university of california, san diego

UNIVERSITY OF CALIFORNIA, SAN DIEGO SAN DIEGO STATE UNIVERSITY

Electronic Structure and Interlayer Binding Energy of Graphite

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Chemistry by Constantinos Zeinalipour-Yazdi

Committee in charge: University of California, San Diego Professor John Wheeler Professor Kurt Marti Professor Richard Herz San Diego State University Professor David Pullman, Chair Professor Peter Salamon Professor William Tong 2006

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Copyright Constantinos Zeinalipour-Yazdi, 2006 All rights reserved.

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The dissertation of Constantinos Zeinalipour-Yazdi is approved, and it is acceptable in quality and form for publication on microfilm:

Chair

University of California, San Diego San Diego State University 2006

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DEDICATION

This doctoral dissertation is dedicated to all the people that devote their mind and abilities to obtain a better understanding of the world around us, and especially to my beloved wife, Erika who let me be one of them.

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Ιθάκη (Ithaca)

Σα βγεις στον πηγαιμό για την Ιθάκη, να εύχεσαι νάναι μακρύς ο δρόμος, γεμάτος περιπέτειες, γεμάτος γνώσεις. Τους Λαιστρυγόνας και τους Κύκλωπας, τον θυμωμένο Ποσειδώνα μη φοβάσαι, τέτοια στον δρόμο σου ποτέ σου δεν θα βρείς, αν μέν' η σκέψις σου υψηλή, αν εκλεκτή συγκίνησις το πνεύμα και το σώμα σου αγγίζει. Τους Λαιστρυγόνας και τους Κύκλωπας, τον άγριο Ποσειδώνα δεν θα συναντήσεις, αν δεν τους κουβανείς μες στην ψυχή σου, αν η ψυχή σου δεν τους στήνει εμπρός σου.

When you set out on your journey to Ithaca, pray that the road is long, full of adventure, full of knowledge. The Lestrygonians and the Cyclops, the angry Poseidon -- do not fear them: You will never find such as these on your path, if your thoughts remain lofty, if a fine emotion touches your spirit and your body. The Lestrygonians and the Cyclops, the fierce Poseidon you will never encounter, if you do not carry them within your soul, if your soul does not set them up before you.

Να εύχεσαι νάναι μακρύς ο δρόμος. Πολλά τα καλοκαιρινά πρωϊά να είναι που με τι ευχαρίστησι, με τι χαρά θα μπαίνεις σε λιμένας πρωτοειδωμένους· να σταματήσεις σ' εμπορεία Φοινικικά, και τες καλές πραγμάτειες ν' αποκτήσεις, σεντέφια και κοράλλια, κεχριμπάρια κ' έβενους, και ηδονικά μυρωδικά κάθε λογής, όσο μπορείς πιο άφθονα ηδονικά μυρωδικά· σε πόλεις Αιγυπτιακές πολλές να πας, να μάθεις και να μάθεις απ' τους σπουδασμένους.

Pray that the road is long. That the summer mornings are many, when, with such pleasure, with such joy you will enter ports seen for the first time; stop at Phoenician markets, and purchase fine merchandise, mother-of-pearl and coral, amber and ebony, and sensual perfumes of all kinds, as many sensual perfumes as you can; visit many Egyptian cities, to learn and learn from scholars. Always keep Ithaca in your mind. To arrive there is your ultimate goal. But do not hurry the voyage at all. It is better to let it last for many years; and to anchor at the island when you are old, rich with all you have gained on the way, not expecting that Ithaca will offer you riches.

Πάντα στον νου σου νάχεις την Ιθάκη. Το φθάσιμον εκεί είν' ο προορισμός σου. Αλλά μη βιάζεις το ταξίδι διόλου. Καλλίτερα χρόνια πολλά να διαρκέσει· και γέρος πια ν' αράξεις στο νησί, πλούσιος με όσα κέρδισες στον δρόμο, μη προσδοκώντας πλούτη να σε δώσει η Ιθάκη.

Ithaca has given you the beautiful voyage. Without her you would have never set out on the road. She has nothing more to give you.

Η Ιθάκη σ' έδωσε το ωραίο ταξίδι. Χωρίς αυτήν δεν θάβγαινες στον δρόμο. Αλλο δεν έχει να σε δώσει πια. Κι αν πτωχική την βρεις, η Ιθάκη δεν σε γέλασε. Ετσι σοφός που έγινες, με τόση πείρα, ήδη θα το κατάλαβες η Ιθάκες τι σημαίνουν.

And if you find her poor, Ithaca has not deceived you. Wise as you have become, with so much experience, you must already have understood what Ithacas mean.

Κωνσταντίνος Π. Καβάφης (1911) Constantine P. Cavafy (1911)

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TABLE OF CONTENTS

Signature Page …...……………………………………………………………….... iii Dedication ...…….……………………………………………………………….…

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Epigraph …………………………………………………………………………....

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Table of Contents ....………………………………………………………………..

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List of Abbreviations ....……..…………………………..…………………………

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List of Figures ....………………………………………………………………...…

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List of Tables and Graphs …………………………………………………………

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Acknowledgements …………………………………………………………..…… xvii Curriculum Vitae ………………….....……………………………………....…… xxi Abstract .….………….………………………………………………………....… xxv Introduction: Structure and Symmetry of Graphite ……….…………………..….

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1 Graphite ……..………...…...……..……………………….. 2 Mineralogy ……..…………..……..……………………….. 3 X-Ray Diffraction ………..……………….……………….. 4 Electron and Neutron Diffraction ……………….….….…. 5 Raman Spectroscopy ……………...…………...………….. 6 Low Energy Electron Diffraction ……………...….….…… 7 Scanning Tunnelling Microscopy ……...…………...…….. 8 Atomic Force Microscopy ………………………………... 9 Conclusions ……..…………….…..………………………. References ………………...………..……...…………...…………………

4 5 6 16 20 21 22 23 24 26

Chapter 1:

Instrumentation and Methods …...………..……………...………..

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1.1 1.2

Introduction ……...………………………………………………... Scanning Tunneling Microscopy (STM) ...…………..………...… 1.2.1 STM Principle of Operation …...………..…...………….... 1.2.2 Theory of Scanning Tunneling Microscopy ………..….… 1.2.3 Simulation of STM Image …...…………………………… 1.2.4 Isolation from Thermal and Mechanical Drifts ..….………

29 30 32 34 37 38

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1.2.5 1.2.6

Tunnel Tips and Sample Preparation …...……………...… 38 STM Description and Operation .…...…...………...…...… 40

1.3

X-Ray Diffraction (XRD) ..………………...………...…...……… 1.3.1 The Bragg Law …..…...………………………………...… 1.3.2 X-ray Radiation Source .………………………...…..….… 1.3.3 X-Ray Diffraction Pattern …….....……………………..… 1.3.4 Sample Preparation ……………...………………...…...… 1.3.5 Peak Shape Functions ..……………...………………...…. 1.3.6 Powder XRD 2θ Pattern Fitting Routine …...……………. References ………………………..……...………...…...………………… Chapter 2:

Computational Methods …..…….………..…………………….…

2.1 2.2 2.3 2.4 2.5 2.6

Introduction ………………………………………..………...…… The SCF Hartree-Fock Method …..…………………………….… Basis Set Saturation …..….……………………...…………….….. Symmetry Unique Atoms ….....……..……………………………. Locating Local Minima in Geometry Relaxations ...……...………. Calculating Van der Waals Interactions ….….……………………. 2.6.1 Basis Set Superposition Error …………………………….. 2.6.2 Electron Correlation ………..…...……………………...… 2.6.3 Zero Point Energy (ZPE) …..…………...……………...…. References ……...……………...……………...………………………….. Chapter 3:

44 44 45 48 49 50 53 55 57 57 58 64 66 68 70 70 72 74 76

Origin of Contrast in STM Images of Graphite ..…...….…………

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3.1 3.2 3.3

Abstract ……………………...……………………..…………….. Introduction …...…………...…………………………………...… Results and Discussion ……...………………...………………….. 3.3.1 Thermal or Mechanical Drift ..……..…..………………… 3.3.2 Tip Convolution and Multiple Tip Effect ……..…………. 3.3.3 Tip-Induced Electric Field Effects ……………………….. 3.3.4 Computational Methods ……...………...………………… 3.3.5 Electron Density at the Fermi Level …………...…………. 3.3.6 Experimental vs Simulated STM Image of Graphite ......… 3.4 Conclusions …...……..…………………………………………… References …………………......……………………………………….…

77 78 81 82 85 87 89 91 94 97 99

Chapter 4:

Experimental/Theoretical Powder X-Ray and Single-Crystal Electron Diffraction Patterns of Graphite ………………….……. 101

4.1 4.2

Abstract …......…………………………...…………………….… 101 Trigonometric Equation for BLA in Graphite .…...……...……… 102 4.2.1 Effect of Compression Along b Vector on Powder XRD ... 105

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4.2.2 Effect of Compression Along b Vector on Powder XRD ... 4.2.3 Effect of Increase in θ on Powder XRD …..……...……… 4.3 Transmission Electron Diffraction Photographs of Graphite ....…. 4.4 Experimental Powder XRD of Graphite ……......………….……. 4.5 Conclusions …………………………………………………....… References ...……………………………………...………………...…….

108 111 114 120 124 125

Chapter 5:

ab-initio Study of π-π Interactions Between Polycyclic Aromatic Hydrocarbons ……...…………………………..……… 126

5.1 5.2 5.3 5.4

126 127 129 133 133 138 140 142 144 146

Chapter 6:

Design of a New Optical Differential Reflectance Apparatus to Study the Binding Energy of PAHs to Graphite …......……….. 149

6.1 6.2

149 151 151 152 153 153 156 160 163 164

Abstract ……….…………………………………………………. Introduction ...…..………………………………………………... Computational Details ..……………………………...…………... Results and Discussion ...…...……………………………………. 5.4.1 Potential Energy Function Fit ...…………………..….….. 5.4.2 Static Polarizability Computations .....…………………… 5.4.3 Convergence of the Interaction ……..……...……………. 5.4.4 Comparison of MP2 to Molecular Mechanics Data …..…. 5.5 Conclusions …...………………...……………………………...… References …….....……………….………………………………………

Abstract …………...……………………………………..………. Introduction …………..………………………………………….. 6.2.1 Optical Differential Reflection (ODR) Spectroscopy ….... 6.2.2 Determination of Interlayer Binding Energy of Graphite ... 6.3 Description of Our Apparatus ……....……………………………. 6.3.1 Laser Optics for ODR ……………......……...……...……. 6.3.2 UHV Chamber and Gas Line ……………………….....… 6.4 ODR Adsorption/Desorption Curves of Water from Graphite ….. 6.5 Future Work ………….………………………………………….. References ………...………………………………………………….……

Appendix ……………………………………………………...……………..…... 166 1 2 3 4 5 6 7 8

Scanning Tunneling Microscopy Image Simulator ……..…...……… Routine to Obtain Electron Density (ρ) from ψν …………………….. Routine to Obtain Coordinates After Translation or Rotation …..…... Routine That Scans 1D PES Using Molecular Mechanics ………….. Fractional Coordinates of Graphite With Various BLAs ……...….…. XRD Pattern of Mixed HOPG/Silicon Powder ………………...……. Experimental XRD Reflections for Graphite/Silicon Powder ……...… Corrected Graphite Reflections and Miller indices …………..………

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166 172 174 177 180 181 182 183

LIST OF ABBREVIATIONS

AFM: Atomic Force Microscopy AO: Atomic Orbital BLA: Bond Length Alternation BSSE: Basis Set Superposition Error B3LYP: Becke three parameter Lee Yang Par functional CCSD(T): Coupled Cluster Single and Doubles with perturbative Triples CVD: Chemical Vapor Deposition DFT: Density Functional Theory DNA: Desoxy ribo Nucleic Acid FWHM: Full Width at Half Maximum GGA: Generalized Gradient Approximation HF: Hartree-Fock HOCO: Highest Occupied Crystal Orbital HOPG: Highly Oriented Pyrolytic Graphite HOMO: Highest Occupied Molecular Orbital HREELS: High Resolution Electron Energy Loss Spectroscopy MM: Molecular Mechanics MO: Molecular Orbital MP2: Møller-Plesset 2nd order perturbation theory MWNT: Multiwall Nanotube LDA: Local Density Approximation

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LDOS: Local Density of States LEED: Low Energy Electron Diffraction LUCO: Lowest Unoccupied Crystal Orbital LUMO: Lowest Unoccupied Molecular Orbital ODR: Optical Differential Reflectance PAH: Polycyclic Aromatic Hydrocarbon PES: Potential Energy Surface SCF: Self-Consistent Field SCI: Science Citation Index STM: Scanning Tunneling Microscopy STO: Slater Type Orbital SWNT: Single Wall Nanotube TDS: Thermal Desorption Spectroscopy TEM: Tunneling Electron Microscope TPD: Temperature Programmed Desorption UHV: Ultra High Vacuum VdW: Van der Waals XC: Exchange Correlation XRD: X-Ray Diffraction ZPE: Zero Point Energy

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LIST OF FIGURES

Figure 1:

Single crystal of natural graphite ……………………………….……

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Figure 2:

Hexagonal crystal structure of graphite proposed by Hull ……...……

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Figure 3:

Rhombohedral crystal structure of graphite proposed by Debye et al.

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Figure 4:

Hexagonal crystal structure of Graphite, belonging to D 4 6h − P 63 / mmc space group …………………………………………………….....…. 10

Figure 5:

Rhombohedral crystal structure of graphite belonging to D53d − R 3m space group ………………………………………………………..… 12

Figure 6:

Rhombohedral crystal structure of graphite showing the hexagonal unit cell ………………………………………………………………

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Schematics of the possible arrangements of the π-electrons and the new unit cell as proposed by Hoerni and Weigle ……………………

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Figure 7: Figure 8:

Orthorhombic crystal structure of graphite proposed by Linus Pauling, belonging to the D232h − Fmmm space group ……………… 19

Figure 9:

(a) Schematic of 1st and 2nd surface layer in graphite …………….… (b) Experimental constant current STM image of HOPG ………...… 22

Figure 1.1: Schematic showing common modes of STM operation …………...… 33 Figure 1.2: Schematic showing the basic components of Beetle-STM head …..... 41 Figure 1.3: Triple helix sample holder …………………………………………… 43 Figure 1.4: Schematic showing the geometric derivation of the Bragg law …….... 44 Figure 1.5: Simplified schematic of an X-ray tube …………………………….... 45 Figure 1.6: X-ray emission spectrum of Molybdenum ..………………………… 46 Figure 1.7: Schematic representation the two types of diffraction patterns……… 48

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Figure 1.8: Comparison between normalized Gaussian and Lorentzian distributions …………………………………………...…………….. 51 Figure 2.1: Schematic representation of symmetry unique atoms in (a) coronene, and (b) a hydrogenated analog of coronene …...……….…………… 66 Figure 2.2: Schematic representation of symmetry elements (a) of coronene, and (b) hydrogenated analog of coronene …...…………………….…… 67 Figure 2.3: One-dimensional PES of face-to-face benzene dimer calculated using MP2/cc-pVDZ ……………………………………………………...… 72 Figure 3.1: Schematic representation of the atom assignment in hexagonal or rhombohedral graphite ...……………………………………………… 79 Figure 3.2: Quinoid structure of graphite previously proposed by Linus Pauling ………………………………………………………………………... 81 Figure 3.3: Experimental constant-current STM images of graphite ...………..… 83 Figure 3.4: Effect of scanning direction on STM image of HOPG taken in air …. 86 Figure 3.5: Schematic representation of the model used to study the effects of the tip-sample electric field …………………………………………...… 88 Figure 3.6: Schematics of four layer slab unit cell …………………………...….. 90 Figure 3.7: Isosurfaces of crystal orbitals at the Fermi level ……………………. 92 Figure 3.8: Comparison between experimental and simulated STM images of graphite ……………………………………………………………… 93 Figure 3.9: Crystal orbitals near the Fermi level of 4 layer slab model of graphite ….……………………………………………………………………. 94 Figure 3.10: Comparison between experimental and simulated STM images of graphite ……………………………………………………………… 96 Figure 4.1: Hexagonal and orthorhombic unit cell of graphite ………………… 102 Figure 4.2: Orthorhombic unit cell showing trigonometric variables used in derivation ………………………………………………………….… 103

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Figure 4.3: Fractional coordinates of symmetry unique atoms in orthorhombic graphite ……….……………………………………………...……… 104 Figure 4.4: Simulated XRD peaks of [2,0,2n] family of planes keeping a and θ constant ……………………………………………………………… 107 Figure 4.5: Simulated XRD peaks of [2,0,2n] family of planes keeping b and θ constant .………………………………...…………………………… 110 Figure 4.6: XRD pattern for hexagonal graphite and orthorhombic graphite with BLA 0.4Å …………………………………………………………… 113 Figure 4.7: Simulated transmission electron photographs of hexagonal graphite along various lattice vectors ……………………………..………..… 116 Figure 4.8: Simulated high beam energy (1000eV) single-crystal transmission electron diffraction photograph of hexagonal graphite …...………… 117 Figure 4.9: Various crystallographic planes perpendicular to the basal plane of graphite ……………………………………………………………… 118 Figure 4.10: Angular dependence of θ1 and θ3 as a function of BLA ……………. 119 Figure 4.11: Calibration curve for powder XRD reflections ………………...…... 121 Figure 4.12: High resolution (0.001°) powder XRD reflection of graphite, obtained with Kα Cu radiation ……...………………………………. 123 Figure 5.1: Schematic representation of PAH superstructures ……………..…… 131 Figure 5.2: BSSE corrected 1D PES of benzene with various PAHs …………… 132 Figure 5.3: Fitting parameter B as a function of the number of carbon atoms in the supermolecule ………………………………………………...… 137 Figure 5.4: Fitting parameter C as a function of the number of carbon atoms in the supermolecule ………………………………..…………….…… 137 Figure 5.5: Polarizability volume product as a function of the number of carbon atoms in the supermolecule ………………………………………… 139 Figure 5.6: Schematic representation of benzene with n-polyacene superstructures ……………………………………………………… 140

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Figure 5.7: BSSE corrected dispersion interactions for n-polyacene-benzene supermolecular series …………………………………….………… 141 Figure 5.8: Comparison among ab-initio derived and MM binding energies as a function of carbon atoms in the supermolecule ……………..……… 143 Figure 6.1: Schematic diagram of optical setup in Optical Differential Reflectance experiment ……..……………………………….…..… 154 Figure 6.2: Schematic representation of the UHV chamber constructed to be used for ODR studies ………………………………………..……… 157 Figure 6.3: Schematic representation of gas line ………………………....…….. 158 Figure 6.4: Adsortption/desorption of water onto graphite …………………...… 159 Figure 6.5: Kinetically different regions seen in our ODR adsortption/desorption signal for water/graphite system ……………………………………. 161

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LIST OF TABLES AND GRAPHS Table 1:

Fractional coordinates of carbon atoms in the primitive hexagonal unit cell as proposed by A.W. Hull …………………………………..…… 7

Table 2:

The fractional coordinates of carbon atoms in hexagonal graphite proposed by Bernal ……………..………………………………….… 10

Table 3:

The fractional coordinates of carbon atoms in the hexagonal unit cell of the rhombohedral structure of graphite ………………......…..…… 13

Table 4:

Fractional coordinates for carbon atom positions in unit cell of orthorhombic graphite …………………………………………….…. 18

Graph 1.1: Graph showing the number of articles published every year dealing with STM ……………………………………………………………. 30 Graph 1.2: Plot of tunneling current as a function of tip to surface separation for graphite (0001) surface …...…………………………………….…… 36 Table 1.1:

Characteristic emission lines of various materials commonly used in X-ray tubes ...………………………………………………………… 46

Table 2.1:

Benzene Hartree-Fock calculations using various basis sets ……...… 64

Table 2.2:

DFT computations of 4-layer graphite slab ………………...………... 65

Table 2.3:

Summary of total energy and cartesian gaussian basis functions used for the optimization of the molecular series of C6n2H8n ……...……… 69

Table 4.1:

Bond length alternation and unit cell parameters for the various structures examined a and θ constant ………….……………...……. 105

Table 4.2:

Bond length alternation and unit cell parameters for the various structures examined b and θ constant ………….………………..…. 108

Table 4.3:

Reflection angles and highest peak-weighted intensities of orthorhombic graphite with various degrees of BLA ………....…… 111

Table 4.4

Reflection angles and highest peak-weighted intensities of orthorhombic graphite with various degrees of BLA ……………… 114

Table 4.5:

Unit cell refinement results for powder XRD of HOPG …………… 122

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Table 5.1:

Fitting parameters and R-squared values for exponential–6 potential energy function ………………………………………………….… 135

Table 5.2:

Lennard-Jones (9,6) –potential well depth (ε) and separation at V=0 of supermolecules …………………………………………..……… 135

Table 5.3:

Static polarizability components and polarizability tensor of supermolecules using HF/cc-pVDZ ………………….…………..… 138

Table 5.4:

Binding energy as a function of the number of carbon atoms of supermolecules as calculated from MM3 and MP2 …………...…… 143

Table 6.1:

Previous computational and experimental determinations of the interlayer binding energy of graphite …………………………...…. 153

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ACKNOWLEDGEMENTS

Ithaca, the poem by Cavafy, describes perfectly my perception of my graduate years. In ancient Greek mythology Ithaca was the hometown of Ulysseus, and his return to it was his ultimate desire. A journey that turned out to be long and difficult, but full of new experiences and knowledge gained along the way. He arrived as an old man with white hair but full of wisdom. My Ithaca for the past few years has been to obtain a doctoral degree in Chemistry, but I have tried to keep in mind the advice given by Cavafy. I tried to be open to new knowledge, made efforts to understand some of the harder concepts in Chemistry, read many scientific articles from the first half of the 20th century and tried to make every step of the learning process an enjoyable experience. I think it is fair to say that graduate student life is the last chance to follow your ideas without restrictions, and to develop one’s unique way of providing answers to scientific issues, but also to questions that emerge from everyday life. Of course there have been many people responsible for making my graduate studies an enjoyable experience. I thank my research advisor Professor David Pullman, who was patient enough to explore the workings of my contentious mind, often suggesting more elegant solutions to the problems at hand. It was nice to see that this battle was not fought alone and that there was a critical eye to monitor the formulation of my ideas. Many thanks, to his wife, professor Peterson for her advice and guidance throughout my doctoral studies.

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I owe a great deal to the members of my doctoral committee who throughout my studies have extended my involvement in Chemistry and have supported me through their own research and knowledge. Since two out of the three main topics of my doctoral dissertation were completely new topics for me, there is a large number of Professors at SDSU and UCSD with whom I have interacted to obtain a better understanding of these topics. I will only mention some of them without meaning that the support of others is not appreciated. My greatest appreciation goes to Dr. Cooksy for his good sense of humor and for the lengthy discussions we had about various aspects of ab-initio computations. He was always interested in the questions that emerged from this work, and he was able to provide me with good guidelines on how to provide answers. Professor Wheeler and Marti are appreciated for the discussions we had about the electronic structure of graphite. Joan Kimbrough and Professor Huxford are appreciated for sharing their knowledge about X-ray crystallography with me. It was also amusing to listen to the stories Professor Hellberg had to say about his graduate years and other aspects of his life, and I would like to see myself become like him when I grow older. Many thanks to Professor K.C. Nicolaou for providing me with a copy of his doctoral dissertation and his general concern about the progress of my thesis. Also many thanks to my fellow research group members, especially David Fee, with whom I had many useful discussions about politics and religion. Benno Spingler, the department technician, did an excellent job building a support for the gas mixing line of the UHV chamber I constructed. A special acknowledgement is extended to Greg Gidofalvi of the University of Chicago for providing me with a critical review of

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my complete thesis during his short stay in San Diego. Our common interest in quantum mechanics and our lengthy discussions about problematic issues of my research were always very helpful to me. I thank my dear parents that regardless of the ten thousand miles distance between San Diego and Cyprus, they were always by my side through their continuous encouragement and silent concern. I was blessed to have a plethora of advice from their experiences as graduate students. I thank my brother Demetris, who recently obtained a doctoral degree in computer science and already has it posted next to my father’s doctoral degree in architecture. I would also like to thank my brother’s wife Christina, for proofreading my introduction and making the amusing comment, “Me with a degree in French literature, could have written this!”. My long standing friends, Antreas, Christos, Demetris, Kyriacos, Neofitos and Savvas, whose loyal friendship and general tendency to express with their high school mentality in our summer reunions, were always relaxing and greatly appreciated. I thank San Diego State University for financially assisting me with a tuition waiver for the course of my studies and the Scholarship office for awarding me with various scholarships. Support has also been forthcoming from the College of Sciences, in the form of travel grants, which allowed me to attend a number of conferences and therefore bring this research to the attention of a broader audience. Without doubt there are errors and omissions in this work for which I apologize, and also in some places simplifications that were intentionally made to make this thesis readable, even by non-chemists. I felt that by doing this, I was making

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a stronger connection of my research to the outside world, a component crucial in one’s healthy scientific research. No doubt the one person whom I must thank to the greatest extent is my wife Erika. She proof read my manuscript and made significant corrections to my writing style. As an example, she advised me to use “we” and “our” rather than “I” and “mine”, and helped me understand how the significant concepts of my research should be presented. Her caring and nourishing character and her persistence in finding solutions to problems apart from chemistry have always been a great source of inspiration to me. My journey to Ithaca would certainly not have been as joyful as it was, if she were not by my side.

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CURRICULUM VITAE

EDUCATION 2006 Ph.D. Chemistry (computational/physical chemistry) (SDSU/UCSD) Advisor: Professor David Pullman “ Simulation of powder and single crystal X-ray and electron diffraction patterns” “Differential reflectance spectroscopy of isothermal desorption of PAHs from graphite” “ Simulation of constant height and constant current STM image of HOPG” 2003 M.A. Chemistry San Diego State University (SDSU) Advisor: Professor David Pullman “Study of ion induced defects on graphite using STM” 1999 B.Sc. Chemistry University of Cyprus (UCY) Advisor: Professor Aggelos Efstathiou “Catalytic reduction of nitrogen oxides (NOx) using lanthanum and calcium oxides.”

RESEARCH EXPERIENCE Physical Chemistry Laboratory, San Diego State University, CA, USA 2000-2005

Construction, operation and maintenance of a 16-node Beowulf PC cluster using Rocks. Electronic structure calculations using quantum mechanical methods. Construction of UHV chamber to study dynamics of induced damages to HOPG and temperature programmed desorption of PAHs from HOPG using differential reflectance spectroscopy. Installation and operation of a large number of UHV and laser spectroscopy components and surface imaging using scanning tunneling microscopy. Heterogeneous Catalysis Laboratory, University of Cyprus, Nicosia, Cyprus 1999

Catalytic reduction of nitrogen oxides (NOx) under highly oxidative conditions using lanthanum and calcium oxides.

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PUBLICATIONS AND PRESENTATIONS “Correlation between molecular size and dispersion interaction in PAHs”, Zeinalipour-Yazdi, C.; Pullman, D., Chem. Phys. Lett., in preparation, 2005. “Origin of contrast in STM images of graphite”, Zeinalipour-Yazdi, C.; Peterson, K.I.; Pullman, D. Abstr. Pap. Am. Chem. S., 229: 184-COLL Part 1, 2005. “A second-order perturbation study of the transformation of hexagonal to rhombohedral graphite”, Zeinalipour-Yazdi, C.; Loizidou, E.; Pullman, D. Abstr. Pap. Am. Chem. S., 229: 184-COLL Part 1, 2005. Poster presentation at the ACSESS Outreach Event & Computational Science Curriculum Development Forum 2005. Poster presentation at the 10th Annual Maria Goeppert-Mayer Interdisciplinary Symposium 2005. “An ab-initio study of the electronic structure in large polycyclic aromatic hydrocarbons”, poster presentation at 1st annual university of California surface science and its applications symposium 2004. “Artificial receptor-attached amphiphilic copolymer for barbiturate binding in aqueous media”, Loizidou, E.; Zeinalipour-Yazdi, C.; Sun, L. Biomacromolecules, 5, 1647-1652, 2004. “On the interpretation of graphite images obtained by STM”, ZeinalipourYazdi, C.; Gonzalez, J.; Peterson, K.I.; Pullman, D. Abstr. Pap. Am. Chem. S., 226: 184-COLL Part 1, 2003.

TEACHING EXEPRIENCE Teaching Associate, San Diego State University Advanced Physical Chemistry Lab CHEM417 General Chemistry Lab CHEM201

2002-2006

Teaching Associate, University of California, San Diego General Chemistry CHEM6C

2001-2002

Teaching Associate, San Diego State University

2000-2001

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General Chemistry Lab CHEM100 and CHEM200 As the instructor of general and physical chemistry laboratories, I was solely responsible for the presentation of the experiments, supervision of up to 25 university level students, grading quizzes and lab reports and proctoring exams.

WORK EXPERIENCE Biotronics Ltd, Nicosia, Cyprus, field technician; technical presentations on analytical chemistry instrumentation and validation and maintenance of analytical chemistry instruments; including HPLCs, GCs, spectrophotometers and titrators; 1999-2000 Foodtech Laboratories Ltd, Nicosia, Cyprus, lab technician; Qualitative & quantitative control on food products; Microbiological analysis of food samples. 1999 Unilab Laboratories, Nicosia, Cyprus, lab technician; hematological analysis, radio-immunoassay (RIA); gram stains and cultures. Summer 1999

PROFESSIONAL ACTIVITIES & MEMBERSHIPS Member of Inorganic Chemistry Faculty Search Committee, SDSU 2005 Undergraduate Research Symposium Judge, College of Sciences, SDSU 2005 Member of Analytical Chemistry Faculty Search Committee, SDSU 2004 Member of the American Chemical Society, Division of Colloid and Surface Chemistry, 2004-2006 Undergraduate Research Symposium Judge, College of Sciences, SDSU 2003 Board Member of the School of Pure and Applied Science, University of Cyprus 1997-1998

AWARDS AND SCHOLARSHIPS SDSU Graduate School Scholarship covering tuition and fees Shared Vision SDSU Research Fellowship Peace Village Photo Contest Award (ISC) Harrington Neil Memorial Scholarship Harry Hamber Memorial Scholarship

xxiii

2000-2005 2005 2004 2003-2004 2001-2002

CONFERENCES AND WORKSHOPS 229th ACS National Meeting, San Diego, CA, March 13-17 226th ACS National Meeting, New York City, NY, Sept. 7-11 221st ACS National Meeting, San Diego, CA, April 1-5

Spring 2005 Fall 2003 Spring 2001

COMPUTER SKILLS Languages: Python, Shell Scripts and Fortran (moderate) Computational Chemistry: GAMESS, CPMD, Gaussian, PCGAMESS and NWChem Molecular modeling: ECCE, MOLDEN, Gaussview and MOLEKEL Data analysis: Axum and Igor Other software: Mathematica, Chemdraw, ImageJ and Photoshop

LANGUAGES Greek (native) English (fluent) German (working knowledge)

xxiv

ABSTRACT OF DISSERTATION

Electronic Structure and Interlayer Binding Energy of Graphite by Constantinos Zeinalipour-Yazdi Doctor of Philosophy in Chemistry

University of California, San Diego, 2006 San Diego State University, 2006 Professor David P. Pullman, Chair

This doctoral dissertation addresses the experimental and theoretical elucidation of the geometric and electronic structure and the interlayer binding energy of graphite. The synthetic form of graphite, highly oriented pyrolytic graphite (HOPG), has been used as a standard for scanning tunneling microscope (STM) calibration for over a decade because of the relative ease of imaging in air and vacuum coupled with the known carbon-carbon distances. Most images show only three of the six carbon atoms in a given six-membered ring.

This observation has been

rationalized in several ways, although no entirely satisfactory explanation exists yet. In this work, a new interpretation of the graphite STM image is proposed. This interpretation is based on 1) an analysis of small asymmetries in STM images of HOPG obtained in our lab and from the literature, 2) computations of the electronic

xxv

structure and STM image of graphite using density functional theory (DFT), and 3) high-resolution powder x-ray diffraction (XRD) studies. The possible effects in graphite’s STM image of interlayer interactions motivated the second topic of this thesis. Here, we investigate the π-π interactions between polycyclic aromatic hydrocarbons (PAHs). Second order Møller-Plesset perturbation theory and density functional theory are used to probe the correlation among molecular size, binding energy, and molecular polarizability. Finally, we describe a proof-of-principle for the operation of a new optical differential reflectance (ODR) technique that can be used to provide accurate measurement of the Van der Waals (VdW) interaction between PAHs and graphite.

xxvi

INTRODUCTION Structure and Symmetry of Graphite

In this thesis we study the geometric and electronic structure of graphite using scanning tunneling microscopy (STM), density functional computations and highresolution powder x-ray diffraction (XRD). The principle aim is to provide an answer to what one really sees in the constant-current STM image of highly oriented pyrolytic graphite (HOPG). We suggest a new interpretation to the STM image of graphite that is consistent with experimental findings. To investigate our hypothesis, we analyze atomically-resolved STM images of graphite and also use results from the computations to simulate the STM image of graphite according to the Tersoff-Hamann model. This work also led to a study of the Van der Waals interactions between various polycyclic aromatic hydrocarbons (PAHs) to probe the correlation between the molecular size and the strength of the interaction. Our results indicate that there is a 1

2 linear correlation between, (i) VdW interactions and molecular size, and (ii) molecular size and static polarizability. To collect experimental evidence for this correlation, we designed and showed the proof-of- principle of a new optical surface technique that appears to be promising for the study of adsorption/desorption of molecules from surfaces. Our group will be using this technique in the future to measure the interaction between PAHs and the basal plane of graphite. In this introduction we give a historical review on the crystallographic elucidation of the structure of graphite, highlight various proposed structures, and discuss the crystallographic techniques used by investigators to elucidate the atomic positions in the lattice. In chapter one, we provide an introduction to the STM and XRD and present a detailed description of the instrumentation used. In chapter two we give background information on electronic structure calculations and the general considerations one should make to obtain meaningful results. In chapter three we present experimental and simulated STM images of graphite under various conditions and provide a new explanation for the “missing” atom in graphite. In chapter four we test for the existence of small bond length alternation (BLA) in graphite. Then, with the use of a mathematical model for graphite, we generate structures with various degrees of BLA, which are used to simulate powder XRD patterns and transmission electron diffraction photographs. BLA causes the splitting of certain reflections, which however could not be experimentally observed in high-resolution powder XRDs. In chapter five we study the correlation between molecular size and VdW interaction energy between various Polycyclic Aromatic Hydrocarbons (PAHs). In chapter six, we describe a new variation of the optical differential reflectance (ODR) technique we

3 developed and show that it could potentially be used for the study of VdW interactions between PAHs and graphite. Graphite is a layered material with weak dispersion interactions that bind the basal planes together. The various deformations and distortions that the graphite lattice can undergo under mechanical stress and the absence of a perfect single crystal, generated different crystallographic assignments for the graphite lattice in the past. 12,13

Many articles carrying the title “The Structure of Graphite” deal with the

elucidation of the precise crystal structure, which had been both controversial and difficult.

12,13

We give a historical review of the crystallographic elucidation of the

structure of Graphite. This chapter is not intended to be a comprehensive overview, but rather to highlight various proposed structures for graphite and to point out the crystallographic techniques used by investigators to elucidate the atomic positions in the lattice. Schematics of the proposed structures and crystallographic parameters are given to aid in the understanding of the relevant articles.

4

1

Graphite Graphite is an allotropic form of carbon found both in nature and artificially

produced by man. The word graphite itself originates from the Greek word ‘γράφει’ which means ‘to write’; it is used even today as it was two centuries ago, as an art medium. The ideal structure of graphite consists of layers composed of hexagonally arranged carbon atoms and bound together by weak Van der Waals interactions. The carbon atoms are covalently bonded to three neighboring carbon atoms, through sp2sp2 axial hybrid orbital overlap. Each isolated carbon atom has an electronic structure of the simplified form 1s22s22px12py1, where the plane of the layer is taken as the xy plane. The 1s2 core orbitals remain intact; essentially only the valence electrons undergo hybridization. For this reason the 2s22px12py1 orbitals hybridize into 3 degenerate hybrid sp2 orbital; leaving the lone 2pz electron available for parallel overlap, forming a rather diffuse electron distribution around the carbon rings (πcloud). The layers are bound through weaker dispersion interactions (London forces) that come as a consequence of the polarizable nature of the π-clouds. Ordered phases of graphite come in two different layer arrangements that result in two distinct crystal structures, the hexagonal and the rhombohedral. In both structures the carbon atoms within the graphitic layers are placed in a perfect honeycomb-like lattice having a distance of about 1.42 Å and an interlayer separation of 3.35 Å. The stacking sequence in hexagonal graphite is ABAB whereas in rhombohedral graphite it is ABCA. This historical review aims to provide an extensive list of scientific articles related to the structural elucidation of graphite. These span a time period of more than

5 a hundred years, starting with the macroscopic study of crystal geometries and ending with the microscopic resolution of atoms. The referenced articles in this review are presented in thematological order to emphasize the progress in structural analytical techniques.

2

Mineralogy The concept that the macroscopic geometry of crystals is a result of the

symmetry of the building units originates from Johannes Kepler1, inspired from the study of snowflakes. In the following centuries, the trigonometric parameters to categorize a crystal into one of the 7 crystal systems were formulated.

Figure 1 Single crystal of natural graphite (1mm thick) found in calcite2 in Crestmore, California. Pyramid faces are also present at the periphery of the crystal along the {112l} family of crystallographic zones.

Mineralogists as early as the mid-1800’s knew that graphite belongs either to the hexagonal or trigonal crystal system. H. Kenngott3,4 ascribed graphite to the trigonal crystal system, since it had three identical axis at 120º angles and one almost normal at α = 85°6’. Later, H. Sjögren5,6 assigned graphite to the hexagonal crystal system based

6 on the thermal conductivity and morphology such as twinning, etch figures and percussion. *

3

X-Ray Diffraction In 1912 Max von Laue reported the diffraction of X-rays by crystals and a new

era of crystal structure analysis began. Shortly thereafter P.P. Ewald7,8 confirmed the hexagonal symmetry of graphite by obtaining a Laue photograph in a flake of graphite along the c axis. A few years later, William Henry Bragg and William Laurence Bragg9 (father-son) formulated the mathematical basis of X-ray crystallography and also developed new instrumentation for the technique. With these developments, the crystallographic determination of single crystals became possible. In 1914, W.H. Bragg10 measured the cleavage plane spacing of graphite to be 3.42 Å, close to the current literature value 3.3538 Å. 11 More complete crystallographic analyses came from A.W. Hull12 and V.P. Debye et al.13 in 1917. The two reports were contradictory since Hull12 obtained a hexagonal unit cell, whereas the latter had a rhombohedral unit cell. Hull examined powder samples of both natural and artificial graphite, which were initially powdered to less than 0.1mm in diameter and annealed to 3500 °C. The artificial graphite he used was a by-product of a steel making process, produced when high carbon containing molten steal was cooled (Kish graphite). Both natural and artificial graphite

*

Twinning is the symmetrical intergrowth of two or more crystals of the same mineral. Etch figures are depressions produced on crystal faces by the momentary application of minute quantities of solvents that often show a symmetry of the structure present. Percussion is the generation of flakes from a crystal by impact of a sharp tool.

7 were assigned to the hexagonal crystal system. The lattice constants of the primitive hexagonal unit cell (Fig. 2) were a1 = a2 = 2.47Å and c = 6.80Å; the fractional coordinates for all the carbon atoms are depicted in Table 1. Table 1 Fractional coordinates of carbon atoms in the primitive hexagonal unit cell as proposed by A.W. Hull.

A

B

A'

(0,0,0) (1,0,0) (0,1,0) (1,1,0)

(2/3,1/3,8/14) (1/3,2/3,1/2)

(0,0,1) (1,0,1) (0,1,1) (1,1,1) (2/3,1/3,15/14)

(2/3,1/3,1/14) c

A'

B

A a2 a1 Figure 2 Hexagonal crystal structure of graphite proposed by Hull. Stacking sequence is ABAB. Also hexagonal and primitive hexagonal unit cell are shown. There are 2 C atoms in the primitive hexagonal unit cell. Half the carbon atoms (shown in gray) are displaced by 1/14c along the c lattice vector. The remaining (shown in white) are coplanar with the shaded areas in the figure. The puckered layers are not shown for better clarity.

8 This structure, although very close to that of hexagonal graphite, was not entirely correct since only half the carbon atoms were coplanar in the proposed lattice. The remaining carbon atoms were positioned 0.49Å above the basal plane resulting in puckered carbon rings. Deviation from the coplanar configuration of sp2 carbons would generate in principle an energetically unfavorable structure.

A

B

a C

c

A

Figure 3 Rhombohedral crystal structure of graphite proposed by Debye et al., Stacking sequence is ABCA. The rhombohedral supercell contains 8 carbon atoms.

Debye et al. examined several samples of natural graphite that were initially compressed and subsequently exposed to X-rays to obtain diffraction patterns. Their final crystallographic assignment was a structure (Fig. 3) that had a rhombohedral unit cell with a total of eight carbon atoms. The dimensions of the unit cell were a= b = c =

9 4.48Å, α = β = γ = 68º26’. Note that the unit cell can be reduced to that of rhombohedral graphite (Fig 6). The recalculated lattice constants for the irreducible rhombohedral unit cell are a = b = c = 3.63Å, α = β= γ = 40º44’, in good agreement with those known for rhombohedral graphite. However, Hassel et al.14 and Bernal15 questioned the crystal structure assignment of the two previous investigators in two very thorough scientific articles in 1924. Bernal examined both natural and artificial graphite formed during silicon-steel casting. The structure of graphite was assigned to the hexagonal crystal system belonging to the D 4 6h − P63 / mmc space group having planar layers of hexagonal rings stacked in an ABAB sequence (Fig. 4). This crystallographic assignment was later verified by Maquin16,17 and Ott18. Bernal attributed the misinterpretation of Hull to the ‘powder method’, which at the time generated diffraction patterns too complicated to interpret. In addition, using the θ values of Debye et al., he recalculated the diffraction indices based on a more precise value of wavelength of the KaCu radiation and rejected some Kβ lines that were erroneously taken into account in their analysis. This modification resulted in a general agreement of their data, although some lines could not be assigned and were thus attributed to impurities. The dimensions of the unit cell of hexagonal graphite at standard temperature and pressure as taken from Bernal are a1 = a2 = 2.45 ± 0.03 Å, c = 6.82 ± 0.04 Å, α = β = 90º and γ = 120º. The fractional coordinates of all the atoms in the hexagonal unit cell are depicted in Table 2.

10

Table 2 The fractional coordinates of carbon atoms in hexagonal graphite proposed by Bernal (Fig. 4). Each column tabulates the fractional coordinates of the atoms that belong to the same layer.

A

B

(0,0,1/4)

(0,0,3/4)

(1,0,1/4)

(1,0,3/4)

(0,1,1/4)

(0,1,3/4)

(1,1,1/4)

(1,1,3/4)

(1/3,2/3,1/4)

(2/3,1/3,3/4)

c

B

a3 A

a1

a2

Figure 4 Hexagonal crystal structure of Graphite, belonging to D 4 6h − P 63 / mmc space group. Stacking sequence is ABAB. Both hexagonal and oblique subunit of hexagonal unit cell are shown. There are 4 C atoms in the oblique hexagonal cell. There is a 3-fold rotation axis lying along c. The Miller indices for α1, α2, α3 and c are [2 1 1 0] , [ 1 2 1 0] , [ 1 1 20] and [1000] respectively.

11 Nevertheless with the development of more sensitive and precise X-ray diffractometers and other techniques like electron and neutron diffraction, various investigators19,20 found additional weak diffraction lines or peaks that could not be assigned to any of the crystallographic planes of the hexagonal structure of graphite. Initially Finch et al.19 measured electron diffraction images of powder graphite and thin graphite flakes and observed additional diffraction rings that could not be assigned to hexagonal graphite. These extra lines were found in a colloidal graphite film and graphite flakes, but not in a single crystal of natural graphite. Furthermore Taylor et al.20 observed some additional diffraction peaks in XRD images having l indices that are multiples of 2/3. These peaks were centered at the

[10 1 1] reflection and had the indices [10 1 23 ] and [10 1 43 ] . These much weaker peaks disappeared once the sample was digested in a heated mixture of concentrated nitric and sulphuric acid. The authors suggest that dynamical stratification21 of the crystal layers, which are described as the periodic movement of the layers due to vibrations, could cause the appearance of additional diffraction peaks. However, no evidence supporting this view was given. In 1942 Lipson and Stokes22 provided an explanation for the additional lines seen in the XRD pattern. The authors attributed this to stacking faults in the regular ABAB stacking sequence of hexagonal graphite similar to those found in hexagonal cobalt. 23,24 These periodic stacking faults generate an ABCA stacking sequence that is no longer hexagonal but rhombohedral (Fig 5). The lattice belongs to the trigonal 5 rhombohedral crystal system and the D 3d − R 3m space group. The lattice constants

12 were a = b = c = 3.635Å and α = β = γ = 39.49°. There are 2 carbon atoms in the unit cell as shown in Fig 5.

A

B

C

A

Figure 5 Rhombohedral crystal structure of graphite belonging to D53d − R 3m space group. Stacking sequence is ABCA. Primitive rhombohedral unit cell is also shown. There are two symmetry unique carbon atoms in the unit cell.

Due to the great similarity of the two structures, most of the XRD reflections of the rhombohedral structure are also reflections of the hexagonal. The authors showed that the reflections that were common in both structures were slightly higher in intensity. By quantitative comparison of the intensity of these reflections with the ones assigned to hexagonal graphite they were able to determine the percentage of rhombohedral (14%) structure mixed in with the regular hexagonal structure (80%) and the disordered regions (6%). The dimensions of the unit cell expressed in terms of a hexagonal unit cell are a = b = 2.456 Å, c = 10.044 Å, α = β = 90º and γ = 120º (Fig. 5), the fractional coordinates for all the atoms in the hexagonal unit cell are given in Table 3.

13

Table 3 The fractional coordinates of carbon atoms in the hexagonal unit cell of the rhombohedral structure of graphite.

A'

C

B

A

(0,0,0) (1,0,0) (0,1,0) (1,1,0) (1/3,2/3,0)

(2/3,1/3,1/3) (1/3,2/3,1/3)

(0,0,2/3) (1,0,2/3) (0,1,2/3) (1,1,2/3) (2/3,1/3,2/3)

(0,0,1) (1,0,1) (0,1,1) (1,1,1) (1/3,2/3,1)

c

A

B

C

A’ a

b

Figure 6 Rhombohedral crystal structure of graphite showing the hexagonal unit cell. There are six symmetry unique carbon atoms in this unit cell.

14 The great similarity between the hexagonal and rhombohedral crystal structures explains the overlapped diffraction patterns obtained from crystals having both phases intermixed. Further confirmation for the rhombohedral structure of graphite came from electron diffraction images obtained by Hoerni.

25,26

He obtained transmission

images through single crystals that exhibit supplementary spots with indices (11 23 ) and ( 01 23 ) . However, at the time no attempt was made to compare the thermodynamic stability of the two structures or the possible explanations of why the hexagonal phase dominates in the crystal. So the questions the following research attempted to address was to explain the prevalence of hexagonal graphite to that of rhombohedral graphite both in natural and in artificial graphite as well as propose the possible mechanism for the phase transformation. Jagodzinski27 suggested that rhombohedral graphite is the thermodynamically more stable form of graphite and that the hexagonal structure is frozen due to kinetic reasons at room temperatures. He notes that the diffraction spots corresponding to the rhombohedral structure were as intense and sharp as the ones belonging to the hexagonal structure. This indicates that the percentage of disordered regions in the perfect ABAB or ABCA stacking is very small. Bacon28 a year later compared XRD data of natural and artificial specimens of graphite before and after powdering. He noticed that the percentage of rhombohedral graphite could increase upon powdering by as much as 15%. This gave an indication to what the mechanism for the phase transformation was.

15 Nonetheless Bacon did not attempt to elaborate on the relative stability of the two phases. It was F. Laves and Y. Baskin29 who provided an answer to which form of graphite, hexagonal or rhombohedral, was thermodynamically more stable. Their studies showed that hexagonal graphite is thermodynamically more stable than rhombohedral graphite. In an extensive XRD study on single crystals of graphite, they were able to form the rhombohedral phase by applying unidirectional shear stress to the crystal. This mechanical transformation of hexagonal to rhombohedral graphite provided also an elegant explanation why the rhombohedral phase increased after powdering in several previous works. They also indicated that annealing of the rhombohedral phase to temperatures above 1300 ºC transformed the specimens to hexagonal graphite. They explained that translation of the layers along the {112l} family of crystallographic zones did not form the rhombohedral structure in single crystals of graphite that exhibited twinning. Only gliding along one of the six crystallographic axes belonging to the

{10 1 0}

family of zones30 increased the

percentage of the rhombohedral structure in the crystal. They also noticed that in certain crystals unidirectional pressure without gliding did not form the rhombohedral structure even at loads up to 4500 kg/cm3, nor did hydrostatic pressure of 10000 atm result in a modification of the relative percentage of the two phases.

16

4

Electron and Neutron Diffraction During the same period crystal structure determination using electron diffraction

had also become a precise technique. The findings of researchers using electron diffraction for the crystallographic determination of graphite were in good agreement with the findings of X-ray crystallographers. 31,32 It is interesting to note that Hoerni et al.33 detected some other electron diffraction peaks about a hundred times less intense

than the strong [100l ] reflections, which corresponded to indices of

[

1 2

[

1 1 2 2

0l ] and

00l ] . These reflections could not be explained based on either the hexagonal or the

rhombohedral phase of graphite. In an attempt to assign the new lines, a non-uniform distribution of π-electrons was proposed that would generate bonds of alternating single bond double bond character (Fig. 7) and a doubling of the unit cell. Extra diffraction lines in XRD data are accepted with skepticism since in some occasions they may be attributed to experimental errors. Gibson34 for instance reported extra diffraction lines that, after examination of the same specimens, were attributed to a not completely monochromatic X-ray beam,

35,36

which caused the splitting of

Figure 7 Schematics of the possible arrangements of the π-electrons and the new unit cell as proposed by Hoerni and Weigle.

17 several reflection lines that essentially corresponded to the same indices. At the same time a new type of pyrolytic graphite had been synthesized by the pyrolytic decomposition of various hydrocarbons on a hot flat substrate. This generated a highly oriented structure of graphite with a small mosaic spread (Highly Oriented Pyrolytic Graphite, HOPG). Lukesh and Pauling37 after studying this material concluded that graphite can have no more than 2-fold symmetry and assigned it to an orthorhombic unit cell. The lattice parameters were a = 2.456Å, b = 4.254Å c = 6.696Å, α = β = γ = 90º, the space group was D 232h − Fmmm and there were eight carbon atoms in the unit cell. Shortly after, Lukesh38-40 published several c-type precession photographs in which some of the reflection spots had weaker satellite peaks. These additional peaks reduced the symmetry in reciprocal space from six-fold to two-fold. He attributed them to an orthorhombic superstructure in which the b unit cell length was 15 times larger. The dimensions of the superstructure unit cell were a = 2.456Å, b = 36.840Å, c = 6.696Å. 15 years later Linus Pauling41 proposed the 1-4 quinoid structure to be the structure of graphite (Fig 8). This structure had an ABAB stacking sequence and was very similar to the hexagonal structure of graphite. They differ predominantly in that not all carbon-carbon bond lengths and angles were equivalent. In other words, there would be bond length alternation in the structure. According to the 1-4 quinoid structure, every carbon formed one double bond (1.354 Å) and two single bonds (1.453 Å) with bond angles at 112º, 124º and 124º. The new crystal structure no longer had six-fold symmetry in reciprocal space, which provided an explanation for the extra

18 diffraction reflections seen in Laue photographs. The structure belonged to the orthorhombic crystal system, and the dimensions of the unit cell were a = 2.409Å, b = 4.339Å, c = 6.708Å, α = β = γ = 90º. Fractional coordinates for all the carbon atoms in the unit cell are tabulated in Table 4. Table 4 Fractional coordinates for carbon atom positions in unit cell of orthorhombic graphite.

A'

B

A

(0,0,0) (1,0,0) (0,1,0) (1,1,0) (0,1/3,0) (1,1/3,0) (1/2,1/2,0) (1/2,5/6,0)

(1/2,5/6,1/2) (0,1/3,1/2) (1,1/3,1/2) (1/2,1/6,1/2) (0,2/3,1/2) (1,2/3,1/2)

(0,0,1) (1,0,1) (0,1,1) (1,1,1) (0,1/3,1) (1,1/3,1) (1/2,1/2,1) (1/2,5/6,1)

Pauling argued that the basic advantage of the quinoid structure based on theoretical calculations is that it permitted a closer stacking of the layers and stronger VdW attraction between the layers. The net stabilization of the lattice due to interlayer attraction was ~0.19 kcal/mole per carbon atom. The interlayer spacing in the quinoid structure was thus smaller (3.354 Å) than that observed in layers that have completely delocalized π-electrons. This was also consistent with the changes in the interlayer spacing in graphitic carbons that are synthesized by the pyrolytic decomposition of hydrocarbons. The degree of order as well as the interlayer spacing changes from 3.44 Å to 3.354 Å depending on the degree of graphitization.

42

The proposition of the quinoid structure

caused controversial discussions in various circles since such bond length alternation

19 would generate considerable anisotropy of the basal plane of graphite, for which no concrete experimental evidence was available.

c

A

B

A’

b

a Figure 8 Orthorhombic crystal structure of graphite proposed by Linus Pauling, belonging to the D 232h − Fmmm space group. Stacking sequence is ABAB.

A few years later Ergun43 reopened the books on this controversy by analysing a flat stressed crystallized slab specimen of pyrolytic graphite to obtain XRD spectra. He compared the observed geometric factors of the (hk0) reflections to the calculated ones and reached the conclusion that Graphite is hexagonal and not orthorhombic as proposed by Pauling. Nevertheless he comes to agree that bond length alternation is possible, if different bond angles (115º 34’ and 122º 13’) from those initially proposed by Pauling were used, which would make the lattice hexagonal. In other words he

20 suggested a quinoid structure that has a hexagonal lattice. In a short article, Donohue44 points out errors made by Ergun in the calculation of the geometric structure factors (F2). Once the errors were corrected, the results did not support the quinoid structure. Trucano et al.45 determined the crystal structure of Graphite using neutron diffraction. They chose their specimen from a collection of seventy small natural single crystals. A careful analysis of the reflections showed no indication of bondlength alternation in the structure. They report the carbon-carbon bond lengths to be 1.422 ± 0.001 Å, which however are averaged over time and a large number of unit cells.

5

Raman Spectroscopy In contrast to the previous crystallographic techniques that sample

predominantly the bulk Raman spectroscopy is surface sensitive. Raman spectroscopy measurments where initially carried out by Tuinstra et al.46 in 1969 on single crystal graphite, stressed annealed pyrolytic graphite, polycrystalline graphite, and activated charcoal. In the single crystal of graphite, only one fundamental absorption peak was observed at ~1575 cm-1, which was assigned to the degenerate E2g vibrational mode and is the only Raman active mode for hexagonal graphite based on symmetry considerations. Their interpretation was that in the graphite with a smaller degree of effective crystallite size a second fundamental at 1355 cm-1 was present that was attributed to a fundamental vibrational mode A1g. They explain that this second Raman mode becomes active due to the lowering of symmetry near the crystallite boundaries. Nakamizo et al.47 later argued that the intensity of the fundamental at 1360 cm-1 was

21 not directly proportional to the effective crystallite size but was a result of lattice distortion. They measured Raman spectra of natural graphite and other carbonaceous materials including pyrolytic graphite, carbon black, glassy carbon, and carbon. Initially they observed the Raman spectrum of crudely pulverized natural graphite which showed an intense peak at 1580 cm-1 and a weaker one at 1355 cm-1. The same specimen was later ground to less than 1μm, which now had only the 1580 cm-1 peak. The relative intensity of the two bands can also be altered by intense laser irradiation or by the bombardment with energetic particles. 48,49 The appearance of the band at 1355 cm-1 was also observed in single crystal graphite after grinding.

50

Splitting of the 1580 cm-1 is also mentioned in the literature51, which has been attributed to lattice distortion of the hexagonal ring to a quinoid structure. It was soon realized that the relative intensity ratio of the I1360/I1580 could be used as a measure of graphitization of carbonaceous materials with annealing. 52

6

Low Energy Electron Diffraction Another technique that probes essentially the surface layer of materials is low-

energy-electron diffraction (LEED). Wu and Ignatiev53 obtained LEED patterns of the [0001] surface of graphite. They observe three-fold symmetry in the diffraction pattern, in which three out of the six first order diffraction peaks are more intense. In

[

][

]

their case the 000 1 , 00 1 1 , and [0010] are relatively more intense and this is attributed to sampling more A termination of the surface, than B termination. Nonetheless, they mention that by cleaving the surface with scotch tape or a razor blade they increase the number of monatomic steps, and thus obtain six fold

22 symmetric LEED patterns. They also observe contraction of 0.05Å in the interlayer separation between the surface and the second layer. Additionally they emphasize that the surface of graphite cannot be cleaned ion sputtering and annealing since such treatments generate considerable damage to the surface making impossible LEED structural determination.

7

Scanning Tunnelling Microscopy In 1983, with the invention of the Scanning Tunneling Microscope54, a new

technique for real space imaging of the topography and the electronic structure of surfaces, became possible. Graphite was one of the first materials to be studied since it had large atomically flat terraces, and contaminant-free surfaces were easily prepared. In 1987 Binnig et al.55, obtained atomic resolution images of Graphite that showed the hexagonal structure of the underlying atomic positions. However half the atoms were missing from the STM image (Fig. 1.9). (a)

(b)

H

A B

B

H

A B A 1st layer

H

A B

H

A H

H site A site B site

B 2nd layer

Figure 9 (a) Schematic of 1st and 2nd surface layer in graphite. Black line and gray line represent surface and second layer respectively. (b) Experimental constant current STM image of HOPG in our lab. Tunneling current = 1.2 nA and Bias Voltage = 0.1 V. Image was Fast Fourier transformed to eliminate high frequency components due to electronic noise. The relationship between A, B and H lattice sites with respect to the STM image of graphite can be seen. Carbon Atom assignment based on the interpretation proposed by Batra et al.. 56

23 The appearance of only half the carbon atoms was attributed to interlayer interactions that had an effect on the electron density near the Fermi level of the surface atoms. Only carbon atoms without a neighbor in the second layer can be seen in the image. Note that the bright maxima are elliptical rather than circular, suggesting two-fold symmetry of the electronic structure rather than trigonal symmetry. Binnig et al. have attributed this to an asymmetric tip. In chapter 3 we carry out various experiments and computations to test an alternative explanation for this phenomenon.

8

Atomic Force Microscopy A few years after the invention of STM another very similar technique emerged,

Atomic Force Microscopy (AFM).

58

The basic advantage of AFM over STM is that

the surface does not have to be conductive to be imaged. In AFM the probe is a tip usually made from silicon or silicon nitride and has the geometry of an inverse pyramid that is a few nanometers in size. The tip is part of a cantilever that serves as a spring. The cantilever/spring system is brought very close to the sample surface. Van der Waals interactions between the tip and the sample lead to a deflection of the cantilever according to Hooke’s law. For graphite obtaining AFM images can be problematic since it is a very soft material. This means that the AFM tip can cause elastic deformations to the layers. Nonetheless, images of graphite have been reported both under vaccum59 and in air60, in which all the carbon atoms (honeycomb lattice) can be resolved. This is expected since AFM will not only probe the electron density at the Fermi level but also that of lower lying energy levels.

24

9

Conclusions In this historical review, an extensive list of scientific articles related to the

structural elucidation of graphite was presented. These articles span a time period of more than a hundred years starting from studies of the external geometry of crystallites and ending with techniques that have the ability to resolve surface carbon atoms. It becomes apparent that some of the previous controversies surrounding the structure of graphite originate from the fact that specimens of different origin (natural, artificial, pyrolytic, etc) or different sample preparation history (powdered, annealed, acid treated, mechanically deformed, etc) were examined by the various crystallographers. This caused deviations from the thermodynamically stable form of graphite resulting in contradictory observations. After the first structural determination using XRD other crystallographic techniques emerged, such as electron diffraction and neutron diffraction, and provided additional details to the diffraction spectra. Raman spectroscopy revealed the surface vibrational modes that were found to be an effective measure of the lattice order. Also the synergy of more recent surface imaging techniques shed light on the surface structure of Graphite and defects of the lattice. The thermodynamically stable allotrope of graphite is the hexagonal structure with a stacking sequence ABAB. This structure is the most abundant in both artificial and natural forms of graphite. Rhombohedral graphite is also observed in which the stacking sequence is ABCA. Natural single crystals of graphite are expected to have a small percentage of the rhombohedral form that will increase with the application of shear stress through mechanical deformation

25 In the first part of this thesis we investigate some of the unresolved issues in the interpretation of the STM image of graphite. We discuss the proposals given by several groups to explain the image and provide a new interpretation that is supported by additional STM experiments and electronic structure computations. Thus, in chapter one, we discuss the principles of operation of STM and XRD along with detailed experimental information, and in chapter two we provide a thorough discussion of on the computational methods used. In chapter three, we present the STM images of graphite obtained in our lab and discuss our hypothesis concerning the “missing” atom in STM images of graphite. Then in chapter four we examine the possibility of bond length alternation (BLA) in graphite through the use of a mathematical model. We apply this model to simulate powder XRD patterns that are then compared to experimental patterns. In the second part of this thesis we provide a detailed description of the π-π interactions in supermolecules composed of PAHs. Since these are the interactions that govern the interlayer binding in graphite, we wish to obtain a better understanding of the interactions between these molecules. Additionally, we are interested in the correlations among molecular size, polarizability, and the VdW interaction between these PAHs. Therefore, in chapter five, we provide second order Møller-Plesset calculations of the interaction between benzene and fourteen PAHs. In chapter six, we describe the proof-of-principle of a new optical differential reflectance (ODR) technique and show that it could potentially be used for the study of VdW interactions between PAHs and graphite.

26

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CHAPTER 1 Instrumentation and Methods

1.1

Introduction In this chapter we describe the two experimental techniques, scanning

tunneling microscopy (STM) and X-ray diffractometry (XRD). We provide details about the two techniques since we use them extensively to elucidate of the nuclear and electronic structure of graphite. STM has the advantage of revealing the surface structure whereas XRD reveals the structure of the bulk material. First we provide some history about the invention and development of these instruments. Then we provide a description of their principle of their operation. This is complemented with the description of various parts of these instruments. Furthermore we explain the basic equations necessary to analyze STM and XRD data. In general this chapter is intended to provide a simplified introduction to the two techniques. 29

30

Scanning Tunneling Microscopy1,2 (STM)

1.2

The scanning tunneling microscope (STM) is extensively used in both fundamental and industrial research to obtain atomic-scale images of conductive materials. The basic advantage of this technique compared to other crystallographic techniques is that it is a real space technique: It directly measures the threedimensional profile of surfaces. This is useful in the study of ordered and disordered surface phenomena, surface reconstruction, observation of surface defects and characterization of surface roughness. Other suggested uses of STM are molecular manipulation and controlled chemical reactions at surfaces, and these are presently under development.

2500

number of articles

2000 1500 1000 500 0 1980

1985

1990

1995

2000

2005

Graph 1.1 Graph showing the number of articles published every year, since 1981, dealing with Scanning Tunneling Microscopy. The search engine searched for the acronym “STM” or the phrases “Scanning Tunneling Microscopy” and “Scanning Tunneling Microscope” in article titles, abstracts and keywords listed in the Science Citation Index Expanded.

Year The usage and development of the STM during the last two decades was enormous. Graph 1.1 shows the number of articles listed in the Science Citation Index Expanded when the acronym “STM” or the phrases “Scanning Tunneling Microscopy” and “Scanning Tunneling Micro-scope” were searched.

31 The operation of the STM somewhat resembles that of the “stylus profilometer”, invented by Williamson3 in 1967, in which the surface profile was measured by dragging a needle across it. Also, Russel D. Young4,5 in 1972, at the National Bureau of Standards invented a similar device named the “topografiner”. The lateral resolutions achieved in these first attempts to construct a high-resolution surface microscope were a few thousand Angstroms, and was not sufficient for atomic resolution of surfaces. In 1981 Binnig and Rohrer6, at the IBM research laboratory in Zürich, incorporated the idea of vacuum tunneling to their device, making the resolution of atoms in real space possible. They observed monoatomic steps and atomic-sized bumps on the lateral terraces of a gold sample. Measurements of the exponential dependence of the tunneling current on the tip-surface separation I(s) was a first indication of vacuum tunneling. The success of the technique came when they were able to correctly estimate a tunnel-barrier height of ~5eV that indicated vacuum tunneling. Similar I(s) curves were also reported earlier, in a thesis by E.C. Teague7,8, but had went unnoticed by Binnig and Rohrer. Also U.Poppe9 reported vacuum tunneling in his effort to do tunneling spectroscopy.

32

1.2.1 STM Principle of Operation The basic principle of operation of an STM is that an extremely sharp metallic tip is scanned over the surface of a conductive or semiconductive material. The tip is held at a distance of a few Angstroms, resulting in a nonzero overlap between the electronic wavefunctions of the tip and the surface. A small potential difference (~0.1V) between the two is sufficient to generate a quantum mechanical phenomenon called vacuum tunneling. In this phenomenon, electrons have the ability to tunnel in the vacuum gap between the surface and the tip. Electrons prefer to move in regions around the nucleus or in-between atoms when chemical bonds are formed. Judging from the high potential energy an electron has in the vacuum gap, this phenomenon is unexpected. Nevertheless, electron tunneling can generate an electric current in the nanoampere range, which can be amplified and measured. A 3D surface map of a conducting material can be constructed by monitoring and maintaining the tunneling current using rapid response electronic feedback circuits during line scans. Fundamentally, the 3D surface map is map of the electronic structure, which often but not always- coincides with the surface topography (nuclear positions). STMs can be operated in two distinct modes, the constant current and the constant height mode. In the constant current mode, the tip is scanned across a surface maintaining a tunneling current constant (see Fig. 1.1). During the line scans the tip extends or retracts depending on the surface corrugation. Small variations in the length of the tip are accomplished with the use of piezoelectric crystals. Variations in the tip length are measured from the voltage applied to the piezoelectric elements.

33

(a)

(b)

Figure 1.1 Schematic showing common modes of STM operation. (a) Constant current mode. Tip length is varied; changes in the length are monitored while tunneling current is maintained constant. (b) Constant height mode. Tip height is constant; changes in tunneling current are monitored while tip to surface separation is constant.

In the case of an electronically homogeneous surface, constant current essentially means constant tip to surface separation. In the constant height mode, the tip height is held constant while the variations in the tunneling current are recorded. Both modes can generate similar surface images, although the second requires a surface with no protrusions or steps. Also in the constant current mode the tip must be scanned at a slower speed so that the tip can follow the surface electronic corrugation without crashing into the surface.

34 An advantage of STM, in comparison with other surface analytical techniques, is that it can probe both periodic and non-periodic 3-D structures on a surface. It is not necessary to operate under ultra high vacuum conditions as with most surface analytical techniques since it can also be operated in air19 and liquids20. In addition, it can provide information on the electronic structure near the Highest Occupied Crystal Orbital (HOCO) of a solid surface or the Highest Occupied Molecular Orbital (HOMO) of an adsorbed monolayer of smaller molecules. These are also the molecular orbitals that are usually involved in chemical reactions. STM, as an analytical surface tool, can contribute to the understanding of surface chemical reactions and the mobility of various species adsorbed on surfaces. 26 There is a plethora of articles on growth phenomena13,14 and surface reconstruction15 studied using the STM. Modified STMs have also been used successfully in nanolithography22 and to manipulate atoms23,24 and molecules25. Other areas of application include the study of the poisoning or the reaction mechanisms of catalysts and structured features of large biological molecules like DNA16-18 and viruses. 21

1.2.2 Theory of Scanning Tunneling Microscopy An STM image represents the tunneling probability density map of electronic states near the Fermi level. The most common method of simulating STM images invokes the Tersoff-Hamann treatment for the tunneling current. Tersoff and Hamann33 used the Bardeen approximation to derive simplified analytical expressions for the tunneling current that is occurring at small bias voltage (~10mV) and low temperatures. In their analytical derivation of the tunneling current, the surface is

35 treated “exactly” while the wavefunction at the tip is assumed to have spherical symmetry (s-orbital). This approximation is also known as the spherical tip approximation. The analytical expression for the tunneling current I t is given by It =

⎛ 8 ⋅ me ⋅ ϕ ⎞ 8 ⋅ e2 ⋅ π 3 ⋅ V ⋅ D t (E F ) ⋅ R 2 ⋅ =3 G 2 ⋅ exp⎜⎜ ⋅ R⎟⎟ ⋅ ∑ Ψv ( ro ) ⋅ δ ( E v − E F ) , (1.1), 2 = me ⎠ v ⎝

where e is the elementary charge (1.602177·10-19 C), V is the bias voltage (V), D t (E F ) is the density of states of the tip (J-1m-3), R is the radius of the spherical tip (m), = is Planck’s constant (1.05457·10-34 Js), m e is the mass of an electron G (9.10939·10-31 kg), ϕ is the surface work function (J), ro is the center of curvature of the tip, E F is the Fermi level, E ν is the energy of state Ψν , and d is the tip-to-surface separation. Importantly the sum extends only over states near the Fermi level (specifically between E F

and E F -eV). Since the probability density of all

wavefunctions has an exponential decay at large distances from the nuclei, it would be expected that the same trend applies to the tunneling current. This is proven by substituting into Eqn. 1.1, the exponential part of the probability density of a wavefunction. After a few algebraic steps, one obtains that the probability density is proportional to the following expression, − G 2 Ψv ( ro ) ∝ e

8⋅me ⋅ϕ ⋅( R + d ) =

. (1.2)

Therefore, the tunneling current in Eqn. 1.1 exhibits an exponentially decaying dependence on the tip to surface separation (d) and a linear dependence on the bias voltage (V) shown in Eqn. 1.3,

36

It ∝ V ⋅ e

−

8⋅me ⋅ϕ ⋅d =

. (1.3)

Making use of the experimentally determined work function for HOPG(0001) (φ = 4.63eV), we plot the tunneling current as a function of tip to surface separation. This plot is shown in Graph 1.2 and demonstrates the dramatic decrease of the

Tunneling current (arb.units)

tunneling current as a function of the tip to surface separation.

0

1 2 tip to surface distance (Angstroms)

3

Graph 1.2 Plot of tunneling current as a function of tip to surface separation for graphite (0001) surface.

This result suggests that only molecular orbitals (MOs) that lie close to the tip are going to make a significant contribution to the tunneling current. We also know that these MOs need to have certain energies in order to participate in the tunneling process. As mentioned previously, only MOs that have energies in the range (EF – eV) to EF, will contribute to the tunneling current. Efermi is the Fermi energy and is equal to the energy of the HOMO of a graphitic layer, whereas Vbias is the bias voltage applied between the tip and the surface.

37

1.2.3 Simulation of STM Image The use of the Tersoff-Hamann treatment for tunneling currents is ideal when the electronic structure of the tip does not have a significant effect on the resulting STM image. This is the usual case in many systems and turns out to be the case for graphite. However, more involved theoretical treatments are also available in the literature that take the effect of the tip structure into account. Nonetheless, if a lower level of theoretical treatment is sufficient to provide results in agreement with experiment, then it is not necessary to use more involved theoretical treatments. In our work, we did not evaluate the absolute tunneling current since the first term in Eqn. 1.1 involves the density of states of the tip near the Fermi level and the tip radius. These quantities are unknown to us and vary from experiment to experiment since the tunneling tip is produced in an uncontrolled and non-reproducible manner. Nevertheless, the fact that similar images were obtained using Pt/Ir and W tunneling tips is an indication that the general features in the image are determined by the surface electronic structure and not by tip effects. So, the physical quantity used in our STM simulations was the local density of states (LDOS) near the Fermi level ρ(r, E F ) , given by

ρ (r, E F ) ≡ ∑ Ψv ( ro ) ⋅ δ ( E v − E F ) . (1.4) G

2

v

This quantity determines the shape, and relative intensity, and periodicity of the spots in the simulated STM image, whereas the omitted part of Eqn. 1.1 affects only the contrast of the image.

38 The possibility of perturbations caused by the tip’s electric field were taken into account for smaller test cases (as discussed in section 3.3.3) and did not have any great effect on the shape of the orbitals at the Fermi level.

1.2.4 Isolation from Mechanical and Thermal Drifts The most significant obstacle in obtaining clear and real images in STM is drift that is caused by ground and sound vibrations or uneven thermal expansion of the various parts in the STM head. The first attempts of obtaining STM images usually resulted in physical contact of the tip with the surface. Mechanical or thermal drift on the tip or sample also makes straight-line scans impossible. Considering the small tip to surface distance, usually 2-5 Å depending on the tip material and tunneling conditions, even small vibrations are sufficient to introduce imaginary features into the image or result in a tip crash. Vibrational damping was initially done with superconducting magnetic levitation of the STM that was sitting in a superconducting magnetic bowl.10 The levitation was later replaced with eddy-current damping. Another type of STM was constructed in 1985, the “pocket-size STM”, in which all the components were significantly reduced in size to minimize all sources of drift.

1.2.5 Tunnel Tips and Sample Preparation The geometry and material of the tip plays a key role in obtaining atomic resolution. A blunt tip will usually deliver poor images. The lateral resolution of an STM is about 0.2 Å for non-periodic and periodic structures, respectively. Field-

39 emission tips that have diameters as small as 100 Å are not suitable due to their vibrational sensitivity. Tips made from Tungsten or Platinum/Iridium alloy are the most commonly used materials for STM tips. The tips can be produced by mechanically cutting a piece of wire, and tungsten tips can also be made by electrochemically etching in 4M KOH. The surface corrugation can be determined to within ~0.01Å due to the extreme height sensitivity of this technique. The enhanced sensitivity is a result of the exponential decay of electron wavefunctions since only small variations in the tip to sample spacing can cause dramatic changes in the tunneling current. A rule of thumb is that an approach of one Angstrom by the tip results in a ten-fold increase in the tunneling current. Tunneling does not require Ultra High Vacuum (UHV) conditions. Nonetheless, in some systems, UHV conditions are necessary to ensure slower contamination of a clean sample. Additionally various surface preparation methods, like Argon sputtering, high temperature annealing and electron sputtering are only UHV compatible. Some materials like graphite, boron nitride, and tin disulfide have layered structure and can be cleaved with the use of adhesive tape to expose clean surfaces. Other materials that have high degree of roughness can be deposited with the use of Chemical Vapor Deposition (CVD) or other electroplating techniques. These techniques deposit the materials onto flat substrates, such as glass and silicon wafers, which can then be used to obtain STM images of the rough materials.

40

1.2.5 STM Description and Operation The STM used in this work was a “beetle-type”, which is an improved version of the STM head developed by K. Besocke11, and was purchased from Carl Zeiss Jena GmbH.12 In his design, minimizing the dimensions of the elements that are present in the STM head significantly reduces drift and the magnitude of vibrational noise. This STM consists of a base plate on which four piezoelectric crystal tubes (BaTiO3) are mounted as shown in Fig. 1.2. On the central piezoelectric crystal, an electrochemically etched tungsten tunneling wire is mounted. The tip dimensions are approximately 0.25x10.0 mm. The base plate rests on a platform vibrationally isolated by three viton dampers. The HOPG sample is secured with the use of a metal clamp onto the “triple helix” sample holder shown in fig 1.3(a)&(c). The sample holder is made of molybdenum (Mo) metal, coated with a polymer to enhance its gliding on the metal bearings during the sample holder movement. It is UHV compatible and can be heated to 1200ºC for sample annealing. First, the sample is lowered with a manual Z translational stage onto the 3 carrier piezoelectric elements. These are terminated with metallic spheres to ensure electrical contact. The coarse tip approach is actually monitored by looking at the base plate that reflects the sample image. After the sample has been lowered to a few micrometers from the tip, the STM controller takes over and gradually decreases the tip-to-surface separation by monitoring the tunneling current. Lowering of the sample holder is achieved by a screw-type movement as shown in Fig 1.3(b).

41

HOPG sample Triple-helix sample holder Tungsten tip (0.25x10mm) Z piezoelectric tube XY piezoelectric tubes

Base plate

Viton dampers

Platform

Figure 1.2 Schematic showing the basic components of Beetle-STM head (purchased from Carl Zeiss Jena GmbH).

The three XY carrier piezos are simultaneously actuated in a fast clockwise movement in which the metal spheres slide along the declined surface of the sample holder. In this phase, the sample holder itself does not move due to its inertia. In the second phase, the piezos return to their initial configuration in a much slower movement, causing the triple helix sample holder both to rotate counterclockwise and to approach the tip. The reverse scheme is applied during the sample withdrawal.

42 Approaching the sample is always easier than withdrawing it due to the sample holder mass. These XY piezos are also responsible for the XY movement of the sample during the image acquisition. Once a tunneling current is in the nA range based on a preset tunneling voltage, the automatic approach stops and image aquasition can now be started. The image is acquired by line scans that cover the area under investigation and that are generated by supplying the correct voltages to the three XY piezos perform line scans. An area with square dimensions as large as 5000 Å can be scanned with this instrument. The sample holder can be translated in any direction in the μm range. Also a goniometric knob can adjust the scanning direction. Note that in our STM it is not the tip that makes line scans but it is the relative movement of the sample that generates line scans.

43

(a)

Metal clamp Mo holder HOPG sample Metallic sphere XY Electrodes STM tip Z Electrode

(b)

fast slow

(c)

1.7 cm

Figure 1.3 Triple helix sample holder, (a) side view; The tip to surface separation during scanning is in the range of 2-5 Å. (b) Top view of XY piezoelectric tubes (carrier piezos) during fine Z approach, (c) bottom view; Side in contact with piezoelectric crystals.

44

1.3

X-Ray Diffraction (XRD)

1.3.1 The Bragg Law The fundamental equation in crystallography is the Bragg law, which was published by Sir William Henry Bragg and Sir William Laurence Bragg9 (father-son) in 1913. This equation provided the mathematical basis of X-ray crystallography and made possible the crystallographic determination of the structures of crystals. The law has the following simple form, n ⋅ λ = 2 ⋅ d ⋅ sin θ , (1.5),

where n is a positive integer that is the order of diffraction, λ is the wavelength of the incident X-ray radiation, d is the spacing between a set of crystallographic planes, and θ is the angle between the incident radiation and the set of planes. The implication of

this relation is that the waves that are in phase interfere constructively, generating diffraction peaks only when the path length difference ( 2 ⋅ d ⋅ sin θ ) is an integral number of wavelengths ( n ⋅ λ ).

2 ⋅ d ⋅ sin θ θ

d Figure 1.4 Schematic showing the geometric derivation of the Bragg law.

45 Although Bragg’s law was initially formulated for X-rays diffracted from crystals, it was later realized that it applies to all states of matter and other beams, such as electron and neutron beams.

1.3.2 X-ray Radiation Source A conventional radiation source for X-ray diffractometers is the X-ray tube that can generate X-rays29 of a narrow wavelength range. A simple schematic of such an X-ray source is shown in Fig. 1.5. The tube consists of an anode and cathode held at a high voltage difference (30-150 kV) in a vacuum tube. X-rays

filament cathode

Vacuum tube -

e

anode (target)

high voltage

Figure 1.5 Simplified schematic of an X-ray tube, which emits at a constant wavelength. The voltage difference between anode and cathode ranges between 30-150 kV depending on target material and application. X-rays generated are not entirely monochromatic thus filtered with the use of filters and monochromators.

A heated filament emits electrons by thermionic emission. These emitted electrons are accelerated towards the anode due to the presence of the strong electric field between anode and cathode. The electrons collide with the anode (target) and ionize inner shell (core) electrons of the target atoms. Subsequently in these atoms, higher energy electrons undergo relaxation through radiative emission of X-rays. The

46 most intense X-rays emitted from the source have energies characteristic of the spacing of the energy levels in the emitting atoms. This means that in order to change the wavelength of the radiation a different material has to be used for the anode. A list of heavy metals that are commonly used in X-ray tubes and their characteristic wavelengths are given in Table 1.1. Table 1.1 Characteristic emission lines of various materials commonly used in X-ray tubes. Characteristic wavelengths taken from W.L. Bragg. 27

1

Material

Z

Kα1 (Å)

Kα2 (Å)

Kα1 (Å)

Kβ1 (Å)

β filter

α filter

Molybdenum Copper Iron Chromium

42 29 26 24

0.70926 1.54050 1.93597 2.28962

0.71354 1.54434 1.93991 2.29352

0.7107 1.5418 1.9373 2.2909

0.63225 1.39217 1.75654 2.08479

Nb Ni Mn V

Y Co Cr Ti

Ka is the intensity average of Kα1 and Kα2, Kα1 being twice the weight of Kα2.

Although, as mentioned earlier, the emitted radiation from the X-ray tube has stronger intensity in a very narrow window of wavelengths, it is not entirely monochromatic. This can be seen in an X-ray emission spectrum, such as that of Molybdenum presented in Fig. 1.6.

11.0

Κβ Κα

Intensity (arbitrary units)

15.2

37.2

0.0 0.30

0.95

Wavelength (Å)

Figure 1.6 X-ray emission spectrum of Molybdenum. Voltage of X-ray tube kept at 35 kV. Image reproduced from C.T Ulrey. 28

47 The radiation consists of a broad bell-shaped band (Bremsstrahlung continuum) and some sharper peaks such as, Kα and Kβ. There is also a fine structure to Kα and Kβ, which however, is not resolved in this spectrum. For Molybdenum, Kα is a doublet denoted by the symbols Kα1 and Kα2. The letter K indicates that the characteristic line corresponds to transitions to the n=1 shell. In the same fashion L, M, and N, would correspond to the relaxation to shells where the principal quantum number n is 2, 3, and 4 respectively. The Greek subscript α indicates that the transition originates from an adjacent shell (e.g. M to L or L to K) while β indicates that the transition originates from a non-adjacent shell (e.g. M to K or N to L). Monochromatic radiation can be obtained with the use of filters and monochromators. Filters are usually used to absorb the Kβ radiation, which is less intense than the Kα radiation. Furthermore, monochromators are used to increase the relative intensity ratio between Kα1 and Kα2, in favor of the Kα1 radiation that is again twice as intense as the Kα2. It is interesting to note that the monochromator used for some X-ray sources is a bent single crystal of graphite with a very low mosaic spread (0.4º ± 0.1º) and a lateral grain size of about 3 mm. Instruments that lack a monochromator can subtract the Kα2 peaks during data processing. As mentioned earlier, the wavelength at which the X-ray tubes emit is constant and depends on the material of the anode used in the cathode ray tube. However it has recently become possible to use a continuously tunable X-ray source, the synchrotron accelerator source. In this device a charged particle beam is accelerated through the use of bending magnets in a circular path. The radiation emitted through this process

48 can have shorter wavelengths than conventional X-ray tubes, and thus greater resolution is possible in the diffraction patterns.

1.3.3 X-Ray Diffraction Pattern The X-ray diffraction pattern is produced when a collimated monochromatic Xray beam is directed towards a sample and the reflected radiation (not the transmitted) is collected on a photon detector. Most of the radiation passes through the sample, as in the case of a Röntgen photograph29, but some is scattered in specific directions that are characteristic of the sample. It is both the direction as well as the relative intensity of the scattered radiation that provide information for crystallographic analysis. The diffraction pattern was originally collected on a photographic film, which has now been replaced by photon detectors. There are two types of diffraction patterns depending on the physical state of the sample. A powder diffraction pattern is obtained from polycrystalline or powder samples and single-crystal diffraction pattern from single crystals. The main difference between the two can be seen in Fig. 1.7. (a)

(b)

Figure 1.7 Schematic representations of the two types of diffraction patterns. (a) Powder diffraction pattern. (b) Single crystal diffraction pattern.

49 The powder diffraction pattern is essentially the pattern obtained when the complete collection of single-crystal diffraction pattern of a crystal are superimposed and rotated around the [000] reflection. For completely random powders the powder XRD has a center of symmetry at [000]. In that case a plot of the intensity as a function of two-theta (2θ), which is the angular deviation of the diffraction ring from the incident beam, is sufficient to present all the crystallographic information given by the powder XRD. These plot are called two-theta powder XRDs. The powder method has relatively less crystallographic information than the single crystal method due to this superposition. However it has the advantage that impurity as large as 50% are allowed as long as the impurities is amorphous, such as glass, plastic, or other materials with a known crystal structure. The single-crystal method on the other hand requires the use of single crystals of appreciable size, free from twinning and distortions. The growth of single crystals is usually time consuming and sometimes unfeasible.

1.3.4 Sample Preparation A requirement for the powder method is that there is random orientation of crystallites in the sample.

This can usually be achieved by finely grinding the

substance and passing the powder through a mesh to obtain particles with an average diameter of a few μms. The decrease of average particle size will increase the intensity of the scattered radiation, however, particles smaller than 0.1 μm are not desirable since they cause lower resolution due to peak broadening. If possible, thermal annealing of the sample is also recommended to eliminate particle deformation during

50 grinding. The particles can be deposited on a glass plate by solvent evaporation. This works for most materials but fails for the materials that yield needle-shaped or lamellar crystallites, such as carbon nanotubes and graphite. Here the particles acquire orientations that are random in two dimensions but rather regular in the third. For example graphite powder, after solvent evaporation, obtains positions with their c crystal vector perpendicular to the deposition plate and with random rotation around it (mosaic spread of 180º). This causes the resolution of only the [002], [004] and [006] lines in the diffraction pattern. To overcome this situation, one must disperse graphite powder on a double-sided tape or on a flat layer of Vaseline or Silicone. Also a finer powder with particles in the 10-40 μm diameter range will promote random configuration. We found that with the use of a carbon steel brush (DREMEL 428) operated at 7000 rpm, crystallites in that range could be readily obtained.

1.3.5 Peak Shape Functions The peak shape observed in a diffraction pattern and the full width at half maximum (FWHM) depends on the following three broadening mechanisms: (i) particle size broadening, (ii) strain broadening, (iii) the radiation source beam diameter and spectral distribution, and (iv) strain broadening. Particle size broadening has a reciprocal relationship to the particle diameter, thus crystals with a diameter less that a tenth of a micrometer must be avoided. The effect of particle size broadening is modeled through the Scherrer30 equation,

B=

0.9 ⋅ λ , (1.6), t ⋅ cos θ

51 where t is the grain size, λ is the wavelength of the incident X-ray radiation and θ is the angle between incident radiation and the set of planes. Strain broadening can alter both 2θ and the FWHM. Uniform strain (axial compression of crystal) will affect 2θ in the diffraction pattern whereas non-uniform strain (bending of crystal) affects the FWHM. The former type of strain causes equal deformation in most unit cells, whereas the latter generates a continuous spread of unit cells with small variations in their dimensions. In practice, the spectral distribution of the diffraction peaks can be described by a Lorentzian, a Gaussian, or a Pseudo-Voigt function. The functional form of a normalized Gaussian and Lorentzian distribution are given by, PG ( x, μ , σ ) =

⎛ − ( x − μ) 2 ⎞ 1 ⋅ exp⎜ ⎟ , (1.7), ⎝ 2σ 2 ⎠ σ 2π

PL ( x, μ , Γ ) =

⎞ Γ2 1 ⎛ ⋅⎜ 2 2 ⎟ , (1.8), π ⎝ ( x − μ ) + ( Γ 2) ⎠

respectively, where μ is the average, σ is the standard deviation, and Γ is the FWHM. The graph in Fig 1.8 shows the plot of a Lorentzian function and a Gaussian function.

Distribution

Figure 1.8 Comparison between normalized Gaussian (dashed) and Lorentzian (solid) distributions. The functionals of the two distributions are given in Eqns. 1.7 and 1.8, respectively. The parameters used were Γ = 1, σ = 0.6271 and μ = 0. Note that the area under both curves is 1. x

52 The first has a wider tail and a less rounded appearance near the distribution’s maximum. Nonetheless, peak shapes in most real cases are Voigt distributions, due to inhomogeneous broadening. A Voigt distribution is actually the superposition (convolution) of a Gaussian and a Lorentzian function. If μ is set to zero, it has the form, ∞

∞

⎞ ⎛ − x2 ⎞ ⎛ Γ2 ⎜ ⎟ dx . (1.9) Pv = ∫ PG ⋅ PL dx = exp ⋅ ⎜ ⎟ 2 2 2 ∫ σ 2 π 3 −∞ ⎝ 2σ ⎠ ⎝ x + ( Γ 2) ⎠ −∞ 1

Due to the integral form of this function it is not conveniently implemented in fitting routines. Nevertheless, there are alternative functions that are used, such as pseudo-Voigt functions. Pseudo-Voigt functions are usually a linear combination or a product of a Gaussian and a Lorentzian function. The pseudo-Voigt function used in the present work has the general form of Ppseudo − V = η ⋅ PV + (1 − η) ⋅ PG , (1.10), where η is the contribution of the Lorentzian to Ppseudo − V (0 ≤ η ≤ 1). If particle size broadening and strain broadening are eliminated, then the peakshape function is solely determined by the radiation source. Synchrotron diffraction results in pseudo-Voigt and neutron diffraction in Gaussian shape peaks.

53

1.3.6 Powder XRD 2θ Pattern Fitting Routine In our experimental powder XRD patterns, the reflections consist of two peaks, the Kα1 and Kα2. The peak shape function ( Pfitting ) used in our fitting routine was a linear combination of two pseudo-Voigt functions and a third order polynomial. Pfitting = Ppseudo − V ,1 + Ppseudo − V , 2 + ax 3 + bx 2 + cx + d , (1.11)

Here x is the value of 2θ and Ppseudo − V ,i is given by Eqn. 1.10. The third order polynomial is added to take into account the non-zero background. This is particularly important between 40 and 60 degrees in 2θ, where broad unresolved diffraction from impurities present in the sample holder (amorphous glass) contributes to the signal. It also occurs in situations where the peak tail of an intense reflection is adjacent to a weaker reflection. A nice example of such a case is the stronger [002] (26.5°) of HOPG found adjacent to the weaker [111] (28.5°) reflection of Silicon (see Fig 4.11), which was spiked into the graphite sample to provide an internal calibration of the measured angles. Although we initially tried Gaussian and Lorentzian peak shape functions, the pseudo-Voigt function provided the best fitting results. The restrictions imposed on the fitting of the two peak functions were, (a) The intensity of Kα2 is fixed to be exactly half that of Kα1, (b) the 2θ position32 of the Kα2 peak is determined from the position of the Kα1, and (c) the width of both peaks is assumed to be the same. The custom routine for fitting these reflections in our XRD patterns is implemented in IGOR.

31

It uses the Leavenburg-Marquardt algorithm to

simultaneously fit Pfitting to the experimental peaks. The Leavenburg-Marquardt

54 algorithm minimizes chi-squared ( χ 2 ) to obtain the best fitting parameters. Chisquared is given by 2

⎡ Pexp,i − Pfitting ,i ⎤ χ = ∑⎢ ⎥ , σi i =1 ⎣ ⎦ N

2

where Pexp,i is the experimental intensity value of the ith data point, Pfitting ,i is the intensity value that the fitting function yields for a given set of fitting parameters and σ i is the standard deviation of the data point. The range of 2θ values involved in

fitting for each reflection was approximately ± 1° in 2θ for the weaker reflections and ± 2° for the stronger reflections. The algorithm begins by calculating chi-squared based on an initial guess of the fitting parameters. Subsequently it uses a steepest descent minimization of the error surface with respect to the fitting parameters.

55

References 1.

G. Binnig and H. Rohrer, Surf. Sci. 126, 236 (1983).

2.

G. Binnig and H. Rohrer, Rev. Mod. Phys. 59, 615 (1987).

3.

B. P. Williamson, Proc. Inst. Mech. Eng. London 182, 21 (1967).

4.

R. D. Young, Physics Today, November 42 (1971).

5.

R. D. Young, J. Ward and F. Scire, Rev. Sci. Instrum. 43, 999 (1972).

6.

G. Binnig, H. Rohrer, C. Gerber, E. Weibel, Phys. Rev. Lett. 50, 120 (1983).

7.

E. C. Teague, PhD Thesis, North Texas State University, (1978).

8.

E. C. Teague, Bull. Am. Phys. Soc. 23, 290 (1978).

9.

U. Poppe, Verh. Dtsch. Phys. Ges. 16, 476 (1981).

10.

C. F. Quate, Physics Today August, 26 (1986).

11.

K. Beschoke, Surf. Sci. 181, 145 (1987).

12.

Beetle-STM user manual, Carl Zeiss Jena GmbH (1994).

13.

T. A. Land, T. Michely, R. J. Behm, J. C. Hemminger, G. Comsa, J. Chem. Phys. 97, 6774 (1992).

14.

T. A. Land, T. Michely, R. J. Behm, J. C. Hemminger, G. Comsa, Surf. Sci. 264, 261 (1992).

15.

R. M. Tromp, R. J. Hamers, J. E. Demuth, Science 234, 304 (1986).

16.

P. G. Arscott, G. Lee, V. A. Bloomfield, D. F. Evans, Nature 339, 484 (1989).

17.

D. D. Dunlap and C. Bustamante, Nature 342, 204 (1989).

18.

R. J. Driscoll, M. G. Youngquist, J. D. Baldeschwieler, Nature 346, 294 (1990).

19.

S. I. Park and C. F. Quate, Appl. Phys. Lett. 48, 112 (1986).

20.

J. Schneir, R. Sonnenfeld, P. K. Hansma, J. Tersoff, Phys. Rev. B 34, 4979 (1986).

56 21.

J. G. Mantovani, D. P. Allison, R. J. Warmack, T. L. Ferrell, J. R. Ford, R. E. Manos, J. R. Thompson, J. Microscopy 158, 109 (1990).

22.

J. A. Dagata, W. Tseng, J. Bennett, J. Schneir, H. H. Harary, J. Appl. Phys. 70, 3661 (1991).

23.

M. F. Crommie, C. P. Lutz, D. M. Eigler, Science 262, 218 (1993).

24.

L. Bartels, G. Meyer, K. H. Rieder, Phys. Rev. Lett. 79, 697 (1997).

25.

A. J. Heinrich, C. P. Lutz, J. A. Gupta, D. M. Eigler, Science, 298, 1381 (2002).

26.

B. M. Trafas, D. M. Hill, P. J. Benning, G. D. Waddill, Y. N. Yang, J. H. Weaver, Phys. Rev. B 43, 7174 (1991).

27.

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28.

C. T. Ulrey, Phys. Rev. 11, 401 (1918).

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W.C. Röntgen discovered X-ray radiation in 1895.

30.

P. Scherrer, Göttinger Nachrichten (1918).

31.

IGOR Pro, version 4.0, WaveMetrics, Oregon (2000).

32.

The relationship is used in the fitting routine to determine the 2θ position of the Kα2 with respect to the Kα1 peak is, ⎛λ ⎞ θ 2 = sin −1 ⎜⎜ 2 ⋅ sin θ 1 ⎟⎟ . ⎝ λ1 ⎠

33.

J. Tersoff and D. R. Hamann, Phys. Rev. Lett. 50, 1998 (1983).

CHAPTER 2 Computational Methods

2.1

Introduction Computational chemistry has proven to be a valuable tool to understand the

static and dynamic behavior of interacting molecules in a wide array of disciplines. Using the principles of quantum mechanics we can calculate how a molecule behaves in the presence of another, how chemical reactions occur and which structural changes a molecule undergoes during chemical reactions. This can be extended to complex molecular structures such as the folding of proteins in biological systems. New molecules can be designed with tailored properties, and one can also observe the electronic structure changes during chemical reactions. Industrial applications, such as computer-aided-drug-design have emerged recently due to the great success of computational chemistry methods in research. Geophysicists and material scientists have also greatly benefited from the advances in ab-initio calculations since the

57

58 extreme conditions required in their studies (high pressures and temperatures), are not easily obtained experimentally. In this chapter, we describe one of the earliest electronic structure methods, the Hartree-Fock method. Although many other electronic structure methods have been devised since then, this method is sufficient to unfold the basic ideas of electronic structure computations. A detailed review on computational methods can be found in Szabo and Ostlund2. In section 2.3 we present examples of saturating the basis set in orbital and plane-wave based methods. Subsequent sections (2.4 and 2.5) discuss practical considerations in ab-initio electronic structure calculations, such as the use of symmetry unique atoms and how to efficiently locate local minima in geometry relaxations. In section 2.6 we explain terms such as, basis set superposition error (BSSE), electron correlation and zero point energy (ZPE) that become important in the accurate computation of weaker Van der Waals interactions.

2.2

The SCF Hartree-Fock Method 2,3,4 The purpose of the Hartree-Fock method is to find approximate solution to the

time-independent Schrödinger equation, HΨ = EΨ . (2.1)

An equation of this form is called an eigenvalue equation. By acting with the Hamiltonian H (energy operator) on the N-electron wavefunction Ψ we obtain the energy of the molecule. Hence, the wavefunction is a mathematical function that describes all the electrons and nuclei in a molecule. Unfortunately, solutions to this equation for many-electron systems is a computational problem that is too complex to

59 be solved analytically. Nonetheless, for one-electron systems such as the hydrogen atom or the molecular hydrogen cation H2+, analytical solutions2 are available. In order to obtain numerical solutions certain approximations have been made in the historic course of quantum mechanics that we will point out in the following paragraphs. The complete Hamiltonian operator is, Htot = Te + Tn + Vne + Vee + Vnn , (2.2)

where Te and Tn are the nuclear and electronic kinetic energy operators, and Vne , Vee , and Vnn are the nucleus-electron, electron-electron and nucleus-nucleus interaction operators, respectively. The complete substituted Hamiltonian operator in atomic units (a.u.) is,

Htot

N M 1 N 2 1 M ∇ 2A z A N N 1 M M z A zB = − ∑ ∇i − ∑ − ∑∑ + ∑∑ + ∑ ∑ , (2.3), 2 i =1 2 A=1 M A i =1 A=1 riA i =1 j >i rij A=1 B> A R AB

where ∇ 2 is the Laplacian operator, and M A , z A , riA , rij and R AB , are the nucleus over electron mass ratio, the atomic number of nucleus A, the distance between electron i and nucleus A, the distance between electrons i and j, and the distance between nucleus A and B, respectively. An approximation crucial in solving the Schrödinger equation is the BornOppenheimer approximation, which simplifies significantly the Hamiltonian operator. Due to the large mass of the nucleus, as compared to that of an electron, the nucleus moves much more slowly than electrons. For this reason, the kinetic energy of the nucleus term in the Hamiltonian may be set to zero ( Tn = 0 ). In other words, the nuclei

60 in a molecular structure are treated as stationary points, and only electrons are moving in the potential energy field of the nuclei. Thus the nuclear repulsive potential energy term becomes a constant ( Vnn = const ). The simplified form of the complete Hamiltonian is called the Electronic Hamiltonian and has he following form,

Helec

1 N 2 N M zA N N 1 = − ∑ ∇ i − ∑ ∑ + ∑ ∑ . (2.4) 2 i =1 i =1 j >i rij i =1 A =1 riA

Solving the Schrödinger equation using the electronic Hamiltonian yields the electronic wavefunctions Ψelec and the electronic energies Eelec. The total energy is the sum of the electronic energy and the internuclear repulsion energy, M

M

z A zB . (2.5) A =1 B > A R AB

Etot = Eelec + ∑ ∑

The other component of the Schrödinger equation which has to approximated in some manner is the wavefunction Ψ. Hartree in 1926, approximated the electronic wavefunction ΨHP ( x1 , x2 ,..., x N ) of the molecule as the product of N independent one electron wavefunctions ϕ ( xi ) (spin-orbitals). The spin-orbitals are products of spatial orbitals and spin functions that describe the electronic spatial distribution and spin, ΨHP ( x1 , x2 ,..., x N ) = ϕ ( x1 )ϕ ( x2 )...ϕ ( x N ) . (2.6)

This approximation is not completely correct since electrons are not treated as fermions. For fermions, the wavefunction has to be antisymmetric with respect to particle interchange,

ΨHP ( x1 ,..., xi ,..., x j ,..., x N ) = − ΨHP ( x1 ,..., x j ,..., xi ,..., x N ) . (2.7)

61 Slater and Fock suggested the use of a determinant (Slater determinant) that would enforce the antisymmetry of the many electron wavefunctions,

ΨHP ( x1 , x2 ,..., x N ) =

1 N!

ϕ1 ( x1 ) ϕ1 ( x 2 )

ϕ 2 ( x1 ) … ϕ N ( x1 ) ϕ 2 ( x2 ) ϕ N ( x2 )

ϕ1 ( x N ) ϕ 2 ( x N )

. (2.8)

ϕ N ( xN )

A determinant has the mathematical property to change sign when two rows are interchanged. This can be easily seen in the following illustrative example. For a two-electron system the Slater determinant describing the wavefunction is,

Ψ( x1 , x2 ) =

1 ϕ1 ( x1 ) ϕ 2 ( x1 ) 1 = ϕ ( x )ϕ ( x ) − ϕ1 ( x2 )ϕ 2 ( x1 ) , (2.9). 2 ϕ1 ( x 2 ) ϕ 2 ( x 2 ) 2 1 1 2 2

[

]

Interchanging the labels of the two electrons, making electron ( i.e.1 2

2 and

1) it becomes,

Ψ( x2 , x1 ) =

1 ϕ1 ( x 2 ) ϕ 2 ( x 2 ) 1 = ϕ ( x )ϕ ( x ) − ϕ1 ( x1 )ϕ 2 ( x2 ) . (2.10) 2 ϕ1 ( x1 ) ϕ 2 ( x1 ) 2 1 2 2 1

[

]

Comparison of Eqns. 2.9 and 2.10 yield Ψ ( x1 , x2 ) = − Ψ ( x2 , x1 ) . Another feature of the Slater determinant is that the wavefunction vanishes when two electrons occupy the same spin-orbitals. This can be seen in Eq. 2.11,

Ψ( x1 , x1 ) =

1 ϕ1 ( x1 ) ϕ 2 ( x1 ) = 0 . (2.11) 2 ϕ1 ( x1 ) ϕ 2 ( x1 )

In practical applications the one-electron orbitals are expanded as a linear combination of basis functions ( χ μ ( x ) ), K

ϕi ( x ) = ∑ Cμi χ μ ( x ) , (2.12), μ =1

62 where Cμi is a dimensionless coefficient and K is the number of basis functions used in the expansion. The value of Cμi is a measure of the contribution of a basis function to the one electron orbitals. As mentioned earlier, there are no analytical solutions for systems with more than one electron. However numerical solutions can be provided with the use of the variational principle. The variational principle states that the best approximation to the true ground-state wavefunction of a molecule is the one that has the lowest expectation value (ground state energy) for the electronic Hamiltonian operator. The variational principle also has the desirable feature that it yields an upper-bound to the exact energy within a given basis set, E0 = Ψ0, HP Helec Ψ0, HP

≥ Eexact , and it forms the

basis for approximate electronic structure methods, such as the Hartree-Fock procedure. In order to further simplify the electronic Hamiltonian, the SCF Hartree-Fock method makes use of the Hartree-Fock approximation. Instead of solving the Schrödinger equation using a Slater determinant that specifies all the electrons in a molecule, one can solve N-independent one-electron equations. A direct result of this mathematical approach is that the electron-electron repulsions are no longer evaluated on a one to one basis. Every electron experiences an average repulsive interaction that is a result of the electron density of the remaining N-1 electrons,

63

Helec

⎛ N 2 2⎞ ⎜ ∑ ϕi ( x) − ϕ k ( x) ⎟ N N ⎜ i =1 ⎟ ∇2 N M z i≠k ⎟ , (2.13), ≈ −∑ i − ∑ ∑ A + ∑ ⎜ rik ⎟ i =1 2 i =1 A =1 riA i =1 ⎜ ⎜ ⎟ ui ⎝ ⎠ gi ⎞ ⎛ ∇ i2 ≈ ∑⎜− + ui + gi ⎟ . (2.14) 2 ⎠ i =1 ⎝ N

Helec

This assumption simplifies the N-electron electronic Hamiltonian to a sum of one electron Hamiltonians, which are called the Fock operators hi , hi ≈ −

∇i2 + ui + gi . (2.15) 2

The many-electron Schrödinger equation can be solved as N independent oneelectron equations, which are called the Hartree-Fock equations,

hiϕ i ( x ) = ε iϕ i ( x ) . (2.16) Notice that each Fock operator depends on the density of the other electrons, and hence, the Hartree-Fock (HF) equations have to be solved in a self-consistent fashion. Hence the HF method is often denoted as the self-consistent field (SCF) method, and it consists of the following steps. Make initial guess of spin-orbitals by choosing a basis set and a guess method (i.e. Hückel) Calculate gi which is the average field due to the remaining N-1 electrons seen by each electron. Solve Hartree-Fock equations and obtain new eigenfunctions.

64 if improved energy is the same as the one in the previous step within a preset threshold, then stop SCF procedure. if not, use the current eigenfunctions as an initial guess in the first step.

2.3

Basis Set Saturation The one-electron wavefunctions in ab-initio calculations are usually expanded

using atom centered gaussian-type functions, which are constructed to closely resemble Slater-type orbitals. Slater-type orbitals are spatially close to the region where electron density is expected and they also decay exponentially. Another popular method is to expand the wavefunctions using planewaves in conjunction with pseudopotentials. In principle, the wavefunctions could be expanded in any complete basis set, such as a complete set of the powers of x (x, x2, x3, x4, … ). However, expanding the wavefunctions using the powers of x would require a substantially greater number of basis functions than using gaussian-type orbitals. Table 2.1 Hartree-Fock calculations of Benzene using various basis sets. The geometry was kept to the STO-3G optimized geometry.

Basis set label

Basis functions CPU timea Total energy

ΔΕc

(s)

(Hartreeb)

(Hartree)

STO-3G

36

3

-227.8913

1.5243

3-21G

66

4

-229.4156

1.2438

6-311G

96

13

-230.6594

0.0598

6-31+G(d,p)

144

44

-230.7192

0.0348

6-311++G(d,p)

174

119

-230.7540

0.0102

6-311+G(2d,2p)

216

199

-230.7642

0.0000

6-311++G(2d,2p)

222

229

-230.7642

-

a b c

on a Pentium 4, 3.2 GHz (800 MHz FSB) hyperthread technology, 1GB RAM, NWChem.10 1 Hartree = 627.5094717 kcal/mole

ΔΕ = Esmall basis – Elarge basis

65 The variational flexibility of the one-electron wavefunctions depends both on the choice of spin-orbitals and the number of basis functions. The use of an infinite number of spin-orbitals is known as the Hartree-Fock limit. At this limit, you get an accurate approximation to the one electron wavefunction. The general strategy of choosing a basis set for a particular calculations is to look at the total energy convergence with respect to the basis set size. For atom-centered gaussian-type orbitals, this means that additional polarization and diffuse functions are added to the wavefunctions until convergence of the total energy is achieved within the desired precision. Table 2.1 shows the convergence of the total energy in benzene as a function of gaussian basis functions. Table 2.2 shows such a convergence in a supercell slab of graphite, using a plane-wave basis and core pseudopotentials. Table 2.2 DFT computations of 4-layer graphite slab. The exchange and correlation effects are considered within the generalized gradient approximation (GGA) using the PBE96 exchange-correlation (XC) functional. The electronic core was described by Hamann normconserving pseudopotentials in which the core radii rcl for the various orbitals (l = s, p, d) was set to, 0.80 αο (bohr unit of length), 0.85 αο and 0.85 αο, respectively. The lattice parameters for the slab are a = 20.000Å, b = c = 4.912Å, α = 120º , and β = γ = 90º. The program, NWChem10 was used in all the computations.

k-points

planewaves

KE cutoff (eV)

Total Energy (Hr)

ΔEa (eV)

80x20x20

6976

22.1

-45.30495743

-

88x22x22

9332

26.8

-45.33818951

0.1130

96x24x24

12099

31.8

-45.35422936

0.0546

104x26x26

15385

37.4

-45.36863463

0.0490

112x28x28

19241

43.3

-45.37486024

0.0212

120x30x30

23607

49.7

-45.37680327

0.0066

128x32x32

28738

56.6

-45.37746630

0.0023

136x34x34

34436

63.9

-45.37753171

0.0002

66

2.4

Symmetry Unique Atoms In ab-initio calculations it is essential to reduce the computational time and

resources as much as possible. In high symmetry molecule this is possible with the use of symmetry unique atoms. These are the atoms in a molecular structure that can generate the complete molecule through point group operations. An example of the symmetry unique atoms in coronene and a hydrogenated analog of coronene is given in Fig 2.1. Coronene belongs to a higher symmetry point group (D6h) and thus has a relatively smaller set of symmetry unique atoms. Their symmetry unique atoms are (3C/1H) and (7C/3H), respectively. H

(a)

(b)

H

H

H

H

H

H

H

H

H

H

H H

H

H H

H

H

H

H

H

H

H

H H

H H

H

Figure 2.1 Schematic representation of symmetry unique atoms in (a) coronene, and (b) a hydrogenated analog of coronene. They belong to D6h and D2h point groups, respectively. ) and symmetry unique hydrogen atoms Symmetry unique carbon atoms shown as ( shown in bold.

This means that with a moderate basis set, such as cc-pVDZ1 within the Hartree-Fock formalism (conventional SCF scales roughly as N3.5, where N the number of basis functions), a single-point energy calculation of coronene would be 19 times faster than that for the hydrogenated analog. Thus high symmetry molecules are desired in high-level ab-initio calculations due to the decrease in computational time and memory requirements.

67 Group theory applied to chemical systems states that electrons or better the one electron wavefunctions that exist around two symmetry equivalent atoms should behave in the same fashion during the iterative self-consisted field approximation of the one electron wavefunctions. As a consequence the resulting molecular orbital has the same shape around symmetry equivalent atoms. 2C6, 2C3, C2, S6 and S3

C2(z) C2(y)

3C''2 and 3σd

i

3C'2 and 3σv

i

C2(x)

Figure 2.2 Schematic representation of symmetry elements (a) of coronene, and (b) hydrogenated analog of coronene. Mirror planes σh and (σ(xy) , σ(xz) , σ(yz)), tht belong to the point groups D6h and D2h are not shown for better clarity.

The use of symmetry in ab-initio structure optimizations, however, can occasionally lead to false structural results. Symmetry restrictions may force a structure to reside in a saddle point rather than a local minimum. This happens when other quantum-mechanical phenomena occur that are not obvious from the molecular configuration itself, such as first order Jahn-Teller distortion. In that case, one can use initially symmetry for the optimization, and then use the optimized geometry and one electron wavefunctions for a subsequent optimization with no symmetry restrictions (C1). A common, but more time-consuming approach is to compute the vibrational

68 frequencies, where negative frequencies would be an indication of a saddle point rather than a local minima.

2.5

Locating Local Minima in Geometry Relaxations Structural optimization of high molecular weight molecules can often be hard

to accomplish. Reaching convergence of the total energy depends both on the initial guess for the Cartesian coordinates of a molecular structure and the initial guess for the wavefunction. Our benchmark calculations show that both the wavefunction and the forces exerted on a molecular geometry can reach convergence if a gradual optimization scheme is followed. This optimization scheme should be useful only for geometry optimizations of molecules containing more than 50 carbon atoms. For smaller molecules, convergence is not generally problematic if an initial geometry from a semi-empirical calculation is used. The optimization scheme has the following steps: Design molecular geometry and optimize using one of the semiempirical methods, such as AM1. Use geometry from previous step to carry out HF calculation with a minimal basis set (STO-3G). Use geometry from previous step to repeat HF with a larger basis set (6-31Gd) Use geometry and wavefunctions from previous step, for a density functional calculation (B3LYP estimates atomization energies and bond lengths of close shell systems to about ± 5 kJ/mole and ± 0.003 Å of the experimental values, respectively).

69 Use geometry from previous step and project wavefunctions onto an augmented basis set, such as aug-cc-pVDZ to repeat B3LYP. Our benchmark calculations are tabulated in Table 2.3. Structural optimization of the larger MW molecules in this homologous molecular series C6n2H8n, such as C54H24, C96H32, and C150H40, did not converge, unless the gradual optimization scheme was followed. Table 2.3 Summary of total energy and cartesian gaussian basis functions used for the optimization of the molecular series of C6n2H8n.

Molecular formula

Method / basis set

Total energy (Hartree)

Cartesian gaussian basis functions

H2

HF / STO-3G HF / 6-31G(d) B3LYP / 6-31G(d) B3LYP / cc-pVDZ

-1.1175 -1.1268 -1.1687 -1.1666

2 4 4 10

C6H8

HF / STO-3G HF / 6-31G(d) B3LYP / 6-31G(d) B3LYP / cc-pVDZ

-229.0431 -231.8336 -233.2568 -233.2731

38 106 106 130

C24H16

HF / STO-3G HF / 6-31G(d) B3LYP / 6-31G(d) B3LYP / cc-pVDZ

-907.0169 -918.1318 -923.5571 -923.6387

136 392 392 440

C54H24

HF / STO-3G HF / 6-31G(d) B3LYP / 6-31G(d) B3LYP / cc-pVDZ

-2033.9668 -2058.9261 -2070.9951 -2071.1446

294 858 858 930

C96H32

HF / STO-3G HF / 6-31G(d) B3LYP / 6-31G(d) B3LYP / cc-pVDZ

-3609.7990 -3654.1003 -3675.5515 -3675.8158

512 1504 1504 1600

HF / STO-3G HF / 6-31G(d) B3LYP / 6-31G(d) B3LYP / cc-pVDZ

-5241.8438 -5306.1776 -5337.2564 -5337.6681

790 2330 2330 2450

C150H40

70

2.6

Calculating Van der Waals Interactions In this work we were interested in the π-π interactions between graphene

layers and polycyclic aromatic hydrocarbons (PAHs). Such weak interactions are especially difficult to accurately capture with ab-initio methods since they represent only a small fraction of the total energy of the system. Thus, even small errors in the total energies of interacting molecules that arise due to assumptions introduced in the calculations, may provide a completely inaccurate assessment of intermolecular interactions. There are several general considerations in computations of weak nonbonded interactions among molecules that will be described in the subsequent sections. The first is the basis set superposition error (BSSE), which is essential for correcting the incompleteness of conventionally used basis sets. The second is the use of electron correlation to capture the effects of electron-electron interactions among molecules. The third is the inclusion of the zero point energy (ZPE) of the interacting molecules that in some occasions has proven to be necessary.

2.6.1 Basis Set Superposition Error (BSSE) The BSSE in molecular orbital calculations is an error that arises from the incompleteness of the basis set used in the expansion of wavefunctions. Complete basis sets are impossible to use since they require a near infinite number of basis functions. As mentioned earlier, the total energy of a molecular system does depend on the number of basis functions used to approximate the wavefunctions. If such energies are used for the computation of weak interactions, errors can have

71 magnitudes close to that of the interaction (a few kcal/mole). Generally to calculate the interaction energy between two molecules the following relationship is used, ΔΕ VdW = E AB − E A − E B , (2.17), where E AB is the total energy of the supermolecule, and E A and E B are the energies of molecules A and B, respectively. The non-BSSE corrected VdW interaction may be defined as, - BSSE ΔΕ non = E AB (φ Α & φ B ) − E A (φ Α ) − E B (φ Β ) . (2.18) VdW

This means that when E AB is evaluated, basis functions on both molecules are used in approximating the wavefunctions. This results in an artificial lowering of E AB due to the use of a larger basis set during the supermolecule energy evaluation. Boys and Bernardi11 suggested the counterpoise method to correct for this error. They defined the BSSE corrected VdW interaction energy as, ΔΕ BSSE VdW = E AB (φ Α & φ B ) − E A (φ Α & φ B ) − E B (φ A & φ B ) . (2.19) Thus during the total energy evaluation of molecule A, “ghost” functions of molecule B are also included. “Ghost” functions are basis functions that are not accompanied by an atomic nucleus. Fig. 2.3 shows the differences between the BSSE and the non-BSSE corrected PES surface of the benzene dimer. The BSSE correction results in a decrease in the well depth and an increase in the equilibrium separation. This effect was observed in many other systems of PAHs and will be discussed in further detail in Chapter 5. Other alternative approaches to reduce the BSSE error have also been proposed. For instance Tao et al.12 suggested the use of bond-functions,

72 which are orbitals centred between nuclei, to saturate commonly used basis sets in that region.

Binding energy (kcal/mole)

4 3 2 1 0 -1

Figure 2.3 One-dimensional PES of face-to-face benzene dimer calculated using MP2/cc-pVDZ. Circles (●) represent the BSSE corrected PES and squares (■) the PES without the correction.

-2 -3 -4 2.5

3.5

4.5

5.5

6.5

Distance (Å)

2.6.2 Electron Correlation The correlation energy is usually defined as the part of the total energy of a system that is not taken into account in the Hartree-Fock description of molecules. Although there are many definitions for electron correlation found in the literature the one commonly used is the one by Löwdin5 “The correlation energy for a certain state with respect to a specific Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration.” This does not mean that the Hartree-Fock approximation does not have any electron correlation built in it, but it does not capture the total correlation. The part of

73 the correlation it captures is the so-called Fermi correlation that arises from the antisymmetry of the wavefunction and basically prevents electrons of the same spin to occupy the same region of space. It is noteworthy to mention that the correlation energy is usually of the same order of magnitude as the dissociation energy of a molecule. 6 Generally there are three factors that affect the electron correlation. The most significant is the Coulomb repulsion that has the 1/rij dependence and thus becomes infinite when electrons are in close proximity and zero when their separation becomes infinite. This generates the so-called Coulomb hole in electronic structure computations. There are molecular systems where this is the only type of correlation present apart from the Fermi correlation. The two classical examples are He and molecular Hydrogen. The second type of correlation is termed as the Fermi hole. The third factor that has influence on electron correlation is the point group to which the one-electron wavefunctions are restricted. In this case lower symmetry point groups impose less restrictions onto the one-electron wavefunctions and result in higher correlation effects. In practice the complete electron correlation for molecular systems can only be approximated and there are methods in the literature that seem to do better. The two methods used extensively in the computational chemistry literature is the MøllerPlesset perturbation theory, such as MP2 and Coupled cluster methods, such as the CCSD(T) method. There are also certain schemes one can make to invoke such high level computations feasible. For example one that is extensively employed is to calculate

74 the correlation energy using a small basis set and then add this energy to a lower level calculation with an extensive basis set.

2.6.3 Zero Point Energy (ZPE) In contrast to a classical system the energy of a quantum system even at 0 Kelvin is not exactly zero. This residual energy is called zero point energy (ZPE) or zero point vibrational energy, since practically it is due to vibrations that are present even if the molecule is in it’s ground vibrational state. The model usually employed for their computation is the simple harmonic oscillator model. The second order derivative of the potential with respect to the coordinates in the harmonic oscillator model is the force constant. Having the force constants for all 3N-6 vibrational modes in a molecule (3N-5 for linear molecules), one can calculate the ZPE. Although the zero point energy in molecules is very small compared to the total energy of a molecule, it can have an important effect when it comes to reaction enthalpies and Gibbs free energies.

These thermodynamic quantities arise from the difference

between the energy of the products and reactants, and are thus much smaller in magnitude and can sometimes be comparable to the difference between the ZPE of products and reactants. In practice, conventional ab-initio ZPE energy calculations yield slightly higher values than expected form experiment. Thus usually a scaling factor of 0.8-1.0 is used to lower their magnitude and bring theoretical values in better agreement with experiment. Pople et al.7 suggested scaling of the ZPE by 0.89 by looking at a large number of theoretical and experimental data. The scaling compensates for the error

75 introduced by the use of the simplified harmonic oscillator model, which assumes a parabolic PES. In reality, vibrations are anharmonic, meaning that the PES resembles more a Morse potential, and the vibrational modes are independent from one another.

76

References 1.

Each hydrogen and carbon have (4s,1p) and (9s, 4p, 1d) primitive gaussian functions, respectively.

2.

A. Szabo and N. S. Ostlund, ‘Modern quantum chemistry’, Macmillan, New York (1989).

3.

Wolfrram Koch, Max C. Holthausen, ‘A Chemist’s Guide to Density Functional Theory’, Wiley (2001).

4.

Density Functional theory, Hartree-Fock and the Self-Consistent Field, Peter M.W. Gill. http://maclaurin.chem.nott.ac.uk/ECC.pdf

5.

P.-O. Löwdin, Adv. Chem. Phys. 2, 207 (1959).

6.

D.B. Cook, Mol. Phys. 42, 235 (1980).

7.

J. A. Pople, A. P. Scott, M. W. Wong, L. Radom, Israel J. Chem. 33, 345 (1993).

8.

J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78, 1396 (1997).

9.

D. R. Hamann, Phys. Rev. B 40, 2980 (1989).

10.

R. A. Kendall, E. Apra, D. E. Bernholdt, E. J. Bylaska, M. Dupuis, G. I. Fann, R. J. Harrison, J. L. Ju, J. A. Nichols, J. Nieplocha, T. P. Straatsma, T. L. Windus, A. T. Wong, Comput. Phys. Commun. 128, 260 (2000).

11.

S. F. Boys, F. Bernardi, Mol. Phys. 19, 553 (1970).

12.

F.-M. Tao, J. Chem. Phys. 98, 2481 (1993).

CHAPTER 3 Origin of Contrast in STM Images of Graphite

3.1

Abstract In this work, a new interpretation for the STM image of graphite is proposed.

Our experimental and computational findings indicate that the spots observed in common STM images of graphite may not correspond to carbon positions but to πstates near the Fermi level that are localized above the carbon-carbon bonds. This interpretation is based on (i) the analysis of well resolved STM images of HOPG that exhibit two-fold symmetry rather than trigonal symmetry and (ii) the computation of the STM images of graphite, using density functional theory within the generalized gradient approximation and the Tersoff-Hamann model. Various periodic supercells of graphite are considered, having hexagonal or orthorhombic symmetry, to exclude the possibility of finite size effects. Our computations revealed that the effect of the second layer on the simulated STM image is small, and thus cannot account for the

77

78 resolution of the other atom. The observation of electron density belonging to πorbitals localized on top carbon-carbon bonds, is adequate to provide a compelling, novel interpretation that does not depend on an assymetric tip, multiple tip effects, or the stacking of layers, as was previously suggested.

3.2

Introduction Highly oriented pyrolytic graphite (HOPG), a synthetic form of graphite, has

been used as a standard for STM calibration for over a decade due to the relative ease of imaging in air and vacuum coupled with the known carbon-carbon distances. The vast majority of images reported in the open literature, with only one exception1, show only three of the six carbon atoms in a given six-member ring. They also demonstrate a corrugation amplitude not expected for a topographically flat surface, such as the surface of graphite. Selloni et al.2 reported ab-initio calculations on the electron density of graphite, in which a significant difference between the corrugation amplitude near the Fermi level and the total electron density is observed. Their findings indicate an enhancement of the electron density above B-type carbon atoms (Fig 3.1) that results in 0.15Å larger height corrugation3 at V = 0. One year later, Binnig et al.4 obtained the first actual STM image of graphite, which demonstrated the resolution of every other atom, and is thus in good agreement with the computed electron densities of Selloni et al.. Similar STM images were obtained by Sang-il Park and C.F. Quate5, and Schneir et al.6, by imaging graphite in air and in water, respectively.

79 The two explanations concerning the large corrugation amplitude were given by Tersoff 7 and Soler et al.8. Tersoff argued that the large corrugation may be a result of “imaging a single wavefunction, rather than mapping surface topography” and what one sees is the nodal nature of the wavefunction within the unit cell. Soler et al. attributed the “giant corrugations” to elastic deformations of the graphene surface in response to compressive tip forces. The resolution of only every other atom in the STM image of graphite was attributed to the carbon-site structural asymmetry, as suggested in many theoretical calculations. 2,3,9-11 This structural asymmetry results in two types of carbon sites, A and B, in which the first lacks and the second posseses a neighbor in the second layer (see in Fig. 3.1).

B H

A A

B 2nd layer

H

B A

Surface layer

Figure 3.1 Schematic representation of the atom assignment in hexagonal or rhombohedral graphite. Black line and gray line represent surface and second layer, respectively. The relationship between A, B and H sites with respect to the STM image are given in Fig. 3.3(b).

This structural asymmetry has a direct impact on the electronic distribution of A and B carbons. Based on the current interpretation of graphite’s STM image, B sites have an increased electron density near the Fermi level, causing them to appear as protrusions (bright areas). A sites on the other hand, appear as saddle points, since

80 some of their electron density is shared with the underlying carbon atoms. The minima appear at the H sites which reside in the center of the hexagons. Explanations for the observed elongation in the protrusions based on extensions of this interpretation can also be found in the literature. The elongation reduces the image symmetry to a two-fold symmetric pattern. Binnig et al.2 attributes the deviations from the calculated images to an “asymmetric tip”. Batra et al.5 performed electronic structure calculations on monolayer and three-layer graphite slabs having ideal and slipped configurations. They suggest that the loss of trigonal symmetry is a result of a slipped surface graphitic layer from the ideal ABAB stacking sequence. Another explanation, based on the effect of a multiple-tip on the image, is given by Mizes et al.. 12 They showed that two atomically sharp tips might result in an image that is the superposition of two regular images that are shifted with respect to each other by the distance between the two tips. The different tunneling amplitude at each tip can also generate different kinds of periodic structures very different from the usual constant current STM image of graphite. Hembacher et al.13 carried out a combined low-temperature STM/AFM study and the authors attribute the elongation to a tilted 5dz2-like tip state. We propose that the protrusions that appear in graphite’s STM images may not correspond to carbon positions but to enhanced electron density near the Fermi level that is a result of imaging π-states that are localized above the carbon-carbon bonds. Such an electronic structure is similar to the quinoid structure proposed by Linus Pauling.14, 15 He proposed this structure in order to i) explain some satellite reflections in X-Ray diffraction photographs of graphite that could not account for the six fold14

81 symmetry (see Fig 3.2), and ii) explain the unexpectedly large compressibility measured in the basal plane of graphite. However, the quinoid structure of Pauling exhibits relatively strong bond length alternation (BLA). In chapter four, we obtain high-resolution powder XRDs that confirm that BLA is not present in graphite.

Figure 3.2 Quinoid structure of graphite previously proposed by Linus Pauling. 15 Single bonds have 15% double bond character (1.453Å) and the double bonds have 70% single bond character (1.357 Å).

The lack of bond length alternation agrees with other crystallographic studies, such as powder XRD data16, 17, neutron diffraction data18 and low-energy-electrondiffraction19 (LEED) data, that assign equal bond length to all carbon-carbon bonds.

3.3

Results and Discussion Our approach is to first establish the real symmetry seen in STM images of

graphite by examining on or more high resolution STM image of HOPG obtained under ambient conditions. Then we exclude the possibility of known experimental artifacts, such as drift, tip convolution, multiple tip effects, and tip-induced electric field perturbations that may render our symmetry assignment incorrect. Finally, we present simulated STM images based on electronic structure computations that show excellent agreement with experimental data.

82

3.3.1 Thermal or Mechnical Drift Although theoretical calculations2,4,9,10 yield a trigonal symmetry in the simulated STM image of graphite, all experimental STM images obtained in our lab and most of the images reported in the scientific literature5,

11-13

lack such high

symmetry. This can be seen in Fig. 3.3(a) which is a constant-current STM image of HOPG obtained in air at a tunneling current of 1.2 nA and bias voltage of -0.1 V. The STM protrusions (bright areas in the image) are elongated along the [ 1 100] lattice

direction (zone). This elongation reduces the symmetry of the image to a two-fold symmetric pattern. Similar images were obtained with Pt/Ir tunneling tips at various tunneling conditions. It is interesting to note that such a preferential direction, in which the STM spots are elongated, can also be seen in the first experimental images taken by Binnig et al.. 4 To ensure that the protrusion elongation is not a result of mechanical or thermal drift, we compared the image symmetry to the geometric shape of the STM protrusions. A quantitative comparison between them can reveal if the surface or tip is drifting during the acquisition of the image. First we measure the spot density (ρ) along the crystallographic zones [ 1 100] and [ 1 1 20] (see Fig. 3.3(a)). Qualititavely, a large protrusion density indicates a denser carbon atom chain along the corresponding lattice direction. In the case that drift is occuring in that direction, then the spot density is expected to decrease. For the perfect hexagonal structure of graphite the ratio of spot densities along the [ 1 100] and [ 1 1 20] is calculated to be equal to

3 ≈ 1.732 .

83

[ 1 100]

(a)

[ 1 1 20] [ 1 2 1 0] 97 Å

[2 1 1 0] 97 Å (b)

(c)

H site A site B site

Figure 3.3 Experimental constant-current STM image of graphite. (a) Constant-current STM image of HOPG at a tunneling current of 1.2 nA and bias voltage of - 0.1 V. Image was Fast Fourier transformed to eliminate high frequency components due to electronic noise. [ 1 2 1 0] , [ 1 1 20] and [21 10] are the vectors along which the inverse spot density was measured. [ 1 100] shows the direction in which the spots are elongated. b) carbon atom assignment based on the interpretation proposed by references. 2,4,9,10 (c) carbon atom assignment based on our interpretation. The honeycomb lattice is shifted by half a C-C bond (71pm) in the direction indicated by the arrow.

84 In the atomically-resolved STM image that we obtained in our lab (shown in Fig. 3.3(a)), the ratio of spot densities along the [ 1 100] and [ 1 1 20] was measured to be 1.73 ± 0.01, which sugggests that there is no significant presence of mechanical or thermal drift. To obtain an understanding of the degree of asymerty in the STM spots we measure the asymetry factor that we define as ω=

a , (3.1), b

where a and b are the lengths of the features along [ 1 100] and [ 1 1 20] respectively. In the case of only circular spot, ω is equal to unity. In reality such a spot would be the result of simultaneously having both a centrosymmetric tunneling matrix element at that site and the absence of any type of thermal or mechanical drift during image acquisition. An example of a centrosymmetric tunneling matrix element would be the axial overlap of a pz orbital of the surface with the dz2 orbital of the tip. In section 3.3.2, we will show that in a tungsten-graphite system there is such a centrosymmetric overlap between the surface and tip wavefunctions. Based on the statistical measurment20 of the protrusion asymetry in Fig. 3.3(a) we find that ω ≅ 1.7. This means that that the STM spots are 70% longer in the

[ 1 100]

than the [ 1 1 20] direction.

Since the possibility of mechanical or thermal

drift has already been ruled out as the source of elongation, we conlude that it is native to the electronic structure at the surface. In section 3.3.5 we show through electronic structure computations that this elongation relates very well to the crystal orbitals found close to the Femi level of graphite.

85

3.3.2 Tip Convolution and Multiple Tip Effect In this section, we present experimental evidence that the spot elongation observed in STM images of graphite is not a result of tip convolution or multiple tip effects. Due to the exponential decay of the tunneling current with tip-to-surface separation, small asymetries in the electronic density distribution at the tip apex near the Fermi level are expected to significantly affect the tunneling current in STM imaging. On one hand Binnig et al.4, and more recently Hembacher et al.13, suggested that the protrusion elongation seen in STM images of graphite is a result of a tilt of the 5dz2 orbital of the tip. Park et al.21 reports siumlated STM images in which elongation results from a double-tip. They indicate that such a tunneling tip may generate images that are the superposition of two images separated by the distance of two atoms at the tip apex. In Fig. 3.4 we present experimental STM images of HOPG that were obtained for different scanning directions of the same surface area. The elevation of the graphene step on the left side of the image ensures that we are sampling the same area after changing the scanning direction. It can be seen that the direction of elongation changes according to the scanning direction. This clearly suggests that the elongation is neither the result of tip convolution nor a multipule tip effect since either of these would keep the direction of elongation unaltered.

(a)

(b)

13º

Figure 3.4 Effect of scanning direction on STM image of HOPG taken in air (FFT). (a) It = 1.1nA, Vbias = -0.1V. Image dimensions 38x38Å.(b) same tunneling conditions as previous but CCW rotation by 13º.

86

87

3.3.3 Tip-Induced Electric Field Effects Another effect in STM that may cause perturbations is the tip-induced electric field. The sharpness of the metallic tip and the potential difference between the tip and the sample can generate strong electric fields in the vacuum gap. These electric fields, if sufficiently intense, may cause perturbations in the electronic structure of the tip and the surface. Taking into account the tip induced electric field in theoretical STM computations have brought theoretical computations in better agreement to experimental images, such as in the cases of a monohydride Si(100)-H(2x1)22 surface with missing hydrogen atoms and ethene adsorbed on Si(001)23. The fact that our biasvoltage is an order of magnitude larger than the voltage range used in the theoretical derivation by Tersoff and Haman24 necessitates the consideration of the electric field. The electric field in the proximity of the surface layer is approximately linearly proportional to the bias voltage and inversely proportional to the tip-surface separation. A bias voltage < 2V or > -2V will only marginally polarize the higher energy surface electronic states. However, higher bias voltages (> 2V or < -2V), can result in tip-induced perturbations of the surface electronic states at the Fermi level. Bias voltages in that range are actually used in molecule manipulation using STM probes.25 The tip-induced electric field could have two possible effects in our system: (i) the field may split the energy of degenerate pair (2E2g) of HOMO or (ii) it could cause a larger polarization of one HOMO versus the other, towards the tip. Both phenomena could be plausible explanations to the lowered symmetry observed in experimental STM images of graphite. Nonetheless, if the effect of the tip induced electric field is

88 small, the explicit inclusion of the tip electric field in our theoretical STM simulations can be avoided. This also reduces considerably the complexity of the STM simulation. As shown by Konsek et al.26, the tip-sample system can be modeled by a point charge above an infinite conducting plate. According to this treatment the electric field is given by E=

α (β + α)(2β + α)V

[

β (β + α ) + δ 2

]

3 2 2

, (3.3),

where α is the tip radius, β is the tip-sample separation, V is the bias voltage, and δ is the distance between a point on the surface and the vertical projection of the point charge onto the surface. For α we used the atomic radius of tungsten27 (1.35Å ) and for β the equilibrium separation between a tungsten atom and the hollow side of benzene, β = 5.44Å, obtained using B3LYP/LanL2DZ. The electric field for V = - 0.1 and δ = 0 was 6.58·107 V/m, and the model system to study the effects of the electric field is depicted in Fig 3.5.

+

Figure 3.5 Schematic representation of the model used to study the effects of the tipsample electric field.

89 Our model consists of two point charges (q = ± 0.006762), one above and one below a coronene molecule, that generate an electric field equivalent to the one calculated using Eqn. 3.3. We performed single point energy calculation to obtain the wavefunctions in the absence and presence of the electric field for various tip positions. The superimposed isodensity surfaces of the wavefunctions were identical, and thus, the tip-induced electric field doesn’t seem to affect the STM of graphite. Perturbations of the wavefunctions were observed only when the electric field strength is two orders of magnitude higher than our experimental conditions. This suggests that we can simulate our STM images without the explicit inclusion of the tip and that a simple Tersoff-Hamann approach is suitable for this system.

3.3.4 Computational Methods All computed STM images were obtained within the planewave density functional formalism as implemented in the NWChem28 code. The surface of graphite is modeled with a four layer slab separated by a 10 Å vacuum gap in the direction perpendicular to the basal plane (see Fig. 3.6). The valence one-electron wavefunctions were expanded using a plane wave basis, and periodic boundary conditions were invoked. For the hexagonal supecell a 136x34x34 k-point mesh, which corresponds to a kinetic energy cutoff of 132 Ry and a planewave expansion of 34436 sinusoidal waves was used. This ensures energy convergence to less than 10-3 eV (see Table 2.2). The exchange and correlation effects are considered within the generalized gradient approximation (GGA) using the PBE9629 exchange-correlation (XC)

90 functional. The electronic core was described by the Hamann30 norm-conserving pseudopotentials in which the core radii rcl for the various orbitals (l = s, p, d) were set31 to, 0.80 αο, 0.85 αο and 0.85 αο, respectively. (αο = bohr unit of length) For the hexagonal supercell slab the lattice paramters were a = 20.000Å, b = c = 4.912Å, α = 120º , and β = γ = 90º, and for the orthorhombic supercell slab they are a = 4.912Å, b = 8.508Å c = 20.000Å, and α = β = γ = 90º. All carbon-carbon bonds were set to 1.418Å and the interlayer separation is 3.354Å.

(a)

(b) A B A B

a

b c

(c) ~ 10 Å

.

b

.

c

a Figure 3.6 Schematics of four layer slab unit cell. (a) two adjacent unit cells of the 4 layer slab used in the computations, (b) 2x2 hexagonal supercell of hexagonal graphite, and (c) 2x2 orthorhombic supercell of hexagonal graphite. The axis not shown points into the plane of the paper.

To obtain the simulated STM images we employ the Tersoff-Hamann32 treatment for the tunneling current. The surface is treated exactly, while the wavefunction at the tip is assumed to have spherical symmetry (s-orbital). Since we were interested only in the shape and periodicity of the features in the STM image of

91 graphite, rather than the absolute tunneling current, we used the local electron density near the Fermi level in our simulations. This density is given by G 2 ρ(r, E F ) ≡ ∑ Ψν ( ro ) ⋅ δ(E ν − E F ) , (3.4), ν

G where Ψν ( ro ) is the unperturbed surface eigenstate at the tip is center of curvature, E F G is the Fermi energy and E ν is the eigenvalue of Ψν ( ro ) . A lateral resolution of 0.05 Å

was used in all electron density maps to obtain smoothly varying landscapes. Although the thermodynamically more stable form of graphite is assigned to the hexagonal crystal system, we examined both hexagonal and orthorhombic crystal systems to exclude the possiblity of a finite size effect in the computations. In principle both hexagonal and orthorhombic supercells should yield a similar electron density distribution, if the symmetry of the electronic distribution is trigonal. However, if the electronic density desires to relax to a lower symmetry distribution, such as two-fold symmetry, that would be posssible in the orthorhombic unit cell but impossibel in the hexagonal unit cell.

3.3.5 Electron Density at the Fermi Level Our simulated STM images are given by the electron density close to the Fermi level at a tip-to-surface separation of 2.5Å. In principle, only states with energies in the range EF - eVbias and EF, and which are in close physical proximity to the tunneling tip will contribute significantly to the tunneling current. We were also interested on the effect of the bias voltage, thus we carried out simulations both at low (-0.1 V) and high (-1.0 V) bias voltages. In the case of the 2x2 orthorhombic supercell of graphite a

92 bias voltage of -0.1 and 1.0 V corresponds to five and twenty-two MO’s close to the Fermi level, respectively. Of course the number of occupied states in the specific energy interval depends on the size of the unit cell in these calculations. Larger unit cells are expected to have denser electronic states within a specified energy range. We obtained isosurfaces of the highest occupied crystal orbitals (HOCOs) to study the shape and location of the orbitals close to the Fermi level (see Fig. 3.7(a,b) and 3.8(a,b)). The orbitals HOCO through HOCO-4 appear to be located on carboncarbon bond positions and their shape is similar to the HOMO of ethylene. Crystal orbitals lower in energy (see Fig 3.7(c) and 3.8(c)) occupy the space between the previously mentioned crystal orbitals.

(a)

(b)

(c)

Figure 3.7 Isosurfaces of crystal orbitals at the Fermi level for 2x2 orthorhombic unit cell. (a) and (b) superposition of degenerate pair highest occupied crystal orbitals (HOCO + HOCO-1) in perspective view and view along [0,0,0,1], respectively. Only wavefunctions in the surface layers are shown for better clarity. (c) HOCO-5 crystal orbital appears complementary to the HOCO + HOCO-1indicating a uniform electron distribution around the carbon rings when electron density is summed over all states.

88

(a)

(b)

(c)

Figure 3.8 Crystal orbitals near the Fermi level of 4 layer slab model of graphite. (a) shows the HOCO, (b) the HOCO-1 and (c) the HOCO-5. Note that HOCO and HOCO-1 are degenerate and complimentary.

93

94

3.3.6 Experimental vs Simulated STM Image of Graphite In Fig. 3.9 we present experimental and simulated STM images of graphite obtained at low and high bias voltage. Our simulated images using the orthorhombic supercell (see Fig. 3.9(b)), at both bias voltages, are in good agreement with the experimental images obtained in our lab (see Fig 3.9(a)). Experimental (a)

Simulation

Vbias = - 1.0

Vbias = - 1.0

Vbias = - 0.1

Vbias = - 0.1

(b)

Simulation (c)

Simulation (d)

Figure 3.9 Comparison between experimental and simulated STM images of graphite. (a) Experimental constant-current STM image of HOPG for a bias voltage of -0.1V and -1.0V. (b) Simulated STM image of graphite using 2x2 orthorhombic supercell for a bias voltage of 0.1V and -1.0V.(c) Simulated STM image of graphite using a 2x2 hexagonal supercell and a bias voltage of -0.1 V.(d) This image was obtained from image c by filtering low intensity signal in to enhance the contrast, similar to Tomanek et al..10 Tip-sample separation in simulated images is 2.5 Å, bias voltage is -0.1V and -1.0V for low bias and high bias voltage, respectively. Image dimensions are 4.912Å x 8.508Å.

95 88 However the simulated STM images in which a hexagonal supercell (see Fig. 3.9(c,d)) was used lack good agreement to the experimental image. In a real sample of graphite there are three equivalent directions along which the π orbitals at the Fermi

[

]

level can run. These are the symmetry equivalent crystallographic directions 1 100 ,

[1 010] , or [0 1 10] . Nonetheless, it is possible that small defects in the lattice or even edge effects can slightly favor one of the three directions. We also obtained electron density difference maps near the Fermi level using MP2/aug-cc-pVDz, for coronene and the coronene dimer in a parallel AB configuration. The diffuse basis set and the explicit consideration of dispersion interactions due to polarization effects are expected to correctly capture the effect of the second layer on the electron density of the surface layer. These computations revealed only small differences < 5% in the charge densities at the Fermi level, due to the attractive interlayer interactions. This suggests that the effects of the 2nd layer are negligible. This is in agreement with the experimental findings by Land et al.

33

who

obtained STM images of heteroepitaxially grown monolayer graphene islands from ethylene on Pt(111) identical to those of bulk graphite. STM images of carbon nanotubes34 bare similar features to that of bulk graphite. Additionally, our STM simulations are consistent with the experimental findings that show that the 2nd layer does not have a great effect on the STM. As matter of fact we were able to simulate STM images of a graphite sheet (graphene) that is in very good agreement with STM images of graphite. In Fig. 4.11 we show the STM simulation of such a graphene sheet having a 5x5 orthorhombic supercell for a bias voltage of -0.1V.

90

Figure 3.10 Simulated STM image of graphene at Vbias = - 0.1V. We used a 5x5 orthorhombic supercell, the PBE96 XC functional, a 64x64x64 k-mesh, 37264 planewaves and sampled states in an energy window 0.1eV from the Fermi level. Image dimensions are 12.280Å x 21.270Å. Superimposed on the image the 2x2 orthorhombic supercell of graphite.

96

97 Nonetheless, it must be noted that the 5% change in the electron densities may result in the formation of larger periodic structures35, 36 (superstructures) observed in the STM images of graphite. This could happen if two layers diverge from the perfect stacking sequence and cause the formation of a Moiré pattern. Another feature of our simulated STM image that is in agreement with experimental findings is the relative shift of the STM pattern with respect to the atomic positions in the crystal. Our simulated STM image is shifted by about 70pm in the

[2 1 1 0]

direction. This is in fact in excellent agreement with the 68pm

displacement measured experimentally by Hembacher et al..13 who observed such a relative shift in the STM image of HOPG with respect to the atomic force microscope image (AFM) in a combined low temperature STM/AFM study under ultra high vacuum conditions.

3.4

Conclusions We have presented evidence that the reduced symmetry observed in STM

images of graphite is not a result of (a) thermal or mechanical drift, (b) tip-convolution or double-tip, or (c) tip-induced electric field perturbations, but inherent to the surface electronic structure of graphite. Our DFT computations reveal that the consideration of an orthorhombic supercell may be required to obtain the correct symmetry of the relaxed electron density distribution at the Fermi level. This can be seen in the simulations using the hexagonal supercell of graphite that generate an electronic distribution very different from that of the orthorhombic supercell. Using an

98 orthorhombic unit cell is similar to optimizing a molecule using a lower symmetry point group. Our simulated STM images of graphite, using an orthorhombic supercell, are in good agreement with the experimental images. They exhibit the experimentally observed two-fold symmetry and have relatively large corrugation amplitude, even in the absence of contrast enhancing filters. The distance among protrusions is

3 ⋅ 1.418

Å, also observed experimentally, and they appear elongated. We also show that a honeycomb-like pattern can be obtained with a higher bias voltage. Thus, we conclude that the protrusion in low bias voltage STM images of graphite corresponds to πorbitals localized above carbon-carbon bonds and not carbon atom positions. This means that the origin of contrast in STM images of graphite is native to the electronic structure of this material and not a result of the effect of the second layer and the STM tip. In the following chapter we test if our explanation of the STM image is consistent with the current assignment of graphite to the hexagonal crystal system and if there is a possibility of a small bond length alternation (BLA) in the lattice of graphite that was not observed in crystallographic studies so far.

99

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101

CHAPTER 4 Experimental/Theoretical Powder X-Ray and Single-Crystal Electron Diffraction Patterns of Graphite

4.1

Abstract In this chapter we present experimental and simulated powder X-ray

diffraction (XRD) patterns as well as simulated single-crystal electron transmission photographs of graphite. The aim of the simulations is to quantify whether small bond length alternation (BLA) can generate measurable differences in the diffraction patterns obtained by these techniques. First, we derive a mathematical model for graphite that is used to generate structures with various degrees of BLA. The model has three degrees of freedom (a, b and θ ) that are varied independently to generate structures with various degrees of BLA. Our model indicates that BLA in graphite can be generated in three distinct ways, (i) by changing θ while keeping the unit cell dimensions constant, (ii) by expansion of the unit cell along vector a, or (iii) by the compression of the unit cell along vector b. We observe that BLA greater than 0.003 Å causes splitting of a certain family of reflections in the case of powder XRDs.

101

102 However, BLA less than 0.003 Å causes only peak broadening and may not be readily observed in the analysis of experimental powder XRDs of graphite.

4.2

Trigonometric Equation for BLA in Graphite In this section we derive a trigonometric relationship to quantify bond length

alternation (BLA) in graphite. The existence of BLA will reduce the symmetry of graphite, thus changing the crystal system from hexagonal to orthorhombic. The orthorhombic unit cell contains eight symmetry unique carbon atoms in contrast to the hexagonal unit cell, which contains only four. This lower symmetry unit cell is necessary due to symmetry breaking presumably caused by BLA. Schematic representations of the two unit cells are shown in Fig. 4.1. The unit cell parameters used for hexagonal graphite are those of Trucano and Chen1 which were derived from single crystal neutron diffraction data and refined to the P 63/m m c space group. c

(a)

(b)

a 3

2

1

4

b

3 4

5 2

a

7

8

6

1

b c

Figure 4.1 Hexagonal and orthorhombic unit cell of graphite. (a) Unit cell parameters of hexagonal graphite belonging to the P 63/m m c space group are a = b = 2.464 Å, c = 6.711 Å, α = β = 90º and γ = 120º. (b) Unit cell parameters of orthorhombic graphite belonging to the P m c 21 space group are a = 2.464 Å, b = 4.2678 Å , c = 6.711 Å and α = β = γ = 90º. The Schoenflies symbol for the space group of hexagonal and orthorhombic graphite are D46h and C22v, respectively.

103 In our mathematical model of graphite we assume that puckering of the graphite layers is not allowed. This means that all the carbon atoms are coplanar within each layer. This assumption is reasonable considering that most polycyclic aromatic hydrocarbons have planar structures. To derive a mathematical model, we will consider a two-dimensional orthorhombic unit cell with the following variables, r1 , r2 , θ , a and b (Fig 4.2). a

r1 θ c

r1

r2 x

r2 x

b

Figure 4.2 Orthorhombic unit cell showing trigonometric variables used in derivation.

We the define the bond length alternation, BLA, as the difference between the longer and shorter the carbon-carbon bond BLA = r2 − r1 (4.1). One can write the following simple trigonometric equations, x x cos(180 − θ) = ⇒ r2 = − (4.2), r2 cos θ b = 2r1 + 2x ⇒ r1 = tan(180 − θ) =

b − x (4.3), 2

a 2 a a cos θ ⇒ − tan θ = ⇒x=− . (4.4) x 2x 2 sin θ

104 By substitution of Eqns. 4.2 and 4.3 into Eqn. 4.1 and then replacing x with Eqn. 4.4 we obtain, BLA = −

x b a cos θ b a cos θ , (4.5), − + x ⇒ BLA = − − 2 sin θ cos θ 2 2 sin θ cos θ 2

After a few algebraic steps we obtain the following expression for BLA, BLA =

a (1 − cos θ) − b sin θ . (4.6) 2 sin θ

One can see that BLA for hexagonal graphite is zero by substituting, b = 3a and θ = 120º. From this equation, it is apparent that there are three degrees of freedom for BLA since each variable (i.e. a, b and θ ) can be independently varied to generate BLA. One can also obtain the fractional coordinates of the symmetry unique carbon positions in the orthorhombic unit cell by using the relationships in Fig. 4.3.

C1 (0,0,0) ⎛ r ⎞ C 2 ⎜ 0, 1 ,0⎟ ⎝ b ⎠ ⎛ 1 r +x ⎞ C 3 ⎜ , 1 ,0 ⎟ ⎝2 b ⎠ Figure 4.3 Fractional coordinates of symmetry unique atoms in orthorhombic graphite. Note that x is given by Eqn. 4.4, r1 by Eqn. 4.3 and r2 by Eqn. 4.2.

⎛ 1 2r + x ⎞ C 4 ⎜ , 1 ,0 ⎟ b ⎝2 ⎠

1⎞ ⎛ C 5 ⎜ 0,0, ⎟ 2⎠ ⎝ ⎛ r + 2x 1 ⎞ C 6 ⎜ 0, 1 , ⎟ b 2⎠ ⎝ ⎛1 x 1⎞ C7 ⎜ , , ⎟ ⎝2 b 2⎠ ⎛ 1 r +x 1 ⎞ C8 ⎜ , 1 , ⎟ ⎝2 b 2⎠

We observe that BLA in graphite can be generated by either changing the atomic positions within a fixed unit cell or by changing the dimensions of the unit cell by varying either one of the two basal plane lattice vectors.

105 These three methods appear to have distinct effects on the simulated diffraction patterns of hexagonal and orthorhombic graphite that will be discussed in the following sections.

4.2.1 Effect of Compression Along Vector b on Powder XRD A series of seven powder XRD diffraction patterns are simulated that correspond to structures (A-G) with various degrees of BLA, obtained by decreasing the lattice vector b. The lattice vector a, and the angle θ are kept constant at 2.464Å and 120º, respectively. Substituting θ reduces Eqn. 4.6 to, b = a 3 − 2 ⋅ BLA . (4.7) In Table 4.1 we present the BLA and lattice vector b for the series of graphite structures that were examined. From this table it can be seen that Δ(BLA)=Δb/2. Table 4.1 Bond length alternation and unit cell parameters for the various structures examined having a and θ constant. Note that a = 2.464 Å, α = β = 90º and γ=120º. Fractional coordinates for lattice A-G are given in Appendix 5.

lattice A B C D E F G

BLA (Å) 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150

b (Å) 4.26777319 4.26277319 4.25777319 4.25277319 4.24777319 4.24277319 4.23777319

The simulated XRD diffraction patterns are compared to that of hexagonal graphite. The comparison reveals that the intensity and the peak position of the [0,0,2n] reflections, where n is an integer, remain unchanged. This result is reasonable

106 considering that the [0,0,2n] family of planes corresponds to graphene sheets whose separation is unaltered during expansion/compression along one of the basal plane lattice vectors. Nonetheless, most of the diffraction peaks, especially the ones at greater 2θ, exhibit splitting and shifting by a few tenths of a degree. An example of such a family of crystallographic planes is the [2,0,2n] family of planes (see Fig. 4.4). The reflections that correspond to this family of planes split into doublets. The appearance of this new family of crystallographic planes is denoted by [1,3,2n]. Relative to the original peak the [2,0,2n] and [1,3,2n] diffraction peaks have intensities proportional to 1/3 and 2/3, respectively. In contrast to the [1,3,2n] reflections which move towards higher 2θ values the [2,0,2n] reflections maintain their original 2θ positions. The separation between the two peaks appears to be linearly proportional to the BLA. It must be noted that the two peaks are not resolved if the BLA is less than 0.003Å which results in a single but broader peak. However, BLA of the order of 0.005 Å results in peak separation of 0.2º-0.3º, a difference that can be measured in conventional powder X-ray diffractometers.

107

Α

Intensity ( ·102 cps )

C DE F G

2θ

107

Figure 4.4 Simulated XRD peaks of [2,0,2n] family of planes keeping a and θ constant, where n is a positive integer. Hexagonal graphite shown in black and orthorhombic graphite with BLA of 0.005, 0.0075, 0.010, 0.0125 and 0.015 Å, shown in dark gray and lighter grays, respectively. Note that BLA causes the reflection of hexagonal graphite (black) to split, generating two peaks. The peaks have 1/3 an 2/3 the intensity of the original peak. X-ray radiation is a monochromatic Ka1 Cu source emitting at 1.54060 Å. Lattice parameters shown in table 4.1 and fractional coordinates given in Appendix 5.

108

4.2.2 Effect of Expansion Along Vector a on Powder XRD Again, a series of seven powder XRD diffraction patterns are simulated that correspond to structures (A’-G’), which differ from the perfect structure by small variations in the vector a. The lattice vector b and the angle θ are kept constant at 3 ⋅ 2.464 Å and 120º, respectively. Rearranging Eqn. 4.7, one obtains the lattice vector as a function of the vector b and the BLA, a=

b + 2 ⋅ BLA 3

(4.8).

In Table 4.2 we present the BLA and lattice vector a for the series of graphite structures that were examined. Table 4.2 Bond length alternation and unit cell parameters for the various structures examined b and θ constant. Note that b = 3 ⋅ 2.464 Å, α = β = 90º and γ=120º. Fractional coordinates for lattice A’-G’ are given in Appendix 5.

lattice A’ B’ C’ D’ E’ F’ G’

BLA (Å) 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150

a (Å) 2.46400000 2.46688675 2.46977350 2.47266025 2.47554701 2.47843376 2.48132051

The chief difference of this series of diffraction patterns compared to the case where the b vector is compressed is that reflections corresponding to the [2,0,2n] family of planes shift towards lower 2θ values (see Fig. 4.5). Previously, the [2,0,2n] reflections appeared to have no change in their 2θ values, and only the [1,3,2n] reflections shifted to higher 2θ positions. Here we have the reverse situation, in which

109 the [1,3,2n] reflections remain almost invariant with respect to their 2θ positions but there is a negative shift of the [2,0,2n] reflections. The relative intensities of the [2,0,2n] and [1,3,2n] reflections are 1/3 and 2/3 with respect to the intensity of the original peak, respectively. An additional feature of the two families of reflections is that their difference in 2θ is linearly correlated to the degree of BLA. Our results indicate that BLA generated by expansion of the orthorhombic unit cell of graphite along the lattice vector a will cause the [2,0,2n] reflections to split and shift towards lower 2θ positions. If BLA is greater than 0.003Å then complete resolution of both peaks can occur. Nonetheless, if BLA is less than 0.003Å this will result only in a negative 2θ shift and simultaneous peak broadening of the [2,0,2n] reflections.

110

Intensity ( ·102 cps )

A ’

G F’ E’ D C’ ’ ’

2θ ( ° )

110

Figure 4.5 Simulated XRD peaks of [2,0,2n] family of planes keeping b and θ constant, where n is a positive integer. Hexagonal graphite shown in black and orthorhombic graphite with BLA of 0.005, 0.0075, 0.010, 0.0125 and 0.015 Å, shown in dark gray and lighter grays, respectively. Note that BLA causes the reflection of hexagonal graphite (black) to split, generating two peaks. The peaks have 1/3 an 2/3 the intensity of the original peak. X-ray radiation is a monochromatic Ka1 Cu source emitting at 1.54060 Å. Lattice parameters shown in table 4.2 and fractional coordinates given in Appendix 5.

111

4.2.3 Effect of Increase in θ on Powder XRD Finally a series of seven powder XRD diffraction patterns are simulated that correspond to structures (A”-G”), which depart from the perfect structure by small variations in the angle θ. The ratio between the basal plane lattice vectors is kept constant at b a = 3

3 and a = 2.464Å. In other words, the unit cell dimensions are

constant, but the atomic positions within the crystal are displaced to generate the desired BLA. Only angles slightly larger than 120º are considered since others would generate negative BLA, which is inconsistent with our model. Table 4.3 Bond length alternation and unit cell parameters for the various structures examined, a and b constant. Note that a = b = 2.464 Å. Fractional coordinates for lattice A”-G” are given in Appendix 5.

lattice

BLA (Å)

A” B” C” D” E” F” G”

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150

θ(°) 120.000000 120.057922 120.115844 120.173766 120.231688 120.289610 120.347532

Fig. 4.6 shows the superposition of simulated XRD patterns for hexagonal and graphite with BLA. Our simulations indicate that for BLAs on the order of 0.01 Å no differences are observed in the diffraction patterns of the hexagonal and the graphite with BLA. Differences among the diffraction patterns are only observed when the BLA becomes as large as 0.4Å, which significantly exceeds the bond length difference between a regular double (1.34Å) and single (1.54Å) carbon-carbon bond. This is highly unlikely to occur and even then, new diffraction peaks are not observed nor do

112 we see the complete disappearance of existing peaks. Also the [ 002n] family of diffraction peaks remains unchanged for the same reason as mentioned before. In contrast to the changes in the diffraction pattern observed when unit cell compression/expansion is used to generate BLA, here the simulated diffraction pattern

[ ]

has peaks that both increase and decrease in intensity. For instance, the 1 3 n family of crystallographic planes for which n = 1, 2, or 3, and previously decreased in intensity, appear to have a 40% increase in intensity. Nevertheless, other reflections,

[ ] [ ] [

]

[ ]

such as the 1 20 , 212 , 2 1 4 and the 1 2 6 peaks, exhibit a relative lowering in their intensity by 30%.

Intensity ( ·102 cps )

113

2θ ( ° )

113

Figure 4.6 Simulated XRD pattern for hexagonal graphite (gray) and orthorhombic graphite (black) with BLA 0.4Å. X-ray radiation is a monochromatic Ka1 Cu source emitting at 1.54060 Å. Lattice parameters shown in Table 4.3 and fractional coordinates given in appendix 5.

114

4.3

Transmission Electron Diffraction Photographs of Graphite We also simulated the effects of BLA in the graphite lattice on the single-

crystal transmission electron microscope images along the zone perpendicular to the basal plane [0001]. Similar to the powder XRD simulations, we examined the effect of changes to one of the two basal plane lattice vector (a or b) or the angle θ. Since the changes in the simulated diffraction photographs are not easily resolved by visual inspection, in Table 4.4 we present the relative peak intensities (I/Imax%) and the angles between the lattice directions rather than the actual diffraction photographs.

b and θ constant

a and θ constant

a and b constant

Table 4.4 Reflection angles and relative intensities of orthorhombic graphite with various degrees of BLA. The angles are in degrees and the relative peak intensities are given by (I/Imax%).

BLA

θ1 (°)

θ2 (°)

θ3 (°)

I1

I2

I3

I4

I5

I6

0.0025

30.000

30.000

30.000

3.0

0.4

10.8

3.0

0.4

10.8

0.0050

30.000

30.000

30.000

2.9

0.4

10.8

3.1

0.4

10.8

0.0075

30.000

30.000

30.000

2.9

0.5

10.8

3.1

0.4

10.8

0.0100

30.000

30.000

30.000

2.8

0.5

10.8

3.1

0.4

10.8

0.0125

30.000

30.000

30.000

2.8

0.5

10.8

3.1

0.4

10.8

0.0150

30.000

30.000

30.000

2.7

0.5

10.7

3.2

0.4

10.8

0.0025

29.971

30.000

30.029

3.0

0.4

10.7

3.0

0.4

10.8

0.0050

29.943

30.000

30.057

3.0

0.4

10.7

3.0

0.4

10.8

0.0075

29.914

30.000

30.086

2.9

0.4

10.7

3.1

0.4

10.8

0.0100

29.885

30.000

30.115

2.9

0.4

10.6

3.1

0.4

10.8

0.0125

29.857

30.000

30.144

2.9

0.4

10.6

3.1

0.4

10.8

0.0150

29.828

30.000

30.173

2.8

0.4

10.6

3.1

0.4

10.8

0.0025

29.971

30.000

30.029

3.0

0.4

10.8

3.0

0.4

10.8

0.0050

29.943

30.000

30.057

3.0

0.4

10.8

3.1

0.4

10.8

0.0075

29.914

30.000

30.086

2.9

0.4

10.8

3.1

0.4

10.8

0.0100

29.885

30.000

30.115

2.9

0.4

10.8

3.1

0.4

10.9

0.0125

29.857

30.000

30.144

2.9

0.4

10.8

3.1

0.4

11.0

0.0150

29.828

30.000

30.173

2.9

0.5

10.8

3.1

0.4

11.0

115 For the simulations, a 1000 keV (0.0087Å) collimated electron beam was transmitted through a 100 Å thick single-crystal of graphite. The beam was incident along the [001] (or [0001]) lattice vector, and the diffraction patterns were collected at a distance of 381.4 cm. We also simulated diffraction photographs for hexagonal graphite along other crystallographic directions. The photographs for these simulations are presented in Fig. 4.7(a)-(f). These photographs clearly demonstrate that the reciprocal lattice is inverse of the projection of the real space lattice onto one of lattice planes. This means that it bares only crystallographic information that spans the 2D space. To generate a 3D dimensional lattice, one needs to record a series of such photographs. It is also interesting to note that only certain directions generate completely symmetric diffraction patterns. The vast majority of directions appear to have only the [0000] reflection, since most directions result in destructive interference of the scattered electrons. This phenomenon is expected in relatively small and highly symmetric unit cells, such as the one of graphite, where most directions result in destructive interference of the scattered electrons. This appears to be very different from diffraction patterns obtained from proteins that generate diffraction patterns along many directions. For the subsequent sections the pattern shown in Fig. 4.7(a) will be considered. In this diffraction photograph the reflections are positioned in perfect hexagons around the [000] reflection having 60º angles and identical intensities for the reflections belonging the same family of crystallographic planes (i.e. the ones belonging to the same hexagon).

116

(a)

(b)

(b)

(d)

(e)

(f)

Figure 4.7 Simulated transmission electron photographs of hexagonal graphite along various lattice vectors. Electron beam (100keV) transmitted along (a) [001], (b) [010], (c) [100], (d) [011], (e) [101], lattice vector. (f) stereographic projection given on a Wulf net, indicating 3D orientation of the single crystal with respect to the electron beam.

117

(a)

I2 I3 I5

I1 θ1

I4 θ2 θ3

I6

(b)

Figure 4.8 (a) Simulated high beam energy (1000eV) single-crystal transmission electron diffraction photograph of hexagonal graphite. (b) Stereographic projection given on a Wulf net, indicating 3D orientation of the single crystal with respect to the electron beam. The electron beam impinges the crystal along the [001] lattice vector. The planes corresponding to the diffraction peaks are shown.

118

[110] [-1-10] [220] [-2-20]

[200] [-200]

[1-10] [-110] [2-20] [-220]

[020] [0-20] [040] [0-40]

[130] [-1-30]

a

b

c

[-130] [-310]

Figure 4.9 Various crystallographic planes perpendicular to the basal plane of graphite. The electron diffraction pattern reveals these planes.

119 Our results indicate that generating BLA by varying θ has no effect on the angular relationship of the reflections in the [001] diffraction photograph. This is clearly demonstrated by the invariance of the angle θ1, θ2 and θ3 with BLA (see Table 4.4). However, it has an effect on I1 and I4. The first reflection exhibits a decrease in intensity as a function of BLA, whereas the latter shows an increase. Only small changes are observed in the intensities of the remaining reflections. In contrast, variations in the length of the crystallographic vectors a or b has noticeable effects on the angular relationship of the crystallographic zones. Regardless of whether BLA is generated by compression of the unit cell along the b vector or expansion along lattice the vector a, θ1 decreases and θ3 increases. Both θ1 and θ3 show a linear dependence on the degree of BLA as shown in Fig 4.10. Here, we note that this graph could actually serve as a calibration curve to determine BLA in an experimental transmission electron diffraction photograph.

30.20 30.15

angle θ (°)

30.10 30.05 30.00 29.95 29.90 29.85 29.80 0.000

0.005

0.010

BLA (Å)

0.015

Figure 4.10 Angular dependence of θ1 (circles) and θ3 (squares) as a function of BLA. See Fig 4.8(a) for the definition of θ1 and θ3.

120

4.4

Experimental Powder XRD of Graphite Experimental powder XRD patterns were obtained on a Philips X’Pert MPD

Pro powder diffractometer. The diffractometer is equipped with computer-controlled divergence, anti-scatter and receiving slit assemblies. The radiation source consists of a 2.2 kW ceramic flip-focus Cu target X-ray tube operated at 1.8 kW (45kV,40mA). The goniometer is scanned between 5º and 155º in 2θ, using a step size/time of 0.01º / 6 sec. This setting is expected to provide sufficient signal averaging to increase the signal-to-noise ratio. The X-ray radiation emerges through a fixed size 1ºdivergence slit and irradiates a 3x10 mm area of mixed graphite/silicon powder dispersed on a flat glass slide. The Silicon powder2 serves as an internal reference standard for the crystallographic analysis to correct for sample position and beam penetration. Preparation of the graphite/silicon powder is described in section 1.2.4. The diffracted beams are detected, after passing through a 1mm receiving slit, with the use of a sealed xenon scintillation detector. All measurements are obtained at 298 Kelvin. The experimental powder XRD pattern for the mixed graphite/silicon powder is given in Appendix 6. Although many reflections appear to have a low signal-tonoise ratio, they are sufficient to identify all previously assigned reflections. 6 The low signal-to-noise ratio is attributed to the non-random orientation of the graphite particles in the powder. An implication of this is that crystallographic information derived from intensities of reflections is expected to be erroneous. Thus the unit cell refinement was carried out based on experimental 2θ values.

121 Fitting of the reflections in the powder XRD pattern was carried out using the fitting routine described in section 1.2.6. The fitting routine takes into account the presence of the undesirable Kα2 reflections, deviations of the background signal from the baseline, and the specific line shape in our experiment. The experimental data obtained from our fitting routine for graphite and silicon powder are given in Appendix 7. Subsequently, silicon reflections are plotted versus the reflections given for the standard reference material 640c. 2 This provides a calibration curve (shown in Fig. 4.11) to obtain the corrected 2θ values of graphite reflections via linear interpolation or extrapolation. The corrected graphite reflections along with their assigned Miller indices are given in Appendix 8.

Standard reference material reflections

160 140 120 100 80 60 40

y = 0.9991x + 0.0959

20 0 0

20

40

60

80

100

120

140

160

Figure 4.11 Calibration curve for powder XRD reflections. Plot shows the two-theta reflections of the standard reference material potted as a function of our experimental Silicon reflections.

Experimental fitted Si reflections

Unit cell refinement was carried out using the Rietveld3 method as implemented in the computer program Unitcell.4,5 Using a nonlinear least-squares method, Unitcell minimizes the residuals in the experimentally determined Bragg positions (2θexp – 2θcalc). It also offers regression diagnostics to determine the

122 correctness of the Miller index assignment to the reflections. The unit cell refinement results for the triclinic, the orthorhombic, and the hexagonal crystal systems are given in Table 4.5. Table 4.5 Unit cell refinement results for powder XRD of HOPG. X-ray wavelength used for the refinement was the Ka1 Cu line ( λ = 1.5405929 Å ). Note that α is the angle between the a and b vector, β the angle between a and c vector and γ the angle between the a and b vector. V is the volume of the unit cell.

Crystal system

Triclinic

Orthorhombic

Hexagonal

a (Å)

2.466 ± 0.002

2.466 ± 0.002

2.467 ± 0.001

b (Å)

2.467 ± 0.003

4.273 ± 0.006

2.467 ± 0.001

c (Å)

6.718 ± 0.002

6.717 ± 0.002

6.716 ± 0.002

α(º)

89.97 ± 0.14

90

90

β(º)

90.01 ± 0.06

90

90

γ(º)

120.15 ± 0.25

90

120

V (Å3)

35.35 ± 0.11

70.79 ± 0.09

35.39 ± 0.03

Based on the standard deviations of the lattice angles in the triclinic system, one can assign graphite to the hexagonal crystal system. Our experimental lattice parameters for hexagonal graphite are in very good agreement with the somewhat more precise neutron diffraction unit cell parameters, reported by Trucano and Chen. 1 Fig. 4.12 depicts the [103], [202], and [006] high-resolution (0.001°) powder XRD reflection of graphite. We could not observe the splitting of the [2,0,2n] or [1,3,n] reflections that would clearly support the existence of BLA in the structure. The Ka1 peak of the [103] and [202] reflections (see Fig 4.11(a) and (b), respectively) exhibit excellent fitting to a single gaussian similar to the [006] reflection that is not expected to exhibit splitting in the presence of BLA.

123

Intensity (arb. units)

(a)

Kα1 Kα2

2θ ( ° ) (b)

(c)

Figure 4.12 High-resolution (0.001°) powder XRD reflections of graphite, obtained with Kα Cu radiation. Kα1 and Kα2 are fitted with gaussian functions. (a) is the [103], (b) is the [202], and (c) is the [006] reflection, respectively.

124

4.5

Conclusions In this chapter, we examine the possibility of having a small degree of bond

length alternation present in the structure of graphite. Such a BLA is a possible explanation for the “localized” π oribtals we observe in STM images of graphite. In order to test this possibility, we derived a mathematical model for graphite to generate structures with various degree of BLA. For these structures, various XRD and electron diffraction patterns were computed. We observe that BLA causes small, but measurable, changes in the diffraction patterns. The most significant effect of BLA, obtained by compression/elongation of the basal plane lattice vector, appears in the splitting of certain reflections in powder XRDs. We obtain high resolution XRDs for these reflections and fit peak-shape functions to see whether such splitting occurs. Such splitting could not be observed and thus, we conclude that BLA is not present in bulk graphite, or if it is, then it is less than 0.003 Å. Our unit cell refinement assigns graphite to the hexagonal crystal system with the following unit cell parameters, a = b = 2.467 ± 0.001 Å and c = 6.716 ± 0.002 Å. These results don’t exclude the possibility of small BLA at the surface of graphite, since powder XRD and electron transmission diffractometry sample predominantly the bulk and are not surface sensitive. Scanning tunneling microscopy however, is sensitive only to the surface structure, and it is conceivable that the surface layer has a slightly different structure from the bulk. Nonetheless low energy electron diffraction results by Ignatiev et al.7 indicate that the surface structure of graphite is hexagonal and that no BLA is present.

125

References 1.

P. Trucano and R. Chen, Nature 258, 136 (1975).

2.

Silicon powder, Standard Reference material 640c, National Institute of Standards and Technology, 1 (2000).

3.

H. M. Rietveld, Acta Crystallogr. 22, 151 (1967).

4.

T. J. B. Holland and S. A. T. Redfern, J. Appl. Cryst. 30, 84 (1997)

5.

T. J. B. Holland and S. A. T. Redfern, Minerol. Mag. 61, 65 (1997).

6.

ICDD-PDF2 Powder XRD database Ref. code 25-1402 and 25-0284 for synthetic Silicon and graphite, respectively.

7.

N. J. Wu, A. Ignatiev, Phys. Rev. B 25, 2983 (1982).

126

CHAPTER 5 ab-initio Study of π-π Interactions Between Polycyclic Aromatic Hydrocarbons

5.1

Abstract We compute the one dimensional potential energy surface (PES) of benzene

translated parallel to various polycyclic aromatic hydrocarbons (PAHs) within the framework of second order Møller-Plesset (MP2) perturbation theory. An increase in the molecular weight of the PAHs gives rise to an increase in the binding energy and lowering of the equilibrium distance. This increase in binding energy with increasing supermolecular size has previously been attributed to the extent of geometric overlap between supermolecular subunits. We provide an alternative interpretation in terms of the average molecular polarizability of the individual constituents of the supermolecule. The effect of molecular polarizability on π-π dispersion interactions is carefully monitored by considering face-to-face configurations only.

Additional

frozen core polarizability computations provide insight into which part of the molecular electronic structure is perturbed to the greatest extent during the interaction

126

127 and play a significant role in π-π interactions. We also evaluate the performance of force fields commonly employed to describe dispersion interactions by fitting them to the computed ab-initio data.

5.2

Introduction Dispersion interactions are an important driving force for the supermolecular

complexation and dynamics of PAHs in the condensed phase. These interactions govern the relative geometry and binding energy of the individual components in DNA1,

proteins2,

host-guest

systems3,

and

self-assembled

architectures4 and are responsible for aggregation phenomena5,6.

supermolecular Fundamental

understanding and computational modeling of dispersion interactions in π-π systems of PAHs, which lack other weak interactions such as dipole-dipole, dipole-induced dipole, and hydrogen-bonds, can be essential to other more complex chemical and biological systems where the dispersion interaction is only part of the overall interaction. X-ray diffraction studies of solid benzene7, naphthalene8,9, pyrene10 and coronene11 and other planar PAHs12 show the coexistence of two distinct relative configurations of the molecules. Molecules in these crystals appear in paralleldisplaced and “T-shaped” configurations, and the attractive forces are attributed to π-π interactions and quadrupole-quadrupole13 interactions respectively. Both interactions seem to be equally important in small supermolecules of PAHs, but the former completely dominates in larger systems such as graphite.

128 Weak π-π interactions are generally hard to measure experimentally since they are only a very small fraction of the total energy. Thus, any experiment that will probe these interactions must not excite intramolecular vibrations or induce structural changes whose energies will considerably exceed that of weak interactions. Only recent isothermal desorption spectroscopy of PAHs from the basal plane of graphite have proven successful in quantifying these interactions experimentally14. Furthermore, spectroscopic experiments in supersonic jet expansions have been able to provide significant information on the dominating equilibrium structures of the benzene dimer15-20. The benzene dimer has served as the model system for π-π interactions since its relatively small size makes spectroscopic data easier to interpret and high-level computations feasible. In the literature, extensive experimental15-20 and computational21-32 studies exist that examine the conformation and binding energy of the benzene dimer. Hobza et al.27 reports similar binding energies for the paralleldisplaced and T-shaped configurations, and Tsuzuki et al.32 showed the existence of a possible low energy (~0.2 kcal/mole) interconversion path between the configurations. The face-to-face configuration exhibits binding but is somewhat higher in energy28 than the other conformations. In a recent computational study by Lee et al.33, the linear increase of the binding energy in the supermolecular series of benzene, naphthalene, and anthracene was attributed to the geometric overlap of π-π pairs. We propose a different interpretation for the observed trend and show that the binding energy depends linearly on the average molecular polarizability of the PAHs. Furthermore, the molecular polarizability is a linear function of the number of carbon atoms in each PAH.

129 In this work we compute the basis set superposition error (BSSE) corrected PES of benzene and various PAHs at the MP2 level of theory. Potential energy functions commonly employed in molecular mechanics (MM) force fields are fitted to the ab-initio data to evaluate their accuracy. A simplified relation is derived that describes the PES in these supermolecules as a function of the number of carbon atoms in the supermolecule. We compute the polarizability of the PAHs to examine (a) the correlation of the polarizability to the number of carbon atoms in the PAH, and (b) the part of the PAH electronic structure that is perturbed to the greatest extent. Finally binding energies obtained here are compared to the MM3 force field.

5.3

Computational Details Basis set quality and level of correlation treatment are crucial for

achieving quantitative agreement with experiment. Augmentation of the basis with diffuse functions and the addition of polarization functions have been proven necessary34 to provide adequate variational flexibility to the one electron wavefunctions. High-level correlated methods, such as CCSD(T), are also necessary to quantitatively describe the quantum mechanics of dispersion interactions28. While density functional methods are applicable to the large systems considered here, they underestimate the dispersion interactions between molecules and in most cases are completely repulsive35. Because our primary focus is to obtain qualitative trends in the binding energy with respect to molecular size in large supermolecules for which highlevel correlated methods with large basis sets are prohibitive, we chose to carry out our calculations within the MP2 formalism.

130 All PAHs geometries, except that of circumcoronene, were optimized at the MP236 level within the cartesian correlation-consistent polarized valence double-zeta basis set (cc-pVDZ) by Dunning37. The structure of circumcoronene was optimized with the same basis, but using density functional theory within the generalized gradient approximation using the PBE9638 exchange-correlation functional. The 1s orbitals on carbons were frozen in all single point energy and polarizability calculations as well as geometry optimizations. In the case of the benzene dimer, this had only a minimal effect on the computed binding energy. The root mean square of the energy gradient (GRMS) and energies were converged to 10-5 Hartrees/bohr and 10-9 Hartrees respectively. The BSSE corrected 1D PES scans were computed at the same level of theory by translating the rigid molecules along the supermolecular axis shown in Fig. 5.1. In order to correct the BSSE, we employed the full counterpoise method by Boys and Bernardi.39 All geometry optimizations and potential energy surface scans were carried out using NWChem40 4.6, and static polarizabilities were computed using the finite field method (FF) implemented in GAMESS41. The latter were carried out at electric field strengths of 0.000au, ±0.001au and ±0.002au (0.001au = 5.14 x 108 V/m). For all the molecules the first hyperpolarizability (βιιι) was found to be about four orders of magnitude lower than the polarizability and is thus not reported.

131

A

B

z

D

C

x y

E

F

G

H

I

J

Figure 5.1 Schematic representation of PAH superstructures. Their nomenclature is provided in Table 2.1. The center of mass of benzene is centered above the underlying ring indicated by the dashed lines.

132

3.0 2.0 1.0 0.0 Binding Energy (kcal/mole)

-1.0 -2.0

A

-3.0

B

-4.0

C

-5.0

D E

-6.0

F

-7.0

G

-8.0 -9.0 3.0

4.0

5.0

6.0

7.0

8.0

Distance (Å) Figure 5.2 BSSE corrected 1D PES of benzene with various PAHs. Molecular structures and nomenclature are given in Figure 2.1 and Table 2.1 respectively. Single point energy calculations are carried out in increments of 0.5 and 0.1Å for 4.0Å