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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 COMPUTER SIMULATION OF THE NUCLEATION OF DIAMOND FROM LIQUID CARBON UNDER EXTREME PRESSURES ANASTASSIA SORKIN COMPUTER SIMULATION OF THE NUCLEATION OF DIAMOND FROM LIQUID CARBON UNDER EXTREME PRESSURES RESEARCH THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ANASTASSIA SORKIN SUBMITTED TO THE SENATE OF THE TECHNION | ISRAEL INSTITUTE OF TECHNOLOGY TISHREI, 5767 HAIFA OCTOBER, 2006 THIS RESEARCH THESIS WAS SUPERVISED BY DR. JOAN ADLER AND PROF. RAFAEL KALISH UNDER THE AUSPICES OF THE PHYSICS DEPARTMENT ACKNOWLEDGMENT I wish to express my gratitude to Dr. Joan Adler and Prof. Rafi Kalish for the excellent guidance and support during this research. I am grateful to all Computational Physics Group and especially to my husband Slava for their help during this research period. I am grateful to Prof. Y. Lifshitz and Prof. A. Hoffman for useful discussions. I thank to Prof. A. Horsfield and Prof. M. Finnis for providing us with the OXON package. THE GENEROUS FINANCIAL HELP OF THE TECHNION IS GRATEFULLY ACKNOWLEDGED Contents Abstract xix List of symbols 2 1 Introduction 4 2 Diamond and other allotropes of carbon 6 2.1 The structure of diamond . 6 2.2 The structure of graphite . 7 2.3 Properties of diamond and graphite . 10 2.4 The phase diagram of carbon . 10 2.5 Amorphous carbon and its characteristics . 13 2.6 Lonsdaleite . 16 2.7 Electronic structure of diamond, lonsdaleite and amorphous carbon . 19 2.8 Hydrogen in diamond . 22 2.9 Quantum confinement . 24 3 Diamond synthesis 26 3.1 Natural diamonds . 26 3.2 High pressure High Temperature diamond synthesis . 27 iii CONTENTS iv 3.3 Shock-wave processing . 30 3.4 CVD diamond growth . 33 4 Previous simulations of diamond nucleation 36 4.1 Computer simulation of high pressure high temperature conversion of graphite to diamond . 36 4.2 Computer simulations of the BEN process and \thermal spike" . 41 5 Goal of the research 46 6 Tight-binding model 49 6.1 Advantages and disadvantages of different models to describe the in- teratomic interaction . 49 6.2 LCAO approach . 51 6.3 The bond energy model . 54 6.4 The rescaling functions . 58 6.5 Force calculation . 59 7 Numerical techniques 60 7.1 Equations of motion . 60 7.2 The Predictor-Corrector algorithm . 61 7.3 Periodic boundary conditions . 62 7.4 Initial configuration . 63 7.5 General description of the calculations . 64 7.6 AViz . 65 7.7 Coordination number . 65 7.8 Analysis of errors . 67 CONTENTS v 8 Results: Nucleation of diamond under pressure 69 8.1 Amorphous carbon compressed in all three directions. 70 8.1.1 Computational details . 70 8.1.2 The effects of different densities (pressures) . 71 8.1.3 The effects of different cooling rates . 77 8.2 Amorphous carbon compressed in one direction . 79 8.2.1 Computational details . 79 8.2.2 Samples prepared with fast cooling rate . 80 8.2.3 Samples prepared with intermediate and slow cooling rates . 80 8.2.4 Interesting cases . 82 9 Results: Growth of diamond under pressure 89 9.1 Growth of diamond on cubic diamond seed within compressed amor- phous carbon. 89 9.2 Growth of diamond on diamond layer within compressed amorphous carbon layer . 92 10 Results: Quantum confinement 98 10.1 Quantum confinement effects in cubic nanodiamond cluster. 98 10.2 Quantum confinement in diamond layers located between two layers of amorphous carbon. 103 11 Results: Nucleation in hydrogenated carbon 108 11.1 Computational details . 109 11.2 Structure of hydrogenated amorphous carbon network . 109 11.3 Diamond nucleation in the hydrogenated carbon network . 112 CONTENTS vi 11.4 Varying density and cooling rates. 116 12 Results: Liquid-liquid carbon phase transition 122 13 Summary and discussion 130 A OXON 137 B Computer program of data handling 145 References 148 Hebrew Abstract List of Figures 2.1 Schematic presentation of sp3 (left) and sp2 (right) hybridization. 8 2.2 Diamond lattice (top): view from the <210> direction (left), view from the <100> direction (right). Graphite lattice (bottom): view from the <112> direction (left), view from the <001> direction (right). 9 2.3 P, T phase diagram of carbon reproduced from [3] . 12 2.4 g(r) for an a C sample (up), taken from [6] and g(θ) for a ta C − − sample (down), taken from [7]. A4 and A3 are the contribution of the fourfold and the threefold atoms respectively. 15 2.5 Structures of (a) perfect cubic diamond and (b) perfect hexagonal diamond. These viewpoints show the similarity between these structures. Differences can be seen by careful observation of the hexagons: at these angles each hexagon appears to have 2 short and 4 long bonds. In cubic diamond the short bonds are on opposite side of the hexagons separated by 2 long bonds, whereas in hexagonal diamond either 1 or 3 long bonds separate these 2 short bonds. We note that in fact the hexagons are not in a plane and all bonds are of the same length. The apparent lengths of the bonds are due to the viewing angle. 17 vii LIST OF FIGURES viii 2.6 Radial distribution, g(r), of perfect cubic diamond (top) compared with that of perfect lonsdaleite (bottom). The radial distribution func- tions were calculated for samples containing 64 atoms. 18 2.7 Comparison of the DOS of cubic and hexagonal diamonds, taken from [13] . 19 2.8 The electronic density of states of a-C, taken from [17] . 21 3.1 Schematic representation of the belt apparatus. 30 3.2 Schematic of shock-wave processing of diamond. 32 3.3 Schematic diagram of the microwave plasma CVD apparatus, taken from [51]. 34 4.1 Hexagonal, orthorombic, and rhombohedral phases of graphite. The different stacking of the hexagonal planes are viewed along the c axis (above) and sideways (below), taken from [67] . 38 4.2 Graphite under pressure of 20 GPa. Interlayer distance collapses, new sp3-bonds extend between the graphitic planes. White objects are carbon atoms and yellow iso-surfaces represent charge density of electrons. We see new bonding represented by bonding charge between graphite layers. Taken from [61]. 40 4.3 The sp3 fraction plotted as a function of density calculated by differ- ent methods: OTB-orthogonal tight-binding [6], EDTB- environment- dependent tight-binding [74], NOTB-non-orthogonal tight-binding [69], DFT-ab initio [72] and our previous orthogonal tight-binding simula- tions [75] (indicated by \my simulations"). 43 LIST OF FIGURES ix 8.1 Microscopic structures of amorphous carbon with densities of 3.3 g/cc with 52 % of sp3-bonded atoms (a), 3,7 g/cc with 81 % of sp3-bonded atoms (b) and 4.1 g/cc with 95 % of sp3-bonded atoms (c). Red balls represent fourfold coordinated atoms, blue balls represent threefold coordinated atoms. 72 8.2 Microscopic structures of amorphous carbon with density of 3.9 g/cc with 89 % of sp3-, 10 % of sp2- and 1% of sp-bonded atoms. Red balls represent fourfold coordinated atoms, blue balls represent threefold coordinated atoms and green balls represent twofold coordinated atoms. 74 8.3 The damaged diamond cluster from the sample drawn on Fig. 8.2 generated at a density of 3.9 g/cc from two different view points . 75 8.4 Angular distribution function of the diamond cluster drawn on Fig.8.3 (black thick line) compared with the angular distribution functions of a pure diamond crystal (red line) and of amorphous carbon (blue line). 76 8.5 Radial distribution function of the diamond cluster drawn on Fig.8.3 (black thick line) compared with the radial distribution functions of a pure diamond crystal (red line) and of amorphous carbon (blue line). 77 8.6 Density of states of the diamond cluster drawn on Fig.8.3 (black line) compared to the density of states of a pure diamond (red line). The insert shows a magnified part of the density of states near the band gap. 78 8.7 Sample generated at 3.8 g/cc at fast cooling rate (left) and damaged diamond cluster found within this sample (right). 81 LIST OF FIGURES x 8.8 Radial distribution function, g(r), of the damaged diamond cluster generated at 3.8 g/cc with cooling rate of 1000 K/ps (black line) com- pared to the radial distribution function of a pure diamond crystal (red line). 82 8.9 Angular distribution function, g(θ), of the damaged diamond cluster generated at 3.8 g/cc with cooling rate of 1000 K/ps (black line) com- pared to the angular distribution function of a pure diamond crystal (red line). 83 8.10 Density of states of the damaged diamond cluster generated at 3.8 g/cc with cooling rate of 1000 K/ps. 83 8.11 A graphitic configuration generated at 3.7 g/cc with intermediate cool- ing rate: a) view from the direction parallel to the graphitic planes, b) one graphitic plane, view from the perpendicular direction. Red balls are sp3 coordinated atoms, blue balls are sp2 coordinated atoms and green balls are sp-coordinated atoms. 84 8.12 Radial distribution function of the graphitic structure in the sample generated at 3.8 g/cc subjected to uniaxial pressure (black line) com- pared with the radial distribution function of perfect graphite (red line). 85 8.13 Angular distribution function of the graphitic structure in the sample generated at 3.8 g/cc subjected to uniaxial pressure (black line) com- pared with the angular distribution function of perfect graphite (red line). 86 LIST OF FIGURES xi 8.14 Flexed graphitic configuration generated at 3.7 g/cc with slow cooling rate: (a)-before relaxation, (b)-after relaxation .
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