computer simulation of the nucleation of...

COMPUTER SIMULATION OF THE

NUCLEATION OF DIAMOND FROM

LIQUID CARBON UNDER EXTREME

PRESSURES

ANASTASSIA SORKIN

COMPUTER SIMULATION OF THE

NUCLEATION OF DIAMOND FROM LIQUID

CARBON UNDER EXTREME PRESSURES

RESEARCH THESIS

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

ANASTASSIA SORKIN

SUBMITTED TO THE SENATE OF THE TECHNION — ISRAEL INSTITUTE OF TECHNOLOGY

TISHREI, 5767 HAIFA OCTOBER, 2006

THIS RESEARCH THESIS WAS SUPERVISED BY DR. JOAN ADLER AND

PROF. RAFAEL KALISH UNDER THE AUSPICES OF THE PHYSICS

DEPARTMENT

ACKNOWLEDGMENT

I wish to express my gratitude to Dr. Joan Adler and Prof. Rafi Kalish

for the excellent guidance and support during this research.

I am grateful to all Computational Physics Group and especially to my

husband Slava for their help during this research period.

I am grateful to Prof. Y. Lifshitz and Prof. A. Hoffman for useful

discussions.

I thank to Prof. A. Horsfield and Prof. M. Finnis for providing us with

the OXON package.

THE GENEROUS FINANCIAL HELP OF THE TECHNION IS GRATEFULLY

ACKNOWLEDGED

Contents

Abstract xix

List of symbols 2

1 Introduction 4

2 Diamond and other allotropes of carbon 6

2.1 The structure of diamond . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The structure of graphite . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Properties of diamond and graphite . . . . . . . . . . . . . . . . . . 10

2.4 The phase diagram of carbon . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Amorphous carbon and its characteristics . . . . . . . . . . . . . . . 13

2.6 Lonsdaleite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 Electronic structure of diamond, lonsdaleite and amorphous carbon . 19

2.8 Hydrogen in diamond . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.9 Quantum confinement . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Diamond synthesis 26

3.1 Natural diamonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 High pressure High Temperature diamond synthesis . . . . . . . . . 27

iii

CONTENTS iv

3.3 Shock-wave processing . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 CVD diamond growth . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Previous simulations of diamond nucleation 36

4.1 Computer simulation of high pressure high temperature conversion of

graphite to diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Computer simulations of the BEN process and “thermal spike” . . . 41

5 Goal of the research 46

6 Tight-binding model 49

6.1 Advantages and disadvantages of different models to describe the in-

teratomic interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2 LCAO approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3 The bond energy model . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.4 The rescaling functions . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5 Force calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7 Numerical techniques 60

7.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2 The Predictor-Corrector algorithm . . . . . . . . . . . . . . . . . . . 61

7.3 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . 62

7.4 Initial configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.5 General description of the calculations . . . . . . . . . . . . . . . . . 64

7.6 AViz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.7 Coordination number . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.8 Analysis of errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

CONTENTS v

8 Results: Nucleation of diamond under pressure 69

8.1 Amorphous carbon compressed in all three directions. . . . . . . . . 70

8.1.1 Computational details . . . . . . . . . . . . . . . . . . . . . . 70

8.1.2 The effects of different densities (pressures) . . . . . . . . . . 71

8.1.3 The effects of different cooling rates . . . . . . . . . . . . . . 77

8.2 Amorphous carbon compressed in one direction . . . . . . . . . . . . 79

8.2.1 Computational details . . . . . . . . . . . . . . . . . . . . . . 79

8.2.2 Samples prepared with fast cooling rate . . . . . . . . . . . . 80

8.2.3 Samples prepared with intermediate and slow cooling rates . 80

8.2.4 Interesting cases . . . . . . . . . . . . . . . . . . . . . . . . . 82

9 Results: Growth of diamond under pressure 89

9.1 Growth of diamond on cubic diamond seed within compressed amor-

phous carbon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

9.2 Growth of diamond on diamond layer within compressed amorphous

carbon layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

10 Results: Quantum confinement 98

10.1 Quantum confinement effects in cubic nanodiamond cluster. . . . . . 98

10.2 Quantum confinement in diamond layers located between two layers of

amorphous carbon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

11 Results: Nucleation in hydrogenated carbon 108

11.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 109

11.2 Structure of hydrogenated amorphous carbon network . . . . . . . . 109

11.3 Diamond nucleation in the hydrogenated carbon network . . . . . . . 112

CONTENTS vi

11.4 Varying density and cooling rates. . . . . . . . . . . . . . . . . . . . 116

12 Results: Liquid-liquid carbon phase transition 122

13 Summary and discussion 130

A OXON 137

B Computer program of data handling 145

References 148

Hebrew Abstract�

List of Figures

2.1 Schematic presentation of sp3 (left) and sp2 (right) hybridization. . . 8

2.2 Diamond lattice (top): view from the <210> direction (left), view from

the <100> direction (right). Graphite lattice (bottom): view from the

<112> direction (left), view from the <001> direction (right). . . . . 9

2.3 P, T phase diagram of carbon reproduced from [3] . . . . . . . . . . . 12

2.4 g(r) for an a − C sample (up), taken from [6] and g(θ) for a ta − C

sample (down), taken from [7]. A4 and A3 are the contribution of the

fourfold and the threefold atoms respectively. . . . . . . . . . . . . . . 15

2.5 Structures of (a) perfect cubic diamond and (b) perfect hexagonal diamond.

These viewpoints show the similarity between these structures. Differences

can be seen by careful observation of the hexagons: at these angles each

hexagon appears to have 2 short and 4 long bonds. In cubic diamond the

short bonds are on opposite side of the hexagons separated by 2 long bonds,

whereas in hexagonal diamond either 1 or 3 long bonds separate these 2

short bonds. We note that in fact the hexagons are not in a plane and all

bonds are of the same length. The apparent lengths of the bonds are due

to the viewing angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

vii

LIST OF FIGURES viii

2.6 Radial distribution, g(r), of perfect cubic diamond (top) compared

with that of perfect lonsdaleite (bottom). The radial distribution func-

tions were calculated for samples containing 64 atoms. . . . . . . . . 18

2.7 Comparison of the DOS of cubic and hexagonal diamonds, taken from

[13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.8 The electronic density of states of a-C, taken from [17] . . . . . . . . 21

3.1 Schematic representation of the belt apparatus. . . . . . . . . . . . . 30

3.2 Schematic of shock-wave processing of diamond. . . . . . . . . . . . . 32

3.3 Schematic diagram of the microwave plasma CVD apparatus, taken

from [51]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Hexagonal, orthorombic, and rhombohedral phases of graphite. The

different stacking of the hexagonal planes are viewed along the c axis

(above) and sideways (below), taken from [67] . . . . . . . . . . . . . 38

4.2 Graphite under pressure of 20 GPa. Interlayer distance collapses,

new sp3-bonds extend between the graphitic planes. White objects

are carbon atoms and yellow iso-surfaces represent charge density of

electrons. We see new bonding represented by bonding charge between

graphite layers. Taken from [61]. . . . . . . . . . . . . . . . . . . . . 40

4.3 The sp3 fraction plotted as a function of density calculated by differ-

ent methods: OTB-orthogonal tight-binding [6], EDTB- environment-

dependent tight-binding [74], NOTB-non-orthogonal tight-binding [69],

DFT-ab initio [72] and our previous orthogonal tight-binding simula-

tions [75] (indicated by “my simulations”). . . . . . . . . . . . . . . 43

LIST OF FIGURES ix

8.1 Microscopic structures of amorphous carbon with densities of 3.3 g/cc

with 52 % of sp3-bonded atoms (a), 3,7 g/cc with 81 % of sp3-bonded

atoms (b) and 4.1 g/cc with 95 % of sp3-bonded atoms (c). Red balls

represent fourfold coordinated atoms, blue balls represent threefold

coordinated atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.2 Microscopic structures of amorphous carbon with density of 3.9 g/cc

with 89 % of sp3-, 10 % of sp2- and 1% of sp-bonded atoms. Red balls

represent fourfold coordinated atoms, blue balls represent threefold

coordinated atoms and green balls represent twofold coordinated atoms. 74

8.3 The damaged diamond cluster from the sample drawn on Fig. 8.2

generated at a density of 3.9 g/cc from two different view points . . . 75

8.4 Angular distribution function of the diamond cluster drawn on Fig.8.3

(black thick line) compared with the angular distribution functions of

a pure diamond crystal (red line) and of amorphous carbon (blue line). 76

8.5 Radial distribution function of the diamond cluster drawn on Fig.8.3

(black thick line) compared with the radial distribution functions of a

pure diamond crystal (red line) and of amorphous carbon (blue line). 77

8.6 Density of states of the diamond cluster drawn on Fig.8.3 (black line)

compared to the density of states of a pure diamond (red line). The

insert shows a magnified part of the density of states near the band gap. 78

8.7 Sample generated at 3.8 g/cc at fast cooling rate (left) and damaged

diamond cluster found within this sample (right). . . . . . . . . . . . 81

LIST OF FIGURES x

8.8 Radial distribution function, g(r), of the damaged diamond cluster

generated at 3.8 g/cc with cooling rate of 1000 K/ps (black line) com-

pared to the radial distribution function of a pure diamond crystal (red

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.9 Angular distribution function, g(θ), of the damaged diamond cluster

generated at 3.8 g/cc with cooling rate of 1000 K/ps (black line) com-

pared to the angular distribution function of a pure diamond crystal

(red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.10 Density of states of the damaged diamond cluster generated at 3.8 g/cc

with cooling rate of 1000 K/ps. . . . . . . . . . . . . . . . . . . . . . 83

8.11 A graphitic configuration generated at 3.7 g/cc with intermediate cool-

ing rate: a) view from the direction parallel to the graphitic planes, b)

one graphitic plane, view from the perpendicular direction. Red balls

are sp3 coordinated atoms, blue balls are sp2 coordinated atoms and

green balls are sp-coordinated atoms. . . . . . . . . . . . . . . . . . . 84

8.12 Radial distribution function of the graphitic structure in the sample

generated at 3.8 g/cc subjected to uniaxial pressure (black line) com-

pared with the radial distribution function of perfect graphite (red line). 85

8.13 Angular distribution function of the graphitic structure in the sample

generated at 3.8 g/cc subjected to uniaxial pressure (black line) com-

pared with the angular distribution function of perfect graphite (red

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

LIST OF FIGURES xi

8.14 Flexed graphitic configuration generated at 3.7 g/cc with slow cooling

rate: (a)-before relaxation, (b)-after relaxation . Red balls are sp3

coordinated atoms, blue balls are sp2 coordinated atoms and green

balls are sp-coordinated atoms. . . . . . . . . . . . . . . . . . . . . . 87

8.15 Configuration generated at 3.8 g/cc with intermediate cooling rate.

Graphitic layers alternate with diamond like amorphous carbon layers.

Red balls are sp3 coordinated atoms, blue balls are sp2 coordinated

atoms and green balls are sp-coordinated atoms . . . . . . . . . . . . 88

9.1 Cut of initial diamond configuration; black balls represent the frozen

atoms, white balls represent the moving atoms. . . . . . . . . . . . . 90

9.2 The sample of amorphous carbon with embedded pure diamond cluster

(a) and the diamond cluster (b). . . . . . . . . . . . . . . . . . . . . . 91

9.3 The sample of amorphous carbon with embedded pure diamond cluster

after relaxation at 4.1 g/cc (a) and the diamond cluster (b). . . . . . 93

9.4 Density of states of diamond cluster grew up within an amorphous

carbon network (black line) compared to that of perfect diamond (red

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9.5 Samples of amorphous carbon located between two layers of diamond.

(a) initial sample, (b) sample compressed to 3.9 g/cc. Red balls are

sp3-coordinated atoms, blue balls are sp2-coordinated atoms and green

balls are sp-coordinated atoms. . . . . . . . . . . . . . . . . . . . . . 95

9.6 Damaged diamond cluster found in the compressed sample. . . . . . . 96

LIST OF FIGURES xii

9.7 Radial distribution function of the damaged diamond cluster formed

in amorphous carbon layer located between layers of diamond (black

line) compared with the radial distribution functions of a pure diamond

crystal (red line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9.8 Angular distribution function of the damaged diamond cluster formed

in amorphous carbon layer located between layers of diamond (black

line) compared with the angular distribution functions of a pure dia-

mond crystal (red line) . . . . . . . . . . . . . . . . . . . . . . . . . . 97

9.9 Density of states of the damaged diamond cluster. . . . . . . . . . . . 97

10.1 Amorphous carbon samples with diamond cluster inside, generated at

different temperatures of heating: a) at 12000 K, b) at 14000 K, c)

at 22000 K. Red balls are sp3-coordinated atoms, blue balls are sp2-

coordinated atoms, green balls are sp-coordinated atoms, frozen atoms

are marked by yellow color. . . . . . . . . . . . . . . . . . . . . . . . 100

10.2 Local densities of states of atoms in the sample generated at 13000 K:

within the frozen diamond cluster (atom 1, black line), in the boundary

of diamond cluster (atom 2, red line) and in the amorphous carbon

(atom 3, green line). The insert a) shows the location of the atoms 1,2

and 3, the insert b) shows the magnified part of the density of states

near the band gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

LIST OF FIGURES xiii

10.3 The samples of pure diamond located between two layers of amorphous

carbon, b) the initial sample contained 192 atoms (64 of them are frozen

diamond), a) the sample where one diamond layer was cut out, the new

sample contain 160 atoms, c) the sample where one diamond layer was

inserted in the center of the sample, the new sample contain 224 atoms,

d) the sample where three diamond layers were inserted in the center

of the sample, the new sample contain 288 atoms. Yellow atoms are

initially frozen diamond and inserting diamond layers, green atoms are

amorphous layers (the ”green” part is the same for each sample). . . 104

10.4 Local densities of states of the atoms from sample of 256 atoms, within

the frozen diamond layer (atom 1, black line), near the boundary be-

tween diamond and amorphous carbon layers (atom 2, red line), and

in the amorphous carbon (atom 3, green line). The insert a) shows the

location of the atoms, the insert b) shows the magnified part of the

density of states near the band gap. . . . . . . . . . . . . . . . . . . . 106

11.1 a-C:H structures with different content of hydrogen atoms. The red,

blue and green balls are the carbon atoms with four, three and two

C-C bonds (excluding C-H bonds) respectively. Hydrogen atoms are

represented by large light-blue balls. . . . . . . . . . . . . . . . . . . . 110

11.2 a-C:H structure with 25 hydrogen atoms generated with high (3.9 g/cc)

density and intermediate (500 K/ps) cooling rate (a) and disordered

pure sp3-cluster found within this sample (b). . . . . . . . . . . . . . 113

LIST OF FIGURES xiv

11.3 Radial distribution function of hydrogenated amorphous carbon con-

tained 25 hydrogen atoms generated at 3.9 g/cc with intermediate cool-

ing rate (red line) compared to that for C-C bonds only (black line). . 113

11.4 Radial distribution function of hydrogenated amorphous carbon gener-

ated at 3.9 g/cc contained 5 hydrogen atoms (red line) and contained

25 hydrogen atoms (black line). . . . . . . . . . . . . . . . . . . . . . 114

11.5 Local density of states of the atoms from a-C:H structure with 25

hydrogen atoms: C atom with 4 C-C bonds (black line), C atom with

3 C-C and 1 C-H bond (red line), C atom with 3 C-C bonds (blue line).114

11.6 Density of states of hydrogenated amorphous carbon samples at 3.9

g/cc, with 0 hydrogen atoms (black line), with 5 hydrogen atoms (red

line) and with 25 hydrogen atoms (blue line). The insert shows a

magnified part of the density of states near the band gap. . . . . . . . 115

11.7 a-C:H structure with 10 hydrogen atoms generated with low (3.5 g/cc)

density and fast (1000 K/ps) cooling rate. . . . . . . . . . . . . . . . 117

11.8 Diamond cluster contained 22 carbon atoms found in the sample from

Fig.11.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

11.9 Radial distribution function of diamond cluster inside hydrogenated

amorphous carbon, contained 10 hydrogen atoms generated at 3.5 g/cc

with fast cooling rate (black line) compared with the radial distribution

function of pure diamond (red line). . . . . . . . . . . . . . . . . . . . 118

11.10Angular distribution function of diamond cluster inside hydrogenated

amorphous carbon, contained 10 hydrogen atoms generated at 3.5 g/cc

with fast cooling rate (black line) compared with the angular distribu-

tion function of pure diamond (red line). . . . . . . . . . . . . . . . . 118

LIST OF FIGURES xv

11.11Density of states of diamond cluster inside hydrogenated amorphous

carbon, contained 10 hydrogen atoms generated at 3.5 g/cc with fast

cooling rate (black line) compared with the density of states of pure

diamond (red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

11.12Graphitic configuration contained 10 hydrogen atoms generated at 3.9

g/cc with slow cooling rate (a), one of the damaged graphitic planes (b).120

11.13Radial distribution function of hydrogenated graphite contained 10 hy-

drogen atoms generated at 3.5 g/cc with slow cooling rate (red line)

compared to that for C-C bonds only (black line). . . . . . . . . . . . 121

11.14Angular distribution function of hydrogenated graphite contained 10

hydrogen atoms generated at 3.5 g/cc with slow cooling rate. . . . . . 121

12.1 Snapshots of hydrogenated liquid carbon at 3.9 g/cc with 15 H atoms

at different temperatures in the process of cooling. Red balls represent

fourfold coordinated carbon atoms, blue balls represent threefold co-

ordinated carbon atoms and green balls represent twofold coordinated

carbon atoms. Large light-blue atoms are hydrogen atoms. . . . . . . 123

12.2 Percentage of sp2-coordinated atoms in liquid carbon at 3.9 g/cc and

6000 K as a function of time. . . . . . . . . . . . . . . . . . . . . . . 125

12.3 Snapshot of liquid carbon at 6000 K and 3.9 g/cc. Red balls represent

fourfold coordinated carbon atoms, blue balls represent threefold co-

ordinated carbon atoms and green balls represent twofold coordinated

carbon atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

12.4 Structure of one of the carbon planes in the sample of liquid graphite. 126

LIST OF FIGURES xvi

12.5 Radial distribution function of liquid graphite (black line) compared to

the radial distribution of the liquid sample before the phase transition

(red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

12.6 Angular distribution function of liquid graphite (black line) compared

to the angular distribution of the liquid sample before the phase tran-

sition (red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

12.7 Percentage of sp2-coordinated atoms in liquid graphite as a function of

time in the process of cooling. . . . . . . . . . . . . . . . . . . . . . . 127

12.8 Structure of liquid graphite before cooling (a) and after cooling (b). . 128

12.9 Structure of one of the carbon planes in the sample of cooled liquid

graphite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

List of Tables

2.1 Properties of diamond and graphite. . . . . . . . . . . . . . . . . . . . 10

3.1 Kinetic data of the direct graphite-to-diamond and graphite-to-lonsdaleite

transitions. Note: diamond type-C is cubic diamond, H is hexagonal

diamond (lonsdaleite). . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.1 Coefficient of the Predictor-Corrector algorithm for k = 4 for second-

order differential equation. . . . . . . . . . . . . . . . . . . . . . . . . 62

8.1 Fraction of four-, three-, and twofold coordinated atoms in the entire

amorphous carbon sample subjected to the cooling rate of 500 K/ps.

The number of cases (out of 5) where a diamond cluster containing

more than 20 atoms was generated are given in the last column of the

table. The numbers in brackets are the number of atoms in each such

cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.2 Band gap of the best unrelaxed diamond cluster at each density com-

pared with the band gap of perfect diamond at the corresponding density. 79

xvii

LIST OF TABLES xviii

8.3 Percentage of sp3 coordinated atoms and the structure of three samples

generated at different densities with applying of uniaxial pressure: the

first and the second at 3.8 g/cc, the third at 3.7 g/cc for different

cooling rates under uniaxial pressure. . . . . . . . . . . . . . . . . . . 85

9.1 Fraction of four-, three-, and twofold coordinated atoms in the relaxed

amorphous carbon sample and the number of atoms in the grown dia-

mond cluster N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

10.1 Width of the band gap of the central diamond clusters for the samples

generated at different heating temperature. . . . . . . . . . . . . . . . 101

10.2 Width of the band gap of the entire sample and the central diamond

layer for the samples with 19 % of sp3-coordinated atoms in an amor-

phous layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

10.3 Width of the band gap of the entire sample and the central diamond

layer for the samples with 26 % of sp3-coordinated atoms in an amor-

phous layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

11.1 Average number of differently bonded carbon atoms in the a-C:H sam-

ples generated at 3.9 g/cc and with cooling rate of 1000 K/ps. . . . . 111

11.2 Fraction of carbon atoms with 4 C-C bonds (sp3) in the samples with 5

and 25 hydrogen atoms generated with different densities and different

cooling rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Abstract

The stable solid form of carbon is graphite; diamond is thermodynamically unstable

at atmospheric pressure. High pressure and high temperature must be applied to

enable diamond crystal growth. Cubic diamond grows when hydrostatic pressure is

applied, whereas hexagonal diamond (which is another form of sp3-hybridized carbon)

has been reported to grow when uniaxial pressure is applied.

The aim of our simulations is to clarify conditions of high pressure high temper-

ature nucleation and growth of diamond and hexagonal diamond, in particular, the

influence of different pressures and cooling rates and the role of hydrogen in this

process. We also study interesting aspects of nanodiamond physics such as quantum

confinement, i.e. a shift in energy levels when the material sampled is of sufficiently

small size, as in our diamond samples.

In the present study we simulate the precipitation and growth of diamond clus-

ters inside an amorphous carbon or hydrogenated amorphous carbon network by rapid

quenching of the compressed liquid phase, followed by volume expansion. This pro-

cedure is similar to that occurring during the bias-enhanced nucleation process. Our

computational method is tight-binding molecular dynamics. This method incorpo-

rates electronic structure calculations in the molecular dynamics through an empirical

tight-binding Hamiltonian.

xix

ABSTRACT xx

The simulations of diamond nucleation are carried out under both hydrostatic (in

all three directions) and uniaxial pressure. At fast cooling rates (500-1000 K/ps) and

high densities (3.7-3.9 g/cc), large diamond crystallites (containing up to 120 atoms)

are formed. We find that the probability of precipitation of diamond crystallites

increases with density and with cooling rate. Uniaxial compression of the samples does

not lead to nucleation of the hexagonal form of diamond; all uniaxially compressed

ordered sp3 clusters were identified to be cubic diamond.

The samples of hydrogenated amorphous carbon were prepared in the same way.

The diamond clusters generated inside hydrogenated amorphous carbon network are

smaller and of lower quality than those formed without hydrogen atoms. Hydrogen

atoms are bonded with sp2- and sp-bonded atoms, and are expelled from the sp3

amorphous or diamond clusters.

At slower cooling rates (200-500 K/ps), some samples (both with and without

hydrogen) transformed to graphite with an interplanar distance smaller than that of

perfect graphite. The graphite formed under hydrostatic pressure had planes with

random orientation whereas the planes of graphite formed under uniaxial pressure

were oriented parallel to the direction of compression. We suggest that this graphitic

configuration is formed as a result of a structural phase transition occurring in liquid

carbon under very high pressure.

In order to study the growth of diamond, samples of compressed amorphous carbon

with embedded frozen diamond clusters were generated. We observed new carbon

atoms joining the diamond core as the diamond grew. This epitaxial growth of

diamond is more favorable at higher pressures. Quantum confinement effects were

not found in our diamond clusters or diamond layers surrounded by an amorphous

carbon phase.

LIST OF SYMBOLS 2

List of symbols

sp3 hybridization of one s orbital with three p orbitals

sp2 hybridization of one s orbital with two p orbitals

sp hybridization of one s orbital with one p orbital

a0 lattice constant

a-C Graphitelike Amorphous Carbon

ta-C Diamondlike Amorphous Carbon

a-C:H Hydrogenated Amorphous Carbon

g(r) radial distribution function

g(θ) angular distribution function

MD Molecular Dynamics

DFT Density Functional Theory

LDA Local Density Approximation

LCAO Linear Combinations of Atomic Orbitals

HPHT high presssure high temperature

TNT Trinitrotoluene

CVD Chemical Vapor Deposition

BEN Bias Enhanced Nucleation

TEM Transmission Electron Microscopy

OTB Orthogonal Tight Binding

NOTB Nonorthogonal Tight Binding

EDTB Environment Dependent Tight Binding

LIST OF SYMBOLS 3

ri position of the atom i

vi velocity of the atom i

rij the distance between the atom i and j

∆t time step

Ei site energy of the atom i

Vij bonding energy between the atoms i and j

Ψk,iα Bloch function

ψiα Lowding orbitals

α atomic orbital index

Hiα,jβ hopping integral

|n > single particle eigenfunction

ε(n) eigenvalue

Chapter 1

Introduction

Carbon is unique among the elements in its ability to form strong chemical bonds

with a variety of coordination numbers, including two (e.g. linear chains or carbyne

phase), three (e.g. graphite) and four (e.g. diamond). Diamond is transparent to

light over a wide range of wavelengths, optically isotropic crystalline carbon. It is the

hardest material known. The fact that carbon atoms are relatively small and very

tightly bonded results in high atomic-vibration frequencies. Diamond can therefore

conduct heat very well.

Given the extraordinary set of physical properties diamonds exhibit, large, cheap

diamonds could have a wide-ranging impact in many fields. For the last 50 years

people have been able to make synthetic diamonds that replicate the superlative

physical and chemical properties of natural diamonds. Today, in the year 2006, the

man-made diamond industry is an annual US $1 billion market, producing some 3

billion carats, or 600 metric tons, of synthetic diamond. This should be compared

with the 130 million carats (26 metric tons) mined annually for gem purposes.

4

CHAPTER 1. INTRODUCTION 5

A few diamond-based and diamond-coated products are already in use commercially-

x-ray windows in electron microscopes, strong abrasion-resistant industrial tools, and

diaphragms in stereo speakers, but these represent only a tiny fraction of the antic-

ipated applications. For an environments where high pressures and temperatures,

intense radiation, high salt content, and other adverse conditions can destroy ma-

terials (places like the ocean, space, engines, and nuclear reactors), fabrication of

diamond materials and devices may be justified already, even at the currently high

costs of production. For example, doped with impurities like boron and phosphorus,

diamond has potential uses as a semiconductor. Diamond transistors are functional

at temperatures many times higher than those of silicon and are resistant to chemical

and radiative damage.

Diamond is not thermodynamically stable at atmospheric pressure. High pressure

and high temperature have to be applied in order to permit diamond crystal growth.

The problem in trying to predict conditions of diamond synthesis precisely is a subject

of intensive experimental and theoretical research. Despite great interest in this

problem the phase diagram of carbon is still incompletely known, especially in the

regions of the high temperatures and high pressures, required for diamond synthesis.

This is a direct consequence of the difficulty of performing experiments at these

extreme conditions, and underlines the importance of computer simulations in this

area.

Chapter 2

Diamond and other allotropes of

carbon

2.1 The structure of diamond

Atomic carbon has an atomic number of 6 and a 1s22s22p2 electronic ground state

configuration. The carbon atom’s electronic configuration is believed to change from

its ground state in diamond as follows:

If a carbon atom enters into the structure of diamond its two 2s and 2p electrons

redistribute into four new equal-energy-level orbitals called 2(sp3) hybrid orbitals. It

requires a loss of energy but this effect is compensated by a very profitable covalent

bonding. The angular distribution of the wave functions for these four 2(sp3) orbitals

can be illustrated by drawing four lobes whose axes are at 109◦28′ to each other,

6

CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 7

the axes of these lobes thus extend toward the corners of an imaginary tetrahedron

centered around the carbon atom (Fig.2.1).

Quantum-mechanical calculations indicate that greater overlap between orbitals

results in a stronger covalent bond. The diamond structure represents a three-

dimensional network of strong covalent bonds (Fig.2.2), which explains why diamond

is so hard.

The diamond structure is cubic with a cube edge length of a0 = 3.567 A and

can be viewed as two interpenetrating FCC structures displaced by (1/4,1/4,1/4)a0.

The diamond crystal is highly symmetric with a cubic space group F41/d 3 2/m =

Fd3m = O7h. Since all the valence electrons contribute to the covalent bond, they are

not free to migrate through the crystal and thus, diamond is a poor conductor with

a band gap of 5.48 eV.

2.2 The structure of graphite

In going from its ground state to the graphite structure, a carbon atom’s electronic

configuration is believed to change as follows:

Three of the two 2s and two 2p electrons in carbon’s ground state redistribute into

three hybrid 2(sp2) orbitals which are a mathematical mixing of the s orbitals with

two of the three p orbitals. The angular probabilities for these 2(sp2) orbitals can

be represented by three coplanar lobes at 120◦ to each other (Fig.2.1). The fourth

electron of the original two 2s and two 2p electrons fills that p orbital which does not


participate in the 2(sp2) hybrid, the lobe for this p orbital being perpendicular to the

plane defined by the three 2(sp2) orbitals.

In the graphite structure, overlap occurs between the 2(sp2) orbitals of neighboring

atoms in the same plane. For such neighbors a side-to-side overlap also occurs between

their unhybridized p orbitals. A resultant side-to-side bonding known as π-bonding

results between these neighbors. The electrons participating in this π-bonding seem

able to move across these π-bonds from one atom to the next. This feature explains

graphite’s ability to conduct electricity along the sheets of carbon atom parallel to the

(0001) direction. The in-plane nearest-neighbor distance is 1.421 A. Normal to (0001),

adjacent sheets of carbon atoms are held together by weak Van der Waals bonds and

separated by a distance 3.40 A (Fig.2.2). This gives softness to the structure [1, 2].

The crystal structure is described by a hexagonal lattice with the D46h (P63/mmc)

space group.

Figure 2.1: Schematic presentation of sp3 (left) and sp2 (right) hybridization.��

sp2 �� sp3 �� !��"��!�� #��$%��&�'#� � ��


Figure 2.2: Diamond lattice (top): view from the <210> direction (left), view fromthe <100> direction (right). Graphite lattice (bottom): view from the <112>

direction (left), view from the <001> direction (right).� ��!�� !' � �� <100> ��

�� <210>

�� &�� !��&�� '��

<001> ��

<112>�� &�� & �� $�� $ �


Property Graphite Diamond

Lattice constant (RT) [A] 2.462 6.708 3.567

Bond length (RT) [A] 1.421 1.545Atomic density [cm−3] 1.14 ×1023 1.77 ×1023

Thermal conductivity [W/cm-K] 30 0.06 25Debye temperature [K] 2500 950 1860

Electron mobility [cm2/V-sec] 20×103 100 1800Hole mobility [cm2/V-sec] 15×103 90 1500

Melting point K 4200 4500Band gap [eV] -0.04 5.47

Table 2.1: Properties of diamond and graphite.� � �� #$ � � � & $

2.3 Properties of diamond and graphite

Due to the high anisotropy in the graphite structure as compared to that of diamond,

the electronic, mechanical and optical properties of these two phases of carbon are

very different. In Table 2.1 some properties of diamond and graphite crystals are

presented. In the column related to graphite, the in-plane properties appear on the

left and the transverse ones (between planes) on the right.

2.4 The phase diagram of carbon

The stable bonding configuration of carbon at ambient conditions is graphite, as

shown in Fig.2.3, with an energy difference between the graphite and the diamond

of ≈ 0.02 eV per atom. Due to the high energetic barrier between the two phases

of carbon, the transition from diamond to the stabler phase of graphite at normal

conditions is very slow. This transition can occur more rapidly when diamond is


exposed to ion bombardment, or high temperature, for example.

There are two main methods to produce synthetic diamond from graphite. The

original method is High Pressure High Temperature (HPHT) which is the most widely

used method because of its relatively low cost. It uses large presses that can weigh

a couple of hundred tons to produce a pressure of 5 GPa at 1,500 degrees Celsius to

reproduce the conditions that create natural diamond inside the Earth. Another tech-

nique of HPHT synthesis of diamond from carbonaceous materials makes use of the

short time compression and high temperatures achievable during detonation. Various

types of carbonaceous precursors can be used in this detonation process, including

graphite, carbon black, fullerenes, organic substances, but amongst these graphite is

the most widely used. The second method, using chemical vapor deposition or CVD,

was invented in the 1980s, and does not require high presure and high temperature

conditions to create diamond crystallites. In this method a carbon plasma is created

on top of a substrate onto which the carbon atoms deposit to form diamond. The

topic of diamond synthesis is dicussed in detail in the following chapter.

Bridging between the two main allotropes of carbon (diamond and graphite)

lie a whole variety of carbon materials which include, among others, amorphous

sp2 bonded carbon (such as thermally evaporated carbon), micropolycrystalline sp2

bonded graphite (such as glassy carbon), nanodiamond films, and amorphous sp3

bonded carbon (sometimes referred to as amorphous diamond), which is structurally

analogous to amorphous Si and is formed during low energy carbon ion deposition.

Another polymorphic form of carbon was discovered in 1985. It exists in discrete

molecular form, and consists of a hollow spherical cluster of carbon atoms. Each

molecule is composed of groups of sixty and more carbon atoms that are bonded

to one another forming both hexagonal and pentagonal geometrical configurations.


Figure 2.3: P, T phase diagram of carbon reproduced from [3][3]� � $%�� $%�� P,T


C60 is known as buckminsterfullerene, named in honor of R. Buckminster Fuller, who

invented the geodesic dome. In the solid state, the C60 units form a crystalline struc-

ture and pack together in a face-centered cubic array [4]. The discovery that carbon

could form stable, ordered structures other than graphite and diamond stimulated re-

searchers worldwide to search for other new forms of carbon. The Japanese scientist

Sumio Iijima discovered fullerene-related carbon nanotubes in 1991. The bonding in

carbon nanotubes is sp2, the tubes can therefore be considered as rolled-up graphitic

sheets [5]. Carbon nanotubes exhibit extraordinary strength and unique electrical

properties, and are efficient conductors of heat, that make them potentially useful in

a wide variety of applications in nanotechnology, electronics, optics, and other fields

of materials science.

2.5 Amorphous carbon and its characteristics

Generally we can characterize amorphous structures by a high degree of short range

order and absence of long range order. From the energetic point of view, atoms in an

amorphous structure are not bonded ideally, and they are subject to intensive stresses

and distortions. The energy of an amorphous solid is thus higher than that of a pure

crystal.

There are two specific amorphous form of carbon: diamond-like amorphous carbon

(ta − C) and graphite-like amorphous carbon (a − C). These two structures can be

distinguished clearly by their macroscopic and microscopic properties. The former has

higher density, is transparent and much harder than the latter. From the microscopic

point of view, the ratio of fourfold, diamondlike bonds to threefold, graphite-like

bonds (sp3/sp2) will determine the kind of structure we obtain. This ratio is strongly


affected by the way the amorphous solid is prepared and depends on temperature and

pressure.

In order to describe an amorphous structure the following characteristics can be

used: coordination number, radial distribution function, and angular distribution

function. The coordination number z is the number of nearest neighbor atoms. The

radial distribution function g(r) is a generalization of the coordination number. In-

stead of looking at the first nearest neighbors only, one now counts the number of

atoms that lie at the distance r from a specific atom, averaging over all the atoms

of the lattice. When normalized g(r) is precisely the probability of finding a neigh-

boring atom at distance r. It is clear that for a perfect lattice, g(r) will give delta

functions at characteristic distances of the lattice. The g(r) function, as a coordina-

tion number, can be very useful for a description of more complicated structures. For

example, short-range order is expressed by one or two broad peaks at the shortest

distances, following by a quite flat tail, which is characteristic to the g(r) of amor-

phous structure. For the a−C structure, for instance, the first peak is centered near

the graphite bond length (1.42 A) and is broad enough to include the diamond bond

length (1.54 A), so that many bonds, in the graphite-like structure, can be specified

as diamond-like bonds (see Fig.2.4). The liquid phase exhibits a very similar form,

except that the peaks are broader and shallower than in the amorphous case [8].

The bond angle distribution function g(θ) is defined for angles between nearest

neighbors atoms. For a diamond crystal, g(θ) is a delta function centered at θ =

109.47◦. For an amorphous crystal, g(θ) is centered at an angle close to the tetrahedral

angle for the ta − C structure and to θ = 120◦ for the a − C structure. Large angle

distortions occur in these structures, as is indicated by the significant width of the

bond angle distribution.


Figure 2.4: g(r) for an a− C sample (up), taken from [6] and g(θ) for a ta− Csample (down), taken from [7]. A4 and A3 are the contribution of the fourfold and

the threefold atoms respectively.� � A3

�� A4�[7]� � $%� � �� $�� $ � ta-C � � � �

g(θ) ��[6]� � $%� � �� a-C � � � �

g(r)� �� $ � � � �� &�� !� �� $ � � � �!$ �


2.6 Lonsdaleite

It is interesting that sp3 bonded carbon was found to exist not only as cubic crystals

but also as hexagonal crystals (Lonsdaleite). Lonsdaleite was first identified from

the Canyon Diablo meteorite at Barringer Crater (also known as Meteor Crater) in

Arizona in 1967. It is believed to form when meteoric graphite falls to Earth. The

great heat and stress of the impact transforms the graphite into diamond, but retains

graphite’s hexagonal crystal lattice. Later lonsdaleite was grown in the laboratory

[9].

For a long time, hexagonal diamond has been formed artificially only by static

and shock wave compression of well-crystallized graphites [9, 10]. Recently it was

shown that hexagonal diamond can be obtained also from cubic diamond [11].

Cubic and hexagonal diamond, both being composed of sp3 bonded carbon atoms,

have a rather similar structure, differing only in the stacking order of the sp3 bonded

carbon layers. The angles between C-C bonds is 109 degrees and the interatomic

distance is 1.54 A for both these forms of crystalline diamond. The difference between

c-D and h-D is only apparent when looking at the longer range structural properties

of C atoms in the crystals. Figs.2.5 and 2.6 show the structures of ideal c-D and h-D

crystals and their radial distribution functions. The similarity between these should

be noted.

In contrast to the high pressure high temperature (HPHT) cubic diamond growth

achieved under hydrostatic pressure, hexagonal diamond was observed to grow when

uniaxial pressure was applied to liquid carbon during its solidification [12]. Lons-

daleite is fundamentally less stable than diamond, therefore the hardness of lonsdaleite

to be slightly less than that of diamond.


Figure 2.5: Structures of (a) perfect cubic diamond and (b) perfect hexagonal diamond.

These viewpoints show the similarity between these structures. Differences can be seen by

careful observation of the hexagons: at these angles each hexagon appears to have 2 short

and 4 long bonds. In cubic diamond the short bonds are on opposite side of the hexagons

separated by 2 long bonds, whereas in hexagonal diamond either 1 or 3 long bonds

separate these 2 short bonds. We note that in fact the hexagons are not in a plane and all

bonds are of the same length. The apparent lengths of the bonds are due to the viewing

angle.�� !"�$#��$��%'&( ��)*�+� ��,�-� .0/1��2�� ,�$�3�4��15�� ,��6�7�8��!'&*1��9�� :;�,&*�<��1=��,�$��>� �� !'&*1@?BAC�8�� )��!'�+��:D�,&FEG�H�� .0/1��<�,)*��IJ��K� �L��K��.0/1��8�M�N A-��*!'%'��2�� !'&*1��9�� :;�,&*�2��,�K� .0/1��9�� ,�$��(OD�QP ��)�� !'�� :;�,&R?SA-��*!;%T��8�U��$�+��&*��OD!'��Q��,:D��2��!D�1��+�,)��*VW�X!'� �H��$�� K��Y��K��.0/1��6�L��:D�,&*�+� ��)�� !'�X��:D�,&(Z[A-��

� �$��\�


Figure 2.6: Radial distribution, g(r), of perfect cubic diamond (top) compared withthat of perfect lonsdaleite (bottom). The radial distribution functions were

calculated for samples containing 64 atoms.64�� $�� $ � �� ' � � � � � � � �� g(r) ��

�� g(r)� ��


Figure 2.7: Comparison of the DOS of cubic and hexagonal diamonds, taken from[13]

[13]� � $ � � �� ' � � � � � � � �� DOS

� �� DOS� ��

2.7 Electronic structure of diamond, lonsdaleite

and amorphous carbon

The density of states of cubic and hexagonal diamonds calculated by the local density

approximation (LDA) method is shown in Fig.2.7. It is seen that the spectrum of

the density of states of cubic diamond consists of the valence and conduction bands

separated by an energy of 5.5 eV. The valence band is fully occupied, leaving the

conduction band empty. Thus diamond is typical of the group IV semiconductors.

The band gap is indirect because the wave vector at which the valence band is a

maximum does not coincide with the wave vector where the conduction band is a

minimum.


Amorphous carbon can form a large number of different bonding types. Hence,

the electronic structure of amorphous carbon is governed by the relative importance

of three and fourfold sites. A purely four-fold coordinated model of amorphous carbon

[14] predicts only sp3 bonding to occur which gives a large gap in the electronic density

of states. The electronic structure predicted by this model is similar to a broadened

diamond-carbon density of states. It is now clear that this is not the correct model

for diamond-like amorphous carbon and later tight-binding calculations [6, 7] (see

Chapter 6) have found states which close the gap and have been associated to 3-

fold coordinated atoms exhibiting sp2(π) bonding. The total number of states in

the gap, which appear due to sp3 bonds, increase when sp2 orbitals are introduced

into the simulation. However, some models [15] produced from the Tersoff potential

have a significant density of states near the Fermi level. This is in contradiction

to experimental and ab initio (see below) calculations [16] which show only a small

density of states at the Fermi level.

The electronic structure of amorphous carbon simulations performed by an ab

initio method [17] is shown in Fig.2.8. The part of the density of states corresponding

to sp3(σ) bonding is very similar to a broadened diamond-like electronic structure.

Most of the states around the Fermi level are found to be π-like in nature leaving no

band gap. Therefore the optical properties of amorphous carbon will be dominated

by the sp2(π) bonded sites. This structure is similar to an ab initio calculation on

diamond-like amorphous carbon performed by Drabold [18], where 3-fold sites are

found to group in pairs. Due to the lack of clustering of sp2 sites, it follows that it is

the intermediate range correlations of the sp2 sites which will have profound effects

on the optical spectrum.


Figure 2.8: The electronic density of states of a-C, taken from [17][17]� � $ � � a− C

�� !�� $ � �!� ��


2.8 Hydrogen in diamond

Hydrogen in semiconductors has attracted much attention the last two decades, and

appears to induce fundamental changes in the electronic properties of the host ma-

terial. It was found, for example, that dangling bonds existing on grain boundaries,

and that point defects can interact with hydrogen and be neutralized. The presence

of hydrogen could therefore, in that manner, reduce the density of state in amorphous

diamond [19].

In CVD grown diamond, the abundance of hydrogen is due to the growth con-

ditions themselves, since its presence in the ambient plasma is required to promote

diamond bonding over graphite bonding. Landstrass et al. have experimentally shown

[20] that the behavior of diamond subjected to the action of hydrogen from a hydro-

gen plasma is very similar to that of diamond film. In that case, hydrogen passivates

electrically active defects, resulting in a substantial reduction in the resistivity.

Calculations with ab initio method [21] show that the most important interstitial

sites of hydrogen in diamond structure materials are: the T site which lies equidistant

from four carbon sites and possesses Td symmetry, the H-site which lies midway

between two T-sites and posesses D3d symmetry and the bond-centered (BC) site is

the mid-point between two carbon atom site (D3d symmetry). The Bond Center (BC)

site was found to be lower in energy than the tetrahedral (Td) site and the H-site

[21, 22].

The location of H in semiconductors (C, Si) is best determined experimentally by

electron paramagnetic resonance (EPR) [23] or by the measurements of the location of

muonium (the light pseudoisotope of H) in the crystal [24]. The EPR signal in CVD

diamond [23] reveals the presence of dangling bonds associated with hydrogen atoms.


Muon spin resonance (µSR) measurements [24] indicate that two paramagnetic forms

of muonium exist: the “normal” muonium (Mu), with an isotropic hyperfine interac-

tion, and the “anomalous” muonium (Mu?), with an anisotropic hyperfine interaction.

For Si, it is well accepted [25] that Mu resides on a Td site, and Mu? on a BC site.

The experimental situation for muons in diamond is less clear, with the location of

Mu? not unambiguously established [26].

A new hydrogen site was found by D. Saada et al. [27] and O. Hershkovitz [28]

using tight-binding techniquies. This structure was labelled equilateral triangle (ET)

due to the two sets of three equivalent sites around the C-C bond that the H atom

could adopt. The length of C-H bond is 1.08 A, that is closer than BC site. The ET-

site was predicted to be 1.4 eV lower in energy than the BC-site. However, ab-initio

calculations of Goss [21] and our previous ab-initio calculations [29] carried out at at

0 K shows that this site is unstable. Hydrogen atoms initially placed at the ET-site

migrated to the BC-site

The electronic states induced by hydrogen depend on which interstitial site is

occupied. Simple molecular bonding arguments can explain the position of the energy

levels obtained. In the case of hydrogen in a bond center site in diamond, the bonding

states of the carbon atoms in diamond can couple to the 1s state of hydrogen to form

one occupied state in the valence band, and one corresponding unoccupied state in

the conduction band. The remaining antibonding states of the carbon atoms create

defect levels in the upper part of the energy gap [25].


2.9 Quantum confinement

Quantum confinement is the change of electronic and optical properties when the

material sampled is of sufficiently small size - typically 10 nanometers or less. The

bandgap increases as the size of the nanostructure decreases. Specifically, the phe-

nomenon results from electrons and holes being squeezed into a dimension that ap-

proaches a critical quantum measurement, called the exciton Bohr radius.

The first experimental evidence of quantum confinement effects in clusters came

from crystalline CuCl clusters grown in silicate glasses [30]. Spectroscopic studies on

these clusters clearly indicated an up to 0.1 eV blueshift of the absorption spectrum

relative to the bulk. In the case of CdS clusters, the absorption threshold is observed

to blueshift by up to 1 eV or more as the cluster size is decreased [31]. When the size

of the cluster is smaller, its band gap is larger, consequently the first absorption peak

is shifted closer to the blue.

A recent study [32] of the X-ray absorption spectra in nanodiamond thin films

with grain diameter from 3.5 nm to 5 µm showed that the C 1s core exciton state

and conduction band edge are shifted to higher energies with decrease of the grain

size especially when the crystallite radius is smaller than ∼1.8 nm. The conduction

band of nanodiamonds with radius R > 1.8 nm, when the crystalline contains more

than 4300 C atoms, remain more or less bulklike.

Recently Raty et al [33] presented ab initio calculations based on density-functional

theory (DFT) in order to investigate quantum confinement effects in hydrogenated

nanodiamonds. They detected a rapid decrease of the DFT energy gap from a value

of 8.9 eV in methane to 4.3 eV in C87H76. The last value is very close to that of

the bulk diamond (4.23 eV), obtained using the same method. This indicates that in


contrast to Si and Ge where quantum confinement effects persists up to 6-7 nm, in

diamond there is no detectable quantum confinement for sizes larger than 1-1.2 nm.

In addition the authors predicted a slight influence of surface structure reconstructed

by hydrogen atoms on the optical properties.

Chapter 3

Diamond synthesis

3.1 Natural diamonds

As seen in the phase diagram of carbon (see Fig.2.3), at ordinary pressures graphite

is the stable form at all temperatures while diamond is theoretically stable only at

high pressures. These pressures are found deep within or under the Earth’s crust at

depth of about 150 km, where pressure is roughly 5 GPa and the and the temperature

is around 1200 ◦C.

Diamonds are carried to the surface by volcanic eruptions. In order to retain its

structure and avoid diamond being transformed into graphite by the high temper-

ature, diamond must be cooled while still under pressure. This would occurs if it

moved rapidly upward through the Earth’s crust. A rapid ascent is also necessary to

minimize any possible reaction with the surrounding, corrosive, molten rocks.

26

CHAPTER 3. DIAMOND SYNTHESIS 27

3.2 High pressure High Temperature diamond syn-

thesis

Since 1814, when the English chemist H. Davy proved conclusively that diamond is

a crystalline form of carbon, many attempts were made to synthesize diamond by

trying to duplicate nature. Synthetic diamond was first produced on February 16,

1953 in Sweden by the ASEA, Sweden’s major electrical manufacturing company using

a bulky apparatus designed by Baltzar von Platen and the young engineer Anders

Kampe (1928-1984) [34]. Pressure was maintained within the device at an estimated

83,000 atmospheres (8.4 GPa) for an hour. A few small crystals were produced. The

discovery was kept secret. A year later on December 16, 1954, Tracy Hall et al of

General Electric managed to repeat that feat and published their results in Nature

[35] and that result is today the officially recognized first synthesis of diamond.

The equation of graphite-to-diamond phase boundary was determined by Berman

[36] who succeeded in summarizing extensive thermodynamic data which include the

heat of formation of graphite-diamond, the heat capacity of graphite as a function of

temperature, and the atomic volume and coefficient of thermal expansion of diamond.

The data are taken from a temperature range most applicable for diamond synthesis

(600-1700 ◦C). The equation has the following form:

P (kb) = 12.0 + 0.0301T (◦C) (3.1)

This equation is different from a more general phase boundary proposed by Kennedy

and Kennedy [37] with the form:

P (kb) = 19.4 + 0.0250T (◦C) (3.2)

It should be noted that Eq.3.1 underestimates the transition pressures relative to


Eq.3.2 at low temperature (e.g. 13 Kb instead of 20 Kb at room temperature) as the

slope (dP/dT ) of the phase boundary tends to decrease with decreasing temperature.

Although thermodynamically feasible at relatively low pressure and temperature,

the direct transformation graphite-diamond faces a considerable kinetic barrier since

the rate of transformation apparently decreases with increasing pressure. This kinetic

consideration supersedes the favorable thermodynamic conditions and it was found

experimentally that very high pressure and temperature (>130 kb and >3300 K) were

necessary in order for the direct graphite-diamond transformation to proceed at any

observable rate [38]. These conditions are very difficult and costly to achieve. For-

tunately, it is possible to bypass this kinetic barrier by the solvent-catalyst reaction.

It establishes a reaction path with lower activation energy than that of the direct

transformation. This permits a faster transformation under more benign conditions.

As a result, solvent-catalyst synthesis is readily accomplished and is now a viable

and successful industrial process. The solvent-catalyst are the transition metals such

as iron, cobalt, chromium, nickel, platinum and palladium. These metal-solvents

dissolve carbon extensively, break the bonds between groups of carbon atoms and

between individual atoms, and transport the carbon to the growing diamond surface

[39]. Bundy and co-workers showed that graphite can be artificially converted to

diamond under pressure of 7 GPa (∼ 70 000 atm) with temperatures of about 6000

K in the presence of catalysts [35]. It was later shown that diamond can be synthe-

sized from graphite by direct conversion under high pressure and high temperature

(HPHT) even without using catalysts (above 10 GPa and 2000 ◦C) [3, 40, 41, 42].

There are two main press designs used to supply the pressure and temperature

necessary to produce synthetic diamond. These basic designs are the belt press and

the cubic press. The original GE invention by H. Tracy Hall, uses the belt press


(see Fig.3.1), where upper and lower tungsten-carbide anvils supply the pressures

of between 50000 and 70000 atm inside a cylindrical capsule. A carbon source is

placed at the top of the capsule, while a metal slug is placed in the center. Tiny

grit-sized synthetic diamond seed are then placed at the bottom of the capsule. The

pressed capsule is then heated to temperatures of 1200-1500◦C by passing an electric

current through heaters inside it. Usually the top of the capsule is held at a higher

temperature than the bottom to create a temperature gradient of a few tens of degrees

inside it. Once the ideal pressure-temperature conditions are reached, carbon from

the source in the upper part of the capsule dissolves in the metal and carbon is driven

to the bottom of the capsule. These conditions make the carbon precipitate out

of the solution onto the seed crystal, in the form of diamond. The second type of

press design is the cubic press. A cubic press has six anvils which provide pressure

simultaneously onto all faces of a cube-shaped volume [43].

In 1967 Bundy and Casper [9] carried out experiments on the direct conversion

diamond to graphite under compression. They observed a rapid increase of the elec-

trical resistivity of well-crystallized graphite when it was compressed to above 15

GPa. At room temperature, this change was reversible upon the release of pres-

sure. However, when the sample was heated above 1000◦C under pressure Bundy

and Casper observed an irreversible increase of electrical resistivity. In the recov-

ered sample observed some additional diffraction lines, which could be indexed as

a “hexagonal diamond” (or lonsdaleite) structure. Since then only a few successful

experiments of the direct static compression high-pressure conversion of graphite to

hexagonal diamond have been made [12].

Data from the literature on the kinetics of the direct graphite-to-diamond and


Figure 3.1: Schematic representation of the belt apparatus.belt ��

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graphite-to-lonsdaleite transitions are summarized in Table 3.1.

3.3 Shock-wave processing

Another technique of HPHT synthesis of diamond from carbonaceous materials makes

use of the short time compression and high temperatures achievable during detona-

tion. In 1961, graphite was converted directly to diamond by this technique using the

shock compression and high temperatures obtained during explosion that create, for

a few microseconds, a pressure of about 35 GPa [47].

A schematic of the process is shown in Fig.3.2. A mixture of graphite and nodular

iron is placed inside a 25 cm diameter cavity in a lead blocks. A flat metal plate,

uniformly coated with TNT on the back side, is placed in front of the cavity. The

TNT is detonated and the plate impacts the cavity at a peak velocity of 5 km/s.


Ref. C source p(kbar) T(◦C) diamond type F E (eV/atom)

[44] sp-1 graphite 150 1300 C 0.3 4.63

[44] sp-1 graphite 150 2100 C 0.3 5.43

[44] sp-1 graphite 150 1700 C 0.7 5.43

[44] sp-1 graphite 150 2800 C 0.7 6.91

[45] sp-1 graphite 150 3300 C 0.7 5.34

[45] spectroscopic C 150 2400 C 0.1 7.42

[9] well-crystallized graphite 130 1000 H 0.3 2.41

[9] well-crystallized graphite 130 1000 H 0.7 2.28

[40] amorphous C 180 1900 C 0.1 6.67

[41] glassy C, graphite 140 3000 C 0.1 9.54

[12] kish graphite 300 25 H 0.1 0.93

[46] fullerute C60 200 25 C 0.1 0.91

Table 3.1: Kinetic data of the direct graphite-to-diamond andgraphite-to-lonsdaleite transitions. Note: diamond type-C is cubic diamond, H is

hexagonal diamond (lonsdaleite).H� �� ' � C �

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A peak pressure estimated at 300 kbar and a temperature of approximately 1000 K

are maintained for a few microseconds. The formation of diamond is assisted by the

presence of iron solvent-catalysts.

The diamonds produced by this method were found to be largely influenced by the

structure and size of precursors as well as by the physical process employed. It was

shown that an increase in pressure during the explosion leads to a higher diamond

content in the detonation soot [48, 49]. After explosion the system passes through the

pressure-temperature region in the carbon phase diagram where the graphitic phase

is preferable. To enhance the growth of diamond relative to that of graphite, the time

spent in this region must be minimized. Hence, the role that the cooling rate after

detonation plays in the final formation of the solid carbon structure is crucial. Titov

et al [50] have shown that an increase of the cooling rate led to preferred precipitation


Figure 3.2: Schematic of shock-wave processing of diamond.shock-wave

$ � �� ' �� &�'

of diamond due to the reduction of the time the system spends in the undesirable

region of the phase diagram.

Hexagonal diamond can also be formed by the shock compression method [10].

The conditions of cubic diamond and hexagonal diamond shoch-wave synthesis are

quite similar. This is very surprizingly, becase hexagonal diamond is metastable with

respect to cubic diamond [9, 52]. The relevance of lonsdaleite in the graphite-to-

diamond conversion appears to be twofold. First the mutual orientation of graphite

and cubic diamond before and after the conversion is consistent with the presence of

hexagonal diamond as an intermediate phase [52]. Secondly, in recent x-ray difraction

experiments hexagonal diamond was observed at high pressure, but after heating and

quenching to room conditions, only cubic diamond could be retreived [12].


3.4 CVD diamond growth

Chemical vapor deposition of diamond growth typically occurs under low pressure (1

to 27 Pa) and involves feeding varying amounts of gases into a chamber, energizing

them and providing conditions for diamond growth on the substrate. The gases always

include a carbon source, and typically include hydrogen as well, though the amounts

used vary greatly depending on the type of diamond being grown. Energy sources

include hot filament, microwave power, and arc discharges, among others. Schematic

diagram of the microvave plasma CVD apparatus is drawn on Fig.3.3. The energy

source is intended to generate a plasma in which the gases are broken down and more

complex chemistries occur. The actual chemical process for diamond growth is still

under study and is complicated by the very wide variety of diamond growth processes

used.

It is difficult for diamond to nucleate on mirror-polished Si and silicon carbide

because of their surface free energy and lattice constant are very different to those

of diamond. In 1991, Yugo et al [51] obtained diamond nucleation with a density of

about 109-1010 cm−2 on a mirror-polished Si by “bias-enhanced nucleation”. In this

method ions from the growth plasma, which contains H+, C+ and other positively

charged C-H radicals, are accelerated by several hundred volts towards a negatively

biased substrate.

The nucleation mechanism has been widely studied and different models have been

proposed. Yugo et al [53] and Gerber et al [54] suggested a shallow ion implantation

model in which the sp3 bonded clusters, formed by low-energy ion implantation,

function as the nucleation precursors. The negative bias caused the positively charged

ions in the growth chamber to accelerate towards and bombard the substrate, these


Figure 3.3: Schematic diagram of the microwave plasma CVD apparatus, takenfrom [51].

[51]� � $ � �CVD

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leads to sub-plantation of carbon and hydrogen ions into sub-surface regions.

Lifshitz et al [55] suggested that once a high concentration of sp3 bonded amor-

phous C clusters is reached in this way, some clusters can crystallize to form perfect

diamond crystallites that can be several nano-meters in size. It was shown by high

resolution TEM (Transmission Electron Microscopy) that prior to the formation of

the diamond, growth of graphitic planes, perpendicular to the substrate surface takes

place [56]. These can be densified by subsequent ion bombardment and can eventu-

ally form diamond crystallites. The physics of this process is related to the stopping

process of implanted ions in matter. It is well-known that during the slowing down of

ions, many atoms of the stopping medium are displaced, resulting in the formation of

a ”thermal spike”. This is a few nano-meters in size and lasts for about a pico-second,


i.e. during this time a small region of the material experiences very high tempera-

tures and local high pressures. Hence the biased enhanced nucleation process thermal

spikes within a dense C cluster can drive the system into the HPHT region in the C

phase diagram in which diamond formation is favored, resulting in the formation of

tiny diamond clusters.

The model of Y. Lifshitz et al suggests that hydrogen can play an important

role in the CVD nano-diamond film formation. Hydrogen bonding in a dense amor-

phous carbon matrix which thermodynamically stabilizes diamond nuclei. Hydro-

gen bombardment of the growing film stabilizes the diamond phase by preferential

displacement of carbon atoms bonded in non-diamond configuration and growth of

diamond crystallites of 2-5 nm in size. Sh. Michaelson and A. Hoffman [57] provided

experimental evidence for this mechanistic model.

Chapter 4

Previous simulations of diamond

nucleation

4.1 Computer simulation of high pressure high tem-

perature conversion of graphite to diamond

There are have been only a few computer simulations of the process of nucleation

of diamond from graphite and others precursors. Several first principle (ab initio)

[58] computer simulations have been carried out to model the conversion of graphite

to diamond under different conditions [59, 60, 61, 62]. This model based on density

functional theory [63] is very accurate. In this model, the local density approximation

(LDA) is used for the exchange-correlation interaction. In most calculations, plane

waves are used as a basis for the electronic wave functions, and pseudopotentials (for

example [64]) describe the interaction between the valence electrons and the ionic core.

The ionic cores are considered in their ground state at any moment for a particular

instantaneous ionic configuration, and the electronic and ionic degrees of freedom

36

CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 37

can therefore be separated. The atoms are considered to be classical particles, this

means that the Newton’s equations can describe their motion. However, first-principle

studies are at present still limited by their heavy demand on computational effort.

When graphite transforms into diamond the interplanar distances shorten under

pressure. Each carbon atom with three neighbors must bridge across the matching

atom in the adjacent layer to form a new bond. Graphite layers can be stacked

up in two different sequences: AB...(2H) or ABC...(3R). The former is known as

hexagonal graphite: and the latter, rhombohedral graphite (see Fig.4.1). Only half

of the amount of atoms in a layer of hexagonal graphite is matched with that in the

adjacent layer. Thus, for 2H graphite to transform directly into diamond, it must first

resequence to form 1H with AAA... or 3R with ABC... [65]. Such resequening can

take place by sliding specific layers in one bondlength without long range diffusion.

Such sliding is thermally activated, but it can be aided by applying shearing stress,

or by contact with a catalyst metal [66].

The mechanism of the direct transformation of rhombohedral graphite (3R) to

diamond was studied by Fahy et al by ab initio total energy calculations [62]. When

distance between the basal planes of rhombohedral graphite collapses from 3.35 to

2.07 A, a new sp2π bonds extends from one plane to another, the planes begin to

pucker, the angle between this sp2π bonds increases rapidly from original 90◦ to ap-

proach 109.47◦, characteristics for sp3 bond. Calculations of Fahy et al predicted

that the activation energy for such transition mechanism is 0.33 eV. The authors also

examined the behavior of rhombohedral graphite under hydrostatic pressure. They

found that rhombohedral graphite transforms to diamond at 80 GPa. Later the au-

thors simulated the transformation of 1H (AAA...) graphite to hexagonal diamond


Figure 4.1: Hexagonal, orthorombic, and rhombohedral phases of graphite. Thedifferent stacking of the hexagonal planes are viewed along the c axis (above) and

sideways (below), taken from [67]$ �� !��' � �� $%��!�� $ �� $ �� ' � � � � ��!�� #$ � � � �

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[67] with the same transformation path. They showed that the energy barrier along

the transformation is only slightly higher than the energy barrier of rhombohegral

graphite-to-diamond transformation. However, the AAA... stacking has higher en-

ergy, than the rhombohedral stacking and hexagonal graphite (2H), therefore the

transformation of hexagonal graphite to hexagonal diamond can not occur by the

way described in this study.

Scandolo et al [60] found that under hydrostatic pressure (i.e. identical pressure

applied in all directions) of 30 GPa the graphite-to-diamond transformation proceeds

through sliding of graphitic planes into an unusual orthorombic stacking (see Fig.4.1),

from which an abrupt collapse and buckling of the planes leads to both cubic and

hexagonal forms of diamond. The authors noted that the simulations at different

compression rates suggests that the formation of cubic diamond is favored at the

highest pressures (300 GPa/s).


In order to clarify the difference of the transition probability between graphite-to-

cubic diamond and graphite-to-hexagonal diamond transformations, Tateyama et al

[61] investigated their transition states under pressure of 20 GPa (see Fig.4.2). The

authors showed that the activation barrier to convert graphite into cubic diamond is

lower than that to form hexagonal diamond. Tateyama et al suggested that, whenever

collective sliding of the graphitic planes is allowed, the transformation to cubic dia-

mond is favored, and the hexagonal diamond can be obtained only then such sliding

is prohibited. This helps to explain the experimental results of Bundy and Kasper

[9] and Yagi [12], in their experiments when a well-crystallized graphite sample was

compressed, the collective sliding of the graphitic planes was inhibited due to the

large sample size, so that the graphite-to-hexagonal diamond transformation would

be expected. Anisotropic compression along the c axis of graphite would also sup-

press the layers sliding, because its leads to to a stronger interlayer interaction with

decreasing interlayer distance at the initial stage of transformation. On the other

hand, if the collective slide is allowed due to the small size of the crystal, which is

probably the case of experiment with a polycrystalline graphite by Endo et al [68],

the graphite-to-cubic diamond transformation with smaller activation energy would

take place.


Figure 4.2: Graphite under pressure of 20 GPa. Interlayer distance collapses, newsp3-bonds extend between the graphitic planes. White objects are carbon atoms and

yellow iso-surfaces represent charge density of electrons. We see new bondingrepresented by bonding charge between graphite layers. Taken from [61].� �� sp3 �� &

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4.2 Computer simulations of the BEN process and

“thermal spike”

Several major obstacles have previously hampered the elucidation of the diamond

nucleation mechanism. Firstly, small diamond clusters of ∼30 atoms cannot be ob-

served by experimental techniques, leaving simulation as the only means to observe

them. Secondly, the very low probability of the formation of a perfect diamond cluster

requires a large number (>104) of cell calculations for it to be observed, calculations

that cannot be performed by currently available computers. Yao et al [69] succeded

to overcome these problem by suggesting that the formation of a perfect diamond

cluster among many other faulty sp3 clusters is statistically possible.

Yao et al [69] have simulated the bias enhanced nucleation (BEN) of diamond,

which is the initial stage of CVD diamond growth, by non-orthogonal density-functional-

based tight-binding molecular dynamics computations. The method used a minimal

basis two-center approach to density-functional theory (DFT) for deriving total en-

ergy and interatomic forces. These calculation showed that diamond nucleation in the

absence of hydrogen can occur by precipitation of diamond clusters in a dense amor-

phous carbon matrix generated by subplantation. In the process of ion bombardment

the kinetic energy of a moving ion is partially transferred to host atoms by elastic

collisions. Hence a cascade evolves resulting in the formation of a highly disrupted,

very hot, region inside the solid named a ”thermal spike”. It can be viewed as the

short term local melting of the implantation affected region. In covalently bonded

solids this melting is followed by a rapid quenching of the liquid phase. Yao et al

found that a defective diamond cluster, containing 41 carbon atoms formed within

the region where high concentrations of carbon atoms are present in an amorphous


structure, generated by quenching the random liquid carbon phase at a density of 3.5

g/cc. Once the diamond clusters are formed, they can grow by thermal annealing

consuming carbon atoms from the amorphous matrix. The simulations of the first

stage of diamond nucleation process in a dense amorphous hydrogenated carbon (Lif-

shitz et al [55]) also showed that a diamond-like sp3 cluster could spontaneously form

in hydrogenated amorphous carbon network (25 % of hydrogen atoms) generated at

a density of 3 g/cc. The hydrogen is concentrated in the more porous parts of the

cell and decorates the surface of the sp3 clusters, mainly forming sp3 C-H bonds.

Kohary et al [70] also simulated the ion bombardment process during BEN by the

same method. A heated a-C:H layer was bombarded with methyl and acetylene ions

of different energies. The number of broken C-H bonds increased continuously with

the bombarding energy of the projectiles, and the excess hydrogen tended to form H2

molecules. The bombardment of the projectile atoms caused structural rearrangement

in the substrate the total sp3 content in the film increased, while the total sp2 content

decreased by the same magnitude.

In many other computer simulations amorphous carbon networks were generated

under conditions close to that occurs within the “thermal spike” with different den-

sities (for example [6, 71, 72, 73, 74] and our previous simulations [75]). All these

simulations agree with the facts that: (i) the percentage of sp3 coordinated atoms in

an amorphous carbon network increases with density, (ii) the percentage of sp3 coor-

dinated atoms in an amorphous carbon network increases with cooling rate. Fig.4.3

shows the sp3-bonded atoms fraction as a function of density for a number of these

calculations in comparison.

Wang and Ho [6] have investigated the structures of amorphous carbon over a


Figure 4.3: The sp3 fraction plotted as a function of density calculated by differentmethods: OTB-orthogonal tight-binding [6], EDTB- environment-dependent

tight-binding [74], NOTB-non-orthogonal tight-binding [69], DFT-ab initio [72] andour previous orthogonal tight-binding simulations [75] (indicated by “my

simulations”).� $ � � � � $ � � �� $ � �!� �� $%�� !& sp3 ��

�[74] environment-dependent tight-binding EDTB

�[6]�� tight-binding OTB

$ � � � � � � $ � ��!� � ��' �� [72] ab initio DFT�[69]

�� tight-binding NOTB�“my simulations”

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wide range of densities (from 2.2 to 4.4 g/cc) generated by rapid quenching of liquid

carbon phase by performing orthogonal tight-binding simulations. Here the electron

wave functions are expanded in terms of a basis set of valence electrons wave functions,

rather than plane waves, controlling the attractive part of the potential, while the

repulsive one is treated empirically. As will be shown below the conditions of their

simulations are very similar to those reported here. However, in contrast to our

findings and in spite of the fact that the percentage of sp3 coordinated atoms was

found by Wang and Ho to increase with density reaching 89 % at a density of 4.4 g/cc,

the authors could not identify any ordered sp3 clusters in their samples. Later the

authors [76] repeated their simulations for hydrogenated amorphous carbon network.

They found that hydrogen tends to break carbon-carbon bonds in ta−C. Reduction

of C-C coordination makes ta − C softer and induced more electronic states in the

energy-band region.

O. Hershkovitz [28] simulated hydrogenated amorphous carbon/diamond com-

posite by the same method. He found that the hydrogen diffused quickly out from

diamond to the region which was low in sp3 bonds. The total fraction of sp3 bonded

carbon atoms not a monotonous function of hydrogen concentration. As the number

of H atoms increased, numbers of sp3 bonds increased and numbers of sp2 bonds

decreased till saturation of the H achieved and a decrease of sp3 bonds begins. When

visualized, one can see that new the sp3 bonds form near previously formed sp3 bonds,

which suggests growth of the diamond cluster.

Finally, A. Sorkin [75, 77] carried out orthogonal tight-binding simulations of

very hot layers of amorphous carbon surrounded of a cold crystal diamond layer

or a cold diamond-like carbon layer. These hot layers mimic the “thermal spike”

induced by heavy ion irradiation in diamond. If the temperature of heating was


lower than the characteristic temperature T ∗ , the central hot layers reconstruct the

diamond structure after annealing. The formation of ta−C structures was observed

in these layers after annealing. Most of the structures of amorphous carbon generated

in this simulation were highly inhomogeneous. The tendency of segregation of the

threefold and fourfold coordinated atoms was observed due to favorable π states in

the graphitelike network.

The samples using in this study, are, of course, smaller than those in the labora-

tory. The latter are almost “infinite” and much larger than can be simulated today.

There is an extensive literature on the effect of finite size in phase transitions, where

because these are cooperative effects dependent on correlation length there is a beauti-

ful scaling theory describing the approach to infinite size. A similar but less dramatic

situation for finite samples of diamond interface was described by Rosenblum et al

[78], who showed that sample size strongly influenced results for lattice mismatch at

a diamond/substrate interface. More recently in a earlier paper of ours [77] the effect

of system size on “temperature” was discussed. We observed that temperature, which

in the simulations is essentially kinetic energy depends on and scales with the system

size.

Chapter 5

Goal of the research

The purpose of this study is to investigate the process of precipitation and growth of

diamond under high pressure and high temperature. As we have shown in previous

chapters, previous experimental and computational studies do not provide a detailed

picture of the complicated processes of nucleation of diamond. For example, computer

simulations still did not yield satisfactory and full answers, as to when cubic diamond

forms, and when lonsdaleite forms. Another intriguing question is the role of hydrogen

in the nucleation process.

In order to study these and other questions we will simulate nucleation and growth

of diamond occurring under high pressure and high temperature within an amorphous

carbon network (with and without hydrogen). This amorphous carbon network will

be generated by the fast quenching of a compressed liquid carbon sample. This pro-

cedure is similar to that occurring within a “thermal spike” during the bias-enhanced

nucleation process. We hope to observe ordered diamond clusters inside the amor-

phous carbon samples. In order to clarify the conditions of the diamond nucleation

we will vary the direction and magnitude of pressure, time (by changing of cooling

46

CHAPTER 5. GOAL OF THE RESEARCH 47

rate) and hydrogen contents in our samples.

We are able to simulate conditions of diamond nucleation inside a real “thermal

spike”. We can artificially generate very high pressures by changing the material

density (shortening the bond lengths) and very high temperatures by imparting high

kinetic energies to the atoms in the simulated sample. The densities at which our

simulations will carried out (3.5-4.1 g/cc) can reasonably be locally attained within

the thermal spike. The times which we choose to perform MD computations (a few

pico-seconds) and the sample sizes (containing a few hundred atoms) will be similar

to those of the thermal spike.

The basic stages of our study are:

(i) The simulation of nucleation of diamond (or hexagonal diamond) inside an amor-

phous carbon network under different pressures (differing in both direction and mag-

nitude). The amorphous carbon samples will be generated by rapid quenching of a

compressed liquid phase with different cooling rates. The pressure applied will both

uniaxial (in one direction) and hydrostatic (in all three directions), its magnitude will

be varied in the range 3.5-4.1 g/cc. When the samples generated by these procedures

will contain some crystallites embedded in amorphous C, their structures will ana-

lyzed according to their content of differently coordinated atoms and by their radial

and angular distribution functions. These will be compared with those of the respec-

tive perfect crystal structure. The electronic structure of the clusters will also studied

and compared with the electronic structure of the related perfect crystal.

(ii) The simulation of epitaxial growth of diamond on a diamond core inside an

amorphous carbon network. The sample of diamond cluster surrounded by an amor-

phous carbon phase will be generated by heating and fast cooling of the carbon

envelope while keeping the inner diamond cluster frozen. Then this structure will

CHAPTER 5. GOAL OF THE RESEARCH 48

compressed to different densities and relaxed at relatively low temperatures. We will

observe the growth of the diamond cluster during of the relaxation procedure. The

epitaxial growth of diamond on diamond layer will also be simulated. Here one di-

amond layer will be frozen, while the surrounding layers will be melted and cooled.

Then the samples will be compressed and repeatedly heated and cooled. The struc-

tural characteristics of these samples will then be studied.

(iii) The possible quantum confinement effects in our diamond clusters will be

measured. We will generate diamond clusters and diamond layers with different sizes

surrounded by an amorphous carbon phase in the way described in the previous

item. The band gap inside the crystallites will computed automatically during the

simulations (in order to calculate the attractive part of the tight-binding potential we

need to calculate the electronic structure of the sample).

(iv) Finally, in order to study the role of hydrogen in the process of diamond

nucleation a dense hydrogenated amorphous carbon network with different hydrogen

contents will be generated in the way described in item (i). The structure of the

samples, in particular, the location of hydrogen atoms will studied. The influence of

hydrogen on electronic structure both of the entire samples and diamond crystallites

formed inside these samples will checked.

We believe that the simulations listed above will help supplement our knowledge

about the processes that occur on the atomic level during bias-enhanced nucleation

and growth of diamond.

Chapter 6

Tight-binding model

6.1 Advantages and disadvantages of different mod-

els to describe the interatomic interaction

Several models are available to describe the interatomic interaction in carbon struc-

tures: ab initio techniques, calculations based on empirical interatomic potentials and

tight-binding approximation. The most accurate is the ab initio model [58] based on

density functional theory. Here the motion of the atomic core is treated classically,

while the electron wave functions are represented in terms of large basis set of plane

waves, keeping the energy of the whole system closed to a minimum with respect to

the wave functions. This technique is very accurate but numerically intensive and

unable to describe the dynamics of thousands of atoms in a reasonable simulation

time. A number of our attemts to use this method with ABINIT package [79] in our

calculations failed. For example, in order to generate the sample of amorphous carbon

containing 216 atoms, we need 10000-30000 MD steps, while ABINIT processes only

one MD step for 100-atoms sample during a number of hours. This method is also

49

CHAPTER 6. TIGHT-BINDING MODEL 50

not suitable for studying the quantum confinemrnt effects in our samples, because

our minimal samples contained 192 atoms which is much larger than ABINIT is able

to handle in reasonable time with current Technion facilities.

On the other hand, there have been numerous efforts to find an accurate model

for the interatomic interaction in carbon empirically, for example, the work of Tersoff

[80]. Another example is the Brenner potential [81], which was developed for describ-

ing of intramolecular chemical bonding in a variety of small hydrocarbon molecules.

Although simulations with classical potentials are fast, these empirical potentials do

not always give correct descriptions for properties that are not explicitly included in

the fitting database. Electronic structure information cannot be obtained, nor can

we expect these classical potentials to accurate describe phenomena where quantum

mechanical interference effects are essential.

The methods mentioned above satisfactorily describe various forms of carbon and

transitions from one form to another. However their disadvantages exclude their use

in the present study. Therefore for our calculation we have chosen the tight-binding

molecular dynamics techniques [82] which will be described in this chapter in more

detail.

Tight-binding molecular dynamics is a useful method for studying the structural,

dynamical, and electronic properties of covalent systems. The method incorporates

electronic structure calculation into molecular dynamics through an empirical tight-

binding Hamiltonian and bridges the gap between ab initio molecular dynamics and

simulations using empirical classical potentials. This model is less accurate than the

ab initio but much less computationally expensive. Moreover, the electronic density of

states is obtained automatically in a process of calculation. Another advantage of the

tight-binding model is its transferability, i.e. the parameters of the model have been


chosen to describe successfully different carbon polytypes: diamond, graphite, linear

chains, fullerenes, as well as a disordered carbon structures like liquid and amorphous

carbon phases.

The method was intensively tested [83]. It has been shown that the energies, vi-

brational and elastic properties for differently coordinated crystalline structures (di-

amond, graphite, linear chain) calculated with the tight-binding potential are in very

good agreement with first-principle calculations and experimental data. Simulations

of liquid and amorphous carbon indicate that the potential is reliable for describing

low-coordinated carbon systems over a wide range of bonding environments. The

description of the more than fourfold-coordinated structures such as simple cubic,

β-tin or bcc structures, is only qualitative with this potential, however since they

do not exist in nature, this is not relevant here. The reliability of this potential for

fullerene and single-wall carbon nanotube calculations was also tested successfully by

comparison with first-principles results [83]. It should be noted that the potential has

a very short cutoff distance of 2.6 A, which makes it innacurate for describing the

interaction between graphite layers under ambient conditions.

In order to carry out the tight-binding calculations we will use OXON (Oxford

Order N package). This is a set of programs for running atomistic static and dynamics

simulations using potentials which are based on tight-binding methods.

6.2 LCAO approach

In atoms the electrons are tightly bound to their nuclei. If the atoms are so close

that their separations become comparable to the lattice constant in solids, their wave

function will overlap. We will approximate the electronic wave functions in the solid


by linear combinations of the atomic orbitals. This approach is known as the tight-

binding approximation or Linear Combinations of Atomic Orbitals (LCAO)

approach.

In covalently bonded semiconductors the valence electrons are concentrated mainly

in the bonds. Therefore the valence electrons wave functions should be very similar to

bonding orbitals found in molecules. In addition to being a good approximation for

calculating the valence bond structure, the LCAO method has the advantage that the

band structure can be defined in terms of a small number of overlap parameters. The

overlap parameters have a simple physical interpretation as representing interactions

between electrons on adjacent atoms.

While the method has been utilized by many authors, the approach we will de-

scribe follows that of Chadi and Cohen [84].

The position of an atom in the primitive cell denoted by j will be decomposed into

rjl = Rj + rl, where Rj denotes the position of the jth primitive cell of the Bravais

lattice and rl is the position of the atom l within the primitive cell. Let hl(r) denotes

the Hamiltonian for the isolated atom l with its nucleus chosen as the origin. The

Hamiltonian for the atom located at rjl will be denoted hl(r−rjl). The wave equation

for hl is given by

hlφml(~r − ~rjl) = Emlφml(~r − ~rjl), (6.1)

where Eml and φml are the eigenvalues and eigenfunctions of the state indexed by

m. The atomic orbitals φml are known as Lowdin orbitals [85]. They have been con-

structed in such a way that wave functions centered at different atoms are orthogonal

to each other. Next we assume that the Hamiltonian for the crystal H is equal to

the sum of the atomic Hamiltonians and a term Hint which describes the interaction

between the different atoms. We further assume the interaction between the atoms to


be weak so that H can be diagonalized by perturbation theory. In this approximation

the unperturbed Hamiltonian H0 is simply

H0 =∑

j,l

hl(~r − ~rjl) (6.2)

and we can construct the unperturbed wave functions as linear combinations of the

atomic wave functions. Because of the translational symmetry of the crystal, these

unperturbed wave functions can be expressed in the form of Bloch functions:

Φml~k

=1√N

∑

j

exp(i~rjl ∗ ~k)φml(~r − ~rjl), (6.3)

where N is the number of primitive unit cells in the crystal. The eigenfunctions Ψk

of H can be written as linear combinations of Φmlk:

Ψk =∑

ml

CmlΦml~k. (6.4)

To calculate the eigenfunctions and eigenvalues of H, we operate on Ψk with the

Hamiltonian H = H0 + Hint. From the orthogonality of the Bloch functions we

obtain a set of linear equations in Cml:

∑

ml

(Hml,m′l′ − E~kδmm′δnn′)Cm′l′(~k) = 0, (6.5)

where Hml,m′l′ denotes the matrix element < Φml~k

|H|Φm′l′~k

> and E~kare the eigen-

values of H. When we substitute the wave function Ψml~k

defined in (6.3) into (6.5)

we obtain

Hml,m′l′(~k) =N

∑

j

N∑

j′

exp[i(~rjl − ~rj′l′) ∗ ~k]N

∗

< Ψml~k

(~r − ~rjl)|H|Ψm′l′~k

(~r − ~rj′l′) >= (6.6)N

∑

j

exp[i(~Rj + ~rl − ~rl′) ∗ ~k] ∗

< ψml~k

(~r − ~rjl)|H|ψm′l′~k

(~r − ~rj′l′) >


Instead of summing j over all the unit cells in the crystal, we will sum over the nearest

neighbors only. In the diamond crystal this means j will be summed over the atom

itself plus four nearest neighbors.

The matrix elements < ψml~k

(~r − ~rjl)|H|ψm′l′~k

(~r − ~rj′l′) > can be expressed in

terms of overlap parameters for two diamond atoms. As it will be shown below, for

a homopolar molecule, there are only four nonzero overlap parameters. The band

structure can now be obtained by diagonalizing the Hamiltonian for different values

of k.

6.3 The bond energy model

The model shown in the preceding section enables the calculation of the energy bands

of a system of well defined configuration. Thus, the parameters suitable for the

diamond lattice, for instance, cannot be used for calculations in graphite. To describe

systems with a wide variety of coordinations with the same tight binding parameters,

a total energy scheme has to be employed, which accounts for interactions other than

the single-electron one’s, with an explicit dependency on the interatomic distances.

The tight binding model has been developed on the basis of two major approx-

imations. The first to be considered is the adiabatic approximation [86], which is

based on the fact that electrons move typically ∼ 102 − 103 faster than the ions. The

latter can thus be considered in their ground state at any moment for a particular

instantaneous ionic configuration, and the electronic and ionic degrees of freedom can

therefore be separated. The second approximation consists in reducing the N-body

problem to a one-electron scheme, where each electron moves independently of the

others, and experiences an effective interaction due to the other electrons and to


the ions. Within these approximations, the one-particle electronic part of the total

Hamiltonian can be written in the form

H = Te + Uee + Uei, (6.7)

where Te is the kinetic energy operator of the electrons, Uee and Uei are the electron-

electron and electron-ion interactions respectively. Following the notation of Horsfield

et al [87], the single-particle Schrodinger equation is

H|n〉 = ε(n)|n〉, (6.8)

where |n〉 is a single particle (doubly occupied) eigenfunction, and ε(n) is the cor-

responding eigenvalue. It has to be mentioned that the k dependency of |n〉 and

ε(n) does not appears explicitly in the notation for clarity (see equation (6.3)). We

shall return to this point below. The eigenfunctions are expanded in an atomiclike

(Lowdin) orbitals set

|n〉 =∑

iα

C(n)iα |iα〉 (6.9)

where i is a site index and α an orbital index. It has to be noted that the basis used

to expand the wave functions may be non-orthogonal. However, in the present work,

orthogonal basis functions are used. The influence of this choice on the results will

be discussed further.

Taking into account the orthonormality of the eigenstates, the eigenvalues and

eigenstates of the Hamiltonian are therefore found by solving the matrix equation

∑

jβ

Hiα,jβC(n)jβ = ε(n)C

(n)iα , (6.10)

where


Hiα,jβ = 〈iα|H|jβ〉 (6.11)

are the matrix elements and

∑

iα

C(n)iα C

(m)iα ≡

∑

iα

〈n|iα〉〈iα|m〉 = δn,m. (6.12)

The off-diagonal matrix elements Hiα,jβ = 〈iα|H|jβ〉, for iα 6= jβ, are called

hopping integrals, and the on-site elements Hiα,iα are the atomic orbital energies. In

the tight binding approach, these hopping integrals and the on-site matrix elements

are constants to be fitted on the basis of the following approximations:

(i) Only atomic orbitals whose energy is close to that of the energy bands on is

interested in, are used [88]. This is the minimal basis set approximation. Thus, for

instance, only the 2s (one orbital) and 2p (three orbitals: px, py, and pz) orbitals

are considered in the case of diamond and 3s and 3p orbitals for silicon, to describe

the occupied (valance) bands. For these two materials there are 16 possible hopping

integrals. However, it can be shown [89] that only hopping integrals between orbitals

with the same angular momentum about the bond axis, are non-vanishing. There

remain therefore just four nonzero hopping integrals, labeled (ssσ), (spσ), (ppσ), and

(ppπ). σ stands for orbitals with 0 angular momentum about the bond axis and π

for orbitals with angular momentum ± 1. The dependence of these hopping integral

in the distance between the atoms will be considered further.

(ii) One considers only hopping integrals between two atoms separated by a dis-

tance shorter than a suitable cutoff. Obviously, to reduce the number of parameters to

be fitted, a cutoff which includes the nearest neighbors is appropriate. However, the

orthogonalized functions (Lowdin) extend further than those (non-orthogonal) from


which they are derived, because the orthogonalization procedure involves orbitals

from nearby atoms. Thus, interactions extending beyond first nearest neighbors have

to be taken into account when an orthogonal basis is used.

Considering the approximations above, the off-diagonal elements of the Hamilto-

nian matrix Hiα,jβ = 〈iα|H|jβ〉 (for iα 6= jβ) are fitted to electronic band structure

of the equilibrium crystal phase, as calculated by more accurate first-principle mod-

els [90]. Sets of hopping integrals can thus be obtained for each crystalline structure

considered.

The tight-binding expression for the binding energy of a system with N atoms

[83] is given by :

Ebinding = Ebs + Erep = 2∑

n(occ.)

ε(n) + Erep (6.13)

where Ebs is the band energy and Erep is the repulsive potential, given as a sum

of pair potentials. ε(n) are the eigenvalues obtained from the diagonalization of the

Hamiltonian matrix. Within the adiabatic approximation, the electrons are assumed

to be in their ground state, so that all the states below the Fermi level are occupied,

and the summation that appears in the band energy is made over these occupied k

states. Erep accounts for the ion-ion repulsion, for the double counting of electron-

electron interactions that appears in the band energy, for the repulsion of overlapping

orbitals due to Pauli’s principle and for the exchange-correlation energy related to

the N-body electronic interaction. The form of the repulsive energy Erep proposed by

Xu et al [83] and used in the present research is

Erep =∑

i

f

∑

j

φ(rij)

, (6.14)


where f is a functional expressed as a 4th-order polynomial, φ(r) is a pairwise po-

tential between atom i and atom j, and described below, and rij is the interatomic

distance between the atoms.

6.4 The rescaling functions

As mentioned above, the elements of the Hamiltonian matrix are fitted to first-

principle calculations for different equilibrium structures [62]. To describe the prop-

erties of non-equilibrium structures, as amorphous solids or liquids, the hopping in-

tegrals and the repulsive energy should be rescaled with respect to the interatomic

distance. The rescaling functions proposed by Goodwin et al [91] greatly improve

the transferability of the tight binding model to structures not included in the pa-

rameterization. These functions are now widely used, in the slightly improved form

proposed by Xu et al [83]

h(r) = h0(r0/r)n exp{n[−(r/rc)

nc + (r0/rc)nc]}, (6.15)

for the rescaling of the hopping integrals, and

φ(r) = φ0(d0/r)m exp{m[−(r/dc)

mc + (d0/dc)mc ]} (6.16)

for the repulsive potential. In the rescaling functions found by Goodwin et al, the

parameters nc and rc were the same as mc and dc respectively. All the parameters

appearing in the rescaling functions are obtained by fitting first principle results of

energy versus nearest-neighbor interatomic distance for different crystalline phases,

given equilibrium sets of hopping integrals for these structures. In this way, the tight

binding model is transferable to different atomic environments.


6.5 Force calculation

We can express the forces acting on the atoms in a compact form, by first defining

the density matrix

ρiα,jβ =∑

n(occ.)

C(n)iα C

(n)jβ (6.17)

The cohesive energy thus becomes

Etot = 2∑

iα,jβ

ρjβ,iαHiα,jβ + Urep (6.18)

The forces acting on the atoms are then obtained by differentiating the cohesive

energy with respect to atomic positions, that is

Fk = −∂Etot

∂rk

(6.19)

= −

2∑

iα,jβ

ρjβ,iα

∂Hiα,jβ

∂rk

+∂Urep

∂rk

. (6.20)

Chapter 7

Numerical techniques

7.1 Equations of motion

Once the specific form of the potential is established, the forces between the atoms can

be computed from the gradient of the potential. We solve the differential equations:

Fiα = md2riα

dt2= − ∂Ei

∂riα

(7.1)

viα =driα

dt(7.2)

in order to obtain the position ri and the velocity vi of each atom of mass m as a

function of the time t. i is the atom in consideration and α the coordinates x, y

and z. As explained before, in the case of the tight binding model, a force emerging

from the electronic part of the total energy should also be calculated. However, like

in the classical models, in the quantum or semi-classical approaches, the atoms are

considered as classical, so that also in these cases, the Newton equations have to be

solved.

60

CHAPTER 7. NUMERICAL TECHNIQUES 61

The first step of the calculation consists of determining the neighbors of each

atom within the limit of the force range. Then, for each atom the force applied by

its neighbors is computed and added all together to get the total force on the atom.

From the forces evaluation, the Newton’s equations (7.1) and (7.2) are solved for

coordinates x, y and z and velocities vx, vy and vz. For this purpose, the predictor-

corrector algorithm was used [92].

7.2 The Predictor-Corrector algorithm

We would like to solve the second-order differential equation

r = f(r, r, t) (7.3)

where f is related to the forces in equation (7.1) by F = mf . The first step of this

algorithm consists in evaluating the atomic positions and velocities at time t + ∆t

from the positions and the velocities at time t− i∆t, where i = 0, ..., k − 2, k being

the order of the predictor part. The extrapolation is given by

ri(t + ∆t) = ri(t) + ri(t)∆t + ∆t2k−1∑

i=1

αif(t + [1 − i]∆t), (7.4)

for the atomic positions, and

ri(t)∆t = ri(t + ∆t) − ri(t) + ∆t2k−1∑

i=1

βif(t + [1 − i]∆t) (7.5)

for the velocities. The coefficients αi and βi satisfy the equation

k−1∑

i=1

(1 − i)qαi =1

(q + 1)(q + 2), (7.6)

k−1∑

i=1

(1 − i)qβi =1

q + 2q = 0, ..., k − 2.


k = 4 (× 1/24) 1 2 3αi 19 -10 3βi 27 -22 7γi 3 10 -1δi 7 6 -1

Table 7.1: Coefficient of the Predictor-Corrector algorithm for k = 4 forsecond-order differential equation.

$ � �� !�"� $ � � �� k = 4� � � predictor-corrector ��

� � �� !��' �

These predicted values are then corrected from the value of f at a time t + ∆t (cal-

culated from the predicted values themselves), using the expressions

ri(t+ ∆t) = ri(t) + ri(t)∆t + ∆t2k−1∑

i=1

γif(t+ [2 − i]∆t), (7.7)

for the atomic positions, and

(∆t)ri(t) = ri(t + ∆t) − ri(t) + ∆t2k−1∑

i=1

δif(t + [2 − i]∆t) (7.8)

for the velocities. The coefficient γi and δi satisfy similar equations as αi and βi. The

coefficient used in the present work are for k = 4, and are given in table 7.1.

The predictor-corrector algorithm gives very accurate positions and velocities and

is therefore suitable in very “delicate” calculations. However, it is computationally

expensive and needs significant storage.

7.3 Periodic boundary conditions

Our goal of is to describe the first stage of diamond nucleation. This process involves

a number of tens atoms only. However nucleation is a bulk process, therefore an in-

fluence of the surface should be excluded. So our sample should be very large. Usual


macroscopic system contains an order of 1024 particles. Obviously, this can’t be done

by the molecular dynamics technique with any currently envisaged computer. The

maximum number of atoms that participate in the present simulation is 241. Conse-

quently, placing the boundary atoms at some fixed sites will irremediably influence

the atoms in the bulk after a short time, giving rise to undesired results.

One way to overcome this problem is to use periodic boundary conditions. When

this is applied, a particle that crosses a face of the simulation box, is reinserted at

the opposite face. The primary simulated box is then periodically replicated in all

directions to form a macroscopic sample. Thus, the neighbors that surround it and

the forces applied on it would be different than those in the case of fixed boundary

conditions.

7.4 Initial configuration

To start the molecular dynamics simulation, we should assign initial positions and

velocities to all atoms in the system. In some cases the appropriate choice of initial

conditions is very important, because the results of computer simulations can be

strongly affected by this choice.

But in our simulations, the choice of initial conditions does not affect the structures

of amorphous carbon. The initial configuration of the system is chosen to be a perfect

diamond crystal, which is heated up to a high temperature, at which it melts. The

simulation of the liquid carbon phase is carried out until the equilibrium state is

reached. So in the amorphous carbon structures obtained by cooling of the liquid

carbon phase memory about the initial state is completely lost.


The initial velocities can be chosen randomly with a Maxwell distribution:

f(v) ∼ e−mv

2

kT (7.9)

where f(v) is probable number of molecules which have velocities from v to v + dv.

Then the velocities can be rescaled to be related to the ambient temperature T . The

following relation should hold:

3

2NkT =

∑

i

1

2mv2

i (7.10)

7.5 General description of the calculations

The Oxford Order N (OXON) package (for details see [94], the previous chapter and

Appendix A ) was used in the present work. This is a set of programs for carrying

out atomistic static and dynamic calculations using potentials which are based on

tight-binding methods.

The tight-binding method was employed in the calculations to describe interac-

tions between carbon atoms. The Γ point Brillouin zone sampling was used for the

electronic calculations. With this method, the molecular dynamics technique was

applied to calculate the positions and the velocities of the atoms as function of the

time. In the MD calculation, Newton’s equations of motion were solved using the

predictor-corrector algorithm. The MD time step was 10−15 s. A large change in

volume accompanies the sp2 - to sp3 bond conversion, but the volume of the dam-

aged region in diamond is restricted by the surrounding diamond lattice, therefore

simulations are carryed out at constant volume.

In order to calculate the ratio of sp2 to sp3 bonded atoms, as well as the radial

and angle distribution functions of the structures of amorphous carbon generated a


FORTRAN program was written (see Appendix B).

7.6 AViz

Visualization was essential for development of this project. Our computational physics

group developed the Atomic Visualization package AViz [95]. This is a very pow-

erful visualization tool which helps to enhance the 3D perception. It includes a lot

of various options, which let one to rotate the still sample, change relative sizes of

atoms, create animations and movies, add and remove the bonds and borders of the

sample, use color coding, slice of the sample and much more.

The Atomic Visualization package (AViz) was used extensively in all stages of

this work. A visualization of our amorphous carbon samples with color coding for

different atomic bonding helped indentify clusters of either sp2 or sp3 coordinated

atoms, diamond crystallites and graphite-like planes. The animated vizualizations

were created to keep track of the diamond nucleation.

7.7 Coordination number

As explained in chapter two, amorphous carbon solids (ta − C as well as a − C)

contain both fourfold atoms sp3 and treefold atoms sp2. Each sp3 bonded atom has

four nearest neighbors separated by a distance of approximately 1.54 A. Each sp2

bonded atom has only three nearest neighbors separated by a shorter distance. Thus

the method of distinguishing between sp3 and sp2 sites used in this work is based on

determination of the coordination number of each atom.


In order to define the coordination number, we have calculated the radial distri-

bution function g(r) of the structures of amorphous carbon created after cooling of

a liquid phase. We will return to a detailed discussion of g(r) in the next chapter.

Independently of density and cooling rate, g(r) of all the samples exhibits a clear gap,

centered at about 1.9 A, separating the first and the second peak. All atoms within

the sphere of radius 1.9 A are thus assumed to comprise the first nearest neighbor-

hood of a given atom. Therefore, the number of neighbors of each atom within a

distance of 1.9 A determines the coordination number.

In order to count the coordination number and to calculate the distances between

all pairs of atoms for g(r) the following procedure of determining of neighbors was

used. An integer number (from 1 to 241 for the largest sample) is assigned to each

atom that will permit identification of all atoms whenever needed. Then, for each

atom number, i, its distance rij to atom number j is calculated, for j running over

all the atoms of the crystal. If rij is lower than the distance of 1.9 A, the label of the

atom j is stored in the list of nearest neighbors of the atom i. In order to calculate

g(r) the distances rij are accumulated in a separate file. Afterwards, the number of

the bond lengths restricted to the inteval from r to r+dr was summed up and divided

by the number of atoms in the system.

The file of the nearest neighbors was also used for calculating an angular distribu-

tion function g(θ). Here, the angles between the atom i and each pair of its nearest

neigbors are accumulated in the list of angles.


7.8 Analysis of errors

The possible errors that can influence the results in computer simulation are: sta-

tistical errors, finite-size effects, unreliable generator of random numbers, unaccurate

potential and numerical techniques, unsufficient time of simulation to reach equilib-

rium state.

The main source of systematic errors could be the interatomic potential, which is in

our case a tight-binding potential. It was shown in many previous simulations [6, 96]

that this potential is very reliable and gives an adequate description of amorphous

phases of carbon. However, Marks et al [97] found that the tightbinding method

yields a slightly lower sp3 fraction at high densities in comparison with empirical

potentials and the presence of singly coordinated atoms at low densities. However,

these comments are not relevant in the present calculations. The simulations are

carried out at sufficiently high densities and the tight-binding potential is thought to

be more accurate than an empirical potential (see Fig.4.3).

The predictor-corrector technique is sufficiently accurate. The generator of ran-

dom numbers which is used in the present simulations was checked earlier to be

reliable for this type of simulation [94, 98]. So the contribution of these two factors

to the total error is negligible.

Early attempts to prepare samples of amorphous carbon by quenching from the

liquid carbon, when the liquid was not in an equilibrium state led to erroneous results

[75]. The analysis of these samples did not give any reliable results, the obtained data

were scattered and statistical analysis could not help to reveal any regularity of the

observed data. So in order to obtain the adequate results it was very important to

reach equilibrium and in all subsequent simulation the approach to equilibrium in


the liquid phase was controlled thoroughly by monitoring of the total energy of the

system.

In order to gather a better set of statistics, we should repeat our simulation

with another initial conditions (initial velocities, seeds of random number generator).

However, computer simulations based on the tightbinding method are very time and

memory consuming, so we repeat our simulations only five times. Thus the statistical

error of these calculations gives a large contribution to the total error. The average

statistical error is calculated according to:

∆x =s√k

(7.11)

where s is a standard deviation:

s =

√

√

√

√

√

√

k∑

i=1(xi − x)2

k − 1(7.12)

and x is a simple arithmetic mean value:

x =1

k

k∑

i=1

xi. (7.13)

Chapter 8

Results: Nucleation of diamond

under pressure

In this chapter we will describe our results for nucleation of diamond under high

pressure in amorphous carbon without hydrogen. Hexagonal diamond was observed

to grow when uniaxial pressure was applied [9]. In other experiments hexagonal

diamond was considered as intermediate phase in the process of graphite-to-diamond

conversion [52]. Hence, in order to clarify the conditions of formation of cubic and

hexagonal diamond, our simulations will include cases with pressure applied both

hydrostatically and uniaxially.

69

CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 70

8.1 Amorphous carbon compressed in all three di-

rections.

8.1.1 Computational details

The samples used in the first stage of these calculations are initially arranged as a

perfect diamond crystal of a size 3 × 3× 3 unit cells (i.e. 216 carbon atoms) with

a density of 3.3 g/cc. The sample was melted at a temperature of 8000 K during

5, 10, 15, 20 and 25 ps. The times are sufficient to ensure the liquid phase reached

equilibrium. Thus we generated five different liquid carbon samples. In order to

generate amorphous carbon networks with different densities each of the five liquid

configurations were isotropically compressed by changing the volume of the unit cell

to 3.3, 3.5, 3.7, 3.9 and 4.1 g/cc. Pressures at 8000 K for the densities of 3.3, 3.5,

3.7, 3.9 and 4.1 g/cc correspond to 60, 80, 110, 140, 250 GPa respectively. The

compressing followed by a rapid cooling to 300 K with cooling rates of 500 K/ps, 25

simulations in all.

In order to simulate an expansion, samples at 3.7, 3.9 and 4.1 were then ho-

mogeneously expanded to reduce the density to 3.5 g/cc. Then the samples were

“annealed” by repeatedly heating to 1000 K and then cooling to 300 K during 10

ps. This cooling time is physically realistic and provided a range of conditions within

a real “thermal spike” [98]. As a result of this repeated “annealing” cycle only a

few atoms slightly changed their position or their hybridization (for example in the

sample of 3.9 g/cc the percentage of the sp3 coordinated atoms changed from 88 to

86 %).


The samples generated by these procedures contained some crystallites embedded

in amorphous C. These structures were analyzed according to their content of differ-

ently coordinated atoms and by their radial and angular distribution functions. The

band gap inside the crystallites was computed automatically during the simulations

(in order to calculate the attractive part of the tight-binding potential we need to

calculate the electronic structure of the sample).

8.1.2 The effects of different densities (pressures)

The structures obtained by the processes described above were found to depend on

both pressure and cooling rate. For the cooling rate of 500 K/ps ever increasing

fractions of sp3 bonded carbons were found with increasing pressure as summarized

in Table 8.1(see also Fig.8.1). The structures thus created were highly inhomogeneous

and even the relatively low density samples contained large sp3 and sp2 clusters. The

clustering of the threefold atoms is favored by the delocalization of the π states. It

is worth mentioning that the percentage of the sp3 coordinated atoms found in our

samples is slightly higher than that found by Wang and Ho [6], in their samples

generated by applying a rather similar procedure, however, they did not identify any

diamond crystallites.

All sp3 clusters in the lowest density samples (3.3 g/cc) were completely disor-

dered. The largest sp3 cluster found contained 36 atoms of amorphous carbon. The

sp3 clusters in the samples generated at 3.5 g/cc contained several small groups of

5-10 carbon atoms which formed an ordered structure. Each atom has four neighbors

located exactly in the corners of a tetrahedron i. e. at 109.47 ◦ and with the same

bondlength (1.54 A) as in diamond (or in hexagonal diamond). These are clearly

precipitates of nano-diamond.


Figure 8.1: Microscopic structures of amorphous carbon with densities of 3.3 g/ccwith 52 % of sp3-bonded atoms (a), 3,7 g/cc with 81 % of sp3-bonded atoms (b) and

4.1 g/cc with 95 % of sp3-bonded atoms (c). Red balls represent fourfoldcoordinated atoms, blue balls represent threefold coordinated atoms.

sp3 �� 52 %� ��&�� 3.3 g/cc

�� $ � ��!�!� ��!� � ��

4.1 g/cc��#$ � �!� �� (b) sp3 �� 81 %

� ��&�� 3.7 g/cc��#$ � �!� �� (a)

� �� ' � sp2 �� ' � sp3 �� (c) sp3 �� 95 %

� ��&��

�

� � �!�� ' � sp �� &��


numberDensity (g/cc) Fourfold(%) Threefold (%) Twofold (%) of diamond

clusters

3.3 52±4 44±4 4±4 0

3.5 72±3 24±5 4±2 0

3.7 83±6 17±3 0 2 (22, 31)

3.9 88±2 11±2 1±1 4 (26, 45, 90, 120)

4.1 95±3 5±3 0 3 (21, 21, 25)

Table 8.1: Fraction of four-, three-, and twofold coordinated atoms in the entireamorphous carbon sample subjected to the cooling rate of 500 K/ps. The number of

cases (out of 5) where a diamond cluster containing more than 20 atoms wasgenerated are given in the last column of the table. The numbers in brackets are the

number of atoms in each such cluster.��!� � �� &��

� �� ' � �� 500 K/ps

�� !� � � �� $ � �� ' � � �� ' � ��!' � �

�

�� 20 � ��$ � � � ��&!� � � � � � � � � � � �� & �� $%� �

�

� � � ��

At 3.7 g/cc the ordered diamond clusters grew in size and contained 20-30 atoms in

2 cases out of 5, these structures are large enough to be identified by visual inspection

(using the AViz) as cubic diamond, rather than hexagonal diamond. At a density of

3.9 g/cc in 4 cases out of 5, a nucleation of cubic diamond structures with more than

30 atoms occurred (see Fig. 8.2). Two very large clusters contained 90 and 120 atoms.

The largest cluster of 120 atoms is shown in Fig. 8.3. At a density of 4.1 g/cc we

found that in three cases ordered diamond crystallites containing 20-25 atoms were

formed.

The orientation of the diamond clusters relative to the walls of the simulation box

was found to be arbitrary. Several samples contained more than one small diamond

cluster (10-15 atoms) with different orientations. In light of the above, it seems that

the density of 3.9 g/cc is the most favorable density for the precipitation of diamond.


Figure 8.2: Microscopic structures of amorphous carbon with density of 3.9 g/ccwith 89 % of sp3-, 10 % of sp2- and 1% of sp-bonded atoms. Red balls representfourfold coordinated atoms, blue balls represent threefold coordinated atoms and

green balls represent twofold coordinated atoms.�� 89 %

� ��&�� 3.9 g/cc�� $ � ��!�!� ��!� � �� #��

� � � � �� ' � sp3 �� sp ��

�� 1 % �� sp2 �� 10 %�sp3 ��

�

� � �!�� ' � sp �� &�� ' � sp2 ��


Figure 8.3: The damaged diamond cluster from the sample drawn on Fig. 8.2generated at a density of 3.9 g/cc from two different view points

�� & �� 3.9 g/cc$ � �!� �� $%� ��

��

� ��


0 50 100 150 200angle (degrees)

0

100

200

300

400

g3

Figure 8.4: Angular distribution function of the diamond cluster drawn on Fig.8.3(black thick line) compared with the angular distribution functions of a pure

diamond crystal (red line) and of amorphous carbon (blue line).� �� g(θ)

� � � �� !�� g(θ)

� � � � & � � � ��!� � ��

The structures obtained were investigated by performing statistical analyses of their

radial and angular distributions. The peak of the angular distribution function of

the biggest cluster was found to be significantly narrower and higher than that of

amorphous sp3 bonded carbon (Fig. 8.4), indicating the high degree of order within

the crystallite. The first peak of the radial distribution function (Fig. 8.5) is located

at 1.545 A, which is close to the bondlength of unrestricted cubic diamond which is

1.54 A.

The electronic structure of our samples was automatically obtained in the process

of the tight-binding simulation. The band gaps in the center of the diamond clusters

(before the relaxation process, i.e. at 3.9 g/cc) were found to be slightly narrower

than the band gap of perfect cubic diamond at the corresponding density (see Table


1 2 3distance (Angstrom)

0

50

100

150

200

g(r)

Figure 8.5: Radial distribution function of the diamond cluster drawn on Fig.8.3(black thick line) compared with the radial distribution functions of a pure diamond

crystal (red line) and of amorphous carbon (blue line).� � �� g(r)

� � � �� !�� g(r)

�

� � � � & � � � ��!� � ��

8.2 and Fig. 8.6). The reason for this may be the influence of amorphous envelope of

the clusters.

8.1.3 The effects of different cooling rates

Several simulations in which the slower cooling rate (200 K/ps) was applied were

performed for the samples homogeneously compressed in all three directions. Most of

the samples thus generated were amorphous carbon with all the sp3 clusters found to

be disordered. In one case for a sample with a density of 3.7 g/cc and one case for a

sample with a density of 3.9 g/cc graphitic configurations with random orientations

of the graphitic planes with respect to the walls of the cell were formed as will be

discussed below.


Figure 8.6: Density of states of the diamond cluster drawn on Fig.8.3 (black line)compared to the density of states of a pure diamond (red line). The insert shows a

magnified part of the density of states near the band gap.$ � �!� ��

� �� !�� $ � �!� ��

$ � ��!� �� $%� � ��!� � ' � � � � � ��

� � � ��

�

�!$ � � � � � � � �� !��


Density (g/cc) cluster (eV) diamond (eV)

3.5 4.1 5.4

3.7 5.0 5.6

3.9 5.4 5.9

4.1 5.1 7.4

Table 8.2: Band gap of the best unrelaxed diamond cluster at each densitycompared with the band gap of perfect diamond at the corresponding density.�� !�� !� � � �� $ � ��!� ��&�� !$ � ��

�

� �� $%� � $ � ��!��

8.2 Amorphous carbon compressed in one direc-

tion

8.2.1 Computational details

In order to investigate the possible formation of hexagonal diamond during the cool-

ing of liquid carbon, the samples were subjected to the procedure described above

but with the application of uniaxial pressure. Three liquid samples containing 216

atoms at 3.5 g/cc and 8000 K were generated as described in the subsection 8.1.1 with

melting times of 5, 10 and 15 ps. The uniaxial pressure was simulated by shortening

all bond components in the z direction. Then each sample was compressed in one

direction (z) to bulk densities of 3.6, 3.7 and 3.8 g/cc by shortening of z-edge of the

computational box from 10.65 A to 10.3, 10.0 and 9.7 A correspondingly. As this

takes place, all bond components in the z direction were shortened. These densities

were chosen from consideration of the computational convenience. Then the com-

pressed samples were quenched to 300 K with different cooling rates (200 (slow), 500

(intermediate) and 1000 (fast) K/ps), 27 simulations in all. After cooling the samples

were expanded in the z direction to restore their cubic form (at 3.5 g/cc) and finally


relaxed during 25 ps.

8.2.2 Samples prepared with fast cooling rate

Samples that were prepared by applying the fast (1000 K/ps) cooling rate were found

to be amorphous carbon containing from 67 % to 82 % of sp3 coordinated atoms,

depending on the pressure. The average sp3 fraction varied from 70 % for the density

of 3.6 g/cc to 76 % for 3.8 g/cc. These fractions are somewhat smaller than those ob-

tained by the application of the hydrostatic pressure quoted above. All the structures

were found to contain diamond clusters of different sizes and quality. No preferable

orientation of the cluster relative to the direction of compression could be observed.

All the clusters were identified, by careful visual inspection of the crystallites viewed

from different directions, to be cubic (not hexagonal) diamond. The best quality,

largest diamond cluster (∼ 40 atoms) was generated in the sample at 3.8 g/cc (see

Fig.8.7). This cluster exhibits radial and angular distribution functions which are

close to those of cubic diamond (see Fig.8.8 and Fig.8.9). The band gap in the center

of the cluster is 5.1 eV (see Fig.8.10) which is somewhat smaller than that of diamond

(5.4 eV).

8.2.3 Samples prepared with intermediate and slow cooling

rates

The structures generated at intermediate (500 K/ps) and slow (200 K/ps) cooling

rates fall into two groups. The first was found to contain diamond clusters embedded

in an amorphous carbon network (similar to that obtained at the fast cooling rate)

with a somewhat smaller fraction of sp3. For example for 3.7 g/cc at the slow cooling


Figure 8.7: Sample generated at 3.8 g/cc at fast cooling rate (left) and damageddiamond cluster found within this sample (right).

� � � � � � �!�� 3.8 g/cc$ � �!� �� 1000 K/ps

� � �!� � � �� $ � � � � � �

�

�� $%� ��

rate an the sp3 fraction was 66 %, whereas at fast cooling rate it was 73 %.

The second group of generated structures were found to be damaged graphite with

interplanar distances shorter than in perfect graphite (see Fig.8.11). The formation of

this graphitic structure is more probable at the slow cooling rate. The orientation of

the graphitic planes was found to be parallel to the direction of compression (z) for the

most cases. Only in one case (at 3.8 g/cc with the slow cooling rate) the angle between

graphitic planes and the z direction was 45◦. The interplanar distance varied from

2.1 A (for the samples compressed to 3.8 g/cc) to 2.4 A (for the samples compressed

to 3.6 g/cc). The radial distribution function of the graphitic structure generated

at 3.8 g/cc is drawn in Fig.8.12 in comparison with radial distribution function of

perfect graphite at the same pressure. The average distance to first nearest neighbors

(first peak of the radial distribution function) in the structure is shorter than that


1 1.5 2 2.5 3distance (A)

0

5

10

15

20

25

30

g(r)

Figure 8.8: Radial distribution function, g(r), of the damaged diamond clustergenerated at 3.8 g/cc with cooling rate of 1000 K/ps (black line) compared to the

radial distribution function of a pure diamond crystal (red line).� � � � � � � �

1000 K/ps� � ��

3.8 g/cc$ � ��!�� $%� � � � � � � � �� g(r)

�

� � � �� g(r) � � � ��

of perfect graphite (1.39 A compared with 1.42 A) because of the higher density of

the graphitic structure. The peak of the angular distribution function (Fig.8.13) is

broader than in perfect graphite, but is located at 120◦, as in graphite. Typical trends

in our simulation can be seen on Table 8.3, where the structures of three samples:

two at 3.8 g/cc and one at 3.7 g/cc, cooled at different cooling rates are presented.

8.2.4 Interesting cases

It is worth to mention about some particularly interesting cases that occurred in the

simulations. In one of the simulations carried out at 3.7 g/cc and at slow cooling rate,

the graphitic planes, after cooling, were flexed, presumably by the extreme pressure

(Fig 8.14). The mean bondlength in the graphitic structure was 1.41 A, which is close


50 100 150angle (degrees)

0

10

20

30

40

g3

Figure 8.9: Angular distribution function, g(θ), of the damaged diamond clustergenerated at 3.8 g/cc with cooling rate of 1000 K/ps (black line) compared to the

angular distribution function of a pure diamond crystal (red line).� � �� 1000 K/ps

� � �!� � � �� $ � � 3.8 g/cc$ � �!� �� !�� g(θ)

�

� � � � � � � �� g(θ) �

-40 -30 -20 -10 0 10energy (eV)

0

0.1

0.2

0.3

0.4

0.5

DO

S

Figure 8.10: Density of states of the damaged diamond cluster generated at 3.8 g/ccwith cooling rate of 1000 K/ps.

� 3.8 g/cc$ � ��!�� 1000 K/ps

� � �� $%� � � � � � � � �� $ � ��!�


Figure 8.11: A graphitic configuration generated at 3.7 g/cc with intermediatecooling rate: a) view from the direction parallel to the graphitic planes, b) one

graphitic plane, view from the perpendicular direction. Red balls are sp3

coordinated atoms, blue balls are sp2 coordinated atoms and green balls aresp-coordinated atoms.

� �� &�� (a)� ��

3.7 g/cc$ � ��!�!� �� $%� � � � ��!��

sp3 ��

� ��!� � � �� &�� (b)� � ��!��

�

� � �!�� ' � sp �� &�� ' � sp2 �� ' �


Cooling rate sample 1 (3.8 g/cc) sample 2 (3.8 g/cc) sample 3 (3.7 g/cc)

fast 82 % (diamond cluster) 80 % (diamond cluster) 76 %

intermediate 80 % (diamond cluster) 78 % graphite

slow 61 % graphite graphite

Table 8.3: Percentage of sp3 coordinated atoms and the structure of three samplesgenerated at different densities with applying of uniaxial pressure: the first and thesecond at 3.8 g/cc, the third at 3.7 g/cc for different cooling rates under uniaxial

pressure.$ � $ $ � � � � $ � � � ��!�� $%� � � �� $%� � �� sp3 �� 3.8 g/cc

$ � �!� �� $%� �� & � � ��

�

� � �� 3.7 g/cc$ � �!� �� $ �

1 2 3distance (Angstrom)

0

50

100

150

200

g(r)

Figure 8.12: Radial distribution function of the graphitic structure in the samplegenerated at 3.8 g/cc subjected to uniaxial pressure (black line) compared with the

radial distribution function of perfect graphite (red line).� � �� & � � �� $ � $ 3.8 g/cc

$ � �!� �� !� � � � � � ��g(r)

�

� � � � � � � � � �� g(r)

�


50 100 150angle (degrees)

0

50

100

150

g3

Figure 8.13: Angular distribution function of the graphitic structure in the samplegenerated at 3.8 g/cc subjected to uniaxial pressure (black line) compared with the

angular distribution function of perfect graphite (red line).� � �� & � � �� $ � $ 3.8 g/cc

$ � �!� �� !� � � � � � ��g(θ)

�

� � � �� g(θ)

�

to the bondlength of perfect graphite 1.42 A. After relaxation to the density of 3.5

g/cc, at room temperature, the planes straightened out. Another interesting structure

was generated at 3.8 g/cc subjected to the intermediate cooling rate. Graphite could

not grow throughout the entire sample at this high density, however the layers of

the less dense graphite perpendicular to the direction of compression were found to

alternate with denser diamondlike amorphous structures (Fig. 8.15).


Figure 8.14: Flexed graphitic configuration generated at 3.7 g/cc with slow coolingrate: (a)-before relaxation, (b)-after relaxation . Red balls are sp3 coordinatedatoms, blue balls are sp2 coordinated atoms and green balls are sp-coordinated

atoms.�� !�

(a)� � � �� !� � � ��

3.7 g/cc$ � �!� �� $ � � � � � &!� � � ��!��

sp2 �� ' � sp3 ��

� ��' � �� (b)� � ��!��' � ��

�

� � �!�� ' � sp �� &�� ' �


Figure 8.15: Configuration generated at 3.8 g/cc with intermediate cooling rate.Graphitic layers alternate with diamond like amorphous carbon layers. Red balls

are sp3 coordinated atoms, blue balls are sp2 coordinated atoms and green balls aresp-coordinated atoms

� ��!� � $ � � ��!��

�� 3.8 g/cc

$ � ��!�� $%� � � �� ' � sp2 �� ' � sp3 ��

�

� ��

�

� � �� ' � sp �� &��

Chapter 9

Results: Growth of diamond under

pressure

In this chapter we will describe our simulations of growth of diamond on a diamond

seed in an amorphous carbon network under high pressure. We will check two cases:

first, when the seed is a very small diamond cluster and the second, when diamond

grows on a diamond surface (growth of diamond on diamond layer).

9.1 Growth of diamond on cubic diamond seed

within compressed amorphous carbon.

We next studied the growth of diamond clusters embedded into amorphous carbon

networks. A cubic sample with a density of 3.5 g/cc was initially arranged as a perfect

diamond crystal of 3 × 3 × 3 diamond unit cells (216 atoms). The interesting feature

of this simulation was that the 8 central atoms were frozen, i.e. the motion of these

atoms was forbidden. The initial geometry of the sample is shown in Fig. 9.1.

89

CHAPTER 9. RESULTS: GROWTH OF DIAMOND UNDER PRESSURE 90

Figure 9.1: Cut of initial diamond configuration; black balls represent the frozenatoms, white balls represent the moving atoms.

� �� ' � � ��

�"$ � � $ � � � � � � � � � �� $ �

�

� �� ' � � ��

The sample was then melted at 35000 K. Once the liquid phase reached equilibrium,

the cells containing free-to-move atoms were cooled to the room temperature of 300

K at a cooling rate of 1000 K/ps. As a result an amorphous carbon network with

embedded perfect diamond cluster was generated. This sample contained 47 % of sp3-,

49 % of sp2- and 4 % of sp-coordinated atoms. It is worth mentioning that atoms near

the frozen diamond could not move as quickly as the atoms near the edges of the box.

Therefore a number of the atoms close to the diamond core returned to their initial

diamond positions in the process of cooling. Hence after cooling the central diamond

cluster contained 12 atoms (only 8 atoms were frozen). The sample of amorphous

carbon with the diamond core is drawn in Fig. 9.2.

In order to generate configurations with different densities the initial sample was

isotropically compressed by changing the volume of the unit cell to 3.5, 3.7, 3.9 and


Figure 9.2: The sample of amorphous carbon with embedded pure diamond cluster(a) and the diamond cluster (b).

(b)� � �!�� !�� (a)

� �� !� � � � � � � �!��

4.1 g/cc. After that the compressed configurations were relaxed at 1000 K during

15 ps. The previously frozen atom were “released” in the process of relaxation, this

means that the motion of all atoms was free.

In the process of relaxation new atoms joined the diamond core and the diamond

cluster grew. The fractions of differently coordinated atoms in the relaxed samples

and the number of atoms in the diamond cluster are summarized in Tab.9.1. The

percentage of fourfold coordinated atoms monotonically increased with density. The

number of atoms joined to the diamond core also increased as the density increased,

the size of the new diamond cluster reached 22 and 20 atoms at densities of 3.9 and

4.1 g/cc correspondingly (see Fig. 9.3). The growth was observed in all directions. In

spite of the fact that the new cluster shows structural properties (radial and angular

distribution functions) identical to those of diamond, the band gap inside the cluster

was found to be 4.4 eV (Fig. 9.4), which is lower than that of diamond (5.4 eV).


Density (g/cc) Fourfold(%) Threefold (%) Twofold (%) N

initial 47 49 4 12

3.5 45 48 7 14

3.7 46 48 6 14

3.9 58 35 7 22

4.1 67 32 1 20

Table 9.1: Fraction of four-, three-, and twofold coordinated atoms in the relaxedamorphous carbon sample and the number of atoms in the grown diamond cluster

N .�� !� � �� &!�#�� !� � � ��

� (N)�� $ � � � � � � � � �!�� ' � � � ��' � ��

This discrepancy may be explained by the very small sizes of our cluster and by the

influence of the stressed amorphous carbon network.

9.2 Growth of diamond on diamond layer within

compressed amorphous carbon layer

A sample of amorphous carbon sandwiched between two layers of diamond was taken

from my MSc thesis [75]. The sample contained 192 atoms (2 × 2 × 6 diamond unit

cells). The density was 3.5 g/cc. The sample was generated by heating, following by

fast quenching of the central layer of the sample, while the upper and lower layers

(32 atoms in each of them) were kept frozen. These layers remained perfect diamond,

while the central layer transformed to amorphous carbon. The central amorphous

layer of the resulting sample contained 75 % of sp3-, 23 % of sp2- and 2 % of sp

coordinated atoms.

In order to study the growth of diamond under high pressure we compressed the

sample in the z-direction by shortening its z-edge from 21.3 to 20.5 A, this corresponds


Figure 9.3: The sample of amorphous carbon with embedded pure diamond clusterafter relaxation at 4.1 g/cc (a) and the diamond cluster (b).

� ��' � �� !�� !�� 4.1 g/cc$ � ��!��

� (b)� � �� !�� (a)

Figure 9.4: Density of states of diamond cluster grew up within an amorphouscarbon network (black line) compared to that of perfect diamond (red line).

$ � ��!�� !� � �� !� � � � � � � � � � � � � �� $ � ��!�

�

� � � � � � � � � ��


to the density of 3.9 g/cc, (as this takes place all z-components of interatomic bonds

inside the sample were shortened too). Then the amorphous layer was repeatedly

heated up to 40000 K and then cooled with cooling rate of 1000 K/ps, while the

diamond layers remained frozen.

As a result of these procedures, the fraction of sp3-coordinated atoms in the amor-

phous layer increased to 88 %, the fraction of sp2-coordinated atoms decreased and

the sp-bonded atoms disappeared (see Figs. 9.5). sp3-bonded atoms formed a cluster

contained ∼25 atoms, which was identified as damaged diamond (see Fig.9.6). Figs.

9.7 and 9.8 show the radial and angular distribution function of this cluster, they are

close to those of pure diamond. The band gap in the center of this cluster is 3.8 eV

(see Fig. 9.9).


Figure 9.5: Samples of amorphous carbon located between two layers of diamond.(a) initial sample, (b) sample compressed to 3.9 g/cc. Red balls are sp3-coordinated

atoms, blue balls are sp2-coordinated atoms and green balls are sp-coordinatedatoms.� ��$%� � $ � � � � � � (a)

� � � � � � ��#$ � ��&��#��$%� � �� !�� !� � ��

sp2 �� ' � sp3 �� 3.9 g/cc

$ � ��!�� ' � � � � � � � � (b)�

� � � &�� ' �


Figure 9.6: Damaged diamond cluster found in the compressed sample.' � � � � � � � � � � $%� ��!�� !��

Figure 9.7: Radial distribution function of the damaged diamond cluster formed inamorphous carbon layer located between layers of diamond (black line) compared

with the radial distribution functions of a pure diamond crystal (red line)��#$ � ��&�� "$ � � �� $%�� !� � �� $ ��&�� $ � �� !�� g(r)

�

� � � �� g(r) � � � ��


Figure 9.8: Angular distribution function of the damaged diamond cluster formed inamorphous carbon layer located between layers of diamond (black line) compared

with the angular distribution functions of a pure diamond crystal (red line)��#$ � ��&!�#�"$ � � �� $%��!�� !� � �� $%��&�� $%� ��!�� !�� g(θ)

�

� � � � � � � �� g(θ) � � � ��

Figure 9.9: Density of states of the damaged diamond cluster.� � � � � � �� $ � ��!�

Chapter 10

Results: Quantum confinement

In this chapter we study quantum confinement effects in nanodiamond cluster, i.e

dependence of electronic properties (width of a band gap) on size in these nanostruc-

tures. In the present simulation in order to exclude the surface reconstruction effects

or need to terminate the dangling bonds with hydrogen, the diamond clusters and

the diamond layers will be surrounded by an amorphous carbon phase. This type of

passivation has not been previously used for a quantum confinement investigation of

nanoparticles, althouth this situation is more realistic than nanodiamonds in vacuum.

10.1 Quantum confinement effects in cubic nan-

odiamond cluster.

To study quantum confinement effects in nanodiamond clusters, we built a number

of diamond clusters embedded in amorphous carbon network in the same way as

described in section 9.1. The samples were generated at different temperatures. These

samples contained 216 carbon atoms at density of 3.5 g/cc. Initially, the samples were

98

CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 99

arranged as perfect diamond. Then an inner cubic cluster of 8 diamond atoms was

frozen, i.e. the motion of atoms in the cluster was forbidden. The remaining outer

envelope was heated up to 10000-25000 K. Once the liquid phase reached equilibrium,

the temperature was immediately decreased to room temperature with a cooling rate

of 500 K/ps.

The structure of the resulting samples depended on the heating temperature. In

the process of heating the inner frozen diamond cluster exerted a force in the direction

of returning adjacent atoms to their initial positions. Therefore these atoms could

not move as quickly as the atoms at the edges of the sample. Hence after cooling

atoms close to the diamond cluster could return to their initial pure diamond position.

As the temperature of heating increased, the atoms could escape further from their

initial diamond position, and the possibility of returning to their diamond position

decreased. When the heating temperature was lower, more carbon atoms returned to

their initial diamond position after cooling, and the remained diamond cluster grew

to a larger size. Hence by changing the heating temperature we could change the size

of the remaining diamond cluster.

For example at a heating temperature lower than 12000 K, the entire sample

reconstructed its diamond structure. At 12000 K, only a few atoms at the edges of

the sample changed their positions and bonding state from sp3-, to sp2. At 25000 K

the entire sample except for the 8 frozen atoms, turned into amorphous carbon with

56 % of sp3-, 43 % of sp2 and 1 % of sp-bonded atoms. Clearly, the percentage of

sp3-bonded atoms in the amorphous envelope decreased as the heating temperature

increased (see Fig. 10.1).

To study the quantum confinement effects, the local density of states of the central

diamond cluster was measured. The results do not show quantum confinement effects:


Figure 10.1: Amorphous carbon samples with diamond cluster inside, generated atdifferent temperatures of heating: a) at 12000 K, b) at 14000 K, c) at 22000 K. Redballs are sp3-coordinated atoms, blue balls are sp2-coordinated atoms, green balls

are sp-coordinated atoms, frozen atoms are marked by yellow color.� $ � � � � $ � � � � � �!� � � � �� $ � � � �� !� � � � � � � �� !� � ��

sp2 �� ' � sp3 �� (c) 22000 K

�(b) 14000 K

�(a) 12000 K

�

� � � �� ' � � ��

� � �!�� ' � sp �� &�� ' �

when the size of the inner diamond cluster decreased, the band gap of the frozen

diamond cluster also decreased (see Tab. 10.1). It is worth mentioning that in the

smallest clusters the band gap is not a very smooth function of a diamond cluster

sizes, the reason being the different structure and different fractions of differently

bonded atoms in the amorphous region adjacent to the cluster.

The local density of states was also measured for other atoms. The band gap

decreased from the center of the diamond cluster to its boundary and further to the

amorphous carbon phase near to the edges of the sample. Fig.10.2 shows local density

of states of three atoms in the sample generated at 13000 K, one of them (number 1)

inside the diamond cluster, the second (number 2) inside the diamond cluster close to

the boundary with an amorphous phase, and third (number 3) inside an amorphous

carbon phase. The band gap of the atom number 1 is 4.6 eV, the band gap of the


Heating Number of atoms sp3− Bandtemperature in the diamond cluster fraction gap (eV)

10000 216 100 5.412000 209 95 5.413000 62 70 4.614000 53 57 3.815000 22 50 0.517000 15 49 1.220000 8 54 0.822000 8 52 025000 8 56 1.1

Table 10.1: Width of the band gap of the central diamond clusters for the samplesgenerated at different heating temperature.

$ � � � � � �� $%� � � �� &��!� � � � � � � � �� $ � � � � � � � � � �

atom number 2 is 2.3 eV and the band gap of the third atom is 1.1 eV. We can see

as the location of atoms changes from near the diamond core to the amorphous edges

of the sample new electronic states appears inside the diamond band gap, until the

band gap shrinks to the band gap of amorphous carbon.

We explain the absence of the quantum confinement effects by the influence of an

amorphous carbon envelope. At high temperatures of heating the diamond cluster is

so small, that each atom of the cluster bonded with an atom from amorphous carbon

envelope, which can be sp2-bonded atoms. This decreases the quantum confinement

effects in these samples. The second reason could the presence of very large stresses

in our configuration.


Figure 10.2: Local densities of states of atoms in the sample generated at 13000 K:within the frozen diamond cluster (atom 1, black line), in the boundary of diamondcluster (atom 2, red line) and in the amorphous carbon (atom 3, green line). The

insert a) shows the location of the atoms 1,2 and 3, the insert b) shows themagnified part of the density of states near the band gap.

� � � � � � �!�� $ � � 13000 K � �� $%� � � � � �� $ �� $ � �!� ��

� � $%� � � � � � � � � �� !� � �� !��

2�1�� $%� � �� (a)

� ' � � � � � ��

� � � �!� � � �� !� � ��

� �� $ � �!� �� $%� � ��!� (b)� ' � � � � � �� 3 ��

�

� � � � � � ��


10.2 Quantum confinement in diamond layers lo-

cated between two layers of amorphous car-

bon.

Two different samples of nanodiamond sheets surrounded by amorphous carbon sheets

were generated and described in the MSc thesis of Anastassia Sorkin [75]. These

samples with a density of 3.5 g/cc were initially arranged as a perfect diamond crystal.

Their sizes are 2 × 2 × 6 (192 atoms), and 2 × 2 × 7 (224 atoms) diamond unit cells.

The 64 central atoms of each sample were frozen, i.e. the motion of these atoms was

forbidden. The remaining upper and lower layers were heated up to temperatures of

14000-30000 K. Once the hot liquid layers reached equilibrium, they were cooled to

the room temperature of 300 K by a cooling rate of 10 K/fs. After cooling the hot

layers remained partially or entirely amorphous with the presence of three- and two-

fold coordinated atoms in the structure. In this way the layer of diamond located

between two layers of amorphous carbon was constructed.

The thickness of this diamond frozen layer in all three samples was 7.1 A (64

atoms). In order to study the quantum confinement effects and exclude the influence

of an individual structure of amorphous phase we changed the thickness of the dia-

mond layers,while keeping the amorphous part unchanged. To enlarge the thickness

of the pure diamond layer we inserted new diamond layers into the center of the di-

amond layer as shown in Fig.10.3. To decrease the thickness of the diamond layers,

one or two of them were cut out from the sample.

For example, the sample containing 192 atoms, was cut down to 160 atoms, and

enlarged to 224 and 256 atoms. This sample contained 19 % of sp3-coordinated atoms


Figure 10.3: The samples of pure diamond located between two layers of amorphouscarbon, b) the initial sample contained 192 atoms (64 of them are frozen diamond),a) the sample where one diamond layer was cut out, the new sample contain 160atoms, c) the sample where one diamond layer was inserted in the center of thesample, the new sample contain 224 atoms, d) the sample where three diamond

layers were inserted in the center of the sample, the new sample contain 288 atoms.Yellow atoms are initially frozen diamond and inserting diamond layers, greenatoms are amorphous layers (the ”green” part is the same for each sample).

� � � � (b� � �� $ � ��&��#��$%� � �� $ � �� $ � ��&��

��#$�� &�� (a� � � � � � � � � � � � � � � � � 64

� � �� 192� ��&�� $%� � $ � �

� � � � � ��#$�� &�� (c� � �� 160

� ��&�� $ � � � � � � �

� � ' ��& �� $ � ��&��#��$%� � � � � � (d� � �� 224

� ��&�� ' ��& � � �� ' � � ' ��& � � � � ��

�

� �� 288� ��&��

��& � � � $ � � � $ � � � �!� � $ � ��&�� !�� ' � $ � � �� $ � ��&��


Number Thickness of Band gap of Band gap inof atoms the diamond layer (nm) the entire sample (eV) the diamond layer (eV)

160 0.9 2.15 5.4192 1.25 2.33 5.4224 1.6 2.38 5.4288 2.3 2.38 5.4

Table 10.2: Width of the band gap of the entire sample and the central diamondlayer for the samples with 19 % of sp3-coordinated atoms in an amorphous layers.��

19 %� � �� $%��&�� $%��&�� & � � � ��

$ � � �� $ � ��&�� sp3 ��

in amorphous layers. Tab. 10.2 presents the density of states of each of the resulting

samples with different thickness of the diamond layer and local density of states of the

atoms in the central diamond part. The results do not show quantum confinement

effects, i.e. the band gap of the atoms inside the diamond layer does not increase

with decreasing of the thickness of the diamond layer. In contrast to that the band

gap of the entire sample increased as the thickness of the diamond layers increased,

and after 1.5 nm, stabilized on 2.38 eV, while the band gap in the central diamond

layers does not depend on the size of the diamond layers and remains 5.4 eV, as in

perfect diamond.

The second sample contained 26 % of sp2-coordinated atoms which was enlarged

from 224 to 256 and 288 atoms, shows the same results (see Tab.10.3).

Fig.10.4 shows the local density of states of different atoms in the sample with

256 atoms and 26 % of the sp3-bonded atoms in amorphous layers. As in the case

with diamond cluster the band gap shrinks from 5.4 eV in the central diamond layer

(which is equal to that of pure diamond) to 0.9 eV in the upper or lower edges of the

sample which is equal to the band gap of the entire sample.


Figure 10.4: Local densities of states of the atoms from sample of 256 atoms, withinthe frozen diamond layer (atom 1, black line), near the boundary between diamond

and amorphous carbon layers (atom 2, red line), and in the amorphous carbon(atom 3, green line). The insert a) shows the location of the atoms, the insert b)

shows the magnified part of the density of states near the band gap.� � � � � � $%��&�� $%� � � �� 256

� ��&!� � � � � �� #$ �� $ � ��!�� $%� � � � � � � � � �

�� !� � �� !� � � � � � � $ ��&!� � ��

2�1�� $%� � �� (a)

� ' � � � � � ��

� � � �!� � � �� !� � ��

� �� $ � �!� �� $%� � ��!� (b)� ' � � � � � �� 3 ��

�

� � � � � � ��


Number Thickness of Band gap of Band gap inof atoms the diamond layer (nm) the entire sample (eV) the diamond layer (eV)

224 1.03 0.87 5.4256 1.4 0.89 5.4288 1.74 0.89 5.4

Table 10.3: Width of the band gap of the entire sample and the central diamondlayer for the samples with 26 % of sp3-coordinated atoms in an amorphous layers.��

26 %� � �� $%��&�� $%��&�� & � � � ��

$ � � �� $ � ��&�� sp3 ��

We can derive the following conclusion from the above calculations. The diamond

clusters of 8-60 atoms surrounded by an amorphous carbon phase and diamond layers

with thickness of 1-2 nm sandwiched between an amorphous carbon layers are not

suitable for studying quantum confinement effects. The results disagree with the ab

initio calculations of Raty et al [33], who found quantum confinement effects in very

small (a few atoms) diamond clusters passivated by hydrogen. But in our calculation

we used a different passivation of dangling bonds (amorphous carbon). It worth

mentioning that both the samples of Raty et al (275 atoms) and our samples are

significantly smaller than those used in the experimental confirmation of the quantum

confinement effects in nanodiamonds of Chang et al [32] (4300 carbon atoms).

Chapter 11

Results: Nucleation in

hydrogenated carbon

It was experimentally shown that the optimal hydrogen concentration for diamond

nucleation in an a-C:H matrix during BEN is 25 % [55]. Y. Lifshitz et al suggests

that the role of hydrogen in diamond nucleation is threefold: (1) enhancement of the

probability of spontaneous nucleation of a diamond cluster and its stabilization, (2)

assistance of annealing of defects in faulty diamond crystallites and (3) preferential

displacement of a-C atoms by energetic hydrogen species leading to diamond crystal-

lite growth. In this set of simulations we study nucleation of diamond in hydrogenated

amorphous carbon network. The calculations are carried out with different content

of hydrogen. Density and cooling rate will be also varied.

108

CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 109

11.1 Computational details

The samples used in this stage of our calculations were initially arranged as a perfect

diamond crystal of a size 3 × 3× 3 unit cells (i.e. 216 carbon atoms) with a density

of 3.3 g/cc. The diamond was melted at a temperature of 8000 K during 5, 10, 15,

20 and 25 ps (as in Chapter 8). The 5 liquid carbon samples were rapidly cooled

with a cooling rate of 500 K. In order to generate the samples of amorphous carbon

with different densities each of the configurations were isotropically compressed by

changing the volume of the unit cell to the densities of 3.5 and 3.9 g/cc. 5-25 hydrogen

atoms were placed in each sample. Pairs of neighboring bonded carbon atoms were

randomly chosen and the hydrogen atoms were accommodated on the bond line in

an equal distance of these carbon atoms. The volume of the simulation box remained

unchanged, hence the resulting density of the sample slightly increased as the number

of H atoms increased. Then the hydrogenated samples were repeatedly heated up to

6000 K and cooled to room temperature with three different cooling rates: fast (1000

K/ps), intermediate (500 K/ps) and slow (200 K/ps).

11.2 Structure of hydrogenated amorphous carbon

network

The typical structures of resulting a-C:H samples generated at initial density of 3.9

g/cc with fast cooling rate with different content of hydrogen atoms are presented

in Fig. 11.1. Detail counts of differently bonded atoms as a function of hydrogen

concentration in these samples are listed in Tab.11.1. It is clearly seen that hydrogen

atoms systematically reduce the number of sp3-bonded carbon atoms with 4 carbon


Figure 11.1: a-C:H structures with different content of hydrogen atoms. The red,blue and green balls are the carbon atoms with four, three and two C-C bonds(excluding C-H bonds) respectively. Hydrogen atoms are represented by large

light-blue balls.� ��

�

$ � � � � � � �� $%� � ��& � a-C:H��

� �� & $ � �� &�

� � $ � � � � �� & � � � �� ' � � ��& � ��' � � � � � � ��

�

� � �� ' � � ��

neighbors. For example in the sample with 5 hydrogen atoms there are 186 fourfold

carbon atoms with four carbon neighbors, while in the sample with 25 hydrogen

atoms there are 151 such atoms. At the same time the number of carbon atoms

with 3 carbon and 1 hydrogen neighbors increased from 5 to 21 when the number of

hydrogen atoms increased from 5 to 25. It is clear that the hydrogen atoms breaks

the C-C bonds and replaces carbon atoms. It is interesting to note that the total

fraction of sp3-bonded atoms (C atoms with 4 C-C bonds, C atoms with 3 C-C and 1

C-H bonds and C atoms with 2 C-C and 2 C-H bond) is not a monotonous function

of hydrogen concentration. Only a few fivefold carbon (C with 4 C-C and 1 C-H bond

and C with 3 C-C and 2 C-H bonds) atoms were found, they appeared only in the

samples with 25 hydrogen atoms.


Number of H atoms in the a-C:H sample 0 5 10 15 25

Number of C atoms with 4 C-C bonds and 0 C-H bonds 190 186 172 150 151Number of C atoms with 3 C-C bonds and 0 C-H bonds 24 22 32 47 31Number of C atoms with 2 C-C bonds and 0 C-H bonds 2 3 2 4 5Number of C atoms with 4 C-C bonds and 1 C-H bonds 0 0 0 0 2Number of C atoms with 3 C-C bonds and 1 C-H bonds 0 5 9 13 21Number of C atoms with 3 C-C bonds and 2 C-H bonds 0 0 0 0 1Number of C atoms with 2 C-C bonds and 1 C-H bonds 0 0 1 2 4Number of C atoms with 2 C-C bonds and 2 C-H bonds 0 0 0 0 1

Full number of sp3-bonded C atoms 190 191 181 163 172

Full number of sp2-bonded C atoms 24 22 33 49 35

Full number of sp-bonded C atoms 2 3 2 4 5

Table 11.1: Average number of differently bonded carbon atoms in the a-C:Hsamples generated at 3.9 g/cc and with cooling rate of 1000 K/ps.

3.9$ � ��!�� $%� � a-C:H ��

� � � �� !� � � � � � � �� ' �� !� � � �� g/cc

We can see that the structures contain large sp3 clusters that do not contain

hydrogen, hydrogen escapes from these pure sp3-clusters (see Fig. 11.2). This sp3-

clustering can be considered to be an initial stage of precipitation of diamond clusters.

Hydrogen atoms decorate the surface of the sp3-clusters and stabilize them.

The radial distribution function of the sample contained 25 hydrogen atoms is

drawn in Fig.11.3. On the same figure we draw the radial distribution function of

C-C bonds only. The entire radial distribution function has two low additional peaks

located at ∼1.13 and ∼1.86 A, these are the distances from H atoms to nearest and

second nearest carbon atom respectively. The high peaks located at 1.52 A and 2.4

A represent the nearest carbon neighbors and second nearest carbon neighbors for

carbon atoms. We did not find a detectable dependence of C-C and C-H bondlengths

on the number of hydrogen atoms in the sample. The average C-C bondlength and

average C-H bondlength were equal for all hydrogenated amorphous carbon samples


generated at 3.9 g/cc and the fast cooling rate. We can see this in Fig.11.4, where

the radial distribution functions of the hydrogenated amorphous carbon contained 5

and 25 hydrogen atoms are drawn. The locations of all peaks are coincident.

The density of states of the samples generated at 3.9 g/cc with fast cooling rate

is drawn on Fig. 11.6 for different number of hydrogen atoms. The presence of

hydrogen decreases the band gap. For example, without hydrogen, an amorphous

carbon sample with 90 % of sp3 coordinated atoms shows a band gap of 3.7 eV, while

the band gap of the hydrogenated amorphous carbon with 5 hydrogen atoms and 86%

of sp3-bonded carbon atoms is 1 eV only. The samples with 10-25 hydrogen atoms

did not show any band gap.

We also calculated local densities of states of differently bonded atoms. Fig.11.5

shows the local densities of states of three atoms in the sample containing 25 hydrogen

atoms, two of them sp3-bonded (one with 4 C-C bonds and the second with 3 C-C

and one C-H bond), and third is sp2-bonded with three carbon neighbors. Both of

sp3-bonded atoms show the broad band gap despite the fact that one of them has

C-H bond. The band gap of the threefold atom is almost zero. Hence we can deduce,

that hydrogen bonded with sp3 coordinated atoms don’t reduce the band gap of the

sample significantly. The band gap in the samples with large numbers of hydrogen

atoms closes due to increase of sp2-bonded carbon atoms.

11.3 Diamond nucleation in the hydrogenated car-

bon network

We found only a few cases when sp3-clusters generated within our hydrogenated

amorphous carbon samples resembled diamond. These damaged diamond clusters


Figure 11.2: a-C:H structure with 25 hydrogen atoms generated with high (3.9 g/cc)density and intermediate (500 K/ps) cooling rate (a) and disordered pure sp3-cluster

found within this sample (b).�� 3.9 g/cc

$ � �!� �� $%� � � �� 25� ��&!� � a-C:H ��

� � � ��

(b)� � � � � � $%� �� !� � �� sp3 �� (a)

�� !� � �

Figure 11.3: Radial distribution function of hydrogenated amorphous carboncontained 25 hydrogen atoms generated at 3.9 g/cc with intermediate cooling rate

(red line) compared to that for C-C bonds only (black line).� � �� 3.9 g/cc

$ � �!� �� $ � � � �� 25� ��&�� a-C:H ��

��g(r)

�

� � � � � � � � � ��C-C ��

��!� � ��g(r)

� � � ��


Figure 11.4: Radial distribution function of hydrogenated amorphous carbongenerated at 3.9 g/cc contained 5 hydrogen atoms (red line) and contained 25

hydrogen atoms (black line).��

g(r) ��

3.9 g/cc$ � ��!�� $%� � � �� 5

� ��&�� a-C:H ��

g(r)�

� � � � � � � � � �� 25� ��&�� a-C:H

Figure 11.5: Local density of states of the atoms from a-C:H structure with 25hydrogen atoms: C atom with 4 C-C bonds (black line), C atom with 3 C-C and 1

C-H bond (red line), C atom with 3 C-C bonds (blue line).� � � � �� 25

� ��&!� � a-C:H �� #$%�� $ � �!� ��

�� C-C� ��!� � � � � ��

C� � � � ��

C-C� ��!� � � ��

C�

� � � � & � � �(sp2)

� ��C-C

� ��!� � � � � �� C� � � � ��

C-H


Figure 11.6: Density of states of hydrogenated amorphous carbon samples at 3.9g/cc, with 0 hydrogen atoms (black line), with 5 hydrogen atoms (red line) and with25 hydrogen atoms (blue line). The insert shows a magnified part of the density of

states near the band gap.� �� 0

� ��&�� 3.9 g/cc

$ � ��!�!� � �� $ � � a-C:H �� $ � �!� ��

� � ��

� � � � & � � � � �� 25� ��&�� 5

� ��&!� � � � � � � � � � �

�

� � � � � � �� !�� $ � ��!� �� $%� � ��!� � ' � � �


were smaller than those in the sample without hydrogen (see Chapter 8). The clus-

ters appeared in the samples with 5-15 hydrogen atoms at all densities and cooling

rates, no diamond clusters were found in the samples containing 25 hydrogen atoms.

Fig. 11.7 shows the sample with 10 hydrogen atoms generated with low density (3.5

g/cc) at fast (1000 K/ps) cooling rate. A diamond cluster containing 22 carbon atoms

was found within this sample (see Fig.11.8). Note that the cluster is free from hy-

drogen atoms. Radial and angular distribution functions of the clusters are drawn on

Fig.11.9, and Fig.11.10. The first peak of the radial distribution function is located

at 1.53 A, which is very close to that of diamond 1.54 A. The peak of the angular

distribution function is located at 111◦, this is also close to diamond one of 109◦. The

density of states inside this cluster (see Fig.11.11) shows a band gap of ∼4 eV, which

is narrower than the band gap of perfect diamond at 3.5 g/cc (5.4 eV). The reason

for this may be the strong influence of an outer amorphous envelope.

11.4 Varying density and cooling rates.

We explored the influence of different cooling rates and different densities on the

structure of our hydrogenated amorphous carbon samples. As in the case of amor-

phous carbon without hydrogen the fraction of sp3-bonded atoms decreased as the

density decreased (see Tab.11.2). Hence in the samples with lower density the prob-

ability of forming large sp3 clusters was lower. A systematic influence of cooling rate

on sp3-content and probability of diamond nucleation was not found. However as in

Chapter 8, in the structures generated at intermediate (500 K/ps) and slow cooling

rates (200 K/ps), graphitic configurations formed (see Fig.11.12). These graphitic

configuration appeared more frequently at low density (3.5 g/cc). The orientation of


Figure 11.7: a-C:H structure with 10 hydrogen atoms generated with low (3.5 g/cc)density and fast (1000 K/ps) cooling rate.

�� 3.5 g/cc$ � �!� �� $%� � � �� 10

� ��&!� � a-C:H ��

�

�!� � � � � � ��

Figure 11.8: Diamond cluster contained 22 carbon atoms found in the sample fromFig.11.7.

� 11.7� � �� $%� � ��!�� 22

� ��&�� !��


Figure 11.9: Radial distribution function of diamond cluster inside hydrogenatedamorphous carbon, contained 10 hydrogen atoms generated at 3.5 g/cc with fastcooling rate (black line) compared with the radial distribution function of pure

diamond (red line).$ � ��!�� $%� � � � �� 10

� ��&�� a-C:H �� !�� g(r)

�

� � � � � � � �� g(r) � � � �� !� � � ��

3.5 g/cc

Figure 11.10: Angular distribution function of diamond cluster inside hydrogenatedamorphous carbon, contained 10 hydrogen atoms generated at 3.5 g/cc with fastcooling rate (black line) compared with the angular distribution function of pure

diamond (red line).$ � ��!�� $%� � � �� 10

� ��&�� a-C:H �� !�� g(θ)

�

� � � � � � � �� g(θ) � � � �� !� � � ��

3.5 g/cc


Figure 11.11: Density of states of diamond cluster inside hydrogenated amorphouscarbon, contained 10 hydrogen atoms generated at 3.5 g/cc with fast cooling rate

(black line) compared with the density of states of pure diamond (red line).� �� 10

� ��&�� a-C:H �� !�� $ � ��!�

� �� $ � ��!�� 3.5 g/cc

$ � ��!�� $%� �

�

� � � � � � � � � ��

the graphitic planes was random.

Fig.11.13 shows the radial distribution function of the entire sample compared

with that of C-C bonds only for the graphitic sample contained 25 hydrogen atoms.

An average C-H bondlength in the graphitic configurations varied from ∼1.02 A,

∼1.1 A, as the density increased from 3.9 g/cc to 3.5 g/cc. An interplanar distance

was shorter than in perfect graphite: 2.1 A and 2.5 A for the samples generated at

3.9 and 3.5 g/cc respectively, but our densities are much higher than that of perfect

graphite (2.2 g/cc). The peak of the angular distribution function of the graphitic

configuration is located at 120◦ (see Fig.11.14) which is typical of perfect graphite.


Number of Density fast intermediate slowH atoms (g/cc)

5 3.5 68,77,63,71,71 69,68,graphite,61,58 73,74,67,graphite,655 3.9 83,80,88,86,86 80,90,77,80,86 78,85,57,88,8225 3.5 51,54,67,62,57 51,61,61,56,51 67,graphite,49,53,graphite25 3.9 70,73,67,69,67 71,72,68,69,71 75,71,67,73,70,graphite

Table 11.2: Fraction of carbon atoms with 4 C-C bonds (sp3) in the samples with 5and 25 hydrogen atoms generated with different densities and different cooling rates.� � �� &�� a-C:H ��

�� sp3 � � � � � ��&!� � ��!� � � ��

� �� $%� � ��!�!� � �� $%� � � �� 25 ��

Figure 11.12: Graphitic configuration contained 10 hydrogen atoms generated at 3.9g/cc with slow cooling rate (a), one of the damaged graphitic planes (b).

� � �� 3.9 g/cc$ � ��!�� $ � � � � �� 10

� ��&�� !� � � � ��

� (b)� �"� � � �� (a)

� � ��


Figure 11.13: Radial distribution function of hydrogenated graphite contained 10hydrogen atoms generated at 3.5 g/cc with slow cooling rate (red line) compared to

that for C-C bonds only (black line).�� 3.5 g/cc

$ � ��!�� $%� � � � �� 10� ��&�� !�� g(r)

�

� � � � � � � � � ��C-C ��

�� g(r)

� � � �� !� � �

Figure 11.14: Angular distribution function of hydrogenated graphite contained 10hydrogen atoms generated at 3.5 g/cc with slow cooling rate.

�� 3.5 g/cc$ � ��!�� $%� � � �� 10

� ��&�� !�� g(θ)� � �� !� � �

Chapter 12

Results: Liquid-liquid carbon

phase transition

A number of the simulations carried out for this thesis showed that under extreme

pressures after cooling of the very hot liquid carbon, a graphitic phase can form

(see Chapters 8,11). We observed such graphitic configurations under application

of both hydrostatic and uniaxial pressure, and also in hydrogenated carbon. This

graphitic phase appeared more often at slow and intermediate cooling rates than at

fast cooling rates. Snapshots of the configurations made during the cooling process

showed that the graphitic phase formed at early stages of the simulation while the

temperature was still very high. For example, in the simulation of hydrogenated

amorphous carbon with 15 hydrogen atoms slowly cooled from 6000 K to 300 K, the

configuration contained planes built mostly of sp2-bonded carbon atoms was formed

at 4650 K (see Fig.12.1).

This appeared to suggest the presence of a structural phase transition in liquid

carbon which can occur at high temperatures and high pressures. In order to check

122

CHAPTER 12. RESULTS: LIQUID-LIQUID CARBON PHASE TRANSITION 123

Figure 12.1: Snapshots of hydrogenated liquid carbon at 3.9 g/cc with 15 H atomsat different temperatures in the process of cooling. Red balls represent fourfold

coordinated carbon atoms, blue balls represent threefold coordinated carbon atomsand green balls represent twofold coordinated carbon atoms. Large light-blue atoms

are hydrogen atoms.� �� $ ��& � � $ 3.9 g/cc$ � �!� �� 15

� ��&!� � �� $%� � � $ � � � ��$

sp �� &�� ' � sp2 �� ' � sp3 ��

�

� � �� ' � � �� & $ � �� &

�

� � �!�� ' �

this assumption 5 simulations of liquid carbon at 6000 K and at density of 3.9 g/cc,

which correspond to pressure of 180 GPa were carried out. Indeed in one simulation

out of 5 the liquid phase transformed to mostly sp2-bonded carbon planes after 35 ps

of simulation. The configurations in the other simulations appeared to be disordered

liquid carbon up to 60 ps of simulations (two weeks of calculations). We did not ob-

served a phase transition in two additional constant-temperature simulations carried

out at 5000 K and two simulations carried out at 7000 K.

We observed carefully the process of the transition occuring in liquid carbon in

the successful simulation. During the first 28000 fs the configuration held at 6000

K remained liquid carbon with a low content of sp2-coordinated atoms (∼ 20-25 %).

This value is in good agreement with tight-binding simulations of liquid carbon at this

density carried out by Morris et al [99]. Then the sp2-fraction suddenly increased


(see Figs.12.2 and 12.3) up to 45-50 % during 500 fs. In the second step of this

transition the sp2-fraction continued to increase slowly and a carbon atoms began

to form planes as in graphite. Finally the fraction of sp2-coordinated atoms in this

hot phase stabilized on 60-70 %. The atoms continued to move very quickly, they

changed their location within a plane constantly and sometimes jumped to another

plane. But the planes did not disappear (at least during the 30 ps of our simulation)

despite the very high temperature. The structure of one of such plane is shown in

Fig. 12.4. Most of the atoms in this plane formed hexagons as in graphite, but many

sp3-bonded atoms remained in the structure. Therefore we called this structure liquid

graphite. Radial (see Fig. 12.5) and angular (see Fig. 12.6) distribution functions

of this liquid graphite were broad as in liquid carbon. The first peak of the radial

distribution function moved to the shorter distances during the transition, which

notes to the increasing fraction of sp2 bonded atoms. The final location of the peak

was at ∼1.43 A, which is close to graphitic bondlength (1.42 A). It is worth to

mention that the first peak of the radial distribution function corresponded to in-

plane interatomic distances. The interplanar distance was ∼2.2 A, which is much

shorter than in uncompressed graphite (3.5 A). This distance integrated with the

distance to second nearest in-plane neigbors to one broad second peak of the radial

distribution function. The peak of angular distribution function locates at graphitic

angle of 120◦.

This graphitic structure also did not disappear after cooling of the structure to

room temperature. In contrast, the in-plane structure became more ordered and

turned out to be damaged graphite. The sp2-fraction continued to increase in the

process of cooling and reached almost 100 % (see Figs.12.7, 12.8 and 12.9).


Figure 12.2: Percentage of sp2-coordinated atoms in liquid carbon at 3.9 g/cc and6000 K as a function of time.

6000 K � � � � ��!� � � 3.9 g/cc$ � �!� �� sp2 �� $ � � $

Figure 12.3: Snapshot of liquid carbon at 6000 K and 3.9 g/cc. Red balls representfourfold coordinated carbon atoms, blue balls represent threefold coordinatedcarbon atoms and green balls represent twofold coordinated carbon atoms.

sp3 �� 6000 K � � � � � �!� � � 3.9 g/cc

$ � ��!�� #$ �� $ � � � ��$

�

� � �!�� ' � sp �� &�� ' � sp2 �� ' �


Figure 12.4: Structure of one of the carbon planes in the sample of liquid graphite.�� !��

Figure 12.5: Radial distribution function of liquid graphite (black line) compared tothe radial distribution of the liquid sample before the phase transition (red line).� �� !� � �� g(r)

� � � �� g(r)� � � � � � � �


Figure 12.6: Angular distribution function of liquid graphite (black line) comparedto the angular distribution of the liquid sample before the phase transition (red line).� �� !� �� g(θ)

� � � �� g(θ)� � � � � � � �

Figure 12.7: Percentage of sp2-coordinated atoms in liquid graphite as a function oftime in the process of cooling.

� � �� $ � �� sp2 �� $ � � $


Figure 12.8: Structure of liquid graphite before cooling (a) and after cooling (b).� (b)

� � �� (a)� � �� !� �� !��

Figure 12.9: Structure of one of the carbon planes in the sample of cooled liquidgraphite.

� � ��


A structural first order phase transition in liquid carbon was previously experimen-

tally suggested by M. Togaya [100] and later predicted by Glosly and Ree [101] in their

calculations with the Brenner bond order potential. This phase transition is charac-

terized by sudden transition from sp2-coordinated liquid carbon to sp3-coordinated

liquid carbon at a pressure of 5-15 GPa and temperature about 6000-9000 K. Re-

cently, Wang et al [102] carried out ab initio calculations of carbon phase diagram

and did not find evidence for a first order liquid-liquid phase transition at these con-

ditions. Morris et al [99] also simulated liquid carbon at 6000 K in a wide range of

densities (1.5-4.2 g/cc). They did not observe the graphitization during 700 fs, while

our simulations lasted up to 60 ps.

The question about the presence of structural phase transition in liquid carbon

remains controversial. The above calculations could shed light to this problem. How-

ever, as we found, at a temperature range of 5000-7000 K the probability of the

successful transition is not large (one out of 5 two weeks simulations), therefore fur-

ther calculations of this process would require more powerful computers.

Chapter 13

Summary and discussion

In this thesis we presented the simulations of the process of high pressure high tem-

perature diamond nucleation by rapid cooling of the compressed liquid phase. The

conditions applied to the simulations performed here are ”unrealistic” in the sense

that they cannot be reached in the laboratory in any controlled way. However similar

conditions may prevail within the ”thermal spike” caused by the energy transferred

to the atoms of the slowing down medium during the stopping process. The local

temperatures within the volume contained in the spike (when converted to an effective

temperature from the kinetic energies imparted to recoiling host atoms) may reach

thousands of degrees, resulting in local melting of the material. The lifetime of this

molten state is of the order of pico-seconds. It is likely, though difficult to estimate

qualitatively, that at the non-equilibrium conditions occuring during the thermal spike

high, non-equilibrium, pressures may prevail. Hence the results obtained here may

simulate those obtained during bias-enhance-nucleation, i.e. they can be compared

to results obtained from experiments in which energetic carbon ions are used to cre-

ate diamond nano-crystallites that serve as nucleation centers for subsequent CVD

130

CHAPTER 13. SUMMARY AND DISCUSSION 131

diamond growth.

In the Chapter 8 of this thesis we have described the simulation of the formation

of diamond under compression (both isotropic and uniaxial) by rapid quenching of

liquid carbon with different densities and different cooling rates. The samples gen-

erated in this way were predominantly ta − C amorphous carbon with ordered sp3

clusters inside. The clusters were identified as cubic diamond. These clusters were

characterized by computing the radial and angular distribution functions, which were

found to be close to those of perfect cubic diamond at the same density. The band

gap inside the diamond crystallites was found to be somewhat narrower than that

of perfect cubic diamond. At slower cooling rates (200 K/ps) graphitic clusters were

formed.

It is interesting to note that the application of uniaxial pressure did not lead to the

formation of lonsdaleite crystallites, but as described above, all samples appeared to

be cubic diamond. However the differences between lonsdaleite and cubic diamond are

small and difficult to observe in a small crystallites. The diamond clusters obtained in

hydrogenated amorphous carbon are smaller than in the cases of amorphous carbon

pure from hydrogen. We also observed growth of diamond on small diamond cluster

embedded in amorphous carbon network.

As it was mentioned above, the present calculations resemble the simulations

of Wang and Ho [6, 7]. However no diamond clusters were found in their work

at a densities higher than 3.5 g/cc. We can see two reasons for that: first, the

precipitation of diamond clusters is a random process and if the number of realizations

is small, the probability of generating an amorphous carbon sample with diamond

cluster inside is not large. The second reason is that it is difficult to identify a

small diamond crystallite in a large amorphous sp3 bonded carbon network without


modern vizualization tools. Sometimes, in order to locate the diamond structure the

sample has to be sliced and the crystallite can be identified only from specific viewing

directions. The AViz [95] (Atomic Visualization package) which we used in this work

possesses all the possibilities to identify the diamond clusters (rotation, slicing of the

samples, color labelling of different types of atoms).

The following trends can be deduced from the present results. (i) The probability

of precipitation of diamond crystallites increases with density. (ii) The probability

of diamond precipitation increases as the cooling rate increases. At slower cooling

rates some samples transform to graphite. (iii) No hexagonal diamond was found

even when uniaxial pressure was applied.

These trends are in qualitative agreement with experimental results of the bias-

enhanced nucleation picture [69] and also with the trends observed in detonation

diamond nucleation [50], where increasing pressure (density) and faster cooling rates

leads to a higher diamond fraction in the detonation soot. In contrast, the probability

of transformation to graphite increases with slower cooling rate. In the cases when

the samples were compressed in one direction, the orientation of graphitic planes is

parallel to the direction of compression. For the case of homogeneous compression

in all three directions the orientation of the graphitic planes obtained for the slow

cooling rate is random.

Yao and co-workers [69] have observed, in high resolution TEM studies on bias

enhanced nucleation of diamond, that well ordered graphitic planes, oriented per-

pendicular to the ion-implanted surface seem to be the precursor of the formation

of nano-diamond nuclei. They proposed that some of the incoming carbon atoms

are channeled between these planes. Further implantation of carbon atoms leads to

the densification of the carbon phase resulting in the formation of a dense volume of


amorphous sp3 bonded carbon atoms that eventually coalesce to form diamond nano-

crystallites. These resemble the graphitic phase and diamond nucleation described

in the Chapter 8. The densities required to lead to the formation of graphite and

diamond in our simulation differed from those observed by Yao et al . However, as dis-

cussed above, this discrepancy may be attributed to the finite size effects observable

in simulations of small samples.

In the second stage of this study (Chapter 9) we simulated growth of diamond

on diamond clusters and on diamond surfaces occurring within amorphous carbon

networks under high pressure. The sample containing diamond seed embedded in

an amorphous carbon network was generated by heating following by quenching of

the sample while the central diamond core was kept frozen. Then the samples were

compressed and annealed at 1000 K during 15 ps. We observed as in the process of

annealing new carbon atoms joined the diamond core and the diamond seed grew.

We found that the growth of diamond is more favorable at higher pressures. The

growth of the diamond seed occurred in all directions. The attempt to grow diamond

on compressed diamond layer surrounded by amorphous carbon layers also was suc-

cessful. The diamond cluster containing ∼30 atoms grew on the diamond surface

after heating of the sample up to 40000 K and following rapid cooling. In contrast

to our observations, Yao et al [69] found in his orthogonal tight-binding calculations

that the growth is energetically more favorable and efficient for the less dense matrix.

However the conditions of the simulations of Yao et al (range of densities, the sizes

of diamond seed and amorphous matrix) differed from those of ours.

Further (Chapter10), we studied the quantum confinement effects in diamond clus-

ters and diamond layers surrounded by amorphous carbon network generated by the

same way as the samples from Chapter 9. We did not find the quantum confinement


neither in the diamond clusters nor in the diamond layers. The band gap of atoms

located inside the diamond clusters and the diamond layers remained unchanged as

the size of this crystallite decreased. Furthermore, the band gap of the entire sample

decreased as the size of the diamond crystallite or the width of the diamond layer

decreased. The local densities of states of atoms inside the diamond phase but close

to the boundary with amorphous envelope showed new states appearing inside the

diamond band gap. The band gap of atoms inside the amorphous envelope is close to

that of entire sample. Based on these calculations we can conclude that the diamond

crystallites are not suitable for studying of quantum confinement effects when their

surface is passivated by amorphous carbon. This type of passivation has not been

previously used for a quantum confinement investigation of nanoparticles. Moreover,

there is as yet no experimental evidence of quantum confinement in diamond on these

size scales (a few atoms). Our samples are significantly smaller than those used in

the experimental confirmation of the quantum confinement effects in nanodiamonds

of Chang et al [32] (4300 carbon atoms).

In the next stage of our study (Chapter 11) we simulated precipitation of diamond

crystallites in compressed hydrogenated amorphous carbon. Samples of hydrogenated

amorphous carbon were prepared by heating and rapid cooling (as in Chapter 8). We

varied the hydrogen contents in the sample as well as the pressure (density) and

cooling rates. Most of the samples generated in this way were ta−C:H, i.e. mostly

sp3-bonded. We found that most of the hydrogen atoms bonded mainly with carbon

atoms which have three carbon neighbors. The number of carbon atoms with four

carbon neighbors smoothly decreased as the hydrogen content increased, while the

total number of sp3-coordinated carbon atoms was is not a monotonic function of hy-

drogen concentration. All the samples contained large pure sp3-clusters, the hydrogen


atoms having been expelled from them. Some of the sp3-clusters were identified as

damaged diamond. The probability of precipitation of diamond crystallites in our

simulations decreased as the hydrogen content increased. The diamond clusters gen-

erated inside hydrogenated amorphous carbon network are smaller and of the worse

quality than those formed without hydrogen atoms. This is in disagreement with ex-

perimental results of Y. Lifshitz at al [55], who argued that the probability of diamond

nucleation in hydrogenated matrix is higher. But our densities are much higher and

hydrogen content is lower than in experimental results and simulations of Y. Lifshitz.

As in the samples from Chapter 8 the sp3 content in our sample increased as the

density increased. At slow cooling rates the graphitic configuration also appeared in

hydrogenated carbon samples.

The appearance of graphite configuration in the samples from Chapters 8 and 11,

was unexpected for us. We found that this graphitic phase formed at early stages

of cooling, when the temperature was still very high. We proposed that a structural

phase transition occurs in liquid carbon under high pressure and high temperature.

In order to explore this assumption we carried out a number constant-temperature

simulations of liquid carbon under high pressure (Chapter 12). Indeed, in one sample

held at 6000 K and 3.9 g/cc, the atoms formed graphitic planes. This transition

was sharp enough, the sp2 fraction jumped from 25 to 55 % during 500 ps. The

structural first order phase transition in liquid carbon under high pressure is not a

new phenomenon. It was previously experimentally suggested by M. Togaya [100] and

later predicted by Glosly and Ree [101] in their calculations with the Brenner bond

order potential, but still remains controversial. Unfortunately, the probability of the

phase transition under our conditions is very low, the search of the exact conditions

and other characteristics of the transition require very long calculations on powerful


computers, and could not be carried out in the framework of this thesis.

Appendix A

OXON

OXON (Oxford Order N package) [94] (see also Chapter 6) is a set of programs for

carrying out atomistic static and dynamics calculations using potentials which are

based on tight-binding methods.

Before running this program, files containing tight-binding parameters and molec-

ular dynamics parameters need to be prepared. A ”CONSTANTS” file contains

tight-binding parameters for atoms and bonds. All molecular dynamics and k-space

parameters are inserted in ”INPUT”.

The running OXON program creates an ”OUTPUT” file that contains general

information about the final configuration of a simulation. It also contains the averages

of certain quantities from a molecular dynamics run (such as average energy and

average temperature).

Typical samples of the programs ”CONSTANTS”, ”INPUT” and ”OUTPUT” are

listed below.

137

APPENDIX A. OXON 138

########################################################################################

# C O N S T A N T S

########################################################################################

#A bond type is defined in terms of two atoms labelled by their chemical symbols.

#The atomic number runs from 1 to 103. All lengths are in Angstrom. The parameters are

#based on the Goodwin, Skinner and Pettifor form for scaling (Goodwin L, Skinner A J,

#Pettifor D G 1989, Europhys Lett vol 9, p 701):

# S(r) = (r0/r)^n * exp(n*((r0/rc)^nc - (r/rc)^nc))

#A cubic tail is automatically added between r1 and rcut (see Xu C H, Wang C Z, Chan C T

#and Ho K M, 1992, J.Phys:Condens Matt, vol 4, p 6047).

#

#NOTE: putting n=0 results in only an exponential term being used

# (S(r) = exp((r0/rc)^nc - (r/rc)^nc))

# putting nc=0 results in a pure power law being used.

#

#The same scaling form is used for the hopping integrals and the repulsive term.

#

#The parameters for the repulsive term are given first.

#The parameters for the hopping integrals are given a pair of angular momenta at a time.

#The angular momenta are indexed according to their position in the list in BOP.atomdat.

########################################################################################

H H

########################################################################################

# l1 l2 r0 rc r1 rcut n nc sigma pi delta

########################################################################################

2.3010 0.3561 1.0600 1.2200 1.0200 0.8458 0.0546

1 1 2.1393 0.7103 1.1000 1.2200 0.4495 1.5650 -0.441 0.000 0.000

########################################################################################

H C

########################################################################################


########################################################################################

1.0840 1.5474 1.5500 1.8500 1.4080 3.5077 11.4813

1 1 1.0840 1.2011 1.5500 1.8500 .5663 3.1955 -6.523 0.000 0.000

1 2 1.0840 1.2011 1.5500 1.8500 .5663 3.1955 6.811 0.000 0.000

########################################################################################

C C

########################################################################################


########################################################################################

1.6400 2.1052 2.5700 2.6000 3.3030 8.6655 8.18555

1 1 1.5363 2.1800 2.4500 2.6000 2.0000 6.5000 -5.000 0.000 0.000

1 2 1.5363 2.1800 2.4500 2.6000 2.0000 6.5000 4.700 0.000 0.000

2 1 1.5363 2.1800 2.4500 2.6000 2.0000 6.5000 4.700 0.000 0.000

2 2 1.5363 2.1800 2.4500 2.6000 2.0000 6.5000 5.500 -1.550 0.000

##############################################################################

# I N P U T #

##############################################################################

#DEFINING THE TYPE OF ENERGY EVALUATOR.


#

#POTFLG is set so as to determine how the energies and forces will be

#evaluated. The options are:

# 0 => Mixed Potentials

# 1 => Empirical Potential

# 2 => Bond Order Potential

# 3 => Cluster Method

# 4 => K Space

# 5 => Global Density of States Method

# 6 => Parallel Bond Order Potential

# 7 => Fermi Operator Expansion Method

# 8 => Density Matrix Method

# 9 => Parallel K Space Method

###############################################################################

POTFLG

4 0

##############################################################################

#SETTING UP THE UNIT CELL.

#

#A is the directions of the three primitive lattice vectors. The coordinates

#are cartesian, and the lengths should be normalised to 1. The program will

#perform the normalisation if they are not normalised here.

#

#LEN is the lengths of the three primitive lattice vectors. The units are

#Angstroms. The primitive translation vectors are generated by multiplying

#these lengths into direction vectors (A above).

#

#ND is the number of mobile atoms. A mobile atom is one that can be moved, and

#which contributes to the total energy, and for which forces are calculated.

#These are to be contrasted with the inert atoms (see below).

#

#D is the positions of the mobile atoms in the unit cell in units of lattice

#vectors. The format is ’l,m,n,s,a,b’ where ’l,m,n’ are the coordinates, ’s’ is

#the chemical symbol, ’a’ is an additional force flag, and ’b’ is a block index.

#If XFRCFLG = 1, then ’a’ determines what extra forces are to be used,

#according to the following table:

# a=0 ==> no extra forces are added.

# a=1 ==> a local thermostat is applied. This thermostat requires a

# relaxation time TAU (see below). The temperature is given by

# TEMP.

# a=2 ==> the atom is not allowed to move.

# a=3 ==> piston dynamics are applied (see XFRCFLG for parameters).

#If MVFLG=1 and RLXFLG = 4 then the value of ’b’ determines which block an

#atom belongs to for the block relaxation. Atoms in the same block are moved

#together.

#If POTFLG=0 then the value of ’b’ determines which potential is used to

#evaluate the energy of which atom. The values of ’b’ is the block to which the

#atom belongs. Each block is then treated with its own potential.

#The coordinates ’l,m,n’ must lie in the range (0,1).

#The chemical symbol consists of two characters in the usual way. The program

#is case sensitive, so the first character must be uppercase and the second


#one lower case. For elements like hydrogen, that have only one letter in the

#chemical symbol, the symbol must be upper case. The chemical symbol must

#always be followed by a space.

#

##############################################################################

A

1.0000 0.0000 0.0000

0.0000 1.0000 0.0000

0.0000 0.0000 1.0000

LEN

3.5500 3.5500 3.5500

ND

8

D

0.0000 0.0000 0.0000 C 0 1

0.25000 0.25000 0.25000 C 0 1

0.0000 0.50000 0.50000 C 0 1

0.25000 0.75000 0.75000 C 0 1

0.50000 0.0000 0.50000 C 0 1

0.75000 0.25000 0.75000 C 0 1

0.50000 0.50000 0.0000 C 0 1

0.75000 0.75000 0.25000 C 0 1

##############################################################################

#BUILDING ATOM LISTS.

#

#RCUT is the cutoff radius used to set up the neighbor lists. It is the radius

#about any atom that defines the volume in which the nearest neighbors of that

#atom are taken to be. The distance is in Angstroms. The second number is the

#cutoff for the extended EAM terms.

#

###############################################################################

RCUT

3.0 3.0

##############################################################################

#GENERAL ATOM MOVER PARAMETERS.

#

#MVFLAG is the flag that dictates the type of atomic mover to be used:

#-2 : Fit Tight Binding parameters. (K-space and Tersoff potential only)

#-1 : Convergence test.

# 0 : Interactive analysis of a configuration (no atomic motion).

# 1 : Structural relaxation. (Select relaxation method with RLXFLG below)

# 2 : NVE Molecular Dynamics.

# 3 : NVT Molecular Dynamics.

# 4 : NPT Molecular Dynamics.

# 5 : Simulated annealing.

# 6 : Elastic constants.

# 7 : Diffusion barrier height.

# 8 : Scan energy of a cluster through a volume.

# 9 : Produce force constant matrix. (*** Under development ***)

#10 : Gamma surface in units of mJ/m^2.

#11 : NVE Molecular Dynamics with shear in x direction.


#

#MXITER is the maximum number of iterations alloed when moving atoms.

#

##############################################################################

MVFLAG

3

MXITER

1

##############################################################################

#SIMULATED ANNEALING.

#

#TMAX is the highest (initial) temperature used in a simulated annealing run.

#The temperature is in Kelvin.

#

#TMIN is the lowest (final) temperature used in a simulated annealing run. The

#temperature is in Kelvin.

#

#NOTE: the constraints can be applied here as in the static relaxation. See

#CNST_N and CNST_A above. Further, equilibration at TMAX can be accomplished

#by setting NEQUIL (see below) to some appropriate value. The MD time step

#(DT) must also be set. See below.

##############################################################################

TMAX

7000.0

TMIN

300.0

##############################################################################

#GENERAL MOLECULAR DYNAMICS PARAMETERS.

#

#DT is the time step for Molecular Dynamics simulations in units of fs

#(1fs = 10^-15s).

#

#TEMP is the temperature to be used for NVT and NPT MD simulations. Units are

#Kelvins.

#

#TTOL is the tolerance in the temperature. When the temperature fluctuates

#by more than TTOL*TEMP from TEMP during NVT simulations, then the atomic

#velocities are automatically rescaled to regain the correct temperature.

#

#PRESS is the pressure to be used in NPT MD simulations. The units are bars.

#

#MPISTON is the mass of the piston used in NPT calculations. At low pressures

#a value of 1 should work. At higher pressures larger values should be used to

#prevent the temperature and pressure from behaving erratically.

#

#NEQUIL is the number of iterations to be used to equlibrate the system. No

#configurations or averages are recorded during this time, but the trace and

#monitor still operate.

#

#TAU is the relaxation time (in fs) for the local thermostat that can be

#to single atoms. This only applies for MVFLAG=2.


##############################################################################

DT

1.0

TEMP

300.0

TTOL

0.1

PRESS

1.0

MPISTON

100

NEQUIL

7000

TAU

0.01

##############################################################################

#K-SPACE PARAMETERS.

#

#KDMETH is only applicable to the parallel K-space program (MVFLAG=9)

# First entry : 0 = Equal division of k-pts over nodes.

# 1 = User-specified division of k-pts.

# Remaining entries : Number of k-pts per node.

#

#KBASE determines the basis used for the positions of the k-points.

# 0 = cell basis in units of reciprocal lattice vectors.

# 1 = Cartesian basis

#

#NK is the number of k points used.

#

#KPTS is the set of k points.

#

#WTK is the weight assigned to each k point. This allows for symmetry.

##############################################################################

KDMETH

0

2 2

KBASE

0

NK

1

KPTS

-0.25 -0.25 0.0

WTK

1.0

*******************************

* O U T P U T *

*******************************

======================================================

Number of atoms in central cell = 8

Total number of atoms = 1530


Average number of nearest neighbours = 29.000

Total number of electrons per unit cell = 32.000

======================================================

K # 1 : ( -.25000 , -.25000 , .0000 )

Eigenvalues and occupancies :

-31.030 2.0000 -24.924 2.0000 -24.853 2.0000

-22.026 2.0000 -19.692 2.0000 -17.062 2.0000

-16.971 2.0000 -15.951 2.0000 -15.047 2.0000

-15.018 2.0000 -14.408 2.0000 -14.329 2.0000

-14.294 2.0000 -12.352 2.0000 -11.689 2.0000

-9.9815 2.0000 -2.0429 .0000 -.60099 .0000

-.46360 .0000 .44601 .0000 .49811 .0000

1.7156 .0000 2.2240 .0000 2.3709 .0000

2.4050 .0000 2.4399 .0000 2.6573 .0000

2.7120 .0000 3.9395 .0000 3.9999 .0000

4.0396 .0000 6.3107 .0000

========================================================

Fermi energy = -3.0000 eV

Electron temperature = .10000E-01 eV

========================================================

Occupancy = 32.000

On site energy = -258.250345952951 eV

Bond energy = -301.003764326435 eV

Band structure energy = -559.254110279387 eV

Repulsive energy = 201.134883471566 eV

Atomic band energy = -300.606552698095 eV

Electron entropy = .00000000000000 eV

Kinetic energy = .271435500000000 eV

Total energy = -57.2412386097251 eV

Average energy = -7.15515482621564 eV

Total energy (kT=0) = -57.2412386097251 eV

Average energy (kT=0) = -7.15515482621564 eV

========================================================

Consistency check:

Sum of eigenvalues = -246.976000000000

Sum of onsite energies = -246.976000000000

Number of electrons = 32.0000000000000

Occupancy = 32.0000000000000

===============================================

Atomic Parameters.

==================

Z Charge Es Ep Ed C1 C2 C3 C4

C 4.0 -12.7430 -6.0430 .0000 .5721 -.0018 .0000 .0000

Bond energy parameters.

=======================

Z1 Z2 Vsss Vsps Vpss Vpps Vppp Vsds Vdss Vpds Vdps Vpdp Vdpp Vdds Vddp Vddd Phi0

C C -5.000 4.700 4.700 5.500 -1.550 .000 .000 .000 .000 .000 .000 .000 .000 .000 8.186

Bond scaling parameters.

=======================

C C

l1 l2 r0 rc r1 rcut n nc


1.640 2.105 2.570 2.600 3.303 8.665

1 1 1.536 2.180 2.450 2.600 2.000 6.500

1 2 1.536 2.180 2.450 2.600 2.000 6.500

2 1 1.536 2.180 2.450 2.600 2.000 6.500

2 2 1.536 2.180 2.450 2.600 2.000 6.500

Band structure forces (eV/A):

.66179 .16067 -1.6859

-.36606 -.27064 1.1928

-.52157 -.58550E-01 -.47108

.21865 -.25303 .82561

.53022 -.82085E-01 -1.4633

.17561 .25867E-01 1.1794

-.38430 .15084 -.37880

-.31434 .32693 .80128

Pair potential forces (eV/A):

-.60261 -.43317 .33655

.23577 .19549 .20215

.51438 .37568E-01 -.83413

-.17055 .62625 .32456

-.52435 .30045E-01 .39152

-.45920 -.81165E-01 .23235

.46041 -.39831 -.98191

.54617 .23295E-01 .32891

Total forces (eV/A).

.59178E-01 -.27250 -1.3494

-.13029 -.75143E-01 1.3950

-.71964E-02 -.20982E-01 -1.3052

.48098E-01 .37322 1.1502

.58696E-02 -.52041E-01 -1.0718

-.28359 -.55298E-01 1.4117

.76109E-01 -.24748 -1.3607

.23182 .35022 1.1302

Atomic positions and velocities.

C 3.5484 .49929E-02 .68704E-02 -.16602E-02 .52841E-02 .68428E-02

C .89071 .88868 .88005 .34140E-02 .12411E-02 -.74510E-02

C .25796E-02 1.7752 1.7787 .27685E-02 .19416E-03 .34457E-02

C .88383 2.6572 2.6626 -.39294E-02 -.55339E-02 .63702E-03

C 1.7746 .67373E-03 1.7756 -.44104E-03 .70862E-03 .16118E-03

C 2.6674 .88818 2.6547 .52146E-02 .71440E-03 -.78668E-02

C 1.7759 1.7797 .32683E-02 .10024E-02 .49627E-02 .29708E-02

C 2.6565 2.6554 .88823 -.63690E-02 -.75711E-02 .12603E-02

=================================

Total CPU time = .14013E-44s

=================================

Appendix B

Computer program of data

handling

This Fortran program calculates the number of sp3 and sp2 coordinated atoms in an

amorphous carbon samples. This program uses as output coordinates of atoms which

can be taken from the ”OUTPUT” file of an OXON program. It calculates the number

of neighbors which separated by a distance no longer than 1.9 A for each atom, the

radial and angular distribution functions of diamond clusters found in the sample.

Then the number of neighbors is written in the ”COORDNUM” file. The data can

be used for preparing of XYZ sample for AVIZ visualization with color coding for

different atomic bonding. The typical sample of the XYZ file is also presented in this

section. The radial and angular distribution functions are written in the ”RADIAL”

and ”ANGULAR” files respectively.

145

APPENDIX B. COMPUTER PROGRAM OF DATA HANDLING 146

program CLUSTERS

real ax(250),ay(250),az(250)

character ff*2

character gg*1

integer i,j,cl,mb,mbb,aa,bb,cc,p,ii

integer drdf,dadf

integer jj,kk,l,num

integer k(250),neib(250),misp(500),misp1(500)

real x,y,z

real sqrr(250,250)

real dx(250),dy(250),dz(250)

real bond,dd,co,co1,co2,ang,di

cl=0

rcut=1.9

open (1,file=’OUTPUT’,status=’old’)

open (2,file=’CLUSTER’,status=’new’)

read (1,300) gg

read (1,100) qq,ww,ee



read (1,301) ff

mb=0

mbb=0

num=0

drdf=100

dadf=50

dd=4.0/drdf

di=180.0/dadf

do 10 i=1,216

read (1,100) x,y,z

if (x.ge.0.0) then

if (x.le.10.65) then

if (y.ge.0.0) then

if (y.le.10.0) then

if (z.ge.0.0) then

if (z.le.10.0) then

cl=cl+1

write (2,117) i,cl,x,y,z

endif

endif

endif

endif

endif

endif

10 continue

close(1)

close(2)

open (3,file=’CLUSTER’,status=’old’)

open (4,file=’COORDNUM’,status=’new’)

open (6,file=’NEIGHBORS’,status=’new’)

open (7,file=’ANGLES’,status=’new’)


do 15 i=1,cl

read (3,117) n,cl,ax(i),ay(i),az(i)

write (4,100) ax(i),ay(i),az(i)

15 continue

do 20 i=1,cl

k(i)=0

do 30 j=1,cl

if (i.ne.j) then

dx(j)=(ax(i)-ax(j))

dy(j)=(ay(i)-ay(j))

dz(j)=(az(i)-az(j))

sqrr(i,j)=sqrt(dx(j)*dx(j)+dy(j)*dy(j)+dz(j)*dz(j))

if (sqrr(i,j).le.4.0) then

mbb=mbb+1

endif

if (sqrr(i,j).le.rcut) then

k(i)=k(i)+1

neib(k(i))=j

mb=mb+1

write (6,*) mb,i,j,sqrr(i,j)

endif

endif

30 continue

if (k(i).ge.2) then

do 400 j=1,k(i)-1

jj=neib(j)

do 700 l=j+1,k(i)

kk=neib(l)

co1=(dx(jj)*dx(kk)+dy(jj)*dy(kk)+dz(jj)*dz(kk))

co2=sqrr(i,jj)*sqrr(i,kk)

co=co1/co2

ang=acos(co)*180/3.14159

num=num+1

write(7,118) num,i,jj,kk,ang

700 continue

400 continue

endif

20 continue

close(3)

close(4)

close(6)

close(7)

do 40 i=1,drdf

misp(i)=0

40 continue

do 45 i=1,dadf

misp1(i)=0

45 continue

open (10,file=’COORDNUM’,status=’old’)

open (11,file=’RADIAL’,status=’new’)

open (12,file=’ANGLES’,status=’old’)


open (13,file=’ANGULAR’,status=’new’)

do 50 i=1,mbb

read (10,*) aa,bb,cc,bond

p=int(bond/(dd))+1

misp(p)=misp(p)+1

50 continue

do 70 i=1,num

read (12,118) ii,aa,bb,cc,ang

p=int(ang/(di))+1

misp1(p)=misp1(p)+1

70 continue

do 60 i=1,drdf

write (11,270) i*dd,misp(i)/2

60 continue

do 80 i=1,dadf

write (13,*) i*di,misp1(i)

80 continue

close(10)

close(11)

close(12)

close(13)

100 format (2x,f24.21,f24.21,f24.21)

270 format (1x,f7.3,i4)

300 format (/A1)

301 format (/A2)

118 format (1x,I4,1x,I3,1x,I3,1x,I3,1x,f20.10)

117 format (I3,2x,I3,2x,f24.21,f24.21,f24.21)

end

This is a sample of XYZ-file for amorphous carbon containing 8 atoms. Threefold

coordinated atoms are denoted by a symbol ”C3”, fourfold coordinated atoms are

denoted by a symbol ”C4”.

8

#Am

C4 8.292579650878906250000 6.949803829193115234375 9.274624824523925781250

C3 5.658650875091552734375 5.958342075347900390625 0.095109775662422180176

C3 8.473687171936035156250 9.957231521606445312500 2.699503421783447265625

C3 10.407974243164062500000 1.722601294517517089844 6.091104507446289062500

C4 9.756943702697753906250 9.259089469909667968750 9.779501914978027343750

C4 5.922797203063964843750 3.246033668518066406250 7.691087245941162109375

C3 5.439003467559814453125 6.295923233032226562500 8.237220764160156250000

C4 10.397355079650878906250 3.916403293609619140625 7.317611217498779296875

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