical
consideration. These two force fields will be used in simulating the epitaxial growth
of graphene on the SiC substrate.
Tersoff or TEA force field has the following general expression,
= ∑ = 1
(3.1)
where denotes the total energy for an atom at site . The location of the site is
denoted . The potential energy, (), arising from the interaction between an atom
at site and another at site , where both are separated by a distance , is assumed
to take the following form in the original Tersoff paper,
() = ()[R() + A()] (3.2)
21
in which is a cutoff function varying continuously from 1 to 0 around the position
( + )/2, where they are defined as per = ()1/2 and = ()1/2.
and respectively define the cutoff distance around ( − )/2 for atoms located
at the first-neighbor shell. Based on the ideas suggested in Tersoff’s original papers
[49, 50], the function is chosen to take the form
() = {
(3.3)
where and stands for C or Si. The first term in Eq. (3.2) (denoted by the subscript
“R”) represents a repulsive part, whereas the second (denoted by the subscript “A”) an
attractive one. R() and A() in the square brackets in Eq. (3.2) are both
expressed in the Morse potential form, namely,
R() = exp[−( − (0)
)]
)].
(3.4)
The coefficients in Eq. (3.4) are defined via = ()1/2 , = ()1/2 ,
whereas coefficients in the exponents are = ( + )/2 and = ( + )/2.
It is reasonable to assume that the interaction among the atoms is effective up
to a distance set by the first-neighbor shell. Such assumption leads an approximated
expression for the coefficient , namely, = 1.
is much complicated quantity. It measures the bond order describing the
coordination of atoms and . Cast in the most general form, it reads
= (1 +
(3.5)
22
where the parameter plays the role of strengthening or weakening the heteropolar
bonds, relative to the value estimated by interpolation, and
= ∑ ()
≠,
Eq. (3.7) contains the three-body interaction function
() = [1 +
] (3.7)
In Eq. (3.7), is the bond angle between bond and any atom at (≠ , ) bonded
with atom forming bond , and constants , , and are accordingly
determined by three-body interactions.
3.1.2 Construction of 6H-SiC substrate
To begin with the MD simulation, a data file containing the details of the
positions of all atoms in the unit cell and the lattice parameters of a 6H-SiC (0001)
crystal has to be first prepared. The information of the crystal structure of the 6H-SiC
(0001) substrate was obtained from the NRL (Naval Research Laboratory) structure
database [45], and is reproduced in Table 3.1.
23
Table 3.1: LAMMPS input requires for the data files which provides information
required for constructing a rhombus shape 6H-SiC substrate. The information was
extracted from http://cst-www.nrl.navy.mil/lattice/struk/6h.html [45].
The numerical values of the primitive vectors 1, 2, 3 (correspond to
a(1), a(2), a(3) in Table 3.1) are, according to the NRL database,
1 = {1.54035000, −2.66796446,0 .00000000},
2 = {1.54035000,2.66796446,0.00000000},
3 = {0.00000000,0.00000000,15.11740000},
in nanometers. In a more conventional notation, the three primitive vectors 1, 2, 3
are denoted ≡ 1, ≡ 2, ≡ 3 respectively. The norm of , , , denoted by
, , , are the three lattice constants defining the SiC unit cell. The value of a can be
easily solved for, as per


⇒ = = 3.08.
, , denote the elemental basis vectors, namely, = {1,0,0}, = {0,1,0}, =
{0,0,1}. Since 6H-SiC belongs to the hexagonal class, by definition, = , = =
90° and =120°, where is the angle between the lattice vectors and , the angle
between the lattice vectors and , he angle between the lattice vectors and .
The lattice parameter is simply the norm of 3, = 15.12, The unit cell for the SiC
crystal so constructed is rhombus in shape. However, LAMMPS does not support
direct input for the angles of hexagonal lattice. The lattice constants a, b, c and angles
in the form of , and need to be converted into LAMMPS-readable
form, lx, ly, lz, xy, xz and yz. The conversion is shown in Equation 3.8.
25
(3.8)
Figure 3.1: Visualization of the SiC unit cell. Si atom is in light blue. The type of atom
can be read off from column 3 in the inset. The -coordinate of each atom, which are
labeled No. 1 to No. 12 in the first column of the inset, are clearly shown in the last
column of the inset.
There is a total of 12 atoms in a unit cell of a 6H-SiC crystal, see Figure 3.1.
Coordinates of each atom in the unit cell are also displayed. The last column in the
figures of Fig. 3.1 refers to the -coordinates of the respective atoms. Vertical distance
between the atoms can be deduced from their -coordinates straightforwardly. The unit
cell when repeated along the -direction and -direction via periodic boundary
condition will form an infinite substrate along these two directions. The resultant
bilayer of Si and C
bilayer of Si and C
bilayer of Si and C
bilayer of Si and C
bilayer of Si and C
26
structure so formed is as that illustrated in Fig. 3.2, in which the structure is viewed
from the sideway (i.e., from the -direction).
Figure 3.2: The unit cell as shown in Fig. 3.1, when repeated along the -direction and
-direction via periodic boundary condition will form an infinite substrate along these
two directions as shown, in which the structure is viewed from the sideway (i.e., from
the -direction).
The structure is terminated at the (0001) surface by a Si atom. It is characterized by
six hexagonal layers repeated periodically in the (0001) direction. Each hexagonal
layer is a bilayer, composing of one layer of Si atoms and the next layer the C atoms.
Later, 6H-SiC (0001) substrates of different thickness (along the -direction) will be
used as input structure in the LAMMPS simulated annealing procedure. This input
structures are to be constructed in the form of supercells that are made up of this unit
cell. The supercells representing the substrates are made up of 12 × 12 × unit cells,
where =1, 2, 3 corresponds to the number of unit cell layer in the substrate. The
substrate assumes an orthorhombic structure with Si-terminated 6H-SiC at one surface
and C at the other. Figure 3.3 illustrates the substrate comprised of 12 × 12 × unit
cells as viewed from the +-direction. A vacuum of thickness 100 Å is created above
27
and below the substrate surface. Periodic condition is also applied along the -
direction. Due to the thick vacuum, interaction between neighboring surfaces along
the -direction can be safely ignored. For = 1, the dimensions of the orthorhombic
substrate are 33.88 × 33.88 × 15.12 Å3. It is found that, as to be shown later, the
thickness will have some effects in growing epitaxial graphene layers on the
substrate. For example, for growing one-layer graphene, = 1 is sufficient and for
two-layer graphene and three-layer graphene, = 2 is needed due to the procedure of
removal of Si atoms as described below.
Figure 3.3: Substrate of Si-terminated 6H-SiC (0001) taken as the initial configuration
input in LAMMPS software for performing simulated annealing. Standard periodic
conditions are applied along the - and -directions.
As it turns out, to successfully simulate epitaxial graphene growth via
simulated annealing requires a somewhat artificial but non-trivial initial condition to
be imposed on the 6H-SiC substrate constructed using the above-mentioned supercell.
Specifically, the Si atom layer on the surface of 6H-SiC substrate (which is labelled
No. 2 in the first column in Fig. 3.1) has to be removed. The vacancy left by the
removed Si atom is replaced by the carbon atom labelled No. 1 in Fig. 3.1, which has
been shifted from its original location. The resultant substrate, as illustrated in Fig. 3.4
is a configuration consists of a carbon-rich bi-layers sitting on the surface. Now, the
28
C-rich bilayer
C-rich monolayers have an intra-layer separation of 0.624 Å (this is the separation
between the C atom No.1 layer and No. 3 layer in Figure 3.4). In this work, call this
the carbon-rich bilayer (or C-rich bilayer).
Figure 3.4: Modified 6H-SiC substrate after removing the Si atom (that labelled No. 2
in Fig. 3.1) and replacing it by the C atom (that labelled No. 1 in Fig. 3.1).
and shifting the C atom (No. 1) from its original position to replace the vacancy left
by the removed Si atom. The label of each layer, according to the numbering scheme
as specified in the inset of Fig. 3.1, are also indicated. This is the template substrate
with thickness = 1 for growing a single layer graphene. The formation of a C-rich
bilayer is a key step for a successful computer-growth of multilayers of graphene, i.e.,
any set of two C-rich monolayers must lie to within 1 Å.
The same tactics is applied for growing two layers of graphene. To this end,
the substrate has to be artificially configured such that two C-rich bilayers are pre-
conditioned to exist on one of its surfaces. This is done by systematically removing
silicon atoms at selected layers and manually relocating the positions the carbon atoms
C atom No. 1
Si atom No. 4 C atom No. 5 Si atom No. 6 C atom No. 7
Si atom No. 8 C atom No. 9
Si atom No. 10 C atom No. 11 Si atom No. 12
C atom No. 3
29
in selected layers. Fig. 3.4, in which the atomic layers are explicitely labelled, is
referred to faciliate the description of the preparation procedure. Explicitly, the
substrate is prepared by knocking off the Si layers labeled No. 4 and No. 6, after which
the -position of the carbon layer labelled No. 5 is translated to that left by the Si layer
labeled No. 6. Concurrently, the C-rich bilayer labelled No. 1, No. 3 is shifted
simultaneously along the z-direction to occupy the z-position that was left by atom
layers No. 4 and No. 5. The resultant unit cell is shown in Figure 3.5, where the
essential separations between the atomic layers are explicated labelled. Note that each
C-rich bilayer is comprised of two C-rich monolayer that are separated within 1 Å.
These two sets of C-rich bilayers are labeled 1, 3 and 5, 7 respectively in Figure 3.5.
When viewed in the direction of -plane, each of the two C-rich bilayers appears to
portray a hexagonal structure with an average C–C bond-length of 2.895 Å.
It turns out that there is another crucial factor that regulates a successful growth
of two-layer graphene, i.e., the adjustment of , the initial distance separating the two
sets of C-rich bilayers. MD simulations were conducted by progressively changing the
distance , and at each stage, the whole system was relaxed to a minimized energy
state. It was consistently found that the same minimized energy state with = 1.65
Å was always obtained whenever is chosen to be in the range of 0.3–1.35 Å. is
defined as the separation between the two C-rich bilayers mentioned above after the
system is energy minimized. Note that the allowed range of is much shorter than the
original separation 4.42 Å (obtained by evaluating the separation in the -position of
atoms labeled No. 3 and No. 6 in the inset of Fig. 3.1), the latter being beyond the
empirical potential cutoff. The descriptions and figures used to illustrate the
preparation of a two C-rich bilayer substrate, as mentioned above, are for a substrate
30
with thickness = 1 . However, without lost of generality, they can be trivially
generalized to 6H-SiC substrate with larger thickness, e.g., = 2 and 3.
A three-layer epitaxial graphene can be grown on a 6H-SiC substrate by using
the two-layer graphene (that has been grown using the method described above) as a
template. To this end a two-layer graphene is first grown. The system is then quenched
to very low temperature (i.e. 1 K). The removal of the silicon atoms at the buffer layer
nearest to the SiC substrate is performed to form a carbon rich layer right below the
two-layer graphene. The system is then annealed again to very high temperature. A
three-layer graphene can then be formed.
Figure 3.5: Preparation of a two C-rich bilayers substrate for growing a two-layer
graphene. This is done by systematically relocating the atoms in the original unit cell
of Fig. 3.1. The essential seperations between the atom layers are labelled. The system
is then energy-minimized. The values in black and red are those before and after energy
minimization. The value = 1.35 is found by manual tunning (see text).
31
The construction of initial configuration, i.e., the rhombus substrate with
carbon monolayers on top of it, is then ready for simulation using LAMMPS.
Simulated annealing method is adopted as it would determine the lowest energy
configuration of the system. Details of the simulation procedure are summarized as
following:
1) The empirical potentials of Tersoff and TEA to describe the interatomic
interactions of C–C and Si–C are employed separately in the simulations
2) For 6H-SiC substrate that contains one or two C-rich bilayers, a relaxation is
performed by using the conjugate gradient minimization technique. The
minimization criterion is set to finish at a maximum of 10000 steps. But the
structure has already reached the minimum energy after 1624 steps.
3) The simulated annealing procedure is proceeded by MD simulation in the NVT
ensemble using the Nose-Hoover thermostat with a time step = 0.5 fs. Next,
for growing one-layer graphene, the temperature of the system is increased at
a heating rate of 1.2 × 1014 K/s (half of this rate for multilayers of graphene)
until = 300 K, and at this temperature, the system is equilibrated for a time
interval of 2×104 t. The stage is now set to raise the temperature to a desired
target value . The selection of a fixed heating rate of 5×1013 K/s is found to
be suitable to the range of temperature investigated in this work for this process.
Equilibrium of the system for a total time steps of 6×104 is performed
immediately at the desired and is followed subsequently by cooling the
system at a rate of 5 × 1013 K/s until = 0.1 K. As for the heating rate, it was
found that graphene will not form at the right temperature if the value used was
too large. Low heating rate, on the other hand, will cause graphene to form at
32
a much later time, rendering a costly ‘waiting time’. A balance has to be struck
when fine tuning what is the right value of heating rate to use. To this end,
some trial-and-error efforts were conducted to determine the optimal heating
rate. A similar concern also arise regarding how long the simulation at the
equilibration of the system has to be run when the desired has been reached.
The graphene will not form when the equilibration time is too short, while a
too lengthy equilibration time will render excessive cost in terms of time
consumed. Fine tuning via trial and error on the total length of time to run at
the equilibration state at a fixe is hence necessitated.
3.2 Melting of Silicene
Silicene is a two-dimensional silicon sheet that maintains a buckling
honeycomb hexagonal structure. To the best of the knowledge of current field, no
successful production of a free-standing silicene in simulation has been observed so
far using silicon empirical potential [46]. In this thesis, effort is made on studying the
melting point of free standing silicene through molecular dynamics approach. For
melting of silicene using molecular dynamics approach, two interatomic potentials
were adopted, namely the Charged Optimized Manybody (COMB) [47] potential and
the modified Stillinger-Weber (SW) [30] potential, which are believed to be able to
create a stable silicene sheet inside a thermalized simulation box before melting could
be initiated.
Generally, COMB potential expression is based on the original Tersoff
potential expression, but with added long-range Coulombic interaction between
charged atoms described through the charge coupling factor, () . It takes the form
() = ∫ 3 ∫ 3ρ (, )ρ(r , q)/
ρ(, ) = ξ
(3.9)
which is Coulomb integral over 1s-type Slater orbitals and ξ is an orbital exponent
that controls the radial decay of density. A self-consistent charge equilibration is also
added by using Rappe and Goddard method [48]. The details of the parameters used
in the COMB potential can be found in Shan et. al. [47].
The SW potential is well known for its wide usage on semiconducting materials
such as silicon. It was first developed in 1985 for various phase of silicon crystal and
was in good agreement with experimental derived physical properties of silicon
crystals. The formalism of SW and its detail parameters can be seen in Stillinger et. al.
[46]. The conventional SW parameter is unable to produce a stable free standing
silicene sheet, so some modifications were made on the silicon parameters in SW
potential by Zhang et. al. [30] (so called optimized SW) which would maintain the
buckling structure and the 2D honeycomb structure of a silicene.
3.2.2 Construction of silicene
To obtain an infinitely large free-standing silicene sheet, a surface bulk Si
structure is first constructed by using a diamond unit cell with a lattice constant of
5.431 Å. This is done by replicating a total of × × diamond unit cells to form
a simulation box, subjected to periodic boundary condition. The structure is then re-
34
oriented such that the (1 1 1) surface is aligned along the -axis. This is done because
the Si diamond lattice in the (1 1 1) direction resembles a silicene sheet with a non-
zero buckling parameter. After re-orientating all atom-containing -planes
perpendicular to the -axis are removed from the bulk until only a single sheet is left.
A total of 43108 atoms are removed in the process. The removal of the silicon atoms
creates a region of vacuum with 41.5 Å thick along the -direction on both sides of the
remaining silicon atom plane, which now resembles a free-standing silicene sheet with
surface area of 130.563 Å × 150.761 Å, consisting of 9600 atoms and a buckling
parameter of 0.44 Å, see Fig. 3.4.
Figure 3.6: A silicene sheet created from diamond structure of silicon in Si (1 1 1)
orientation It is a 2D honeycomb shape similar to graphene (left) with dimensions of
130.563 Å × 150.761 Å and buckling parameter of 0.44 Å (right). The average bond
length of Si-Si is 2.4Å.)
3.2.3 Simulation Details
After the construction of silicene is done, the silicene structure is inserted into
LAMMPS software to perform simulated annealing. The detail procedures are
summarized as follow:
35
1) COMB and SW empirical potentials is employed separately in this simulations
to describe the interatomic interactions of Si-Si.
2) Conjugate gradient minimization technique is used to relax the free standing
silicene. The minimization criterion is set to run up to a maximum of 10000
steps as the stopping criteria. The energy minimization procedure completes
after 3660 steps before hitting the stopping criteria.
3) The simulated annealing procedure is proceeded by MD simulation in the NVT
ensemble using the Nose-Hoover thermostat with a time step = 0.5 fs. The
system is then equilibrated at 1 K for time interval of 5 × 104 and then raise
to = 300 K at the heat rate of 1 × 108 K/s, before the system is equilibrated
for a time interval of 2.5 × 104 .
4) The stage is now set for raising the system's temperature to a large target
temperature (one at which the silicene would melt). To raise the temperature
from 300 K to , a heating rate have to be fixed. A convergence test is
performed by running trial MD melting simulations at various heating rates by
fixing the target temperature = 2000 . From the convergence test, resultant
melting temperature obtained is invariant if the heating rate 2 × 1010 K/s or
larger. The system is heated from 300 K for a total 1.7 × 107 t (equivalent to
85 ns) at the mentioned heating rate until it reaches 2000 K.
5) Evolution of the silicene MD trajectory in the simulation is monitored visually
as well as quantitatively. Radial distribution function, and caloric curve of the
silicene sheet are numerically sampled and measured. In addition, in this work
also measure a numerical descriptor, known as ‘global similarity index’, , to
gauge the melting process. Detailed description of the global similarity index
is discussed in the following topic.
36
Chemical similarity is defined as compounds having underlying microscopic
similarity [49]. However, it is possible to quantitatively define an index such that it
can be used to predict a variety of important properties including biological activity
[50], chemical reactivity and the chemical properties.
The definition of the global similarity index is based on generic chemical
similarity idea for detecting configurational changes along the trajectory during the
heating process. It was first proposed in the work of [51]. The evolution of the global
similarity during the simulated annealing process, which carries statistical information
of geometrical variations in the melting mechanism of the silicene will be elaborated
in Chapter 4.
The functional form of is proposed to take the form of
= 1
, = |√, − √,0| (3.11)
where , and ,0 represent the sorted distance of atoms relative to the average
positions (center of mass) of all the atoms in the cluster for the th (denoted as the
subscript ) and the 0th “frame” (for the simulated annealing process, this is the input
structure), while corresponds to the number of atoms, which is an integer equals to
the number of pairs of ,. The value of = 1 corresponds to totally identicalness
and → 0 for vast difference.
37
Having defined the similarity index in Eq. 3.11 and Eq. 3.12, one can optimize
the sensitivity of the index with respect to degrees of freedom correspond to a few
parameters such as translation and rotation of the particles relative to one another. Note
that the average positions, a.k.a., mean, that enters the definition of the parameter
can be non-uniquely defined. Different definitions of mean capture different aspects
of configurational information contained in the system. Several definitions of mean
are possible, namely (a) arithmetic mean (Eq. 3.13), (b) harmonic mean (Eq. 3.14) and
(c) quadratic mean (Eq. 3.15).
= 1 + 2 + 3 +

(3.14)
However, during a melting process, variations in the configuration of the atoms
are not known. In principle, variation in a certain mode of motion among the
atoms could be more sensitively picked up by certain mean than the other. To cover
all possibilities, the parameter that enters the definition of in Eq. 3.11 and Eq.
3.12 is calculated by averaging overall of the above three types of mean, i.e.,
= 1
COR ∑
COR
COR
(3.15)
38
where COR stands for center of reference, COR = {arithmetic mean, harmonic mean,
quadratic mean}, COR = 3, COR
is global similarity index defined based on a COR
mean. Including all types of COR can in principle increase the sensitivity as well as
accuracy of the index in capturing variations in the geometrical configuration of the
system. Similarity index of the silicene frames in the MD simulation during its heating
process is captured in successive temporal sequence. If the silicene sheet remains intact,
its geometrical configuration should be very close to the one at the initial frame, i.e.
≈ 1. If the silicene melts, is expected become less than 1 and in the extreme case,
becomes 0. In the following the bar symbol in the definition of shall drop when the
index is referred. As it turns out, chemical similarity is rarely used for very large
system [52]. In the present case, the system being simulated consists of 9600 Si
particles. This is considered a moderately large one. Global similarity index (which is
a modified form of chemical similarity index) as defined above has a high sensitivity
to detect even minor distortions in the geometrical configuration of silicene, and hence
a very convenient tool to pinpoint the location of melting point during the temporal
evolution of the MD.
It is commented that caloric curve and the other two quantities, (), , are
two qualitatively different indicators for gauging the melting process. The former is a
thermodynamical quantity whereas embed only geometrical information of how
atoms distribute in 3D space.
3.3 Annealing of ZnO surfaces
3.3.1 Structure of ZnO
39
Crystalline ZnO can exist in various polymorphs. The stability of the
polymorphs is dependent upon the pressure and temperature [53]. These polymorphs
include wurtzite, zinc blende and rocksalt. ZnO in wurtzite structure is the most stable
form at room pressure. The parameters of crystal structure of the wurtzite ZnO unit
cell are shown in Table 3.2. A slab comprising 15×15×3 unit cells was constructed.
The slab was sandwiched between two vacuum layers of thickness 100 , while the
thickness of the slab itself was 15 . The surface area of the slab in the supercell was
1783 2 . The thickness of vacuum was chosen such that the interactions between
adjacent surfaces of two neighboring supercells were negligible so as not give rise to
any significant effect on the simulation outcome. Periodic boundary condition was
imposed in all x-, y- and z-directions. Fig. 3.5 shows the supercell of the ZnO slab. In
all simulations the total numbers of atoms in the simulation box were maintained at
2700. The MD simulation throughout this work was carried out using the MD code
LAMMPS.
Table 3.2: Crystal structure of wurtzite ZnO, as obtained from [54].
40
Figure 3.7: The 15×15×1 supercell of wurtzite ZnO slab used as initial structure in this
MD simulation. Left: Direct surface view from the direction +; Right: Edge-on view.
The (0 0 0 1) surface terminates with oxygen atoms while (0 0 0 1) Zn atoms. In these
figures, the (0 0 0 1) surface is in the direction pointing along +, while the (0 0 0 1)
surface in the - direction.
3.3.2 Simulation Details
1. The step size throughout the simulation is Δ = 0.5 fs. The structure is first
optimized at 0.1 K using the built-in conjugate gradient minimizer. The
convergence criterion is set to 10000 steps (generally, convergence is achieved
in less than 100 steps).
2. The temperature of the system is then heated up to = 300 K in 5000 steps.
The system is further equilibrated for 20000 steps at 300 K. At the end of the
equilibration at , the temperature is raised to a chosen target temperature, T,
at a rate 5 × 1013 K/s. The evolution of the system at annealing temperature is
then followed at constant T for a total of step = 250000.
41
3. The temperature of the system is then quenched from to 0.1 K at a rate of
5 × 1010 K/s. A typical temperature vs. step profile is shown in Fig. 3.8 for
annealing temperature =1000 K.
4. Nose-Hoover thermostat (NVT) is used throughout the simulation to control
the temperature. The damping constant for the thermostat is set to 5 fs. The
total steps in each simulation are dependent upon the target temperature . The
total duration for the simulation ranges from 315 000 steps (0.16 ns) to 387 000
steps (0.20 ns).
Figure 3.8: A typical temperature vs. step profile in the simulation
Independent simulations were also carried out for each target temperature ,
ranging from 300 K to 1300 K at an interval of 100 K. Atoms from the polar surfaces
would leave the surface if the target temperature is sufficiently high. In this case it has
to assure that step that defines the length of the temperature plateau is sufficiently
long such that atom will stop leaving the surface even if the system continues to
42
equilibrate at that annealing temperature. This precaution is necessary so that the
conclusion of the MD outcome remains the same should a larger step is used instead.
After some trial-and-error efforts, it was found that a value of step = 250000 was
optimal for the range of temperatures investigated in this work. At the end of the
simulation, i.e. when the slab has been quenched to 0.1 K from a given target
temperature , the type and total number of atoms leaving the surface were quantified.
After running a series of simulation for various , sufficient data is gathered to plot
the ratio of atoms leaving the surface as a function of annealing temperature.
3.3.3 Empirical Potential
The reliability of the MD results depends crucially on the force field used.
Force field based on the ReaxFF model, first proposed in [55], allows bond formation
and charge transfer to occur among the interacting atoms in the simulation box. In this
work, the ReaxFF for ZnO is adopted. The ZnO reactive force field has been applied
to the calculation of atomic vibrational mean square amplitudes for bulk wurtzite-ZnO
for a temperature up to 600 K and found good agreement with experimental
observations [56]. It has also been applied to study surface growth mechanism for the
(0 0 0 1) surface and water molecule adsorption on stepped ZnO surfaces [57]. The
parameters of the ReaxFF used in this work was the same as that used in [57].
By construct, ReaxFF allows one to obtain the instantaneous information of the
partial charge developed on each atom via the charge derivation method [55] known
as electronegativity equalization method (EEM) [55]. Through ReaxFF the process of
43
equilibration of partial charge distribution among the atoms can be revealed in the MD
simulation. In this simulation, all the atoms in the simulation box were assigned a
partial charge of zero initially. While total charge of the system remains zero, partial
charges among the atoms will redistribute itself as thermal equilibration process
progresses. This work followed the time evolution of partial charge distribution in the
slab when the ZnO is going through the annealing process. The distributions of the
partial charge density in the slab at the end of a MD run were obtained by post-
processing the MD data.
CHAPTER 4: RESULTS AND DISCUSSION
4.1 Epitaxial Growth of Graphene
4.1.1 Binding energy of graphene
Since the Tersoff or TEA potential is empirical, its potential energy surface is
unique for given set of potential parameters. A first indication of the appropriateness
of the two potentials employed here for growing epitaxial graphene is the relative
values of the binding energy (per atom) for an infinite free graphene sheet which is
characterized by a lattice constant 0. Guided by experiments [13], this work chooses
to vary 0 in the range 2.3 ≤ 0 ≤ 3.0 Å. The minimized by conjugate gradient
[58] minimization is depicted in Fig 4.1 which in this work read to yield 0 = 2.53 and
2.56 Å for the Tersoff and TEA potentials, respectively. These values differ from the
recent tight-binding calculation of Reich et al. [59] and the DFT calculation of Gan
and Srolovitz [60] who obtained 0 = 2.468 and 2.471 Å respectively. While the
Tersoff potential appears slightly better in giving the a0 that agrees with ab initio
calculations than TEA potential, one notices, however, that the TEA potential is
relatively more attractive than the Tersoff potential for > 2.6 Å (Fig. 4.1). It is noticed
that in Fig. 4.1 the value of corresponds to Tersoff and TEA potentials crosses at
0 = 2.584 Å, and for 0 > 2.6 Å, for TEA potential is relatively lower in energy.
This characteristic feature of the TEA potential favors graphene formation since the
average C–C bond-length of the initial configuration of C-rich atoms, after relaxation,
is calculated to be 2.78 Å. While this is a first indication that the TEA potential could
be better for describing the high temperature graphene formation, further
corroborations of its suitability in simulation works at different temperatures are
necessary to confirm this feature. This is done in the following.
45
Figure 4.1: The binding energy (per atom) for an infinite free graphene layer
calculated with the Tersoff (solid circles or dashed line) and TEA (open circles or full
line) potentials at different values of lattice constant 0.
4.1.2 One-layer graphene
The 2D structure of graphene is now being examined. Figure 4.2 shows the
one-layer graphene that has been successfully grown on the Si-terminated 6H-SiC
(0001) substrate by employing the Tersoff and TEA potentials separately in MD
simulations. For the temperature range shown, one recognizes easily that the one-layer
graphene first comes into view at 1200 K for the TEA potential and 1400 K for the
Tersoff potential. The annealing temperature 1200 K at which the graphene emerges
is reasonable considering the somewhat mechanical way of the Si desorption in MD
simulations where this work create perhaps much more C atoms than the thermally
decomposed Si in epitaxial experiments. The experimental findings of Hannon and
Tromp [13] who observed graphene formation at step edges in prolonged annealing at
1298 K give further evidence, albeit indirectly. On the other hand, the threshold
graphene formation temperature from using Tersoff potential is relatively higher
46
having a comparatively smaller size graphene sheets with 9 hexagons contrasting to
75 from TEA potential.
Figure 4.2: One-layer graphene overlaid on Si-terminated 6H-SiC (0001) obtained by
the simulated annealing method for (a) Tersoff (second column) and (b) TEA (third
column) potentials. In the second and third columns at the bottom corner on the right,
the integer is the hexagon number. The average distance of separation between the
graphene buffer layer and surface is about 2.43 Å for TEA potential.
47
In this work, the preparation of C-rich substrate for the growth of graphene has
led to graphene formation occurring at an annealing temperature comparatively lower
according to the range of graphitization temperature reviewed by Haas et al. [61]. The
reason is because many processes are going on at high temperature annealing in
ultrahigh vacuum environment such as the rate of Si desorption Si atoms leaving the
surface near the edge region and all of these graphitization conditions were not
considered in this simulation. This MD simulations have in fact been inspired by the
experimental work of Poon et al. [62] who made quantitative studies of the growth
mechanisms of epitaxial graphene.
To quantify further the comparison between these two potentials, this work
depicts the variation of the binding energy (per atom) during the equilibrium
annealing period at 1200 K for Tersoff potential [Fig. 4.3(a)], 1100 K and 1200 K for
TEA potential [Fig. 4.3(b), Fig. 4.3(c)]. One sees readily that at 1200 K the C bilayer
obtained by the Tersoff potential remains in the crystalline state for the whole period
of 6 × 104 time steps. The calculated using TEA potential continues to stay in
crystalline state at 1100 K (Fig. 4.3(b)) which, however, undergoes abrupt decrease in
energy at 1200 K after roughly 4 × 104 time steps (Fig. 4.4). This characteristic trait
is a clear indication that the C bilayer has transformed at 1200 K to a single sheet of
graphene.
48
Figure 4.3: The variation of the binding energy (per atom) Eb plotted against the
equilibrium annealing time steps (Δ = 0.5 fs) at (a) 1200 K for Tersoff potential,
(b) 1100, and (c) 1200 K for TEA potential.
The temperature at which graphene structure emerges at 1200 K for the TEA
potential is encouraging for it implies the reasonableness of the choice of the C–C
interacting potential in simulation studies of graphene growth. In Figs. 4.4 and 4.5, in
this work present furthermore the average bond-length and binding energy (per atom)
for the two potentials. Notice the abrupt declines of and of C–C atoms for the
TEA potential at = 1200 K. In contrast, the same quantities for the Tersoff potential
fall off less sharply and drastically. These scenarios can be understood from Fig. 4.2
where it can be seen that the Tersoff potential leads to more broken graphene sheet
49
morphology with small islands. The changes in and of C–C atoms suggest
furthermore the existence of a threshold annealing temperature, above which one
would anticipate graphene formation. The value of for the TEA potential from Figs.
4.4 and 4.5 falls in the range 1100 K < ≤ 1200 K. On the other hand, for the
Tersoff potential lies in 1400 K < ≤ 1500 K, a range that is different than that
indicated in Fig. 4.2, where one sees a more broken graphene sheet structure that shows
up at 1400 K. The reason for this observation lies in the criterion, in which the C-C
bond length in the MD simulation is set at 1.6 Å , which is in accordance to the
theoretical value [3].
50
Figure 4.4: Comparison of the average bond-length (Å) versus annealing temperature
(in units Kelvin) between results calculated using TEA (open circle) and Tersoff
(solid circle) potentials.
-
Figure 4.5: Same as Fig. 4.4 except for the binding energy (per atom)
51
It is instructive to compare the average bond-length in in Fig.4.3 with the
nearest neighbor distance deduced from the two-dimensional pair correlation
function g(r) which is defined by [64]
() = 1
(4.1)
() describes a chosen -th atom that is placed at a specific position, say the origin,
and with respect to which this work seeks the whereabouts of th atoms which are
present within a shell whose center is at a distance from the central atom . The
is the number of such th atoms within the shell. The calculation of () runs
through the total number of all atoms inside the total area of the simulation with
defining the mean plane number density of carbon atoms. The quantity () is
again coded in LAMMPS software. The display in Fig. 4.5(a) and 4.5(b) shows the
() calculated using TEA and Tersoff potentials, respectively. The difference
between the positions of the first peak of () and averaging over the temperature
range 1200 ≤ ≤ 2000 K yields a value 0.01 Å for the TEA potential whereas for
the Tersoff potential it is 0.04 Å for temperature range 1500 K ≤ ≤ 2000 K. As
for the TEA potential, the small peak begins to form at around 2.5 Å at 1200 K while
it is 1300 K for the case of Tersoff potential. This signifies that the carbon-rich atoms
begin to be torn apart to form smaller graphene island which equilibrium bond length
is at around 1.5 Å.
52
Figure 4.6: (a) Pair correlation function () of carbon atoms obtained usin