Computer Simulation of the Nucleation of Diamond from Liquid Carbon Under Extreme Pressures

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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 interatomic 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 amorphous 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 functions 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 simulations [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) compared 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) compared 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 cooling 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) compared 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 diamond 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 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...... 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 contained 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). . 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 distribution 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 hydrogen 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 coordinated 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 coordinated 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 transition (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 compared 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 diamond 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 amorphous 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 amorphous layers...... 107

11.1 Average number of diﬀerently bonded carbon atoms in the a-C:H samples 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 rôle 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 clusters 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 incorporates 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. 1 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 L¨owding orbitals α atomic orbital index

Hiα,jβ hopping integral n > single particle eigenfunction | (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 materials (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◦280 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 F 41/d 3¯ 2/m =

7 F d3m = Oh. 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 CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 8

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].

4 The crystal structure is described by a hexagonal lattice with the D6h (P 63/mmc) space group.

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

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sp2 sp3 CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 9

Figure 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).

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<100> <210>

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Property Graphite Diamond Lattice constant (RT) [A]˚ 2.462 6.708 3.567 Bond length (RT) [A]˚ 1.421 1.545 Atomic density [cm−3] 1.14 1023 1.77 1023 × × Thermal conductivity [W/cm-K] 30 0.06 25 Debye temperature [K] 2500 950 1860 Electron mobility [cm2/V-sec] 20 103 100 1800 × Hole mobility [cm2/V-sec] 15 103 90 1500 × Melting point K 4200 4500 Band gap [eV] -0.04 5.47

Table 2.1: Properties of diamond and graphite.

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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 CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 11

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 technique 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. CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 12

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

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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 structure 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 CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 14

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 coordination 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 amorphous 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. CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 15

Figure 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.

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A3 A4 [7] ta-C g(θ) [6] a-C g(r)

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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. CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 17

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.

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

calculated for samples containing 64 atoms.

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64 g(r) g(r)

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

[13]

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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. CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 20

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. CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 21

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

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[17] a C − CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 22

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 material. 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 conditions 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 hydrogen 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. CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 23

Muon spin resonance (µSR) measurements [24] indicate that two paramagnetic forms of muonium exist: the “normal” muonium (Mu), with an isotropic hyperﬁne interaction, 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]. CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 24

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 phenomenon results from electrons and holes being squeezed into a dimension that approaches 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 CHAPTER 2. DIAMOND AND OTHER ALLOTROPES OF CARBON 25

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 temperature, 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 K¨ampe (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 CHAPTER 3. DIAMOND SYNTHESIS 28

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 CHAPTER 3. DIAMOND SYNTHESIS 29

(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 electrical 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 CHAPTER 3. DIAMOND SYNTHESIS 30

Figure 3.1: Schematic representation of the belt apparatus.

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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 detonation. 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. CHAPTER 3. DIAMOND SYNTHESIS 31

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 and graphite-to-lonsdaleite transitions. Note: diamond type-C is cubic diamond, H is

hexagonal diamond (lonsdaleite).

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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 rôle 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 CHAPTER 3. DIAMOND SYNTHESIS 32

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

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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]. CHAPTER 3. DIAMOND SYNTHESIS 33

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 CHAPTER 3. DIAMOND SYNTHESIS 34

Figure 3.3: Schematic diagram of the microwave plasma CVD apparatus, taken

from [51].

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[51] CVD

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 amorphous 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 eventually 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, CHAPTER 3. DIAMOND SYNTHESIS 35 i.e. during this time a small region of the material experiences very high temperatures 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 amorphous 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 approach 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 authors simulated the transformation of 1H (AAA...) graphite to hexagonal diamond CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 38

Figure 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]

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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 energy, 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). CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 39

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 diamond 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. CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 40

Figure 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].

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[61] CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 41

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 energy and interatomic forces. These calculation showed that diamond nucleation in the absence of hydrogen can occur by precipitation of diamond clusters in a dense amorphous 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 CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 42 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 densities (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 coordinated 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 CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 43

Figure 4.3: The sp3 fraction plotted as a function of density calculated by diﬀerent 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 simulations [75] (indicated by “my

simulations”).

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“my simulations” [75] tight-binding CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 44 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 CHAPTER 4. PREVIOUS SIMULATIONS OF DIAMOND NUCLEATION 45

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 laboratory. 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 rˆole 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 procedure is similar to that occurring within a “thermal spike” during the bias-enhanced nucleation process. We hope to observe ordered diamond clusters inside the amorphous 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 amorphous carbon network under different pressures (differing in both direction and magnitude). 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 analyzed 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 amorphous 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 diamond 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 structural 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 rôle 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 structures: 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 describing 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 CHAPTER 6. TIGHT-BINDING MODEL 51 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 (diamond, 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 CHAPTER 6. TIGHT-BINDING MODEL 52 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 describe 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 L¨owdin orbitals [85]. They have been constructed 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 is equal to H the sum of the atomic Hamiltonians and a term int which describes the interaction H between the diﬀerent atoms. We further assume the interaction between the atoms to CHAPTER 6. TIGHT-BINDING MODEL 53 be weak so that can be diagonalized by perturbation theory. In this approximation H the unperturbed Hamiltonian is simply H0

= h (~r ~r ) (6.2) 0 X l jl H j,l − 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:

1 Φ = exp(i~r ~k)φ (~r ~r ), (6.3) ml~k X jl ml jl √N j ∗ −

where N is the number of primitive unit cells in the crystal. The eigenfunctions Ψk

of can be written as linear combinations of Φmlk: H

Ψ = C Φ . (6.4) k X ml ml~k ml

To calculate the eigenfunctions and eigenvalues of , we operate on Ψk with the H

Hamiltonian = + int. From the orthogonality of the Bloch functions we H H0 H obtain a set of linear equations in Cml:

(H 0 0 E δ 0 δ 0 )C 0 0 (~k) = 0, (6.5) X ml,m l ~k mm nn m l ml −

where Hml,m0l0 denotes the matrix element < Φ ~ Φ 0 0~ > and E~ are the eigen- mlk|H| m l k k values of . When we substitute the wave function Ψ ~ deﬁned in (6.3) into (6.5) H mlk we obtain

N N exp[i(~rjl ~rj0l0 ) ~k] H 0 0 (~k) = ml,m l X X − ∗ j j0 N ∗

< Ψ ~ (~r ~rjl) Ψ 0 0~ (~r ~rj0l0 ) >= (6.6) mlk − |H| m l k − N exp[i(R~ + ~r ~r 0 ) ~k] X j l l j − ∗ ∗

< ψ ~ (~r ~rjl) ψ 0 0~ (~r ~rj0l0 ) > mlk − |H| m l k − CHAPTER 6. TIGHT-BINDING MODEL 54

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 < ψ ~ (~r ~rjl) ψ 0 0~ (~r ~rj0l0 ) > can be expressed in mlk − |H| m l k − 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 approximations. 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 CHAPTER 6. TIGHT-BINDING MODEL 55 the ions. Within these approximations, the one-particle electronic part of the total Hamiltonian can be written in the form

Hˆ = Tê + Uêe + Uêi, (6.7)

where Tê is the kinetic energy operator of the electrons, Uêe and Uêi are the electron- electron and electron-ion interactions respectively. Following the notation of Horsfield et al [87], the single-particle Schrödinger equation is

Hˆ n = (n) n , (6.8) | i | i where n is a single particle (doubly occupied) eigenfunction, and (n) is the cor- | i responding eigenvalue. It has to be mentioned that the k dependency of n and | i (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

(L¨owdin) orbitals set n = C(n) iα (6.9) X iα | i iα | i 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

H C(n) = (n)C(n), (6.10) X iα,jβ jβ iα jβ where CHAPTER 6. TIGHT-BINDING MODEL 56

Hiα,jβ = iα Hˆ jβ (6.11) h | | i are the matrix elements and

(n) (m) X Ciα Ciα X n iα iα m = δn,m. (6.12) iα ≡ iα h | ih | i

The oﬀ-diagonal matrix elements Hiα,jβ = iα Hˆ jβ , for iα = jβ, are called h | | i 6 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 distance 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 (Löwdin) extend further than those (non-orthogonal) from CHAPTER 6. TIGHT-BINDING MODEL 57 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 Hˆ jβ (for iα = jβ) are ﬁtted to electronic band structure h | | i 6 of the equilibrium crystal phase, as calculated by more accurate ﬁrst-principle models [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 :

E = E + E = 2 (n) + E (6.13) binding bs rep X rep n(occ.)

where Ebs is the band energy and Erep is the repulsive potential, given as a sum of pair potentials. (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

E = f  φ(r ) , (6.14) rep X X ij i  j  CHAPTER 6. TIGHT-BINDING MODEL 58 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 properties of non-equilibrium structures, as amorphous solids or liquids, the hopping integrals 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]

n nc nc h(r) = h (r /r) exp n[ (r/rc) + (r /rc) ] , (6.15) 0 0 { − 0 } for the rescaling of the hopping integrals, and

m mc mc φ(r) = φ (d /r) exp m[ (r/dc) + (d /dc) ] (6.16) 0 0 { − 0 } 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. CHAPTER 6. TIGHT-BINDING MODEL 59

6.5 Force calculation

We can express the forces acting on the atoms in a compact form, by ﬁrst deﬁning the density matrix

ρ = C(n)C(n) (6.17) iα,jβ X iα jβ n(occ.) The cohesive energy thus becomes

E = 2 ρ H + U (6.18) tot X jβ,iα iα,jβ rep iα,jβ

The forces acting on the atoms are then obtained by diﬀerentiating the cohesive energy with respect to atomic positions, that is

∂Etot Fk = (6.19) − ∂rk ∂H ∂U = 2 ρ iα,jβ + rep  . (6.20)  X jβ,iα  − iα,jβ ∂rk ∂rk   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:

2 d riα ∂Ei Fiα = m 2 = (7.1) dt −∂riα

driα v α = (7.2) i dt

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

k−1 r (t + ∆t) = r (t) + r˙ (t)∆t + ∆t2 α f(t + [1 i]∆t), (7.4) i i i X i i=1 −

for the atomic positions, and

k−1 r˙ (t)∆t = r (t + ∆t) r (t) + ∆t2 β f(t + [1 i]∆t) (7.5) i i i X i − i=1 −

for the velocities. The coeﬃcients αi and βi satisfy the equation

k−1 1 (1 i)qα = , (7.6) X i i=1 − (q + 1)(q + 2) k−1 1 (1 i)qβ = q = 0, ..., k 2. X i i=1 − q + 2 − CHAPTER 7. NUMERICAL TECHNIQUES 62

k = 4 ( 1/24) 1 2 3 × αi 19 -10 3 βi 27 -22 7 γi 3 10 -1 δi 7 6 -1

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

second-order diﬀerential equation.

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These predicted values are then corrected from the value of f at a time t + ∆t (calculated from the predicted values themselves), using the expressions

k−1 r (t + ∆t) = r (t) + r˙ (t)∆t + ∆t2 γ f(t + [2 i]∆t), (7.7) i i i X i i=1 − for the atomic positions, and

k−1 (∆t)r˙ (t) = r (t + ∆t) r (t) + ∆t2 δ f(t + [2 i]∆t) (7.8) i i i X i − i=1 − 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 influence of the surface should be excluded. So our sample should be very large. Usual CHAPTER 7. NUMERICAL TECHNIQUES 63 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. CHAPTER 7. NUMERICAL TECHNIQUES 64

The initial velocities can be chosen randomly with a Maxwell distribution:

2 − mv f(v) e 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 1 NkT = mv2 (7.10) X i 2 i 2

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 interactions 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 CHAPTER 7. NUMERICAL TECHNIQUES 65

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 powerful 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. CHAPTER 7. NUMERICAL TECHNIQUES 66

In order to define the coordination number, we have calculated the radial distribution 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 distribution function g(θ). Here, the angles between the atom i and each pair of its nearest neigbors are accumulated in the list of angles. CHAPTER 7. NUMERICAL TECHNIQUES 67

7.8 Analysis of errors

The possible errors that can influence the results in computer simulation are: statistical errors, finite-size effects, unreliable generator of random numbers, unaccurate potential and numerical techniques, unsufficient time of simulation to reach equilibrium 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 random 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 CHAPTER 7. NUMERICAL TECHNIQUES 68 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:

s ∆x = (7.11) √k where s is a standard deviation:

v k u (x x)2 u i u iP=1 − s = u (7.12) t k 1 − and x is a simple arithmetic mean value:

1 k x = x . (7.13) X i k i=1 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 homogeneously 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 %). CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 71

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. CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 72

Figure 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.

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number Density (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 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.

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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. CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 74

Figure 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.

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Figure 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

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CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 76

400

300

200 g3

100

0 0 50 100 150 200 angle (degrees)

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).

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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 CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 77

200

150

100 g(r)

0 1 2 3 distance (Angstrom)

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).

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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. CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 78

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.

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CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 79

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 density

compared with the band gap of perfect diamond at the corresponding density.

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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 (ˆz) 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 compressed 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 ˆz direction to restore their cubic form (at 3.5 g/cc) and ﬁnally CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 80 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 obtained 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 CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 81

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

diamond cluster found within this sample (right).

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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 (ˆz) 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 ˆz 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 CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 82

15 g(r)

0 1 1.5 2 2.5 3 distance (A)

Figure 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) compared to the

radial distribution function of a pure diamond crystal (red line).

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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 CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 83

g3 20

0 50 100 150 angle (degrees)

Figure 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) compared to the

angular distribution function of a pure diamond crystal (red line).

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

with cooling rate of 1000 K/ps.

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3.8 g/cc 1000 K/ps CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 84

Figure 8.11: A graphitic conﬁguration generated at 3.7 g/cc with intermediate cooling 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.

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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 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.

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100 g(r)

0 1 2 3 distance (Angstrom)

Figure 8.12: Radial distribution function of the graphitic structure in the sample generated at 3.8 g/cc subjected to uniaxial pressure (black line) compared with the

radial distribution function of perfect graphite (red line).

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g(r) CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 86

150

100 g3

0 50 100 150 angle (degrees)

Figure 8.13: Angular distribution function of the graphitic structure in the sample generated at 3.8 g/cc subjected to uniaxial pressure (black line) compared with the

angular distribution function of perfect graphite (red line).

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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). CHAPTER 8. RESULTS: NUCLEATION OF DIAMOND UNDER PRESSURE 87

Figure 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.

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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 are

sp-coordinated atoms

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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 frozen

atoms, white balls represent the moving atoms.

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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 CHAPTER 9. RESULTS: GROWTH OF DIAMOND UNDER PRESSURE 91

Figure 9.2: The sample of amorphous carbon with embedded pure diamond cluster

(a) and the diamond cluster (b).

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(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). CHAPTER 9. RESULTS: GROWTH OF DIAMOND UNDER PRESSURE 92

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 relaxed amorphous carbon sample and the number of atoms in the grown diamond cluster

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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 CHAPTER 9. RESULTS: GROWTH OF DIAMOND UNDER PRESSURE 93

Figure 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).

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Figure 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).

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CHAPTER 9. RESULTS: GROWTH OF DIAMOND UNDER PRESSURE 94 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 amorphous 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). CHAPTER 9. RESULTS: GROWTH OF DIAMOND UNDER PRESSURE 95

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-coordinated

atoms.

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CHAPTER 9. RESULTS: GROWTH OF DIAMOND UNDER PRESSURE 96

Figure 9.6: Damaged diamond cluster found in the compressed sample.

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Figure 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)

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Figure 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 diamond crystal (red line)

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Figure 9.9: Density of states of the damaged diamond cluster.

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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: CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 100

Figure 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.

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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 CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 101

Heating Number of atoms sp3 Band temperature in the diamond cluster fraction− gap (eV) 10000 216 100 5.4 12000 209 95 5.4 13000 62 70 4.6 14000 53 57 3.8 15000 22 50 0.5 17000 15 49 1.2 20000 8 54 0.8 22000 8 52 0 25000 8 56 1.1

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

generated at diﬀerent heating temperature.

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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. CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 102

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 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.

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CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 103

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 twofold 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 diamond 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 diamond 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 CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 104

Figure 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).

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CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 105

Number Thickness of Band gap of Band gap in of atoms the diamond layer (nm) the entire sample (eV) the diamond layer (eV) 160 0.9 2.15 5.4 192 1.25 2.33 5.4 224 1.6 2.38 5.4 288 2.3 2.38 5.4

Table 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 amorphous layers.

19 %

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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. CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 106

Figure 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 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.

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256

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CHAPTER 10. RESULTS: QUANTUM CONFINEMENT 107

Number Thickness of Band gap of Band gap in of atoms the diamond layer (nm) the entire sample (eV) the diamond layer (eV) 224 1.03 0.87 5.4 256 1.4 0.89 5.4 288 1.74 0.89 5.4

Table 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 amorphous layers.

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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 rˆole 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 crystallite 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 CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 110

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.

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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. CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 111

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 190 186 172 150 151 Number of C atoms with 3 C-C bonds and 0 C-H bonds 24 22 32 47 31 Number of C atoms with 2 C-C bonds and 0 C-H bonds 2 3 2 4 5 Number of C atoms with 4 C-C bonds and 1 C-H bonds 0 0 0 0 2 Number of C atoms with 3 C-C bonds and 1 C-H bonds 0 5 9 13 21 Number of C atoms with 3 C-C bonds and 2 C-H bonds 0 0 0 0 1 Number of C atoms with 2 C-C bonds and 1 C-H bonds 0 0 1 2 4 Number 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:H

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

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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 CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 112

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 CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 113

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).

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3.9 g/cc 25 a-C:H

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¦ (b) sp3 (a)

Figure 11.3: Radial distribution function of hydrogenated amorphous carbon contained 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).

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3.9 g/cc 25 a-C:H g(r)

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C-C g(r) CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 114

Figure 11.4: Radial distribution function of hydrogenated amorphous carbon generated at 3.9 g/cc contained 5 hydrogen atoms (red line) and contained 25

hydrogen atoms (black line).

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g(r) ¦ 3.9 g/cc 5 a-C:H g(r)

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25 a-C:H

Figure 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).

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25 a-C:H

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C-C C C-C C

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2 (sp ) C-C C C-H CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 115

Figure 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.

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25 5

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CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 116 were smaller than those in the sample without hydrogen (see Chapter 8). The clusters 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 hydrogen 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 amorphous 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 probability 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 CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 117

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.

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3.5 g/cc 10 a-C:H

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

Fig.11.7.

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11.7 22 CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 118

Figure 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).

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10 a-C:H g(r)

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¨ ¥ ¡ ¨ ¡ ¥ !¡

g(r) 3.5 g/cc

Figure 11.10: Angular 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 distribution function of pure

diamond (red line).

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10 a-C:H g(θ)

¡ ¡ ¡ ¡

¨ ¥ ¡ ¨ ¡ ¥ !¡

g(θ) 3.5 g/cc CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 119

Figure 11.11: Density 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).

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10 a-C:H

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3.5 g/cc

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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. CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 120

Number of Density fast intermediate slow H atoms (g/cc) 5 3.5 68,77,63,71,71 69,68,graphite,61,58 73,74,67,graphite,65 5 3.9 83,80,88,86,86 80,90,77,80,86 78,85,57,88,82 25 3.5 51,54,67,62,57 51,61,61,56,51 67,graphite,49,53,graphite 25 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 5

and 25 hydrogen atoms generated with diﬀerent densities and diﬀerent cooling rates.

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a-C:H ¦ sp3

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¦ 25

Figure 11.12: Graphitic conﬁguration contained 10 hydrogen atoms generated at 3.9

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

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3.9 g/cc 10

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(b) (a) CHAPTER 11. RESULTS: NUCLEATION IN HYDROGENATED CARBON 121

Figure 11.13: Radial distribution function of hydrogenated graphite contained 10 hydrogen atoms generated at 3.5 g/cc with slow cooling rate (red line) compared to

that for C-C bonds only (black line).

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3.5 g/cc 10 g(r)

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Figure 11.14: Angular distribution function of hydrogenated graphite contained 10

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

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3.5 g/cc 10 g(θ)

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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 atoms at diﬀerent temperatures in the process of cooling. Red balls represent fourfold coordinated carbon atoms, blue balls represent threefold coordinated carbon atoms and green balls represent twofold coordinated carbon atoms. Large light-blue atoms

are hydrogen atoms.

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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 observed 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 CHAPTER 12. RESULTS: LIQUID-LIQUID CARBON PHASE TRANSITION 124

(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). CHAPTER 12. RESULTS: LIQUID-LIQUID CARBON PHASE TRANSITION 125

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

6000 K as a function of time.

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¦ 6000 K 3.9 g/cc sp2

Figure 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 coordinated

carbon atoms and green balls represent twofold coordinated carbon atoms.

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sp 6000 K 3.9 g/cc

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¦ ¦ 2 sp sp CHAPTER 12. RESULTS: LIQUID-LIQUID CARBON PHASE TRANSITION 126

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

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Figure 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).

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g(r) g(r)

CHAPTER 12. RESULTS: LIQUID-LIQUID CARBON PHASE TRANSITION 127

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

to the angular distribution of the liquid sample before the phase transition (red line).

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g(θ) g(θ)

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

time in the process of cooling.

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¦ sp2 CHAPTER 12. RESULTS: LIQUID-LIQUID CARBON PHASE TRANSITION 128

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

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(b) (a)

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

graphite.

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CHAPTER 12. RESULTS: LIQUID-LIQUID CARBON PHASE TRANSITION 129

A structural first order phase transition in liquid carbon 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. This phase transition is characterized 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 conditions. 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 further 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 temperature 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 create 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 generated 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 CHAPTER 13. SUMMARY AND DISCUSSION 132 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 perpendicular 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 CHAPTER 13. SUMMARY AND DISCUSSION 133 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 successful. 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 clusters 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 CHAPTER 13. SUMMARY AND DISCUSSION 134 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 hydrogen concentration. All the samples contained large pure sp3-clusters, the hydrogen CHAPTER 13. SUMMARY AND DISCUSSION 135 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 generated inside hydrogenated amorphous carbon network are smaller and of the worse quality than those formed without hydrogen atoms. This is in disagreement with experimental 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 CHAPTER 13. SUMMARY AND DISCUSSION 136 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 molecular 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 ######################################################################################## # l1 l2 r0 rc r1 rcut n nc sigma pi delta ######################################################################################## 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 ######################################################################################## # l1 l2 r0 rc r1 rcut n nc sigma pi delta ######################################################################################## 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. APPENDIX A. OXON 139

# #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 APPENDIX A. OXON 140

#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. APPENDIX A. OXON 141

# #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. APPENDIX A. OXON 142

############################################################################## 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 APPENDIX A. OXON 143

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 APPENDIX A. OXON 144

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 (1,100) qq,ww,ee 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’) APPENDIX B. COMPUTER PROGRAM OF DATA HANDLING 147

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’) APPENDIX B. COMPUTER PROGRAM OF DATA HANDLING 148

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”.

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