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Quantum Chemistry Calculations of the Reactions of Gaseous Oxygen Atoms with Clean and Adsorbate-Terminated Si(100)-(2x1)

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QUANTUM CHEMISTRY CALCULATIONS OF THE REACTIONS OF GASEOUS OXYGEN ATOMS WITH CLEAN AND ADS ORBATE-TERMINATED Si(100)-(2X1) By PAULO EMILIO HERRERA-MORALES A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Paulo Emilio Herrera-Morales

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To my late grandmother, Catalina Caal Prado.

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ACKNOWLEDGMENTS I wish to express my sincere appreciation to Dr. Jason F. Weaver (my supervisory committee chair) for all his encouragement and advice throughout my Ph.D. program. I would like to extend my deepest gratitude to the members of my advisory committee (Dr. Gar B. Hoflund, Dr. Vaneica Y. Young, and Dr. Fan Ren) for their assistance, technical support, and guidance. I also thank Dr. David Micha, Dr. Erick Deumens, and Dr. Adrian Roitberg (from the Quantum Theory Project) and Dr. Helena Hagelin-Weaver (from the Chemical Engineering Department) for their valuable support with quantum chemistry matters. I would like to thank my colleagues at Universidad Rafael Landvar in Guatemala, for their support and encouragement. I would like to express my sincere appreciation and gratitude to my parents, Victor Manuel Herrera and Emma Morales de Herrera; and to my sister, Servanda Virginia Herrera, for supporting me and my parents for the duration of this work. Special thanks go to my fiance Sonia Mara Garca, who supported me more than once. She possesses admirable patience. I would like to thank the Quantum Theory Project of the University of Florida for the use of its computational facilities; and Brad Shumbera, for his excellent work in setting up the computer network for our research laboratory. I also want to recognize the other Weaver Research Group members, Alex A. Gerrard, Jau Jiun Chen, Sunil Devarajan and Heywood Kan, for their help with different issues throught the years. iv

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TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................iv LIST OF TABLES ...........................................................................................................viii LIST OF FIGURES .............................................................................................................x ABSTRACT.....................................................................................................................xiii CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Research Objective.................................................................................................1 1.2 The Si(100)-(2x1) Surface......................................................................................1 1.3 Chemistry of Atomic Oxygen on Si(100)-(2x1).....................................................3 1.4 Important Silicon-based Materials for Microelectronics Industry..........................4 2 THEORETICAL BACKGROUND OF QUANTUM CHEMISTRY CALCULATIONS........................................................................................................8 2.1 Introduction.............................................................................................................8 2.2 Energy Functionals and Early Density Functional Theory (DFT).......................10 2.3 Hohenberg-Kohn-Sham Density Functional Theory (KS-DFT)..........................14 2.3.1 Hohenberg-Kohn Lemmas and Exchange-Correlation Functional Definition...................................................................................................15 2.3.2 Kohn-Sham Self-consistent Field Method..................................................17 2.3.3 Exchange and Correlation Functional Approximations..............................18 2.3.3.1 Local density approximation (LDA)..............................................19 2.3.3.2 Density gradient corrections...........................................................20 2.3.3.3 Hybrid methods..............................................................................22 2.3.3.4 Empirical DFT methods.................................................................24 2.3.4 Kohn-Sham Density Functional Theory Computational Chemistry...........25 2.4 Basis Sets..............................................................................................................27 2.5 Geometry Optimization and Transition State Search Calculations......................31 2.5.1 Geometry Optimization of Energy Minima...............................................33 2.5.2 Transition State Searches...........................................................................38 v

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3 NITROGEN ATOM ABSTRACTION FROM Si(100)-(2x1)...................................41 3.1 Introduction...........................................................................................................41 3.2 Computational Approach......................................................................................43 3.3 Results...................................................................................................................45 3.3.1 Bonding Configurations of a Nitrogen Atom on Si(100)-(2x1).................45 3.3.2 Nitrogen Abstraction by a Gas-Phase Oxygen Atom.................................46 3.3.3 Abstraction of N Adsorbed at the Dangling Bond [R(ad)]........................48 3.3.4 Abstraction of the Nitrogen Bonded Across the Dimer [R(db)]................51 3.3.5 Abstraction of the Nitrogen Bonded at a Backbond [R(bb)]......................54 3.3.6 Abstraction from the NSi 3 Structure [R(sat)]...........................................59 3.4 Discussion.............................................................................................................63 4 CHEMISTRY ON Si(100)-(2x1) DURING EARLY STAGES OF OXIDATION WITH O( 3 P)................................................................................................................66 4.1 Introduction...........................................................................................................66 4.2 Theoretical Approach...........................................................................................69 4.3 Results and Discussion.........................................................................................70 4.3.1 Structures with One Adsorbed Oxygen Atom (O 1 -Si 9 H 12 )........................77 4.3.2 Structures with Two Adsorbed Oxygen Atoms (O 2 -Si 9 H 12 )......................80 4.3.3 Structures with Three Adsorbed Oxygen Atoms (O 3 -Si 9 H 12 )....................86 4.3.4 Oxygen Insertion........................................................................................91 4.3.4.1 Thermodynamic considerations.....................................................91 4.3.4.2 Kinetic considerations....................................................................94 5 ATOMIC OXYGEN INSERTION INTO ETHYLENEAND ACETYLENE-TERMINATED Si(100)-(2x1)........................................................100 5.1 Introduction.........................................................................................................100 5.2 Theoretical Approach.........................................................................................102 5.3 Results and Discussion.......................................................................................104 5.3.1 Relative Energies of Oxidized Ethylene-Covered Si(100) Clusters.........110 5.3.2 Relative Energies of Oxidized Acetylene-terminated Surfaces...............115 5.3.3 Oxygen Insertion Mechanisms.................................................................119 5.3.3.1 Thermodynamic considerations...................................................119 5.3.3.2 Kinetic considerations..................................................................120 6 SUMMARY AND CONCLUSIONS.......................................................................124 6.1 Nitrogen Atom Abstraction from Si(100)-(2x1) by Gaseous Atomic Oxygen..124 6.2 Initial Step of Si(100)-(2x1) Oxidation by Gaseous Atomic Oxygen................125 6.3 Oxidation of C 2 H 2 and C 2 H 4 -covered Si(100)-(2x1) by Gaseous Atomic Oxygen................................................................................................................127 vi

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APPENDIX A QUANTUM CHEMISTRY SOFTWARE...............................................................128 A.1 File Format .xyz.................................................................................................133 A.2 File Format .zmt.................................................................................................133 A.3 HyperChem Files (Format .hin).........................................................................136 A.4 Gaussian03 Files (Formats .inp, .log and .chk).................................................138 B EXAMPLE OF GAUSSIAN03 OUTPUT FILE......................................................141 C CALCULATION PROTOCOL FOR INITIAL GEOMETRY FOR GEOMETRY OPTIMIZATIONS...................................................................................................157 D CALCULATION PROTOCOL FOR INITIAL GEOMETRY FOR TRANSITION STATE SEARCHES................................................................................................159 LIST OF REFERENCES.................................................................................................166 BIOGRAPHICAL SKETCH...........................................................................................173 vii

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LIST OF TABLES Table page 1-1 Selected properties of silicon dioxide........................................................................6 2-1 Approximations made to correct Thomas-Fermi-DFT before the development of Kohn-Sham DFT......................................................................................................13 2-2 Variables used to express the approximate exchange-correlation functionals E XC [(r)]...................................................................................................................18 4-1 Suboxide penalty energies for various silicon oxidation states................................75 4-2 Penalty energies of O 1 -Si 9 H 12 isomers.....................................................................78 4-3 Penalty energies of O 2 -Si 9 H 12 isomers.....................................................................81 4-4 Penalty energies of O 3 -Si 9 H 12 isomers.....................................................................87 4-5 Energy barriers for migration of oxygen atoms in O-Si 9 H 12 -O d structures from a dangling bond site to a backbond or dimer bond site in the spin-triplet state..........96 5-1 E 1 (SiSi) structure and energy of adsorption compared to those of reported dimerized C 2 H 4 -terminated Si(100) structures.......................................................107 5-2 A 1 (SiSi) structure and energy of adsorption compared to those of reported dimerized C 2 H 2 -terminated Si(100) structures.......................................................107 5-3 Calculated energy of formation of the different oxidized bridges that form when an oxygen atom inserts into the C 2 H 2 and C 2 H 4 -terminated Si 9 H 12 clusters.............109 5-4 Penalty energies of spin-singlet O-C 2 H 4 -Si 9 H 12 isomers.......................................110 5-5 Penalty energies of spin-triplet O-C 2 H 4 -Si 9 H 12 isomers........................................111 5-6 Penalty energies of spin-singlet O-C 2 H 2 -Si 9 H 12 isomers.......................................115 5-7 Penalty energies of spin-triplet O-C 2 H 2 -Si 9 H 12 isomers........................................116 A-1 Quantun chemistry programs used in this work.....................................................132 viii

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A-2 File formats used for proper handling of the coordinates systems and results in our geometry optimization calculations.......................................................................132 A-3 Gaussian03 input files sections..............................................................................138 C-1 Calculation protocol used to determine the best initial geometry for the geometry optimization, transition state search and frequency calculations...........................157 ix

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LIST OF FIGURES Figure page 1-1 Si(100) -2x1 surface showing the dimer rows...........................................................2 1-2 Metal-oxide-semiconductor field-effect transistor (MOSFET).................................6 2-1 Flowchart of the Kohn-Sham SCF procedure..........................................................26 2-2 Two-dimensional potential energy surface..............................................................31 2-3 Flowcharts for a quasi-Newton algorithm for geometry optimization.....................36 3-1 The Si 9 H 12 cluster used in our UB3LYP calculations..............................................47 3-2 Structural information of N-Si 9 H 12 clusters.............................................................48 3-3 Reaction pathway for the nitrogen abstraction from R(ad)......................................49 3-4 Critical point structures of the nitrogen abstraction from R(ad)..............................50 3-5 Reaction pathway of nitrogen abstraction from R(db).............................................52 3-6 Critical point structures of the nitrogen abstraction from R(db)..............................53 3-7 Reaction pathways for the abstraction of the nitrogen atom inserted in a Si-Si backbond..................................................................................................................56 3-8 Molecular precursors formed after O-chemisorption onto R(bb)............................57 3-9 Structures formed during nitrogen abstraction from MP(bbt).................................58 3-10 Structures involved in the nitrogen abstraction from MP(bbs)................................59 3-11 Reaction pathways for N-abstraction from (and O-atom adsorption on) R(sat)......61 3-12 Structures formed during the direct nitrogen abstraction from R(sat).....................62 3-13 Structures involved in the O-atom chemisorption on R(sat)....................................62 4-1 Structural information and highest occupied molecular orbital plot of clean Si 9 H 12 clusters......................................................................................................................74 x

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4-2 Relative energies of O 1 -Si 9 H 12 isomers....................................................................77 4-3 Structural characteristics of Si 9 H 12 clusters with one oxygen atom.........................79 4-4 Relative energies of O 2 -Si 9 H 12 isomers....................................................................83 4-5 Structural information of dangling bond isomers with two oxygen atoms adsorbed on Si(100).................................................................................................................84 4-6 Structural information of O 2 -Si 9 H 12 isomers............................................................85 4-7 Relative energies of optimized structures for three oxygen atom incorporation into Si(100)......................................................................................................................88 4-8 Structural information of the dangling bond O 3 -Si 9 H 12 isomer DB 3 (OSiOSiOd)...89 4-9 Structural characteristics O 3 -Si 9 H 12 isomers............................................................90 4-10 Qualitative representation of a minimum energy path for oxygen atom incorporation on Si(100)-(2x1) based on the relative energy of the structures........93 4-11 Structural information of transition state structures for oxygen migration in O 1 -Si 9 H 12 structures..................................................................................................95 4-12 Structural information of transition state structures for oxygen insertion in O 2 -Si 9 H 12 structures..................................................................................................98 4-13 Structural information of transition state structure TS 3 (OSiOSiOd-OSiOSiO) ......98 4-14 Proposed preferred path for initial steps of oxidation of Si(100)-(2x1) by oxygen atoms........................................................................................................................99 5-1 Structural information of optimized clusters for olefin-covered Si(100)-(2x1).....106 5-2 Relative energies of spin-singlet O-C 2 H 4 -Si 9 H 12 isomers......................................111 5-3 Relative energies of spin-triplet O-C 2 H 4 -Si 9 H 12 isomers.......................................112 5-4 Structural information of O-C 2 H 4 Si 9 H 12 isomers................................................114 5-5 Structural information of O-C 2 H 2 -Si 9 H 12 isomers.................................................117 5-6 Relative energies of spin-singlet O-C 2 H 2 -Si 9 H 12 structures...................................118 5-7 Relative energies of spin-triplet O-C 2 H 2 -Si 9 H 12 isomers.......................................119 5-8 Structural information of clusters involved in oxygen insertion pathways on C 2 H 2 and C 2 H 4 -terminated Si(100) surfaces...................................................................121 xi

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5-9 Potential energy surfaces for insertion of an oxygen atom on an C 2 H 2 covered Si(100)-(2x1) surface.............................................................................................123 5-10 Potential energy surfaces for insertion of an oxygen atom on an C 2 H 4 covered Si(100)-(2x1) surface.............................................................................................123 A-1 HyperChem 7 interface..........................................................................................129 A-2 Molden visualization interface and main command screens..................................130 A-3 gOpenMol visualization interface..........................................................................131 A-4 Example of a .xyz file describing a Si 9 H 12 O 3 cluster.............................................134 A-5 Z-Matrix file describing a Si 9 H 12+x C y cluster.........................................................135 A-6 HyperChem 7 files describing a Si 9 H 12 O 3 cluster..................................................137 A-7 Gaussian03 file types.............................................................................................140 D-1 Input file for a transition state search that uses the OPT(QST2) option................160 D-2 Diagram of the structures of reactant and product used to search for a transition state structure using OPT(QST2) or OPT(QST3) in Gaussian03..........................161 D-3 Z-matrix (.zmt) file generated by Molden..............................................................162 D-4 Gaussian03 input file for a transition state search where the product structure had to be reordered using Molden.....................................................................................163 D-5 Process used to generate an initial guess for the transition state search using OPT(QST3) in Gaussian03....................................................................................164 xii

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy QUANTUM CHEMISTRY CALCULATIONS OF THE REACTIONS OF GASEOUS OXYGEN ATOMS WITH CLEAN AND ADSORBATE-TERMINATED Si(100)-(2x1) By Paulo Emilio Herrera-Morales May 2005 Chair: Jason F. Weaver Major Department: Chemical Engineering Reactions of gas-phase radicals at solid surfaces are fundamental to the plasma-assisted processing of semiconductor materials. In addition to adsorbing efficiently, radicals incident from the gas-phase can also stimulate several types of elementary processes before thermally accommodating to the surface. Reactions that occur under such conditions may be classified as non-thermal; and examples include atom insertion, direct-atom abstraction and collision-induced reaction and desorption. Indeed, non-thermal surface reactions play a critical role in determining the enhanced surface reactivity afforded by plasma processing. Advancing the fundamental understanding of radical-surface reactions is therefore of considerable importance to improving control in plasma-assisted materials processing. Our study used quantum chemical calculations to investigate the interactions of gas-phase oxygen atoms [O( 3 P)] with clean and adsorbate-modified Si(100)-(2x1) surfaces. We carefully studied reaction pathways for the oxygen insertion, adsorbate abstraction and adsorbate migration of these xiii

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species on reaction with O( 3 P) using density functional theory (DFT) and silicon clusters. Knowledge of these reaction pathways helps determine the viability of certain reactions on Si(100)-(2x1) by gas-phase oxygen atoms in plasma-assisted material processing in general. Another goal of our study was to better understand the chemistry of certain ultrathin silicon-based films. Silicon dioxide has been at the center of the microelectronics industry given its multiple electrical properties. However, as the size of the devices keeps shrinking according to Moores law predictions of number of transistors, the demands imposed on this material necessitate a detailed understading of its chemistry at the molecular level urging researchers to look for several alternatives. Materials such as silicon oxynitride (SiO x N y ) and silicon oxycarbide (SiO x Cy) have been suggested as possible silicon dioxide substitutes as gate dielectric and interface dielectric, respectively. Silicon oxynitride can be obtained by either inserting nitrogen atoms into silicon dioxide or oxygen atoms into a silicon nitride film, resulting in ultrathin layers that enhance properties of pure silicon dioxide as a gate dielectric. Silicon oxycarbide is one of the products that evolved as an alternative for silicon dioxide as a result of the recent trend of organic functionalization of the silicon surfaces (a process by which hydrocarbon molecules are chemisorbed on top of the silicon surfaces to combine the semiconductor properties of silicon with the organic functionality of carbon). Thus we used quantum chemistry calculations to study the initial steps of oxidation of the clean Si(100)-(2x1) surface, of the abstraction of nitrogen atoms, and for insertion of gaseous oxygen atoms on acetyleneand ethylene-terminated Si(100). xiv

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CHAPTER 1 INTRODUCTION 1.1 Research Objective Our main objective was to advance fundamental understanding of the reactions of gaseous oxygen atoms with clean and adsorbate-modified Si(100)-(2x1) surfaces, motivated by an interest in gaining insights into the possible reaction pathways during the initial steps of oxidation. Our work was also motivated by the potential benefits of fabricating ultrathin silicon-based films and by the need to precisely control the properties of such films. Density functional theory (DFT) quantum chemistry calculations were run to investigate the possible reactions. This chapter provides a brief introduction to the Si(100)-(2x1) surface, an overview of the current state of knowledge of the oxidation process of this surface, and a brief description of the industrially important silicon dioxide, silicon oxynitride, and silicon oxycarbide, semiconductor materials whose production can be deeply impacted by knowledge of oxygen-atom chemistry at the molecular level. Density functional theory and the background of the quantum chemistry calculations are explained in detail in Chapter 2. 1.2 The Si(100)-(2x1) Surface Because of its numerous industrial applications and model character for semiconductor surface science, the industrially important Si(100) surface has been widely studied, dating back to the early work of Schlier and Farnsworth [1] It is known that when first formed, the surface reconstructs by forming silicon dimmers, and that the surface adopts a (2x1) orientation (where 2x1 designates the new periodicity of the 1

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2 surface atoms). The surface forms these dimers so it can reduce the number of dangling bonds per surface atom, from two in the bulk-terminated (1x1) surface to one in the reconstructed (2x1), and also because their formation lowers the surface energy by about 1.0 eV [2] (Figure 1-1). Figure 1-1. Si(100) (2x1) surface showing the dimer rows. It is generally accepted [ 3 8 ] that the dimers are asymmetric (buckled) and consist of an sp 2 -like bonded down atom, that moves closer to the plane of its three nearest neighbors; and an up atom that moves away from the plane of its neighbors, and possesses an s-like dangling bond. This rehybridization process from the bulk sp 3 to the sp 2 -like configuration of the silicon dimer atoms is accompanied by a charge transfer from the down to the up atom. To minimize this electrostatic effect (and to relieve local stress) the direction of buckling alternates within the dimer rows. The silicon dimer bonds in Si(100)-(2x1) are similar to a C=C double bond [9] The organic double bond consists of two types of interactions: a bond with symmetry around the axis connecting the two atoms, and a bond with a nodal plane along the axis. Similar interactions exist in the Si-Si dimer bonds, with the difference that the interaction is sufficiently weak that the dimer is not held in a symmetric configuration, thus adopting the buckled asymmetry that constitutes its more recognizable structural feature [10] (Figure 1-1).

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3 1.3 Chemistry of Atomic Oxygen on Si(100)-(2x1) Among the most reliable sources of gas-phase radicals such as ground-state atomic oxygen [O( 3 P)] are plasmas, which typically contain a variety of highly reactive, unstable species (radicals and charged particles) that, in addition to adsorbing efficiently, can also stimulate several types of elementary processes before thermally accommodating to the surface. These elementary processes include atom insertion, direct-atom abstraction, and collision-induced reaction and desorption. It is generally accepted that atom abstraction can occur in several ways, ranging from the Eley-Rideal mechanism (ER) of direct abstraction (in which an atom is abstracted directly from the surface in a single collision with an incident species, without reaching thermal equilibrium with the surface) [11] ; and the Langmuir-Hinshelwood mechanism (LH) of collision-induced reaction and desorption (in which the incident species reaches thermal equilibrium with the surface before reacting with the adsorbate and eventually evolving into the gas-phase) [12] An intermediate reaction mechanism for atom abstraction is the the hot atom mechanism (HA) in which the incident species experiences multiple collisions with the surface, but does not fully thermalize before the reactive encounter [13] To study the aforementioned reaction mechanisms, it is necessary to have a convenient source for all types of radicals. Unfortunately, such a source exists only for hydrogen radicals, which can be produced efficiently by thermal dissociation of molecular H 2 over a hot filament [14] Thus, abstraction reactions on silicon have been widely investigated only for hydrogen atoms [ 15 16 ]; and several kinetic models for the H/Si(100) system have been proposed, most of them favoring the ER mechanism [ 17 18 ], the hot atom precursor scenario [ 19 21 ], or both [ 22 25 ]. Other systems, however (such as those of reactions of oxygen atoms on metal and semiconductor surfaces) have not

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4 been studied as often, despite their industrial and technological importance. Nevertheless, a few studies on oxidation of silicon surfaces by gaseous oxygen atoms have been reported. Engel et al. [ 26 31 ] performed a detailed kinetic study of oxidation of clean Si(100) by gaseous oxygen-atoms and reported that O-atom oxidation is facile compared with that caused by molecular oxygen, and that several monolayers of oxide can be formed efficiently via the direct insertion of O atoms into near-surface bonds. Orellana et al. [ 32 ] studying the effect of the spin-state of the surface on the oxidation of Si(100) by O 2 found that the oxidation proceeds with a high probability along the triplet potential energy surface (PES), and concluded that the spin state of the surface may be an important factor in determining the quality of the Si/SiO 2 interface. Despite much study, the reaction mechanism for abstracting hydrogen atoms from Si(100)-(2x1) is still unclear. So the oxygen-atom chemistry on Si(100) is far from understood. Thus, we used quantum chemical calculations to investigate interactions of the O-atoms with clean Si(100), spin-doublet N-covered Si(100), and spin-singlet and spin-triplet C 2 H 2 and C 2 H 4 -terminated Si(100) surfaces. We focused on determining the reaction pathways leading to oxygen insertion or adsorbate direct abstraction from these surfaces, and on the effects that the spin-state has on the processes. This information may be important for the microscopic understanding of the initial steps of the silicon oxidation process by gaseous oxygen atoms. It may also be relevant for developing improved techniques for the fabrication of ultrathin layers of silicon-based industrial materials. 1.4 Important Silicon-based Materials for Microelectronics Industry Semiconductors are extremely useful for electronic purposes, because they can carry an electric current by electron propagation or hole propagation, and because this

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5 current is generally unidirectional, and the amount of current may be influenced by an external agent [33] The most widely used insulator is silicon dioxide (SiO 2 ), which is also one of the most commonly encountered substances in daily life. Modern integrated circuit industry was made possible by the unique properties of silicon dioxide. It is the only native oxide of silicon that is stable in water and at elevated temperatures, is an excellent electrical insulator, is a mask to common diffusing species, and is capable of forming a nearly perfect electrical interface with its substrate (Table 1-1). SiO 2 has strong, directional covalent bonds, and has a well-defined local structure in which four oxygen atoms are arrayed at the corners of a tetrahedron around a central silicon atom. The Si-O-Si bond angles are essentially the tetrahedral angle (109.4), and the Si-O distance is 1.61 with very little variation. It is these siloxane-bridge bonds between silicon atoms that give SiO 2 many of its unique properties [33] The Si/SiO 2 interface is one of the key components of the commonly used metal-oxide-semiconductor field-effect transistor (MOSFET), the building block of integrated circuits (Figure 1-2). This type of transistor continues to be the predominant device in ultralarge scale integrated circuits (USIC) because it is simple to scale down [34] MOSFETs consists of a source and a drain (regions of doped silicon), a gate dielectric (an extremely thin silicon dioxide layer), and a gate electrode (a layer of polycrystalline silicon), rendered conductive by heavy doping, to bias the gate dielectric. Other MOSFET elements include thin-metal interconnect layers (made of Cu) to connect the transistor electrically to other parts of the circuit, and SiO 2 dielectric layers to provide electrical isolation between the Cu interconnects and other devices. Basically, MOSFETs are switches that allow current to flow from the source to the drain only when the gate electrode supplies the appropriate bias voltage through the gate dielectric [34]

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6 Table 1-1. Selected properties of silicon dioxide. Oxide native to silicon Low interfacial defect density (~10 10 eV -1 cm -2 ) Melting point = 1713C Energy gap = 9 eV Resistivity ~ 10 15 cm Dielectric strength ~ 10 7 V/cm Dielectric constant relative to air () = 3.9 Figure 1-2. Metal-oxide-semiconductor field-effect transistor (MOSFET). Silicon dioxide has a relatively low-dielectric constant ( = 3.9) [33] Thus, since high gate dielectric capacitance is necessary to produce the required drive currents from submicron devices, and because capacitance is inversely proportional to gate dielectric thickness, the SiO 2 layers have been scaled to extremely thin dimensions where the Si/SiO 2 interface becomes a more critical (and limiting) part of the gate dielectric. As a result, 1.0 nm SiO 2 layers are commonly found in modern materials. Such a thin silicon dioxide layer is mostly interface, and it contains about five layers of Si atoms (at least two of which reside at the interface). This has given rise to several electrical and performance problems (including impurity penetration through the dioxide, and a lack of convenient insulating properties) that make it necessary to look for alternate gate dielectric materials. Over the past several years, ultrathin silicon oxynitride (SiO x N y )

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7 films have been incorporated into metal-oxide-semiconductor (MOS) devices as an alternative for pure SiO 2 [33] Incorporating nitrogen into SiO 2 is a relatively simple method for fabricating silicon oxynitride films, since the nitrogen atoms tend to aggregate at the Si/SiO 2 interface, and results in dielectric layers that enhance resistance to gate current leakage and inhibit boron penetration into the dielectric [34] In recent years, surface functionalization also has been intensively investigated [ 35 36 ]. This is the process of depositing layers of organic molecules on semiconductor surfaces to impart some property of the organic materials to the semiconductor device [ 35 36 ]. The organic molecules might be designed to serve in place of gate oxides in metal-oxide semiconductor field-effect transistor (MOSFET) devices, where the higher wire resistance of smaller metal lines and the crosstalk of closely spaced metal increase the interconnect resistance and capacitance product delay. This requires a low-dielectric constant (low k) material as the interlayer dielectric, and low-resistance conductors (such as aluminum). Until recently, silicon dioxide was the material of choice, but the decrease of the device dimensions and the resulting change of aluminum for copper have necessitated the search for alternative materials. Silicon oxycarbide (SiC x O y ), a low k hydrid between organic and inorganic materials obtained by organic functionalization, is one of the most favorable candidates for interlayer dielectric in MOSFETs [ 37 39 ].

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CHAPTER 2 THEORETICAL BACKGROUND OF QUANTUM CHEMISTRY CALCULATIONS 2.1 Introduction From a chemistry and physics viewpoint three types of systems exist [40] : Systems that possess a very small number of particles where the particles do not interact and that can be studied classically and exactly, systems that have many particles and that require statistical methods to give a nearly exact solution of the properties under study, and systems where only a few particles exist, but where the particles interact considerably with one another. The latter cannot be solved exactly (neither classically nor by statistical modeling) so a different strategy is needed. Electron interactions with other electrons and nuclei within a molecule are an example of these complicated systems for which classical and statistical approaches are insufficient. If one considers that the molecule has N electrons with electronic coordinates r i, the total electronic energy of the system is obtained by solving the Schrdinger equation E H where H is the Hamiltonian operator, expressed as in Equation 2-1. ninijjiniiniVHrrr1)(212 (2-1) The three terms on the right-hand side of Equation 2-1 correspond to the kinetic energy, the classical Coulomb electrostatic potential between nuclei and electrons, and the electron-electron interactions, respectively. From these, the electron-electron interaction term highly complicates the calculation, given that it describes the highly coupled 8

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9 motions of the electrons. If one attempts the classical approach, one can ignore the electron-electron interactions completely. This decouples the electronic motions and factorizes the problem into N completely independent problems, one for each electron. However, this approach loses essential elements of the physics of the system. If the statistical approach is attempted there is no need for intuitive simplifications, but a large supercomputer is needed to solve the Schrdinger equation directly. This requires due care in each calculation step, and has the drawback that the result is difficult to interpret and gives little physical insight. Thus, a molecular system is one of those systems for which neither classical nor statistical methods are sufficient. As indicated, these systems require a different approach, and a hybrid between the classical and statistical methods seems the most reasonable. The electron-electron interaction term in Equation 2-1 is replaced by a term that can be solved statistically or computationally, while the first two terms are solved classically. The available methods give a loss of few of the finer details of the system, but they factorize the original problem into N independent one-electron problems that can be solved iteratively (given that usually the one-electron equations depend on the energy, while the energy depends on the one-electron equations) [40] The two more-widely used iterative computational chemistry methods are Hartree-Fock (HF) and Kohn-Sham density functional theory (Kohn-Sham DFT) [ 41 44 ]. Both methods are based on the self-consistent-field (SCF) concept and share many conceptual and computational features. However, they are essentially different because while HF is based on single-electron orbitals ri and many electron wavefunctions () constructed from them, Kohn-Sham DFT is based on the electron density (r) and the fictitious single-particle orbitals ri [ 42 45 ].

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10 This chapter presents the most-common energy functionals (Section 2.2), followed by a detailed description of the Kohn-Sham SCF scheme and its implementation in Gaussian03 (Section 2.3), an introduction to the basis sets (Section 2.4) and an explanation of the geometry optimization schemes used in our study (Section 2.5). Since our main focus is on DFT calculations, we present only the main characteristics of the Hartree-Fock method, and do not discuss the more complex and exact post-Hartree-Fock methods (such as CI and MP2). 2.2 Energy Functionals and Early Density Functional Theory (DFT) It is useful to express the total electronic energy of a molecular system partitioned in energy functionals, as follows E = E T + E V + E J + E X + E C (2-2) given that the computer programs available find the approximate solutions to the Schrdinger equation by calculating each one of the functionals of Equation 2-2 separately. A functional is a mathematical device that maps objects onto numbers, and can be viewed as a function whose argument is also a function. Terms on the right-hand side correspond to the kinetic energy of the electrons, the Coulomb energy of electrons caused by their attraction to the nuclei, the Coulomb energy that the electrons would have in their own field if they moved independently and if each electron repelled itself, a correction for electron exchange, and a correction for electron correlation, respectively. The last two terms can be combined as E XC = E X + E C rendering a correction for the false assumptions involved in E J (i.e., that electrons do not perturb one another at close approach and that their motions are independent). The exchange energy E X is the electron stabilization resulting from the Fermi correlation (i.e.,

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11 the dependence of motion arising from the Pauli Principle); while E C (the correlation energy) arises from the correlation between the motions of electrons with different spins. These five terms have entirely different magnitudes, with E T E V and E J constituting most of E. The trend is as follows: E T > E V > E J >> E X >> E C Additionally (as seen later), the expressions for E V and E J are common for all SCF methods used in quantum chemistry; while the formulas for E T E X and E C are the ones that differentiate those methods [ 40 45 ]. The Hartree-Fock approximation (the first SCF method developed) is based on orbital functionals, which are well-defined procedures that take the orbitals of a system and return an energy. The early orbital functionals that were suggested are the Hartree kinetic functional, the self-interaction-correction functional and the Fock exchange functional. In 1928, Hartree [ 40 45 ] presented a model in which the ith electron in an atom moves completely independent of the other in an orbital ri Since all electrons are represented this way, the system is then a collection of uncorrelated electrons, and the total kinetic energy of the atom E T is the sum of the kinetic energies of the electrons. NiiiHTdErrr22821 (2-3) Molecular orbital (MO) theory originated when Hartrees model was extended to molecules by delocalizing the { ri } over several atoms. However, the model neither gives the exact E T (given that the electrons do not move independently) nor excludes the self-interactions of electrons (i.e., the exchange energy), and an additional self-interaction-correction functional is required. A correction functional was presented in 1930 by Fock [ 40 45 ], who indicated that the Hartree wavefunction violates the Pauli Exclusion Principle because it is not properly

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12 antisymmetric. He then solved the problem by rendering the wavefunction antisymmetric as follows NiNjjijiFXddE212122113021rrrrrrrr (2-4). At this point in the history of quantum chemistry, it was realized that it is cumbersome to work with wavefunctions, given that they have essentially no physical interpretation and have units of probability density to the one-half power. An alternative method was suggested, based on an observable that should allow the construction of the electronic Hamiltonian. Since the Hamiltonian depends on the position and atomic number of nuclei and the total number of electrons N, the best observable for the alternative method is the electron density given that the total number of electrons N is obtained from rrdN (2-5) and, since the nuclei correspond to point charges, the nuclei positions are local maxima of Only the relationship between the atomic number Z A and the electron density was not clear, but it was found that Z A is related to the spherically averaged density. With this solved, Eq. 2-2 for the total electronic energy can be expressed in terms of density functionals rrrrr XCJVTEEEEE (2-6) The simplest approximation for the total electron energy was the approach of Thomas-Fermi [ 46 47 ], who assumed that the system had classical behavior, and then reduced Equation 2-6 to Equation 2-7. rrrr JVTEEEE (2-7)

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13 In Equation 2-7, the Coulomb interaction terms in the right-hand side are given by Equation 2-8 and 2-9. rrRrrdZEnucleiaaaV (2-8) and 21212121rrrrrrrddEJ (2-9) r VE and r JE are extremely useful for modern SCF methods, whether based on HF or DFT theories. Equation 2-8 is exact and is used in all the available SCF methods. Equation 2-9 is an approximation that considers that the electrons move independently and that each electron experiences the field caused by all the electrons (including itself). Thomas and Fermi [ 46 47 ] also derived a kinetic functional based on the uniform electron gas, or jellium, which is an idealized system in the limit of N and V of a system of N electrons in a cubical box of volume V, that has a uniformly distributed positive charge that renders the system neutral. Thomas and Fermi chose jellium because although it is a many-electron system, it is completely defined by its electron density [ 46 47 ]. The functional is as shown in Equation 2-10. rrrdET3/53/226103 (2-10) The Thomas-Fermi kinetic energy functional is the only density functional that has an elegant mathematical derivation, but unfortunately, it is not accurate enough to be chemically useful. Also, it was the first DFT functional as such, because it showed that non-electrostatic energy terms can be expressed in terms of the electron density.

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14 Table 2-1. Approximations made to correct Thomas-Fermi-DFT before the development of Kohn-Sham DFT Functional Author Year Comments Hole function: 2121211212121;2121rrrrrrrrrrrrrrddhddEJ Slater Wigner and Seitz 1930 1933 The hole function h(r 1 ; r 2 ) is centered on electron 1 and a function of electron 2 coordinates. Hole exchange energy of jellium: rrrdEDX3/43/1304323][ Bloch and Dirac 1930 When combined with the work of Thomas and Fermi, is known as the Thomas-Fermi-Dirac approximation. Slater exchange hole function: rrrdESX3/43/151389][ Slater 1951 Completely ignores correlation effects. The exchange hole function at any position is calculated as a sphere of constant potential with a radius that depends on the magnitude of the electron density at that position. X: rrrdEX3/43/1389][ with = 1 for = 1 for E X S51 and 2/3 for E X D30 Gaspar Slater and Wood 1954 1971 = works better than both 1 and 2/3. However, these Thomas-Fermi functionals (known as Thomas-Fermi DFT) are no longer widely used, because they predict that molecules are unstable with respect to the dissociation of their constituent atoms. Table 2-1 shows the approximations made to correct for problems with Thomas-Fermi-DFT before Kohn-Sham density functional theory was developed; of these approximations, X is the only one that still is used in certain scientific endeavours. 2.3 Hohenberg-Kohn-Sham Density Functional Theory (KS-DFT) The early DFT model presented in Section 2.2 did not have widespread use because it resulted in fairly large errors in molecular calculations, and because the theories were

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15 not rigorously founded. This changed in 1964, when Hohenberg and Kohn (HK) [41] presented two key theorems to the scientific community. Before we further describe the HK theorems, we must clarify that we have limited this description to the simplest systems, that is, N non-relativistic, interacting electrons in a non-magnetic state with Hamiltonian rrrr JVTEEEE (2-11) where jjTE221r jjVrvEr and jijiJrrE121r For mathematical reasons, we considered a broad class of Hamiltonians with electrons moving in an arbitrary external potential v(r), besides the physically relevant Coulomb potentials due to point nuclei. 2.3.1 Hohenberg-Kohn Lemmas and Exchange-Correlation Functional Definition The starting point of KS-DFT is the rigorous, simple-existence lemma of Hohenberg and Kohn (HK) [41] who considered that the specification of the ground-state density, (r), determines the external potential v(r) uniquely (to within an additive constant C). Since (r) also determines N by integration, N[(r)] = = N, drcr it determines the full Hamiltonian H (and, implicitly, all the properties determined by H). Hohenberg and Kohn [ 41 45 ] proved that the ground-state density determines the external potential via reductio ad absurdum. They assumed that there were two potentials, v a (r) and v b (r), with ground-state wave functions 0,a and 0,b respectively. They also assumed that both wave functions give rise to the same density (r). Unless [v b (r) v a (r) = constant] 0,a and 0,b cannot be equal since they satisfy different

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16 Schrdinger equations. If one denotes the Hamiltonian and ground-state energies associated with 0,a and 0,b by H a and H b and E 0,a and E 0,b then one has by the minimal property of the ground-state, E 0,b = < 0,b |H b | 0,b > < < 0,a |H b | 0,a > so that rrrrdvvEEbaba0,0,0 (2-12) By interchanging a and b, HK found in exactly the same way rrrrdvvEEabab0,0,0 (2-13) Finally, adding Equations 2-12 and 2-13 leads to the inconsistency E 0,a + E 0,b < E 0,a + E 0,b (2-14) With the help of the existence lemma, HK also showed (in a second lemma) that the density obeys a variational (or minimal) principle. For a given v(r), they defined the following energy functional of (r) rrrrrrFdvEv][)( (2-15) where rrrJTEEF (2-16) One should notice that F is a functional of (r), since the wave function itself is a functional of (r). The variational principle is then given by the expression EEEvvrrrr0)()( (2-17) where 0 (r) and E are the density and energy of the ground-state. The equality in Eq. 2-17 holds only if 0 (r) = (r); in other words, the variational principle states that any calculated energy will be higher than that of the ground-state. Then, Hohenberg and Kohn [ 41 45 ] extracted from F[(r)] its largest and elementary contributions

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17 by writing Equation 2-18, rrrrrrrrr XCTsEddEF'''21 (2-18) where E Ts [(r)] is the kinetic energy of a non-interacting system of electrons with density (r), and the next term is the classical expression for the interaction energy. Equation 2-18 is the KS-DFT definition of E XC [(r)]. 2.3.2 Kohn-Sham Self-consistent Field Method If E XC is ignored in Eq. 2-18, the physical content of the theory becomes identical to that of the Hartree approximation [44] Kohn and Sham (KS) [42] noticed that and transformed the Euler-Lagrange equation associated with the stationarity of E v [(r)] into a new set of self-consistent equations, the Kohn-Sham (KS) equations. 0'''212rrrrrrrjjXCiVdv (2-19a) N[(r)] = Njj12r (2-19b) and rr XCXCEV (2-19c). These equations are analogous to Hartree-Fock equations, although they also include correlation effects. They must be solved self-consistently, like the Hartree-Fock equations, calculating V XC in each cycle from Eq. 2-19c, with an appropriate approximation for E XC [(r)] (Section 2.3.3). Despite the appearance of simple, single-particle orbitals, the KS equations are in principle exact provided that the exact E XC is used in Eq. 2-19c; that is, the only error in the theory is due to approximations of

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18 E XC [(r)]. Once Equations 2-19 are self-consistently solved, the ground-state energy is obtained by rrrrrrrrrrXCvXCjEdVddE1'''21 (2-20) where j and are the self-consistent quantities. The individual eigenfunctions and eigenvalues, j and j, of the Equations 2-19 have no strict physical significance. 2.3.3 Exchange and Correlation Functional Approximations In principle, the KS-DFT exchange-correlation functional accounts for the classical and quantum mechanical electron-electron repulsion and corrects for the difference in kinetic energy between the fictitious non-interacting system and the real electron system. Table 2-2. Variables used to express the approximate exchange-correlation functionals E XC [(r)] Variable Equation Comments Energy density ( XC ) rrrrdEXCXC Function of the electron density, given by the sum of individual exchange and correlation contributions in the system. Effective radius [r s (r)] 3/143rrsr Used when the electron density is expressed for exactly one electron. This electron would be contained within a perfect sphere that has the same density throughout its center. Normalized spin polarization () rrrr where: corresponds to spin-upand to spin-down electrons. Used to express spin densities at any position in open-shell systems. By convention, the spin densities are given by 121rr and rrr

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19 However, there is no known systematic way to achieve an arbitrarily high level of accuracy for this functional. Therefore, an approximate form for the correct exchange-correlation functional must be selected so that the Kohn-Sham differential equations for the orbitals can be obtained by minimization of the total energy in Eq. 2-20. These r XCE approximations can be broadly classified in four major groups: the local density approximation functionals, the gradient-corrected functionals, the hybrid functionals and semi-empirical DFT. Moreover, all these approximate functionals can be expressed in terms of the three variables summarized in Table 2-2. 2.3.3.1 Local density approximation (LDA) The designation local density approximation originally referred to any DFT functional where the energy density at some position r was computed exclusively from the value of the electron density at that particular position (i.e., the local value of ). However, since the only functionals that follow this definition are those derived based on the uniform electron gas electron density, generally the name LDA is applied only to exchange and correlations functionals derived from jellium. In the special case when one needs to calculate the exchange energy density for a spin polarized system, LDA can be modified with Equation 2-21. 122211,3/13/43/4010rrrrXXXX (2-21) where rr3/13/10389X and is analogous to r1X r0X It describes a uniform electron gas composed of electrons with the same spin. This expression is the local spin density approximation (LSDA) [ 40 45 ].

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20 Since there has been little success in finding analytical expressions for r C most of the work available on correlation energy density calculations is approximated by numerical techniques. One of the most widely used functionals is that of Vosko, Wilk and Nusair (VWN) [48] After Ceperly and Alder used Monte Carlo techniques in 1980 to compute the total energy for jellium of various densities and found their correlation energy by subtracting the exchange, VWN developed a spin polarized correlation local density functional analogous to the exchange functional presented in Eq. 2-21 by fitting these computational results. The VWN proposed expression is Equation 2-22. brbcbcxbcbrrxrcbxxbxbrbcbcbcrbrrArsssossssssiC2/12/1212/120222/102002/12/1212/122/124tan422ln24tan42ln2 (2-22) where i = 0 or 1 (analogous to r0X and r1X ), and with different empirical constant sets for and r0C r1C This correlation functional, known as, is exact for jellium but not for molecular systems and it is a good example of how many empirically optimized constants are required in DFT to approximate the unknown exact rVWNCE r XCE Furthermore, the semi-empirical flavor of functionals such as is the main inspiration for the development of the semi-empirical DFT methods (Section 2.3.3.4). rVWNCE 2.3.3.2 Density gradient corrections Because the LDA exchange functionals were derived from a HF density matrix constructed with the plane wave orbitals of the uniform electron gas, they typically underestimate the exchange energy by roughly 10 to 15 %, and, consequently, need a

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21 correction for the non-uniformity of atomic and molecular electron densities. An improvement would be the generation of exchange and correlation functionals that depend on the local value of the electron density, and on the extent to which the density is locally changing (i.e., the density gradient). These types of functionals are called gradient-corrected functionals, and the methodology by which they are formed is known as the generalized gradient approximation (GGA). Most of these gradient-corrected functionals are constructed by adding a correction to an LDA functional energy density expression, that is, Equation 2-23. rrrrxCXLSDACXGGACX///, (2-23) where is the reduced density gradient given by Eq. 2-24, rx rrr3/4x (2-24), and X/C indicates that this could be either an exchange or a correlation gradient-corrected density functional). The reduced density gradient has small values in bonding regions, larger values in core regions and very large values in Rydberg regions of molecules. The most widely used GGA exchange functional is the one developed by Becke in 1988 [49] which combines LSDA exchange with a gradient correction, and yields accurate exchange energies for atoms and correct exchange energy density in Rydberg regions. In its simplest form, the functional is given by Equation 2-25, rdxxxEEDXBX123/43088sinh61 (2-25) where the semi-empirical parameter has a value of 0.0042, obtained by fitting to the E X F30 (Hartree-Fock) exchange energies of six noble gas atoms (He to Rn). Alternative GGA exchange functionals are do not include empirically optimized parameters.

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22 With respect to correlation, rVWNCE predicts correlation energies that are too large, making it a poor starting point for a gradient-corrected functional. Instead, the most popular GGA correlation functional developed by Lee, Yang and Parr (LYP) [50] computes full correlation energies. LYP abandoned jellium in favor of the He atom, the simplest system with a non-vanishing correlation energy. Their approach was based on earlier work of Colle and Salvetti [51] and it was improved for spin-compensated and spin-polarized versions by Miehlich et al. [52] The expression reported by Miehlich et al. for this gradient-corrected correlation functional, denoted as a second-order gradient expansion is Equation 2-26. }323232]9111812518718472[{14,22222222222223/83/83/113/1FLYPCCabdaE (2-26) where 3/113/13/11expdc 3/13/13/11ddc 3/223103FC and the parameters a, b, c and d were obtained from the work of Colle and Salvetti [51] The values for the parameters are a = 0.04918, b = 0.132, c = 0.2533 and d = 0.349. rLYPCE yields better results than rVWNCE and is very important in DFT despite some theoretical deficiencies. 2.3.3.3 Hybrid methods Given that the basis of the Kohn-Sham approximation is the non-interactive reference system, it was suggested that one can control the conversion from a non

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23 interacting to a real interacting system. With the Hollmann-Feynman theorem, the DFT exchange-correlation energy can be computed as shown in Equation 2-27. dVEXCXC10 (2-27) where is the extent of interelectronic interaction, going from 0 (non-interaction system) to 1 (real system). Development of hybrid DFT techniques resulted in methods that are significantly more efficient than ab initio methods of comparable accuracy. The relatively good accuracy and computational efficiency of hybrid methods indicate that further improvements might lead to methods that are both highly accurate and relatively efficient. All these methods include Hartree-Fock exchange because a small exact-exchange component is a necessary constituent of any exchange-correlation approximation aiming for accurate molecular energetic. The methods use the conventional mixing method introduced by Becke [53] which expands the functional with respect to the electron density and its gradient according to Eq. 2-23 and adds adjustable coeffiencts, c i rrdfcEiiBCX,,,,,/ (2-28) an approach that is appealing because its application to a new molecule requires only a single Kohn-Sham calculation. Becke developed the B3LYP exchange-correlation functional [53] which is defined by Equation 2-29. LYPCVWNCBXFXDXJVHTLYPBEcEcEcEcEcEEEE3388230130128311 (2-29) where the coefficients c 1 c 2 and c 3 are semi-empirical coefficients determined by a fit to experimental data. This method cannot be derived rigorously and, at least in its current status, is basically empirical. Nonetheless, it is widely used because its ability to predict

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24 the atomization energies of normal systems is very close to their exact value. Kang and Musgrave [54] proposed a hybrid DFT method in which the exchange functional is a mix of Slater exchange and exact exchange. The KMLYP energy functional is given by Eq. 2-30. LYPCVWNCFXDXJVHTKMLYPbEEbaEEaEEEE11303028 (2-30) where a = 0.557 and b = 0.448 and the subscripts are the same as in Eq. 2-29. 2.3.3.4 Empirical DFT methods These methods increased the emphasis on empirical parameterization, arguing that by modestly increasing parameterization it may be possible to obtain good functionals. Additionally, empirical DFT methods (EDF) looked for an exchange-correlation functional that was optimized for a relatively small basis set and questioned the need for including Hartree-Fock exchange to obtain good agreement with experiment. Hybrid methods suppose that a small exact-exchange component is a necessary constituent of any exchange-correlation approximation aiming for accurate molecular energetics, but this introduces non-local effects and consequent computational complications that EDF methods tried to eliminate. Empirical density functional theory 1 (EDF1) [55] developed a DFT functional by linearly combining several existing functionals. It was optimized to yield accurate thermochemistry when used with the 6-31+G* Pople basis set (Section 2.4). Among other variational schemes, it considered linear combinations of different functionals or linear combinations of the same functionals with different parameters. The component functionals used are those of Becke [49] and Lee, Yang and Parr [50] discussed above. EDF2 [56] was developed following the method used in EDF1 [55] in which a DFT functional is obtained by linearly combining several existing orbital and density

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25 functionals, with a special focus on giving accurate harmonic frequencies when used with the cc-pVTZ [57] basis sets. The expression EDF2 is as shown in Equation 2-31. 1765148833023012EDFCLYPCVWNCEDFXBXDXFxEDFXCEbEbEbEbEbEbEbE (2-31) where b 1 = 0.1695, b 2 = 0.2811, b 3 = 0.6227, b 4 = -0.0551, b 5 = 0.3029, b 6 = 0.5998, and b 7 = -0.0053. 2.3.4 Kohn-Sham Density Functional Theory Computational Chemistry Figure 2-2 summarizes the steps involved in a KS-DFT geometry optimization. For the most part, they are the same as those used in Hartree-Fock theory. First, a basis set is chosen to construct the KS orbitals (Eq. 2-19) and then an initial estimate of the molecular geometry. After that, the overlap integrals and the kinetic energy and nuclear attraction integrals are computed. The latter two kinds of integrals are called one-electron integrals in HF theory to distinguish them from the two-electron Coulomb and exchange integrals. (However, in KS theory all integrals can in some sense be regarded as one-electron integrals since every one reflects the interactions of each one electron with external potentials). To evaluate the remaining integrals, one must guess an initial density, and this density can be constructed as a matrix entirely equivalent to the P HF density matrix, which describes the degree to which individual basis functions contribute to the many-electron wave function, and thus, how important the Coulomb and exchange integrals should be. The elements of P are then computed as indicated in Equation 2-32. occupiediiiaaP2 (2-32) where the coefficients a i and a i specify the contribution of the basis functions to the molecular orbital i and the factor of two appears because this is a restricted calculation,

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26 meaning that it is considering only single wave functions in which all orbitals are doubly occupied [45] Pople and Nesbet [58] presented the equation for P for radicals and excited states, which is known as unrestricted Hartree-Fock and which has one wave function for each electron. Figure 2-1. Flowchart of the Kohn-Sham SCF procedure

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27 Once the guess density is determined one can construct V XC (Eq. 2-19c) and evaluate the remaining integrals in each KS matrix element. New orbitals are determined from solution of the secular equation, the density is determined from those orbitals, and it is compared to the density from the preceding iteration. Once convergence of the SCF is achieved, the energy is computed by plugging the final density into Eq. 2-20. At this point, the calculation proceeds according to the geometry optimization algorithm (i.e., if the geometry does not correspond to the optimal point, a new structure will be found and the KS SCF process will be run again, until the optimum is reached [45] ). The geometry optimization algorithm is described in Section 2.5, but first, we present a brief introduction to the basis sets used in this work. 2.4 Basis Sets As discussed, Equations 2-19 are formally similar to the HF SCF ones, except that the exchange nonlocal one-electron operator is replaced by a local exchange-correlation operator depending on the total electron density. Thus, the KS equations can be efficiently calculated and the single electron wave functions can be represented by several basis functions such as Slater-type orbitals. A basis set is then, a set of mathematical basis functions from which the wave function is constructed in a quantum chemistry calculation. Slater-type orbitals have a number of attractive features primarily associated with the degree to which they closely resemble hydrogenic atomic orbitals. They suffer, however, from a fairly significant limitation. There is no analytical solution available for the general four-index integral given en Equation 2-33 when the basis functions are STOs. The requirement that such integrals be solved by numerical methods severely limits their utility in molecular systems of any significant size.

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28 2122111|12rrddr (2-33) Alternatives to the use of STOs have been proposed. One of them indicates that for there to be an analytical solution of the general four-index integral (Eq. 2-33) formed from such functions is required that the radial decay of the STOs be changed from e -r to e -r2 That is, the AO-like functions are chosen to have the form of a Gaussian function. The general functional form of a normalized Gaussian-type orbital (GTO) in atom-centered Cartesian coordinates is as shown in Equation 2-34. 2222/14/3!2!2!2!!!82,,,;,,zyxkjikjiezyxkjikjikjizyx (2-34) Smalls GTO sets can be used for a wider range of chemical problems but involve some loss of flexibility in the resulting molecular orbitals. The simplest level of basis is minimal and corresponds to one basis function per atomic orbital. The next level is split-valence or valence-multiplein which two basis functions are used for each valence atomic orbital. This second level is known to give a better description of the relative energies and of some geometrical features of molecules. Further improvement of a basis set requires addition of functions of higher angular quantum numbers (polarization functions) [59] The basis functions are generally contracted (i.e., each basis function is a linear combination of a number of primitive Gaussian functions). The contracted GTOs used in DFT to represent the electronic orbital wave functions, often were originally developed for Hartree-Fock calculations. A considerable increase in computational efficiency can be achieved if the exponents of the Gaussian primitives are shared between different basis functions [60] At the split-valence level, this has been exploited by sharing

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29 primitive exponents among s and p functions for the valence functions. In particular, a series of basis was defined and designated K-LMG by Pople and coworkers [ 61 68 ] where K, L, and M are integers. Such a basis for a first-row element (Li to Ne) consists of an s-type inner-shell function with K Gaussians, an inner set of valence sand p-type functions with L Gaussians, and another outer sp set with M Gaussians. Both valence sets have shared exponents. For hydrogen, only two s-type valence functions (with L and M Gaussians) are used. Among this basis sets are 3-21G [ 67 68 ], 4-31G [ 64 69 ], and 6-31G [ 63 65 66 ]. The original 4-31G and 6-31G split-valence basis sets were obtained by optimizing all Gaussian exponents and contraction coefficients to give the lowest spin-unrestricted Hartree-Fock energy for the atomic ground-state. The Pople basis sets have seen widespread use among the scientific community [45] A problem with the calculations based on K-LMG basis sets is that s and p functions centered on the atoms do not provide sufficient mathematical flexibility to adequately describe the wave function for some geometries. This occurs because the molecular orbitals require more mathematical flexibility than do the atoms. Because of the utility of AO-like GTOs, this flexibility is almost always added in the form of basis functions corresponding to one quantum number higher angular momentum than the valence orbitals. Thus, for a first-row atom the most useful polarization functions are d GTOs, and for hydrogen, p GTOs. A variety of molecular properties prove to be sensitive to the presence of polarization functions; for example, d functions on second-row atoms are absolutely required to make reasonable predictions for the geometries of molecules including such atoms in formally hypervalent bonding situations (e.g., siliconates). Because the total number of functions begins to grow rather quickly with the addition of polarization functions, early calculations typically made use of only a

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30 single set. Pople and coworkers introduced first a simple nomenclature scheme to indicate the presence of the polarization functions, the * (star). Thus, 6-31G* implies a set of d functions added to polarize the p functions in 6-31G. A second star implies p functions on H and He (e.g., 6-311G**) [45] However, realizing the tendency to use more than one set of polarization functions in modern calculations, their basis set nomenclature now typically includes an explicit enumeration of those functions instead of the star nomenclature. Thus, 6-31G(d) is preferred over 6-31G* because the former generalizes to allow names such as, for example, 6-31G(3d2fg,2pd), which implies heavy atoms polarized by three sets of d functions, two sets of f functions, and a set of g functions, and hydrogen atoms by two sets of p functions and one of d. Finally, we must indicate that the highest energy MOs of anions and highly excited electronic states tend to be much more spatially diffuse than the molecular orbitals described so far. When a basis set does not have the flexibility necessary to allow a weakly bound electron to localize far from the remaining density, significant errors in energies and other molecular properties can occur. To address this limitation, standard basis sets are often augmented with diffuse basis functions. In the Pople family of basis sets, the presence of diffuse functions is indicated by a + in the basis set name. Thus, 6-31+G(d) indicates that heavy atoms have been augmented with an additional one s and one set of p functions having small exponents. A second plus indicates the presence of diffuse s functions on H [e.g., 6-311++G(2d,p)]. For the Pople basis sets, the exponents for diffuse functions were variationally optimized on the anionic one-heavy-atom hydrides and are the same for 3-21G, 6-31G and 6-311G [45] In this work we use the 3-21G*, the doubleplus polarization 6-31G(d), and the diffuse tripleplus polarization 6-311++G(2d,p) Pople basis sets.

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31 2.5 Geometry Optimization and Transition State Search Calculations Before we discuss how to perform geometry optimizations of molecular systems, we must understand the concept of a potential energy surface (PES) since a potential energy surface describes the energy of a particular molecule as a function of its geometry, and a geometry optimization is a process to find minima on a potential energy surface. Finding the total PES is a very complicated task, mainly because molecules have many atoms and many coordinates that describe their geometry. A simplified three-dimensional PES can be represented as a topographic surface with valleys and saddle points (Figure 2-2). Figure 2-2. Two-dimensional potential energy surface Generally, calculation of most potential energy surfaces is based on the Born-Oppenheimer approximation which specifies that since the electrons are much lighter than the nuclei, the electronic part of the wave function readjusts almost immediately to any nuclear motion. Thus, potential energy surfaces can be obtained by calculating the total electronic energy for all the possible fixed nuclear positions that the molecule may adopt, an approach that is satisfactory for ground-state systems. On a PES, a reaction path is the movement from a valley of reactants to a valley of products, and a

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32 reaction mechanism is the path across the PES. Then, the important points in the calculated PES are the minima and the saddle points, with the equilibrium geometries corresponding to minima and the saddle points to the highest points on a reaction path that requires the least energy to get from the reactants to the products. Both PES minima and saddle points are called stationary points (or critical points), because for them the gradient (first derivative) is zero, and in classical mechanics, negatives of the first derivatives of the potential energy surface are the forces on the atoms in the molecule, as expressed by Eq. 2-34. VEF (2-34) An energy minimum must satisfy two conditions [70] : its first derivative (i.e., gradient of the forces) must be zero (critical point). its second derivative matrix must be positive definite (i.e., all eigenvalues of the Hessian must be positive or all of the vibrational frequencies must be real). Because of the difficulty of finding a complete PES, most quantum chemistry geometry optimization methods are focused on finding equilibrium structures and transition states directly. In this work, we performed geometry optimizations of Si 9 H 12+w O x N y C z (w,x,y,z = 0,1,2, etc.) clusters using the unrestricted hybrid density functional method B3LYP combined with the diffuse tripleplus polarization 6-311++G(2d,p) basis set to expand the molecular orbitals of chemically active atoms (the surface Si atoms, four Si atoms of the first subsurface layer, the carbon atoms, the nitrogen atoms, and the oxygen atoms), and the doubleplus polarization 6-31G(d) basis set to describe the remaining subsurface silicon atoms and terminating hydrogen atoms (Section 2.4). Usually, finding equilibrium geometries and transition structures requires application of unconstrained optimization on the PES. However, we imposed constraints

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33 on the hydrogen atoms that terminate broken Si-Si bonds in the clusters, to mimic the constraints that the rest of the solid should impose on the surface dimer under study. 2.5.1 Geometry Optimization of Energy Minima There are three types of geometry optimization algorithms: Methods that use only the energy, methods that use the first derivatives of the potential energy surface with respect to geometric parameters (i.e., gradient methods), and methods that require second derivatives (i.e., Newton or Newton-Raphson methods). The methods that only use the energy are the most widely applicable, but the slowest to converge, while the second derivative methods converge very fast but, given that analytic second derivatives are not readily available and more costly than the gradient methods. Thus, the gradient algorithms are the preferred methods, and (if analytical gradients are not available) it is usually efficient to calculate them numerically. In this work we performed quantum chemistry calculations with Gaussian03 [71] which uses the gradient method known as the Berny algorithm [ 72 73 ] (Figure 2-3) as the standard method for calculations of both geometry optimizations to a local minimum and transition state searches. Gradient methods, such as the Berny algorithm, approximate the potential energy surface as a quadratic function and calculate the energy with the expressions rHrrgrttEE210 (2-35a) rHgg0 (2-35b) where g is the gradient vector, and H is the Hessian matrix. Since for an energy minimum g = 0, the minimum structure can be obtained by solving linear equations Eq. 2-36a and Eq. 2-36b. 0rrr 00rHgg (2-36a)

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34 01gHr (2-36b) known as the Newton step. These first derivatives can be found analytically or numerically. The rate of convergence of the Berny algorithm depends predominantly on six factors: Initial geometry of the structure to be optimized Coordinate system chosen. Initial guess for the Hessian matrix. Hessian matrix updating method. Step size control. The initial geometry of the structure is a critical issue. It cannot be just any structure that resembles the molecule under study as it would imply that the optimization using a large basis set would require numerous unnecessary steps. Generally, molecular mechanics and semi-empirical methods are used to refine a raw structure and obtain a much better initial guess. In this work, we also ran a DFT/UB3LYP geometry optimization with the 3-21G* Pople basis set after the semi-empirical calculations to get as close as possible to the final minimum energy structure. Early geometry optimization used non-redundant internal coordinates (e.g., the Z matrix internal coordinates), but later it was shown that Cartesian coordinates and a combination of Cartesian and internal coordinates had some advantages for particular systems [74] Cartesian coordinates are the simplest and give an unambiguous representation of the structure, but they are also strongly coupled (i.e., to change a bond length one must change the x, y, and z positions of several atoms). Alternatively, internal coordinates (based on bond lengths, valence angles and dihedral angles) avoid problems with rigid body rotation s and translations. In internal coordinates, the coupling is much smaller than with Cartesian coordinates, and as a result the Hessian is more diagonal.

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35 (Appendix C reviews the protocol used to determine the initial coordinates for our calculations). The rate of convergence of a geometry optimization depends also on the initial guess of the nuclear Hessian since a closer guess to the actual Hessian will translate into a faster convergence. Computationally inexpensive empirical methods to generate initial nuclear Hessian have been proposed by Schlegel [75] and Fischer and Almlf [76] who also compared the effect of different coordinate systems on the geometry optimization. An initial empirical estimate of a diagonal Hessian can be quite satisfactory for redundant internal coordinates and can be readily transformed to other coordinate systems. However, Baker [ 77 78 ] showed that minimizations with Cartesian and redundant internal coordinates were similar if a good initial estimate of the Hessian is used (i.e., a molecular mechanics or semi-empirical Hessian instead of a unit Hessian). The Hessian from an optimization or a frequency calculation previously performed at a lower level of theory is often a very good initial guess for an optimization at a higher level. Some difficult optimizations may require an accurate initial Hessian computed analytically or numerically at the same level of theory as that used for the optimization. The default in Gaussian03 estimates bond stretching force constants from empirical rules, and obtains average angle-bending and torsion constants from vibrational spectra or theoretical calculations, using smaller and less expensive basis sets to calculate the force constants. Finally, since we are interested in constrained structures (given the rigidity that the silicon bulk would impose on the surface dimer chemistry) it is important to know that any components of the gradient vector corresponding to frozen variables are set to zero or projected out, thereby eliminating their direct contribution to the next optimization step.

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36 Figure 2-3. Flowcharts for a quasi-Newton algorithm for geometry optimization Proper update of the Hessian is essential for efficient optimizations, as the quality of the Hessian at each point is critical for the success of the optimization. However, the calculation of the exact Hessian at each point would make the process lengthy and costly. Instead, the Hessian is adjusted in the quadratic approximation represented by Eq. 2-36

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37 so that it fits the gradient g i at the current point r i and the gradient g i-1 at the previous point. This leads to Equation 2-37. grHi (2-37) In Eq. 2-37 and 1iirrr 1 iiggg There are numerous methods to update the Hessian, for example that of Broyden, Fletcher, Goldfab and Shanno (BFGS) [70] which can be written as expressed in Eq. 2-38. rHrHrrHgrggHH1111//ititittii (2-38) Eq. 2-38 is symmetric, positive definite, and minimizes the norm of the change in the Hessian. Schlegel et al. [74] indicate that most modern updating algorithms give similar results. Once there is a new dependable Hessian, a Newton step is taken on the model quadratic surface. The Hessian must be positive definite (i.e., all of its eigenvalues must be positive, for the step to be in the downhill direction). If the structure is far from the minimum (e.g., large gradients) or the potential energy surface is very flat (one or more small eigenvalues of the Hessian), then a simple Newton step may be too large, taking the molecule beyond the region where the model quadratic surface is valid. In this case, a shorter step must be taken. One can limit the step to be no larger than a trust radius which can be adjusted as the calculation proceeds, depending on whether the change in the predicted energy compares well with the actual calculated energy difference. In the Berny algorithm of Gaussian03 the trust radius is updated using the method of Fletcher [79]

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38 2.5.2 Transition State Searches A transition structure is the highest point on the reaction path that requires the least energy to get from the reactant A to the product B. In other words, it is a stationary point that is an energy maximum in one direction and a minimum in all others. For a point to be considered to be a transition state structure the first derivatives must be zero and the energy must be a maximum along the reaction path connecting the valley of reactants with the valley of products on the potential energy surface. The transition state structure must be a critical point of index one (i.e., one of the eigenvalues of the Hessian matrix must be negative and all others must be positive). Energy, structure and vibrational information for transition state structures are obtained from the transition state theory. As the case for minima, transition structures can be found by geometry optimization, although some modifications must be incorporated into the procedure. Initial estimates for transition structures are more difficult to obtain than for equilibrium structures since transition state geometries vary much more than equilibrium geometries. Moreover, molecular mechanics generally cannot handle transition states involving the making and breaking of bonds. As a result, when searching for transition states, many quasi-Newton methods need to start fairly close to the quadratic region of the transition state. Several techniques have been proposed to search for the initial structure of the transition state, including synchronous transit [80] and eigenvector following [81] Gaussian03 combines synchronous transit and quasi-Newton methods to find transition states [ 74 80 82 ]. This method requires two or three structures to start the optimization (one in the reactant valley, the second in the product valley, and an optional third structure as an initial guess for the TS geometry). If there is no third structure, the initial guess is obtained by interpolating between the reactants and products in redundant

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39 internal coordinates. The first few steps search for a maximum along an arc of circle connecting the reactant-like with the product-like structure, and a minimum in all other directions. The initial guess of the points along the reaction path is obtained by interpolating between two input structures; the energy and gradient are calculated at each point and an empirical estimate of the Hessian is obtained also at each point. The highest energy point on the path is chosen to be optimized to the closest TS, dividing the path into two downhill segments. In the remaining optimization steps, a quasi Newton based eigenvector following optimization is guided by the tangent to the arc of circle passing through the presumed TS and minima according to the implementation developed by Baker in 1986 [81] As in the geometry optimizations, the use of a certain coordinate system can improve the transition state search considerably. Ayala and Schlegel [73] showed that the combined used of redundant internal coordinates and the tangent to the quadratic synchronous transit (QST) path improves considerably the transition state optimization, mainly because a better Hessian is obtained in the first few steps and improves the search direction. These redundant internal coordinates are based on the work of Pulay et al.[ 83 84 ], who defined a natural internal coordinate system that minimizes the number of redundancies by using local pseudosymmetry coordinates about each atom and special coordinates for ring structures, and Peng et al. [74] who reduced the number of special cases of these coordinates by using a simpler set of internal coordinates composed of all bond lengths, bond angles and dihedral angles and applied it successfully for transitions state searches. Peng et al. [74] defined their coordinates as follows: First, the interatomic distances are examined to determine which atoms are bonded. Then, a bond angle bend coordinate is assigned for any two atoms bonded to the same third atom.

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40 Special attention must be given to linear bond angles; if the bond angle is greater than ~175, then two orthogonal linear angle bend coordinates are generated. Finally, a dihedral angle coordinate is assigned for each pair of atoms bonded to opposite ends of a bond. If one or both of the bond angles involved in a dihedral angle is linear, then the dihedral is omitted. In addition to the redundant internal coordinates generated automatically, extra stretch, bend and dihedral angle coordinates can be specified in the input. For regular transition state optimizations starting from one structure, the bonds being made or broken need to be specified in the input. Some difficult transition state searches may require an accurate initial Hessian computed analytically or numerically at the same level of theory as that used for the optimization instead of an updated one. Furthermore, Hessian update methods are suitable for finding minima for small and medium-sized molecules, but for difficult cases, such as some of the transition state structures investigated in this work, the Hessian has to be recalculated every few steps (or even every step) instead of being updated. This is equivalent to a Newton or Newton-Raphson algorithm. This is a very expensive method, and by default, Gaussian03 avoids it and updates the Hessian; however, its use has the advantage of obtaining an excellent description of the Hessian at each point of the calculation and also, since a vibrational frequency analysis is automatically done at the converged structure there is no need for an additional frequency job to determine the number of positive eigenvalues or zero point vibrational energies. The protocol that we used for transition state searches is presented in Appendix D.

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CHAPTER 3 NITROGEN ATOM ABSTRACTION FROM Si(100)-(2x1) 3.1 Introduction The reactions of gas-phase radicals at solid surfaces are fundamental to the plasma-assisted processing of semiconductor materials. In addition to adsorbing efficiently, radicals incident from the gas-phase can also stimulate several types of elementary processes before thermally accommodating to the surface, including direct-atom abstraction and collision-induced reaction and desorption. Direct-atom abstraction can occur by an Eley-Rideal mechanism in which an atom is abstracted from the surface in a single collision with an incident species [11] or by a hot atom mechanism in which the incident species experiences multiple collisions with the surface but does not fully thermalize before the reactive encounter [13] Indeed, non-thermal surface reactions such as these play a critical role in determining the enhanced surface reactivity afforded by plasma processing. Advancing the fundamental understanding of radical-surface reactions is therefore of considerable importance to improving control in plasma-assisted materials processing in addition to being of scientific interest. In the present study, we have used quantum chemical calculations to investigate the interactions of an oxygen atom with nitrogen atoms incorporated into the Si(100)-(2x1) surface, focusing on pathways that lead to direct nitrogen abstraction from the surface by the formation of gaseous NO. This investigation is motivated firstly by an interest in determining the viability of direct nitrogen abstraction from Si(100) by gas-phase oxygen atoms, and to gain insights into the possible pathways for this surface reaction. It is 41

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42 further motivated by the potential benefits that may be realized by incorporating ultrathin silicon oxynitride films into metal-oxide-semiconductor (MOS) devices and the need to precisely control the properties of such films. Over the past several years, ultrathin silicon oxynitride (SiO x N y ) films have been incorporated into MOS devices as an alternative for SiO 2 which is no longer suitable as an insulator for MOS devices because of the decreasing dimensions of microelectronic devices imposed by Moores law [ 33 85 ]. Incorporating nitrogen into SiO 2 is a relatively simple method for fabricating silicon oxynitride films, and results in dielectric layers that enhance resistance to gate current leakage and inhibit boron penetration into the dielectric [86] However, as the sizes of semiconductor devices continue to decrease, it is becoming critical to develop processing methods that afford control of film properties and composition at the monolayer level. Low-temperature remote plasma processing offers distinct advantages over other ultrathin silicon oxynitride film preparation methods [33] For example, researchers [ 87 88 ] have reported that the concentration profiles of nitrogen and oxygen atoms within SiO x N y films grown by remote plasma processing on Si(100) can vary significantly with both the feed gas composition and processing protocol that is used. Of particular interest is the observation that the exposure of Si(100) to an N 2 O or N 2 /O 2 plasma produces a film with a highly heterogeneous composition profile in which the nitrogen atoms accumulate at the film-Si interface and the oxygen atoms remain close to the film-vacuum interface. It was suggested that this composition profile results, at least in part, by chemical reactions in which gaseous oxygen atoms efficiently scavenge nitrogen atoms located closest to the film-vacuum interface. In a separate study, Watanabe and Tatsumi [89] also observed a decrease in surface nitrogen concentration after exposing a nitrided Si(100) surface to an oxygen plasma, and asserted that nitrogen atoms are removed from

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43 the surface through reactions with gaseous oxygen atoms. While thermally activated reactions between accommodated oxygen and nitrogen atoms could have taken place at the surface temperature of 750 C used in the work of Watanabe and Tatsumi, the other investigations cited were conducted with the substrate held at 300 C, which is well below the temperature for appreciable thermal decomposition of silicon oxide [30] and nitride surfaces [ 90 91 ]. It is therefore likely that the oxygen-induced removal of nitrogen from the surfaces of these films occurs by non-thermal processes such as Eley-Rideal abstraction. In the present study, we find that direct nitrogen abstraction by gaseous oxygen atoms is energetically favorable when nitrogen is bound at the Si(100) surface in coordinatively unsaturated configurations, but less so when the nitrogen is triply coordinated with surrounding silicon atoms. 3.2 Computational Approach Quantum chemical calculations of oxygen induced abstraction of a nitrogen atom from the Si(100)-(2x1) surface were performed using density functional theory (DFT) and cluster models of the surface. Cluster energies were computed using the unrestricted, hybrid three-parameter Becke method (UB3LYP), which combines the gradient-corrected exchange functional of Becke [ 49 53 ] with the Lee-Yang-Parr (LYP) correlation functional [50] An unrestricted approach was used for all calculations because the reactants, products and transition states are open shell structures. Geometry optimization methods based on the Berny algorithm were used for calculating local minima [ 73 74 ], while transition state structures were obtained using a multiple step procedure. In this approach, an initial transition structure is obtained using the quadratic synchronous transit-guided quasi-Newton method (QSTN) [ 74 82 ]. A frequency analysis is then performed to determine the normal mode of vibration that has the largest imaginary

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44 frequency. In the final step, this normal mode is selected to search for a transition structure using an eigenvector following method. This extensive approach was found to be more robust than the standard search routines available in the Gaussian03 package for identifying transition structures for the open shell systems we investigated. After convergence, a final frequency analysis was performed to confirm that the local minima and transition structures were zero or first order saddle points, respectively [92] Figure 3-1 shows the Si 9 H 12 one-dimer cluster that was used to model the Si(100)-(2x1) surface in these calculations. This cluster model has been widely used to study chemical reactions on Si(100)-(2x1) using DFT [ 10 93 101 ], mainly because it is the smallest structure that adequately represents the main structural characteristics of the Si(100)-(2x1) surface such as the tilted silicon dimer bond and well-oriented sp 3 covalent bonds. DFT calculations using the one-dimer cluster have been found to accurately predict bond energies and barriers for several reactions on Si(100)-(2x1) [ 10 99 ] for which non-local electronic effects [98] are of minor importance. Hydrogen atoms are used to terminate the bonds of the Si cluster (i.e., hydrogen atoms are used as substituents for the bulk silicon atoms that are removed by truncation of Si-Si bonds at the exterior of the cluster). These H atoms preserve the tetrahedral sp 3 bonding environment of subsurface Si atoms, mimic strain that the bulk silicon atoms would impose on the boundary of the cluster, and generally have a negligible effect on the quantum chemistry calculation itself [ 99 102 ]. Geometric constraints are imposed on the hydrogen atoms to improve the simulation of bulk strain effects as follows: All H atoms in place of bulk Si atoms are held fixed in their ideal tetrahedral configurations, while those replacing silicon atoms in neighboring dimers are fixed in positions that mimic the buckled dimer structure [ 96 99 ]. The third and fourth layer Si atoms are not directly constrained, although their

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45 displacements are hindered by the constraints imposed on their neighbors and terminating hydrogen atoms; the fourth layer Si atom is practically held fixed in space due to the limited displacement of the third layer silicon atoms. Finally, all chemically active atoms, which include the silicon dimer atoms, the second layer Si atoms, and the nitrogen and oxygen atoms, are allowed to fully relax during the geometry optimizations. To reduce the computational expense of the calculations, a mixed basis set was used to expand the electronic wave function. A diffuse tripleplus polarization 6-311++G(2d,p) basis set was used to describe the chemically active atoms while the remaining subsurface silicon atoms and terminating hydrogen atoms are described with a doubleplus polarization 6-31G(d) basis set [92] All the structures investigated have a spin doublet multiplicity and the calculations were done using the Gaussian03 program [71] 3.3 Results 3.3.1 Bonding Configurations of a Nitrogen Atom on Si(100)-(2x1) We investigated the pathways for nitrogen abstraction from four different configurations of a nitrogen atom bonded on the Si(100)-(2x1) surface to explore the influence of the local surface bonding environment on these reactions. The N-Si(100) structures that we examined were recently predicted by Widjaja et al. [98] to be possible equilibrium structures resulting from incorporating a single nitrogen atom into the silicon surface. To benchmark our calculations, we optimized the N-Si(100) structures using the same computational procedure as used by those authors. The structures and the corresponding energies of formation that we predicted are shown in Figure 3-2, where the zero of energy is taken to be the energy of the nitrogen atom and silicon cluster at infinite separation. Each structure has a spin multiplicity of

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46 two. The lowest energy structure, which has an energy of formation of 2.69 eV, is obtained when the nitrogen atom bonds to a single dimer atom [R(ad)]. The remaining structures are the N atom bonded with both dimer atoms in an epoxide-like structure [R(db)], the N atom inserted into a Si-Si backbond [R(bb)] and finally the N atom bonded with one dimer atom and two second layer Si atoms [R(sat)]. The geometrical properties of these structures are in excellent agreement with those reported by Widjaja et al. [98] and the energies of structures R(ad), R(db) and R(sat) differ by less than 0.11 eV from their results. However, the energy of formation of structure R(bb) is found to be higher by 0.38 eV from the prediction of Widjaja et al. The differences in the energies predicted in these studies most likely arise from slight differences in the way geometric constraints are imposed in the calculations. It is noted that the energy of structure R(bb) is particularly sensitive to the positions of the terminating hydrogen atoms. Despite this difference, the geometries and trends in the relative energies of the N-Si(100) structures are very close to those reported in the study of Widjaja et al. [98] Finally, each of the structures (Figure 3-2) is predicted to be a local minimum on the doublet N-Si(100) potential energy surface, and the lowest barrier for interconversion between the structures is 0.48 eV [98] These characteristics suggest that at moderate surface temperature each structure could exist in appreciable concentrations during the initial stages of nitridation. 3.3.2 Nitrogen Abstraction by a Gas-Phase Oxygen Atom We investigated reactions between species only in their respective electronic ground-states. All of the reactions occur on a doublet potential energy surface and

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47 Figure 3-1. The Si 9 H 12 cluster used in our UB3LYP calculations. Si atoms are represented by dark spheres and H atoms are shown as light-colored spheres. may be represented by the general equation, O( 3 P) + N-Si(100) (M s = 2) NO( 2 ) + Si(100) (M s = 1), where M s is the initial spin multiplicity of the surface cluster. For each of the N-Si(100) structures investigated, nitrogen abstraction by a gas-phase O-atom was found to be highly exothermic. This conclusion is reached by considering that the bond energy of NO in its doublet ground-state is about 6.5 eV, whereas the energies of formation of the N-Si(100) structures range from about 2.7 to 4.7 eV (Figure 3-2). We initially explored direct pathways for abstraction in which the N-O bond forms and the Si-N bonds break in a single elementary step, as depicted by the reaction equation shown above. Extensive searches revealed a transition structure for only one single-step abstraction process, namely, direct abstraction from the R(sat) structure. For structures R(ad), R(db) and R(bb), the abstraction pathways exhibit energy wells between the reactants and products due to the formation and interconversion of adsorbed NO species. These molecular precursor structures are produced when the O( 3 P) atom attaches directly to a nitrogen dangling bond, yielding an NO species that is bound to the surface.

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48 Figure 3-2. Structural information of N-Si 9 H 12 clusters. A) R(ad). B) R(db). C) R(bb). D) R(sat). All distances are in Terminating hydrogen atoms have been excluded for clarity. Si atoms are shown as light-colored spheres and N atoms as darker spheres. 3.3.3 Abstraction of N Adsorbed at the Dangling Bond [R(ad)] Figure 3-3 shows the predicted pathway by which an oxygen atom abstracts a nitrogen atom that is adsorbed on a single Si atom of the surface dimer [R(ad)]. The reaction involves the initial formation of molecular precursor MP(ad) (Figure 3-4B), which then decomposes to produce a gaseous NO molecule and the bare Si cluster, structure P (Figure 3-4C). The production of MP(ad) is highly exothermic (4.84 eV), since this reaction involves the formation of a strong N-O double bond without cleavage of other bonds in the cluster. The subsequent decomposition of MP(ad) via the formation of a gaseous NO molecule is endothermic by 0.91 eV. A transition structure could not be

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49 found along the path from structure MP(ad) to P so only the thermochemical barrier must be overcome in this step. Although the molecular precursor lies at a lower energy than the abstraction product, a minimum of 3.93 eV of energy would need to be dissipated to the solid for the molecular precursor to stabilize. Moreover, since the NO species couples to the solid mainly through the Si-N bond, it is likely that the Si-N bond Figure 3-3. Reaction pathway for the nitrogen abstraction from R(ad). Energies are in units of eV and the zero of energy is taken to be the ground-state nitrogen atom, the O( 3 P) atom and the singlet Si cluster at infinite separation.

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50 will become highly excited and break during N-O bond formation since only 0.91 eV are needed to detach the NO molecule from the structure. This result therefore suggests that nitrogen abstraction from structure R(ad) will be highly facile, occurring effectively in a single atom-surface collision. Figure 3-4. Critical point structures of the nitrogen abstraction from R(ad). A) R(ad) B) MP(ad). C) P (final product). All distances are in and the terminating hydrogen atoms have been excluded for clarity. The geometric and electronic changes that occur during the reaction shown in Figure 3-3 provide additional insights for understanding this reaction. In reactant R(ad), two unpaired alpha electrons are localized on the N atom, and one beta electron resides mainly on the triply coordinated Si atom of the surface dimer. Since the O( 3 P) atom also

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51 has two unpaired electrons, its interaction with the adsorbed N atom results in the highly exothermic formation of an NO double bond. The N-O bond in structure MP(ad) is close in length to that of the N-O bond of an isolated NO molecule (1.20 vs. 1.15 respectively, Figure 3-4). The predominant change in the relative positions of the Si and N atoms that occurs during the first reaction step is elongation of the Si-N bond from 1.75 to 1.91 which indicates a weakening of the Si-N bond when the N-O double bond of MP(ad) is formed. In the final reaction step, the molecular precursor decomposes by cleavage of the Si-N bond, with one electron transferring to a orbital of the NO molecule and the other remaining at the surface. This remaining electron experiences a weak -interaction with the initially unpaired electron on the opposing dimer atom since the bare Si surface in these calculations is taken to be the singlet, buckled dimer structure that is thought by many to be the ground-state of the Si(100)-(2x1) surface [ 103 104 ]. 3.3.4 Abstraction of the Nitrogen Bonded Across the Dimer [R(db)] Shown in Figure 3-5 is the pathway predicted for the abstraction of a nitrogen atom bonded across the surface dimer [R(db), Figure 3-6(A)]. The first step in this pathway is N-O bond formation, resulting in structure MP(db). The exothermicity of this reaction is 3 eV, which is quite significant but is still lower by 1.84 eV than N-O bond formation in the analogous reaction R(ab) MP(ab) (Figure 3-3). Formation of structure MP(db) is followed by NO migration from the bridging site to the dangling bond site from which NO desorbs. The migration path is predicted to have an energy barrier of 0.48 eV, and the final desorption step presents a thermochemical barrier of 0.91 eV as discussed. Each of these barriers is significantly less than the energy released in the initial formation of the N-O bond. Thus, unless energy is efficiently dissipated away from the initial collision zone, NO desorption can occur rapidly by this pathway as well.

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52 Figure 3-5. Reaction pathway of nitrogen abstraction from R(db). Energies expressed are in eV. The geometric changes that the cluster undergoes during these reactions steps are illustrated in Figure 3-6. N-O bond formation to produce MP(db) causes each Si-N bond to stretch from 1.68 to 1.80 and the Si-Si dimer bond to contract from 2.55 to 2.37 which is indicative of weaker Si-N bonds and a stronger Si dimer bond. Also, the N-O bond in MP(db) is longer than that in MP(ad) (1.27 vs. 1.20 ), suggesting a weaker N-O bond in the former. These changes may be understood by considering the corresponding changes in the alpha spin densities on the surface atoms. In the initial reactant R(db), an unpaired electron is distributed mainly between the Si atoms of the epoxide-like ring. The alpha spin density on these Si atoms decreases to zero upon formation of molecular precursor MP(db), and an unpaired electron is distributed almost evenly between the N

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53 Figure 3-6. Critical point structures of the nitrogen abstraction from R(db). Structural information and relative energy. A) R(db). B) MP(db). C) TS(db-ad). D) MP(ad). All distances are in The terminating hydrogen atoms were excluded for clarity. The final product is shown in Figure 3-4(C). and O atoms in the MP(db) structure. These changes indicate that NO bond formation leading to MP(db) involves a pairing between the lone electron on the reactant R(db) and an unpaired electron on the incident O( 3 P) atom. The second lone electron on the O( 3 P) atom is transferred to a *-like orbital between the N and O atoms. The resulting N-O bond effectively has a bond order of 1.5 and is therefore weaker and longer than the NO double bond of MP(ad). The migration of the NO species from the bridging to the dangling bond position is activated by 0.48 eV, since it involves the simultaneous cleavage of a Si-N bond and the

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54 formation of an NO double bond. In this reaction, an electron from the initial Si-N bond is transferred to the NO species to form the double bond, while the second electron remains localized on the Si atom. The transition structure TS(db-ad) for this reaction is shown in Figure 3-6(C). To reach TS(db-ad) from MP(db), the NO species tilts away from one of the Si dimer atoms, causing the Si-N bond to stretch to 2.44 At TS(db-ad), the NO bond contracted to 1.21 and the remaining Si-N bond is elongated from 1.68 to 1.85 On the path from TS(db-ad) to MP(ad), the structure adopts a more stable configuration as the NO species tilts farther away from the opposing Si dimer atom, and the N-O bond contracts to it final value of 1.20 3.3.5 Abstraction of the Nitrogen Bonded at a Backbond [R(bb)] We found two pathways for nitrogen abstraction from the backbonded position [structure R(bb)] (Figure 3-7). For each path, the oxygen atom is predicted to first bond directly with the nitrogen atom of structure R(bb) to generate molecular precursor MP(bbs) (Figure 3-8B) in which the N-O bond is nearly parallel to the bisector of the Si-N-Si bond angle. In one pathway, the O atom tilts toward the Si atom to form an Si-O bond resulting in structure MP(bbt) which can then decompose by sequential N-Si bond cleavage, forming MP(ad) and then a gaseous NO molecule. The alternative pathway involves direct N-Si bond cleavage, causing the symmetric structure MP(bbs) to convert directly to MP(ad) from which the NO molecule detaches (Figure 3-7, inset). The reaction to produce MP(bbs) from the nitrogen backbonded structure is exothermic by 2.68 eV, and only a small energy barrier of 0.03 eV must be overcome for MP(bbs) to transform into the more stable molecular precursor MP(bbt). An energy barrier of 0.38 eV must be overcome for MP(bbt) to transform to MP(ad), while a barrier of 0.55 eV is predicted for the direct conversion of MP(bbs) to the MP(ad) structure. The

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55 overall reaction is exothermic by 2.25 eV, and the energy released during the initial N-O bond formation is much greater than the barriers that must be surmounted for the NO molecule to desorb from the cluster. The general bonding characteristics of the MP(bbs) structure (Figure 3-8B) are similar to those of the MP(db) structure (Figure 3-6B) discussed above. In particular, formation of the N-O bond in MP(bbs) involves a pairing between lone electrons on the N and O atoms, with the second lone electron of the O atom transferring to a *-like orbital localized along the N-O bond. Despite these similarities, however, structure MP(bbs) is less energetically favorable than MP(db) by 0.61 eV. A comparison of the structures (Figures 3-6B and 3-8B) suggests that MP(bbs) is more strained since its formation is accompanied by an increase of the Si-N-Si bond angle and elongation of the Si-N bonds. The N-O bond in MP(bbs) is also longer than in MP(db), which suggests that additional strain would be imposed if the N-O bond were to achieve its optimum length. One of the pathways for N abstraction involves the conversion of structure MP(bbs) to MP(bbt) (Figure 3-8D), which is exothermic by 0.25 eV and presents a barrier of only 0.03 eV (Figure 3-7). Although the energy barrier is quite small for this reaction, vibrational analysis indicates that structures MP(bbs) and TS(bbs-bbt) do correspond to zero and first order saddle points, respectively. The exothermicity for this reaction step is also relatively small mainly because unfavorable structural changes offset the energy gained in forming the strong Si-O bond. The predominant structural changes include stretching of the N-O bond from 1.32 to 1.51 and an increase of the Si-N-Si bond angle from 132.6 to 164.5. Elongation of the N-O bond indicates a weakening of this bond, and the substantial increase in the

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56 Figure 3-7. Reaction pathways for the abstraction of the nitrogen atom inserted in a Si-Si backbond. Main panel: O + R(bb) MP(bbs) TS(bbs-bbt) MP(bbt) TS(bbt-ad) MP(ad) P. Top inset: O + R(bb) MP(bbs) TS(bbs-ad) MP(ad) P. Energies are in units of eV. Si-N-Si bond angle imparts strain on the Si-Si bonds in underlying layers. Energy is also required for this reaction because an electron must be removed from a -like bond along the Si surface dimer for the Si-O bond to form. The remaining steps in this abstraction pathway include migration of the NO species of structure MP(bbt) to the dangling bond position, and then NO desorption from MP(ad). The migration reaction MP(bbt)

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57 MP(ad) is exothermic by 0.23 eV and has an energy barrier of 0.38 eV (Figure 3-7). The geometry of the transition structure TS(bbt-ad) (Figure 3-9B) reveals that the NO species tilts away from the Si dimer atom, causing the Si-O bond to break and the N-O bond to strengthen and contract early along the reaction path. Concurrently, the Si-N-Si bond angle decreases to allow the Si-Si backbond to begin forming and the lower Si-N bond to break. Formation of the Si-Si backbond and the N-O bond is completed on the MP(ad) side of the barrier. In the second abstraction pathway for N(bb), the symmetric MP(bbs) structure converts directly to MP(ad) by the path shown in the inset of Figure 3-7. This reaction step is exothermic by 0.48 eV and has an energy barrier of 0.55 eV. The corresponding transition structure TS(bbs-ad) for this path is shown in Figure 3-10(B). The bond Figure 3-8. Molecular precursors formed after O-chemisorption onto R(bb). Structural information of the structures involved in the reaction pathway. A) R(bb). B) MP(bbs). C) TS(bbs-bbt). D) MP(bbt). All distances are in The terminating hydrogen atoms were excluded for clarity.

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58 lengths in TS(bbs-ad) are nearly identical to those in TS(bbt-ad) (Figure 3-9B), which is not surprising since the reactions that proceed through these transition states have a common product. Nevertheless, the energies of these transition structures, referenced to the same initial state, differ by 0.42 eV. The most striking structural difference is that in TS(bbt-ad) the NO bond is nearly parallel with the plane defined by the Si-N-Si ring, whereas the NO bond is significantly tilted out of this plane in TS(bbs-ad). This suggests that overlap of the -like orbitals in the NO species with those in the Si-N-Si ring is enhanced in the planar configuration of TS(bbt-ad), thereby lowering the energy of the structure relative to that of TS(bbs-ad). Figure 3-9. Structures formed during nitrogen abstraction from MP(bbt). A) MP(bbt). B) TS(bbt-ad). C) MP(ad). D) P. All distances are in

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59 Figure 3-10. Structures involved in the nitrogen abstraction from MP(bbs). A) MP(bbs). B) TS(bbs-ad). C) MP(ad). D) P. All distances are in 3.3.6 Abstraction from the NSi 3 Structure [R(sat)] Figure 3-11 shows two possible pathways for N abstraction from the NSi 3 structure [R(sat), Figure 3-12(A)] by an incoming O( 3 P) atom, and one pathway for the adsorption of the O atom at a Si dangling bond site. The first abstraction pathway is a direct process wherein the incident O atom removes the N atom from the surface in a single step. This reaction is the only single-step abstraction process that was identified in the study. Direct N abstraction from the NSi 3 structure is predicted to be exothermic by 2.57 eV and presents a relatively small energy barrier of 0.20 eV. The N-O separation at the transition structure is 1.23 (TS(sat-P) (Figure 3-12B), which is only 0.12 longer

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60 than the NO bond of the isolated NO molecule. In addition, each second-layer Si-N bond has broken and two Si-Si backbonds have formed on the path to the transition structure. This indicates that direct abstraction from NSi 3 is a concerted process in which three Si-N bonds break while two Si-Si backbonds and a N-O bond simultaneously form during a single gas-surface collision. This reaction would require the upper Si dimer atom to move toward the second layer Si atoms by a considerable distance, and the N atom to simultaneously move out of the plane defined by these three Si atoms as the O atom approaches the cluster and the N-O bond begins to form. The other possible pathway for N abstraction from the NSi 3 structure is also a concerted process in which the interaction of the O-atom with the cluster produces molecular precursor MP(bbs) (Figure 3-8B), which then decomposes by the pathways shown in Figure 3-7. However, neither an intermediate structure nor a transition structure could be found for the reaction O( 3 P) + R(sat) MP(bbs). Thus, for this reaction to occur, an N-O bond and a Si-Si backbond must form, and a Si-N bond must break during a single collision of the O-atom with the cluster. Although neither of these abstraction pathways is energetically prohibitive, they each involve the formation and scission of multiple bonds in a single gas-surface collision. Moreover, these reactions require that the O-atom come in close proximity to the dangling bond of the Si dimer atom where it could adsorb. Indeed, adsorption of the O-atom at the dangling bond site (Figure 3-11) is predicted to be barrierless and highly exothermic (6.35 eV), and is therefore concluded that to be the predominant outcome of the interaction between an O( 3 P) atom and the NSi 3 structure. As may be seen in Figure 3-12, adsorption of an O( 3 P) atom at the Si dangling bond causes the local NSi 3 configuration to change negligibly, but it does result in contraction of the Si-Si dimer

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61 Figure 3-11. Reaction pathways for N-abstraction from (and O-atom adsorption on) R(sat). Energies are in units of eV and the zero of energy is taken to be the ground-state nitrogen atom, the O( 3 P) atom and the singlet Si cluster at infinite separation. bond. This contraction indicates that O-Si formation alleviates repulsive interactions between the two unpaired electrons that are present on the silicon dimer atoms of the NSi 3 structure. In fact, we find that the energy of O-atom adsorption at the Si dangling bond site of R(sat) is 1.9 eV higher than that for O-adsorption on the clean Si(100)-(2x1) surface. This indicates that significant electronic repulsion exists along the dimer of the R(sat) structure, and that O-atom adsorption is highly favorable on this surface.

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62 Figure 3-12. Structures formed during the direct nitrogen abstraction from R(sat). A) O( 3 P) attacks the nitrogen atom of R(sat). B) Transition structure [TS(sat-P)]. C) NO molecule desorbed into the gas phase and bare Si surface (P). All distances are in The terminating hydrogen atoms were excluded for clarity. Figure 3-13. Structures involved in the O-atom chemisorption on R(sat). Structural information and relative energy. A) O( 3 P) atom approaching structure R(sat) and B) O-atom adsorbed at the Si dangling bond site [O(ad) R(sat)] All distances are in The terminating hydrogen atoms were excluded for clarity.

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63 3.4 Discussion Nitrogen abstraction by a gas-phase O( 3 P) atom is highly exothermic for each of the N-Si(100) structures investigated in this study. However, abstraction is predicted to occur in a single step only for the reaction of O( 3 P) with the coordinatively saturated N-atom of the NSi 3 structure, and (in this case) single-step abstraction appears to be much less probable than O adsorption due to the energetic differences in these reactions and because multiple bonds must break and form for single-step abstraction to occur. For each of the coordinatively unsaturated N-Si(100) structures, N abstraction is predicted to occur by a precursor-mediated pathway that is initiated by the formation of an N-O bond and the release of between 2.7 and 4.8 eV into the surface. Since the subsequent elementary steps leading to NO desorption have barriers that are less than 1.0 eV, NO bond formation provides the system with excess energy that could readily promote local bond rearrangements and ultimately NO desorption. Indeed, nitrogen abstraction by such a pathway would effectively be an Eley-Rideal process since the NO product would evolve into the gas-phase within no more than a few vibrational periods after the initial gas-surface collision. Alternatively, the energy released during NO bond formation could be dissipated away from the surface bonds, allowing the adsorbed NO species to equilibrate in one of the configurations predicted. However, considering the significant amount of energy (~2 to 4 eV) that would need to be channeled away from the initial collision zone for a precursor to stabilize, it is reasonable to expect NO desorption to be the more likely outcome of initial N-O bond formation in these systems. Nevertheless, calculations of the dynamics for abstraction, and particularly the efficiency of energy dissipation to the solid, are needed to explore the propensity for O( 3 P) atoms to abstract nitrogen from Si(100) by the pathways predicted here.

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64 Prior experimental studies have provided indirect evidence for nitrogen abstraction from Si surfaces by gaseous atomic oxygen. For example, exposure of Si(100) to plasmas containing both O and N atoms results in a surface that is depleted of nitrogen [ 87 88 105 ]. Similarly, treating nitrided Si(100) with an oxygen plasma has been reported to lower the surface nitrogen concentration [ 89 106 ]. We recently conducted reactive scattering experiments in UHV to directly examine the abstraction of nitrogen from Si(100) by an atomic oxygen beam, but we did not observe the evolution of gaseous NO using mass spectrometry or nitrogen depletion from the surface [107] However, the absence of measurable nitrogen abstraction in our experiments can be attributed to the bonding state of nitrogen that was investigated. In that work, nitrogen was incorporated into Si(100) by thermally decomposing NH 3 on the surface at a substrate temperature of 900 K; adsorption at this high surface temperature is necessary to ensure the complete desorption of hydrogen. Above about 700 K, however, nitrogen has been found to diffuse into the subsurface region of Si(100) [108] and is apparently inaccessible for direct abstraction by gaseous O-atoms. Our experimental observations therefore suggest that the N-Si(100) structures considered in the present computational study are not stable at temperatures greater than about 700 K. In fact, recent quantum chemical investigations predict that a single nitrogen atom does have an energetic preference to bond in the subsurface of Si(100) [ 109 110 ]. Nevertheless, the N-Si(100) structures investigated in the present study, which were reported originally by Widjaja et al. [98] may be stable at surface temperatures lower than 700 K since, to our knowledge, barriers for the migration of nitrogen into the Si(100) subsurface have not been reported, and may be high enough to enable nitrogen atoms to stabilize at the surface at low to moderate temperatures. Hence, nitrogen

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65 abstraction by gaseous O-atoms may indeed occur during plasma-enhanced oxynitridation of Si(100) if the surface temperature is maintained sufficiently low. Another possibility is that the barriers for nitrogen diffusion to the subsurface are small, which would imply that the N-Si(100) structures that we investigated in this computational study (Figure 3-2) do not exist in appreciable concentrations at low to moderate surface temperature. In this case, a mechanism other than direct abstraction by gaseous oxygen atoms would be needed to explain observations of nitrogen depletion at the Si(100) surface by reaction with oxygen plasmas. Experiments to directly investigate nitrogen abstraction by atomic oxygen will require a method for adsorbing N atoms onto Si(100) at low surface temperature, which should be possible using an active nitrogen source such as gaseous N-atoms or through non-thermal activation of nitrogen-containing adsorbates.

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CHAPTER 4 CHEMISTRY ON Si(100)-(2x1) DURING EARLY STAGES OF OXIDATION WITH O( 3 P) 4.1 Introduction Oxygen adsorption on Si(100)-(2x1) is an important process because it corresponds to the initial stages of the formation of the Si/SiO 2 interface, a system of high technological relevance in microelectronic devices, especially in the metal-oxide-semiconductor field-effect transistor (MOSFET) gate dielectrics. Because of the demands imposed by Moores law, the thickness of the silicon dioxide gate dielectrics grown on Si(100) is crucial for the future of the microelectronic industry. As these layers approach the sub nanometer region (it is now possible to fabricate devices with SiO 2 gate thickness of 1.3 nm [34] which correspond to about ten silicon atoms across the Si/SiO 2 interface) it is important to understand the detailed mechanism of initial oxidation of silicon surfaces. This small thickness is a limit for the use of SiO 2 gate dielectrics given that the concentration of undesirable silicon suboxides (Si + Si 2+ and Si 3+ ) in the Si/SiO 2 interface becomes increasingly greater. Because of this, aggressive research for alternative gate dielectric materials is taking place, but even for these new gate highdielectric materials, knowledge of the initial steps of SiO 2 formation is critical given that many of them still involve one or two monolayers of SiO 2 deposited over the gate channel region. Several methods to fabricate silicon dioxide have been used successfully over the year [ 33 34 ]. However, because of the continuous scaling of the devices, these 66

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67 traditional methods are becoming increasingly less efficient. Furthermore, as the size of the silicon dioxide layers approach molecular and atomic dimensions, it would be ideal to have a fabrication method based on molecular chemistry and low temperature, so the product obtained by the manufacturing process remains in a given configuration and is not affected by thermal changes of the system. One of these methods is plasma processing of silicon substrates. The interactions of radical species such as oxygen atoms, which can be present in large fractions in plasma environment, with silicon surfaces are relatively poorly understood. Engel and coworkers [ 26 31 ] reported detailed kinetics studies of the O atom oxidation of clean silicon using a plasma-based atomic beam source. Also, Yasuda et al. [111] studied the O atom oxidation of hydrogen-terminated silicon using a hot-filament source. These studies revealed that O atom oxidation is favorable compared with that due to O 2 and that several monolayers of oxide can be formed efficiently via the direct insertion of O atoms into near-surface bonds. We explore the structures formed during the initial steps of O( 3 P) incorporation on clean Si(100)-(2x1) using small Si 9 H 12 cluster models and gradient-corrected density functional theory (DFT). From the time when early studies of asymmetric dimer descriptions [112] and initial adsorption of hydrogen on Si(100) [94] were made, the Si 9 H 12 cluster has been proven a very useful tool for performing theoretical studies of local chemistry on Si(100)-(2x1). Recent comprehensive oxidation studies which combine Si 9 H 12 clusters with gradient-corrected density functional theory (DFT) techniques yielded accurate energetics and vibrational frequencies for several oxidation products, although none of them has treated oxidation by O( 3 P) specifically. Most of previous work has focused on the water-induced oxidation of the clean Si(100)-(2)

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68 surface [ 34 95 96 104 113 114 ], and some on oxidation by molecular oxygen [100] What has been found so far using the DFT/cluster approach can be summarized as follows: Initially, water adsorbs dissociatively on top of single silicon dimer and forms H and OH fragments which are most stable when they are bonded with the hydroxyl oriented away from the surface dimer bond [ 95 104 ]. Thermodynamic [96] and mechanistic studies [ 113 114 ] revealed a strong tendency for O to agglomerate on the dimers of Si(100) and predicted that the threeand five-oxygen agglomerated structures were the most stable. Formation of epoxidelike rings upon dehydrogenation of the surface in the event of oxygen agglomeration was also reported. This tendency for O to agglomerate was found to be thermodynamically driven since it is energetically more favorable to have one dimer with n oxygens, for n=2, and (n1) oxygen-free dimers, than it is to have n dimers each with one oxygen. In this work, we performed DFT/cluster calculations to determine minimum energy geometries to elucidate the relative thermodynamic energies of the possible oxidation products that can form by insertion of up to three O( 3 P) atoms, and we use this result to predict a minimum energy reaction path. These oxidized Si 9 H 12 clusters were classified in three different sets of isomers (O 1 Si 9 H 12 O 2 Si 9 H 12 or O 3 Si 9 H 12 ) depending on the number of oxygen atoms adsorbed on the surface. The energy of these isomers was then compared to one another to determine how it was affected by the formation of the two SiO bonds in place of one SiSi bond, by the oxidation state of the surface silicon atoms, by the spin-state of the surface (spin-singlet vs. spin-triplet) and by strain effects (all quantified by using a bond energy model that assumes that they are independent and additive). Finally, we investigated transition states for the insertion process, to develop a model for the preferred mechanism of the initial steps of Si(100) oxidation by O( 3 P).

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69 4.2 Theoretical Approach Our theoretical approach was based on Kohn-Sham density functional theory (DFT) calculations of clusters of silicon that represent the local bonding arrangement of the surface. The silicon cluster used was the Si 9 H 12 (Figure 3-1) which is the smallest structure that appropriately represents the main structural characteristics of the Si(100)-(2x1) surface (i.e., covalent tetrahedral sp 3 arrangement of Si-Si bonds and a tilted dimer). This cluster has twelve H atoms to terminate all dangling bonds resulting from truncation of Si-Si bonds at the exterior of the cluster; these terminating hydrogen atoms preserve the tetrahedral bonding and have a negligible effect on the predicted energies of the different clusters [ 102 115 ]. The clusters were constrained by imposing boundary conditions that mimic the strain that the bulk silicon would impose on the surface dimer under study. In particular, the hydrogen atoms were fixed in their positions along the directions of truncated Si-Si bonds that they terminate. Two types of hydrogen atoms can be found in this cluster. The bulk hydrogen atoms, fixed along tetrahedral directions, and the neighboring dimer hydrogen atoms, constrained in positions that mimic nearest silicon dimers. The third and fourth layer silicon atoms were not directly constrained, but the displacement of these atoms is minimal because they are surrounded by fixed H atoms. All the chemically active atoms including the first and second layer silicon atoms were allowed to relax completely unconstrained. Kohn-Sham density functional theory [ 41 44 ] is used for the electronic structures calculations. Specifically, we use the B3LYP hybrid-gradient-corrected method [ 53 116 ] which calculates the exchange correlation term by means of a linear combination of local, gradient-corrected and exact Hartree-Fock exchange terms with the Becke gradient-corrected term (B88) [49] and the local and gradient-corrected correlations

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70 terms of Vosko-Wilk-Nusair (VWN) [48] and Lee-Yang-Parr (LYP) [50] respectively. The electronic wave function was expanded using mixed Gaussian basis sets. Diffuse tripleplus polarization 6-311++G(2d,p) was used for the oxygen atoms and first and second layer silicon atoms, while subsurface silicon and terminating H atoms were expanded with a doubleplus polarization 6-31G(d) basis set. This approach, used successfully in similar studies [ 97 100 ], focuses basis functions on the chemically active portion of the cluster and accurately describes orbitals involved in the reaction while minimizing the computational expense. All calculations were run using an unrestricted approach to calculate the open-shell structures of the spin-triplet systems. Comparison between restricted and unrestricted calculation results for closed shell systems (i.e., spin-singlet state clusters) showed negligible differences. The commercial program Gaussian03 [71] was used to run all the calculations. 4.3 Results and Discussion A primary goal of this study was to assess the factors which influence the thermodynamic stability of local structures formed in the early stages of Si(100) oxidation. Toward this end, we performed energy minimization for each possible isomer in which one, two or three oxygen atoms are inserted into the first and second layer Si-Si bonds of the Si 9 H 12 cluster. Since these calculations generated many structures, a shorthand notation of capital letters followed by a subscript and a code in parenthesis is used for labeling the clusters to facilitate discussion. The first capital letters which are DB, ME or TSstand for dangling bond, minimum energy, and transition state structure, respectively. Subscripts 1 or 3 appear after the capital letter to indicate whether the structure is a spin singlet or triplet. Finally, the code in parenthesis is based on the notation used by Stefanov and Raghavachari [96] in which SiSi corresponds to a non

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71 oxidized cluster, SiOSi is a cluster with the O atom inserted into the dimer bond, SiSiO2 is a structure with two oxygen atoms inserted into the two backbonds of the same dimer silicon atom, OSiSiO has two oxygen atoms inserted in a backbond pair at the same side of the dimer Si-Si bond, OxSiSiO has the two oxygen atoms in opposite-side backbonds, and SiSiOd is a cluster with an oxygen atom bonded to a single silicon atom of the dimer. Several factors determine the heat of formation of an oxidized cluster, including the Si oxidation states, bond strain and the spin state of the cluster. We investigated clusters in both singlet and triplet spin states since the lowest energy structures of the clean Si(100) correspond to these spin states. The spin has two primary effects on the energy of the cluster. Firstly, a singlet cluster has one extra Si-Si bond compared with a triplet cluster, which tends to lower the energy of the structure. However, the formation of this extra bond also alters the geometric structure of a cluster and hence the strain energy. The strain is typically higher in a singlet cluster than in the analogous triplet structure for the one-dimer model. To directly compare the energies of singlet and triplet clusters, the energy of an oxidized cluster is defined with respect to a common reference state, which was taken to be the energy of the appropriate number of isolated oxygen atoms plus that of the clean (unoxidized) cluster in its singlet ground-state, [structure ME 1 (SiSi)]. The singlet bare cluster was chosen as the reference state because DFT predicts that this structure minimizes the energy of the clean Si(100) surface. The dangling bonds on the Si dimer atoms are unpaired in the spin-triplet state of the bare cluster, [ME 3 (SiSi), Figure 4-1(a)], but they pair to form a weak bond along the dimer in the spin-singlet state, [ME 1 (SiSi), Figure 4-1(B)]. This extra bond in ME 1 (SiSi) shortens and tilts the silicon dimer bond along the ]011[ direction, resulting in an asymmetric structure that is

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72 0.37 eV more stable than its symmetric ME 3 (SiSi) counterpart. The highest occupied molecular orbitals (HOMO) of both ME 1 (SiSi) and ME 3 (SiSi) were also calculated (Figure 4-1). The 0.37 eV energy difference, labeled as E spin reflects the energy gained by the interaction, which has been reported to be 1.24 eV [117] It also reflects the difference in strain energies between the singlet and triplet bare clusters. Since the initial surface is taken to be the singlet bare cluster, the production of an oxidized cluster in the triplet spin state may be considered to occur in two steps, namely, conversion of the bare cluster from the singlet to the triplet ground-state, and then insertion of oxygen atoms into the triplet bare cluster. Thus, when comparing the energies of the oxidized clusters, it is important to recognize that the singlet to triplet conversion step contributes an energy penalty of 0.37 eV to the heat of formation of a triplet cluster. This energy is taken into account in our analysis of other factors that determine the cluster energies as discussed below. We quantified factors which contribute to the energy of an oxidized cluster by invoking a bond energy model, which has previously been shown to provide a reasonable representation of the heats of silicon suboxide formation [118] In the bond energy model, bond strain and distinct Si-O bond strengths for Si +1 Si +2 and Si +3 oxidation states are treated as separable effects that contribute additively to the total heat of formation. In cases for which each oxygen atom in the cluster is inserted into a Si-Si bond, the bond energy model yields the following expression for the energy of an oxidized cluster, strainspinsuboxideSiSiSiOLYPUBEEENE 23 (4-1) where E UB3LYP is the energy of the oxidized cluster calculated by DFT, N is the total number of oxygen atoms in the cluster, SiO is the Si-O bond energy in stoichiometric

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73 quartz crystal and SiSi is the Si-Si bond energy in bulk silicon. In addition, is the total suboxide penalty energy, suboxideE spinE is the 0.37 eV energy difference between the triplet and singlet bare clusters, where is zero for the spin-singlet and one for the spin-triplet clusters, and is the excess energy that is mainly related to changes in strain that result from oxygen insertion into the bare cluster. The suboxide energy penalty is defined by the Equation 4-2. excessE xxSisuboxidexNE (4-2) where is the number of silicon atoms bonded to x oxygen atoms (with x = 1, 2 or 3), and is the energy penalty of a Si atom in the +x oxidation state. xSiN x The concept of suboxide penalty energies was introduced in the bond energy model of Hamman [119] to take into account the effective increase in Si-O bond strength as the Si oxidation state increases. This effect originates mainly from the greater amount of ionic character in Si-O bonds involving Si atoms in higher oxidation states, but quantum resonance has also been suggested to enhance the bond strengths [119] To quantify the differences in suboxide Si-O bond strengths, penalty energies were computed by Hamann [119] and Bongiorno and Pasquarello [118] for each Si suboxide state, under the assumption that the penalty energies can simply be added to the (2 SiO SiSi ) term to compute the heats of formation of suboxide structures. Since the quantity SiO is defined as the Si-O bond energy in crystalline SiO 2 the penalty energies have positive values so that their contribution reduces the energy change in forming Si-O suboxide bonds. Table 4-1 shows the values of the suboxide penalty energies determined in the separate studies of Hamann [119] and Bongiorno and Pasquarello [118]

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74 Figure 4-1. Structural information and highest occupied molecular orbital plot of clean Si 9 H 12 clusters. A) Symmetric spin-triplet surface ME 3 (SiSi). B) asymmetric spin-singlet surface ME 1 (SiSi). ME 1 (SiSi) is calculated to be more stable than ME 3 (SiSi) by 0.37 eV. Silicon and hydrogen are represented by grey and white balls, respectively. Bond lengths are expressed in

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75 Table 4-1. Suboxide penalty energies for various silicon oxidation states. The superscripts denote the number of oxygen atoms bonded to a particular silicon atom. Energies are given in eV. Oxidation state Energy penalty Ref. 118 Ref. 119 Si +1 1 0.50 0.47 Si +2 2 0.51 0.51 Si +3 3 0.22 0.24 The penalty energies determined in these studies by Hamann [119] and Bongiorno and Pasquarello [118] are in good agreement with one another and are significant in value (Table 4-1), which suggests that maximizing the Si oxidation states is an important driving force that determines the types of local structures that form on Si(100) during initial oxidation. We used the bond energy model given in Eq. 4-2 to quantify changes in strain energy resulting from the insertion of oxygen atoms into Si-Si bonds of the bare cluster. We chose to use an average of the suboxide penalty energies reported by Hamann [119] and Bongiorno and Pasquarello [118] in our analysis. This approach appears justified considering the close agreement in the penalty energies determined in those investigations. In addition, the values of SiO and SiSi were taken to be -4.35 and -2.02 eV, respectively, as reported for the Si-O bond energy in crystalline SiO 2 ( quartz, the lowest-energy form), and for the Si-Si bond energy in bulk silicon [ 96 117 ]. Additionally, for spin-singlet structures where the oxygen atom inserts into the silicon dimer, the appropriate value of SiSi in Eq. 4-2 corresponds to the energy for Si-Si bond cleavage (-1.24 eV) [117] These choices introduce some uncertainty in the absolute strain energies that are calculated because SiO and SiSi are experimentally determined values and are, therefore, not subject to the systematic errors that may affect the energies predicted by DFT.

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76 We also investigated clusters in which one of the oxygen atoms adsorbs on a single Si atom of the surface dimer and forms only one Si-O bond, which we label as Si-O d These structures can serve as precursors to the formation of Si-O-Si linkages which form when the adsorbed O atom inserts into Si-Si bonds near the surface. In fact, a recent experimental study by Gerrard et al. [107] shows that gaseous O( 3 P) atoms initially adsorb at surface dangling bond sites on Si(100), forming Si-O d bonds, before incorporating into the solid. Hence, investigating these dangling bond structures is essential for developing a mechanistic understanding of Si(100) oxidation by gaseous oxygen atoms. For the dangling bond structures, the following variation of Eq. 4-2 was used to quantify strain energies, excessspinsuboxidedSiOSiSiSiOLYPUBEEENE 213 (4-3) where all quantities have the same definitions as given above except for SiOd which is the energy of the Si-O d bonds. We find that Si-O d bonds have a more delocalized character than the Si-O bonds in Si-O-Si linkages due to the lone electron from the oxygen atom. One consequence of this delocalization is that the bond energy model with the parameters defined above, yields excessive values for the relative strain energies for the dangling bond structures (Section 4.3.2). In virtue of this, the value of SiOd has been defined by the difference in energy between the DB 3 (SiSiOd) and ME 1 (SiSi) structures. Finally, it is important to point out that charge transfer involved in the incorporation of adsorbed species on Si(100)-(2x1) can exhibit nonlocal effects (i.e., charge transfer to a neighboring dimer on the surface) which may limit the overall usefulness of the O x -Si 9 H 12 clusters in estimating heats of formation. However, Widjaja and Musgrave [100] reported that the electron density in the similar system of O 2

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77 adsorbed on Si(100)-(2x1) remains localized within the one-dimer environment. This observation supports the use of the small Si 9 H 12 cluster for our investigation. 4.3.1 Structures with One Adsorbed Oxygen Atom (O 1 -Si 9 H 12 ) Since we are interested in structures formed during the early stages of oxidation of the Si(100)-(2x1) surface, we considered only oxygen atom insertion into the surface dimer bond, one or more backbonds or a dangling bond. Optimized total energies of the different oxidized clusters are used to evaluate their stability relative to that of the clean clusters. For now, we postpone discussing the mechanistic aspects (i.e., reaction barriers) of these reactions and focus exclusively on the relative thermodynamic stabilities of the different isomers. Figure 4-2. Relative energies of O 1 -Si 9 H 12 isomers. A) DB 3 (SiSiOd). B) ME 3 (SiSiO). C) ME 3 (SiOSi). D) ME 1 (SiOSi). E) ME 1 (SiSiO). Energy reference: ME 1 (SiSi) + O( 3 P) = 0.00 eV. Oxygen and silicon atoms are represented by black and gray balls, respectively. For the minimum energy analysis, the energies of the O 1 -Si 9 H 12 isomers are referenced to the reaction O( 3 P) + Si 9 H 12 (M s = 1) O 1 -Si 9 H 12 (M s = 1 or 3) (4-4)

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78 where M s is the spin multiplicity. Table 4-2 shows the different energies that contribute to the calculated UB3LYP energies of the O 1 -Si 9 H 12 clusters, in order of ascendant stability. The DB 3 (SiSiOd) structure (Figure 4-3A) is the least favorable cluster because it only has one new Si-O bond. According to the bond energy model for dangling bond structures (Eq. 4-3) and the definition of the SiOd energy, there is not any stress associated to DB 3 (SiSiOd). This value is not surprising, since the only difference in the geometric characteristics of the DB 3 (SiSiOd) and ME 3 (SiSi) structures (Figure 4-1A) is an 0.02 elongation of each of the backbonds that are located on the oxygen side of the dimer. Table 4-2. Penalty energies of O 1 -Si 9 H 12 isomers. Contributions of the different factors to the calculated DFT/UB3LYP energies, according to the bond energy model. Energies are given in eV. For all structures: N = 1. Isomers are listed in order of increasing thermodynamic stability. Energy reference: ME 1 (SiSi) + 2O( 3 P) = 0.00 eV. Structure N(2 SiOSiSi ) a SiOd E suboxide (E spin ) E excess E UB3LYP (Eq. 4-3) Dangling Bond Isomers DB 3 (SiSiOd) 0.00 -4.74 0.00 0.37 0.00 -4.37 Minimum Energy Isomers ME 3 (SiSiO) -6.68 N/A 1.00 0.37 -0.28 -5.59 ME 3 (SiOSi) -6.68 N/A 1.00 0.37 -0.50 -5.80 ME 1 (SiOSi) b -7.46 N/A 1.00 0.00 0.56 -5.90 ME 1 (SiSiO) -6.68 N/A 1.00 0.00 -0.49 -6.17 a: corresponds to (N-1) (2 SiO SiSi ) for dangling bond clusters. b: uses a value of -1.24 eV for the SiSi which corresponds to the Si-Si bond In the remaining O 1 -Si 9 H 12 structures, two Si-O bonds are formed at the expense of one Si-Si bond, so each structure has the same suboxide energy penalty. Thus, differences in the relative energies arise from the spin penalty energy and the different amounts of strain in the structures. Firstly, the spin-singlet isomers ME 1 (SiOSi) (Figure 4-3D) and ME 1 (SiSiO) (Figure 4-3E) are predicted to be the more energetically favorable structures, indicating that strain relief in the triplet structures is insufficient to

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79 Figure 4-3. Structural characteristics of Si 9 H 12 clusters with one oxygen atom. A) DB 3 (SiSiOd). B) ME 3 (SiOSi). C) ME 3 (SiSiO). D) ME 1 (SiOSi). E) ME 1 (SiSiO). Oxygen and silicon atoms are represented by black and gray balls, respectively. Bond lengths are expressed in The crystalographic coordinate origin gives the orientation of the clusters. The surface is along the [100] plane.

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80 compensate the spin penalty energy. Structure ME 1 (SiOSi) is more strained, and therefore less favorable than ME 1 (SiSiO) by 0.27 eV, since formation of the epoxide-like ring stretches and weakens the Si-Si dimer bond. This result agrees with observations made by Weldon et al. [ 113 114 ], who were the first researchers to report the epoxide-like ring structures and suggested that they were the thermodynamically favored products after oxygen agglomeration (i.e., after three or more oxygen atoms absorb on the Si(100) surface). For the triplet structures, ME 3 (SiOSi) is more favorable than ME 3 (SiSiO) by 0.22 eV since oxygen insertion breaks the dimer bond and thereby relieves the strain that the dimer bond imposes on the bare cluster. Notice in Figure 4-3(B) that the silicon dimer atoms are separated by 3.00 in the ME 3 (SiOSi) structure. Finally, the bond energy model predicts negative strain energy (i.e. strain relief, for almost all of the O 1 -Si 9 H 12 structures, suggesting that each structure possesses less strain than the corresponding bare cluster). The only exception is the ME 1 (SiOSi) structure, in which the predominant structural change is a significant elongation of the silicon dimer bond to a length of 2.57 that increases the strain in the cluster by 0.56 eV. Nevertheless, the silicon dimer bond is not cleaved in this structure given that the formation of the Si-O bond only disrupts the weak Si-Si interaction along the dimer. This formation does not break the stronger Si-Si bond, although it elongates the bond considerably. 4.3.2 Structures with Two Adsorbed Oxygen Atoms (O 2 -Si 9 H 12 ) Figure 4-4 shows the relative thermodynamic stability of all the isomers with two oxygen atoms (O 2 -Si 9 H 12 ), and the various contributions to these relative stabilities are presented in Table 4-3, in order of ascendant stability. The energies of the isomers are referenced to the reaction expressed in Eq. 4-5.

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81 2O( 3 P) + Si(100) (M s = 1) O 2 -Si(100) (M s = 1 or 3) (4-5) where, as before, M s is the spin multiplicity. As in the case of the O 1 -Si 9 H 12 clusters, the dangling bond structures are the least favorable structures, because they only have three Si-O bonds instead of the four that form on all the other isomers given that one of the oxygen atoms bonds to a silicon dangling bond. An interesting structural effect is predicted by the DFT/UB3LYP calculations for the dangling bond structures which significantly influences the heats of Si-O d bond formation. We find that Si-O d bond Si-Si bond lengths and angles relative to the reactant ME 3 (SiSi). In contrast, formation of an Si-O d bond on the ME 3 (SiOSi) and ME 3 (SiSiO) structures causes one Si-Si bond length to increase, thereby weakening the structure. In particular, with respect to the reactant structures, the dimer Si-Si bond is stretched by about 0.07 in DB 3 (OSiSiOd) and DB 3 (SiSiO2d), and one of the Si-Si backbonds on the O d side of the dimer is Table 4-3. Penalty energies of O 2 -Si 9 H 12 isomers. Contributions of the different factors to the calculated DFT/UB3LYP energies, according to the bond energy model. Energies are given in eV. For all structures: N = 2. Isomers are listed in order of increasing thermodynamic stability. Energy reference: ME 1 (SiSi) + 2O( 3 P) = 0.00 eV Structure N(2 SiO SiSi ) a SiOd E suboxide ( E spin ) E excess E UB3LYP (Eq. 4-3) Dangling Bond Isomers DB 3 (OSiSiOd) -6.68 -4.74 1.00 0.37 0.23 -9.82 DB 3 (SiSiO2d) -6.68 -4.74 1.00 0.37 -0.27 -10.32 DB 3 (SiOSiOd) -6.68 -4.74 1.00 0.37 -0.38 -10.43 Minimum Energy Isomers ME 1 (OSiSiO) -13.37 N/A 2.00 0.00 0.33 -11.04 ME 3 (OxSiSiO) -13.37 N/A 2.00 0.37 -0.26 -11.25 ME 1 (OxSiSiO) -13.37 N/A 2.00 0.00 0.06 -11.31 ME 3 (SiSiO2) -13.37 N/A 1.51 0.37 0.05 -11.44 ME 3 (OSiSiO) -13.37 N/A 2.00 0.37 -0.70 -11.69 ME 3 (SiOSiO) -13.37 N/A 1.51 0.37 -0.57 -12.05 ME 1 (SiOSiO) b -14.15 N/A 1.51 0.00 0.48 -12.15 ME 1 (SiSiO2) -13.37 N/A 1.51 0.00 -0.30 -12.16 a: corresponds to (N-1) (2 SiO SiSi ) for dangling bond clusters. b: uses a value of -1.24 eV for the SiSi which corresponds to the Si-Si bond.

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82 stretched by 0.10 in the DB 3 (SiOSiOd) structure. As expected, this Si-Si bond elongation lowers the heat of Si-O d bond formation. For example, the heat of the reaction ME 3 (SiSi) + O( 3 P) = ME 3 (SiSiOd) is 0.50 eV greater than the heat of the reaction ME 3 (SiSiO) + O( 3 P) = ME 3 (OSiSiOd). This energy difference may be attributed almost entirely to the observed Si-Si bond weakening since the Si-O d bonds in both structures involve a Si +1 species, and the only appreciable structural difference between reactants and products is the longer Si-Si dimer bond in ME 3 (OSiSiOd). It is tempting to conclude that Si-Si bond weakening occurs in the O 2 -Si 9 H 12 structures because the Si atoms in these bonds have higher partial positive charges than in ME 3 (SiSiOd). For example, the dimer bond in ME 3 (SiSiOd) involves an Si 0 and Si +1 species, whereas in ME 3 (OSiSiOd) the dimer Si atoms are both nominally in the +1 oxidation state. However, similar Si-Si bond stretching is not observed in O 2 -Si 9 H 12 structures in which each oxygen atom is present in an Si-O-Si linkage (Figure 4-6). Hence, delocalization of the unpaired electron of the O d atom must play an important role in weakening Si-Si bonds in the O 2 -Si 9 H 12 dangling bond structures. The remaining O 2 -Si 9 H 12 isomers result from the formation of Si-O bonds in siloxane bridges or epoxide-like rings in place of the dimer bond or backbonds (Figure 4-6). The relative energies of these structures are determined by a combination of effects, though the suboxide energy penalty generally has the largest influence. The overall trend is that the structures possessing four Si +1 species are less favorable than those with a combination of two Si +1 and one Si +2 species, and that the spin-singlet structures are more favorable than their spin-triplet counterparts. Indeed, three of the four isomers possessing a Si +2 atom are the most energetically favorable among the two O-atom isomers and, in order of increasing stability, the most favorable structures are ME 3 (SiOSiO) <

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83 ME 1 (SiOSiO) < ME 1 (SiSiO2). The ME 3 (SiSiO2) structure falls out of this trend as it is 0.25 eV less favorable than the ME 3 (OSiSiO) structure which has four Si +1 species and, consequently, should be less stable. After accounting for the suboxide penalties, the bond energy model suggests a substantial 0.75 eV difference in the excess energies of these two structures, which probably arises from a combination of strain effects associated to the bonding site of the oxygen atom. The results suggest that forming two siloxane-bridges with two different silicon dimer atoms is the favored product. Figure 4-4. Relative energies of O 2 -Si 9 H 12 isomers. A) DB 3 (OSiSiOd). B) DB 3 (SiSiO2d). C) DB 3 (SiOSiOd). D) ME 1 (OSiSiO). E) ME 3 (OxSiSiO). F) ME 1 (OxSiSiO). G) ME 3 (SiSiO2). H) ME 3 (OSiSiO). I) ME 3 (SiOSiO). J) ME 1 (SiOSiO). K) ME 1 (SiSiO2). Energy reference: ME 1 (SiSi) + 2O( 3 P) = 0.00 eV. Oxygen and silicon atoms are representend by black and gray spheres, respectively. Another substantial effect of strain is the observation that structure ME 3 (OSiSiO) (Figure 4-6A) is significantly lower in energy than structure ME 1 (OSiSiO) (Figure 4-6E), despite the spin penalty energy that applies to the triplet structure. After taking into account the spin penalty energy, the bond energy model suggests that ME 1 (OSiSiO) has 1.02 eV more strain than ME 3 (OSiSiO). In both of these structures, the formation of two

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84 Figure 4-5. Structural information of dangling bond isomers with two oxygen atoms adsorbed on Si(100). A) DB 3 (OSiSiOd). B) DB 3 (SiSiO2d). C) DB 3 (SiOSiOd). Bond lengths are expressed in Oxygen atoms are represented with black balls. Silicon and hydrogen atoms are represented in gray and white balls, respectively.

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85 Figure 4-6. Structural information of O 2 -Si 9 H 12 isomers. Structures are listed by increasing relative stability of the spin-singlet isomers. A) ME 3 (OSiSiO). B) ME 3 (OxSiSiO). C) ME 3 (SiOSiO). D) ME 3 (SiSiO2). E) ME 1 (OSiSiO). F) ME 1 (OxSiSiO). G) ME 1 (SiOSiO). H) ME 1 (SiSiO2). Bond lengths are expressed in Oxygen atoms are represented with black balls. Silicon and hydrogen atoms are represented in gray and white balls, respectively.

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86 siloxane bridges on the same side of the silicon dimer bond shifts the dimer out of its original symmetry plane, without changing its orientation along the ]011[ direction. However, these structures experience markedly different strain effects due to the different bonding interactions along their respective Si-Si dimers. In ME 3 (OSiSiO), the formation of the two siloxane bridges alleviates strain in the remaining Si-Si backbonds, which each shorten by 0.04 Aside from being shifted out of plane, the dimer bond remains largely unaltered by the formation of ME 3 (OSiSiO). These structural changes are predicted to relieve about 0.70 eV of strain from the cluster, and result in considerable stabilization of the structure, making it the most energetically favorable of all the structures with four Si +1 species. In contrast, the formation of the siloxane bridges significantly weakens the interaction across the silicon dimer in ME 1 (OSiSiO), causing the dimer bond to stretch by 0.16 resulting in an 0.33 eV increase in strain relative to the singlet bare cluster, according to the bond energy analysis. 4.3.3 Structures with Three Adsorbed Oxygen Atoms (O 3 -Si 9 H 12 ) Figure 4-7 shows relative energies of isomers with three oxygen atoms and table 4-4 shows the different contributions to the relative energies of the structures, as determined from the bond energy model. As in the previous two cases discussed above, the energies of the isomers are defined with respect to the reaction described by Equation 4-6. Si 9 H 12 (M s =1) + 3O( 3 P) O 3 -Si 9 H 12 (M s =1 or 3) (4-6) We calculated the relative energies and structural characteristics of the O 3 -Si 9 H 12 isomers (Figures 4-8 and Figure 4-9). We found that the relative energy of the structures is determined by a combination of effects, including strain and suboxide formation.

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87 Table 4-4. Penalty energies of O 3 -Si 9 H 12 isomers. Contributions of the different factors to the calculated DFT/UB3LYP energies, according to the bond energy model. Energies are given in eV. For all structures: N = 3. Isomers are listed in order of increasing thermodynamic stability. Energy reference: ME 1 (SiSi) + 3O( 3 P) = 0.00 eV. Structure N(2 SiO SiSi ) a SiOd E suboxide (E spin ) E excess E UB3LYP Dangling Bond Isomer DB 3 (OSiOSiOd) -13.37 -4.74 1.51 0.37 0.72 -15.51 Minimum Energy Isomers ME 3 (OSiSiO2) -20.05 N/A 2.51 0.37 0.19 -16.97 ME 1 (OSiSiO2) -20.05 N/A 2.51 0.00 0.20 -17.34 ME 3 (OxSiOSiO) -20.05 N/A 2.02 0.37 0.14 -17.52 ME 3 (SiOSiO2) -20.05 N/A 1.74 0.37 0.34 -17.60 ME 1 (OxSiOSiO) -20.83 N/A 2.02 0.00 1.13 -17.68 ME 1 (SiOSiO2) -20.83 N/A 1.74 0.00 1.18 -17.93 ME 3 (OSiOSiO) -20.05 N/A 2.02 0.37 -0.44 -18.10 ME 1 (OSiOSiO) b -20.83 N/A 2.02 0.00 0.25 -18.57 a: corresponds to (N-1) (2 SiO SiSi ) for dangling bond clusters. b: uses a value of -1.24 eV for the SiSi which corresponds to the Si-Si bond. The least favored of the O 3 -Si 9 H 12 isomers are the structures ME 3 (OSiSiO2) and ME 1 (OSiSiO2). These structures have the highest suboxide energy penalty of 2.51 eV, so their energetic positions are governed mainly by the suboxide penalty, with the spin penalty separating the energies of these two isomers. In contrast, however, the lowest energy structures, ME 1 (OSiOSiO) and ME 3 (OSiOSiO) (Figure 4-9D and 4-9F), each have a suboxide penalty of 2.02 eV, which is not the lowest (1.74 eV) among the O 3 -Si 9 H 12 isomers. These two structures owe their high relative stabilities to strain minimization. In fact, according to the bond energy model, only the formation of ME 3 (OSiOSiO) lowers the strain energy, relative to that of the bare clusters; all the other structures have higher strain energies than the bare clusters. And in ME 1 (OSiOSiO), the strain increases by 0.25 eV, but in this cluster only a weak Si-Si bond (1.24 eV) is broken by the oxygen incorporation rather than the stronger crystalline SiSi bond (2.02 eV). It appears that by shifting the dimer out of its original symmetry plane, the cluster is able to accommodate the oxygen atoms and also relieve strain.

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88 Figure 4-7. Relative enegies of optimized structures for three oxygen atom incorporation into Si(100). A) DB 3 (OSiOSiOd). B) ME 3 (OSiSiO2). C) ME 1 (OSiSiO2). D) ME 3 (OxSiOSiO). E) ME 3 (SiOSiO2). F) ME 1 (OxSiOSiO). G) ME 1 (SiOSiO2). H) ME 3 (OSiOSiO). I) ME 1 (OSiOSiO). Energy reference: ME 1 (SiSi) + 3O( 3 P) = 0.00 eV. The energy positions of the remaining clusters are determined by a combination of effects as well. For example, the third and fourth most favorable structures, ME 1 (SiOSiO2) and ME 1 (OxSiOSiO), respectively, are in the same spin state and their strain energies differ by only 0.05 eV. So, the lower energy of ME 1 (SiOSiO2), can be attributed to its lower suboxide energy penalty. Notice, however, that the strain associated to these two structures is considerably high, as a direct result of the fact that the oxygen atom adsorbs by breaking a weak Si-Si bond and forming an epoxide-like ring on top of the dimer bond. This ring imposes considerable strain on the silicon dimer, which stretches from 2.24 to 2.92 in ME 1 (OxSiOSiO), and shrinks in ME 1 (SiOSiO2) from 2.24 to 2.17 The next two isomers, ME 3 (SiOSiO2) and ME 3 (OxSiOSiO), are higher in energy than their spin-singlet counterparts due to the spin energy penalty, and structure ME 3 (SiOSiO2) is lower in energy than ME 3 (OxSiOSiO) by only 0.08 eV. According to the bond energy model, the lower suboxide penalty in ME 3 (SiOSiO2)

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89 Figure 4-8. Structural information of the dangling bond O 3 -Si 9 H 12 isomer DB 3 (OSiOSiOd). Bond lengths are expressed in Oxygen atoms are represented with black balls. Silicon and hydrogen atoms are represented in gray and white balls, respectively. compensates its higher strain energy relative to ME 3 (OxSiOSiO), making ME 3 (SiOSiO2) the more favorable of the two. Finally, each spin-singlet structure is lower in energy than its spin-triplet counterpart, demonstrating the importance of the spin state in determining the relative energies of the O 3 -Si 9 H 12 isomers. Of the possible O 3 -Si 9 H 12 isomers with a single Si-O d bond, we optimized only the DB 3 (OSiOSiOd) structure. This structure was chosen firstly because it is a precursor to the ME 3 (OSiOSiO) structure, which is the most favorable of the triplet O 3 -Si 9 H 12 isomers and is only higher in energy than one other three O-atom cluster, ME 1 (OSiOSiO). In addition, DB 3 (OSiOSiOd) is generated by the adsorption of an oxygen atom at a dangling bond site of ME 3 (SiOSiO) (Figure 4-7C), which is the most favorable of the triplet O 2 -Si 9 H 12 isomers, and is only higher in energy by 0.11 eV than the lowest energy, two O-atom cluster. Assuming that the equilibrium energy of a local structure is important in determining its formation probability, the DB 3 (OSiOSiOd) structure can be expected to be a prevalent precursor in the early stages of Si(100) oxidation since it enables the conversion between the most stable O 2 -Si 9 H 12 and O 3 -Si 9 H 12 isomers.

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90 Figure 4-9. Structural characteristics O 3 -Si 9 H 12 isomers. Isomers are listed by increasing relative stability of the spin-singlet isomers. A) ME 3 (OSiSiO2). B) ME 3 (OxSiOSiO). C) ME 3 (SiOSiO2). D) ME 3 (OSiOSiO). E) ME 1 (OSiSiO2). F) ME 1 (OxSiOSiO). G) ME 1 (SiOSiO2). H) ME 1 (OSiOSiO). Bond lengths are expressed in Oxygen atoms are represented with black balls. Silicon and hydrogen atoms are represented in gray and white balls, respectively.

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91 Oxygen adsorption on ME 3 (SiOSiO) to produce DB 3 (OSiOSiOd) is significantly less exothermic than Si-O d bond formation in the isomers with one and two oxygen atoms. In particular, the heat of the ME 3 (SiOSiO) + O( 3 P) = DB 3 (OSiOSiOd) reaction is 3.46 eV, whereas oxygen adsorption on the triplet bare cluster is exothermic by 4.74 eV. The heat of Si-O d bond formation on clusters containing one Si-O-Si linkage was found to be lower than on the bare cluster due to weakening of an Si-Si bond in the O-Si 9 H 12 -O d structures (Section 4.3.2). There appears to be a similar effect in the DB 3 (OSiOSiOd) structure (Figure 4-8), where the presence of two siloxane-bridges in combination with the Si-O d bond formation weaken the three remaining Si-Si bonds, which is evident by the similar elongation of all of them. This stretching further translates into the reduction of formation energy. Finally, notice that the three spin-singlet most stable structures present an epoxide-like ring. This is in agreement with the observations of Weldon et al. [ 113 114 ], who suggested that the epoxide-like rings were the thermodynamically favored products when three or more oxygen atoms absorb on the Si(100) surface. 4.3.4 Oxygen Insertion 4.3.4.1 Thermodynamic considerations Recent experimental results [107] showed that gaseous O( 3 P) atoms incorporate into the Si(100)-(2x1) surface by first adsorbing at dangling bond sites and then inserting in Si-Si bonds in the near-surface region. Thus, the dangling bond structures serve as precursors for oxygen atom insertion into the Si(100) surface. Because the dangling bond structures are all of triplet spin multiplicity, the insertion of an oxygen atom into a cluster is considered to occur on the triplet potential energy surface. Interconversion between singlet and triplet spin states is not explicitly taken into account in our calculations, but such state-crossing processes are certain to occur during Si(100) oxidation, and may

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92 impose additional energy barriers on the oxidation process. According to our calculations, the most energetically favorable structures exist in spin-singlet states, so the generation of these structures via dangling bond precursors necessitates a triplet to singlet conversion. Nevertheless, the relative energies determined for the oxidized clusters do provide critical information for assessing likely pathways for the initial stages of Si(100)-(2x1) oxidation. Before discussing mechanistic aspects of oxidation, it is useful to consider the likely progression of local structures that will form based on their relative energies. We investigated the thermodynamic feasibility of several insertion reactions in the spin-triplet state, which may be represented by reaction Equation 4-7. O x -Si 9 H 12 (M s =1) + O( 3 P) O x -Si 9 H 12 -O d (M s =3) O x+1 -Si 9 H 12 (M s =3) (4-7) In Eq. 4-7 M s corresponds to the multiplicity state, -O d indicates that the oxygen atom is bonded at a dangling bond, and x = 0, 1 or 2. An O( 3 P) atom will initially adsorb on the clean surface to generate the structure DB 3 (SiSiOd). The adsorbed oxygen atoms can then insert into either the Si-Si dimer bond or a backbond to generate either ME 3 (SiSiO) or ME 3 (SiOSi), with formation of the latter being more exothermic by 0.23 eV. Since the singlet O 1 -Si 9 H 12 isomers are lower in energy, the triplet structures will eventually convert to the spin-singlet state, unless the barriers for state-crossing are prohibitively high. Interestingly, the more thermodynamically favored singlet structure, ME 1 (SiSiO), is the less favored of the triplet structures(i.e., ME 3 (SiSiO) must first form via the DB 3 (SiSiOd) precursor). Based only on the relative energies, structure ME 3 (SiOSi) should form preferentially over ME 3 (SiSiO), and will then convert to ME 1 (SiOSi) to further lower the energy of the system.

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93 Oxygen adsorption onto the O 1 -Si 9 H 12 isomers can result in three distinct dangling bond structures (Figure 4-5). Although the heats of formation of these structures vary by as much as 0.50 eV, oxygen adsorption is non-activated so each dangling bond structure should form with roughly equal probability, assuming that the ME 1 (SiOSi) and ME 1 (OSiSi) reactants are present in equal concentrations. The ME 3 (SiOSiO) structure is the most likely product of the oxygen insertion reactions that can originate from the O 1 -Si 9 H 12 -O d structures. Firstly, the ME 3 (SiOSiO) structure has the lowest energy of all the triplet O 2 -Si 9 H 12 isomers, and can further lower the energy of Figure 4-10. Qualitative representation of a thermodynamic preferred path for oxygen atom incorporation on Si(100)-(2x1) based on relative thermodynamic stability of the structures. Relative energies are not at scale. A) ME 1 (SiSi). B) DB 3 (SiSiOd). C) ME 1 (SiOSi). D) ME 1 (SiSiO). E) DB 3 (OSiSiOd). F) DB 3 (SiSiO2d). G) DB 3 (SiOSiOd). H) ME 3 (SiOSiO). I) DB 3 (OSiOSiOd). J) ME 3 (OSiOSiO).

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94 the system by converting to the ME 1 (SiOSiO) structure through a state-crossing event. In addition, each of the three O 1 -Si 9 H 12 -O d structures can transform directly to the ME 3 (SiOSiO) structure, thereby increasing the likelihood that this structure will be formed. The lowest energy O 2 -Si 9 H 12 isomer is ME 1 (SiSiO2), which is favored by a mere 0.01 eV over ME 1 (SiOSiO). However, the formation of ME 1 (SiSiO2) appears to be less likely than ME 1 (SiOSiO) for several reasons. It can only be produced from the DB 3 (SiSiO2d) isomer via the production of ME 3 (SiSiO2), and the ME 3 (SiSiO2) structure is 0.61 eV higher in energy than the alternate ME 3 (SiOSiO) product. Finally, the most stable triplet O 3 -Si 9 H 12 cluster, ME 3 (OSiOSiO) (Figure 4-9D), can form directly from DB 3 (OSiOSiOd), and ME 3 (OSiOSiO) can then transform into the lowest energy O 3 -Si 9 H 12 isomer via a state-crossing process. These thermodynamic considerations suggest that the progression of structures shown in Figure 4-10 represent the preferred path for the incorporation of three oxygen atoms into a single dimer unit of the Si(100) surface. 4.3.4.2 Kinetic considerations We computed transition structures and energy barriers for oxygen insertion from several dangling bond structures into Si-Si bonds of the cluster. Specifically, transition states for each possible insertion reaction from the one and two O-atom dangling bond structures were calculated to determine kinetically favored pathways at these oxygen concentrations, and also to identify factors that govern the magnitudes of the energy barriers. On the other hand, a transition state was determined for the insertion reaction involving only the most energetically favorable O 3 -Si 9 H 12 isomer [i.e. conversion of DB 3 (OSiOSiOd) to ME 3 (OSiOSiO)].

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95 Figure 4-11. Structural information of transition state structures for oxygen migration in O 1 -Si 9 H 12 structures. (a) TS 3 (SiSiOd-SiSiO). B) TS 3 (SiSiOd-SiOSi). Barriers for these transition states are 0.30 and 0.45 eV, respectively. Bond lengths are expressed in Oxygen atoms are represented with black balls. Silicon and hydrogen atoms are represented in gray and white balls, respectively. We calculated the transition structures for oxygen insertion from the O 1 -Si 9 H 12 dangling bond isomer DB 3 (SiSiOd) into both a backbond and the dimer bond (Figure 4-11). Both transition structures have an epoxide-like ring in which the original Si-O bond is only slightly elongated, a second Si-O bond begins to form (although weak and elongated) and the Si-Si bond length barely changes, suggesting that the barriers for oxygen insertion occur early along the reaction path. The barrier for insertion into the backbond is predicted to be 0.30 eV, whereas the barrier for insertion into the dimer bond is 50% higher at 0.45 eV. This difference indicates that insertion into the backbond is the kinetic preference, even though the overall reaction is less exothermic than insertion into the dimer bond (1.22 vs. 1.47 eV). Interestingly, however, the product of the more

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96 kinetically-favored reaction can transform, via a state-crossing event, into the lowest energy O 1 -Si 9 H 12 isomer, ME 1 (SiSiO). Table 4-5. Energy barriers for migration of oxygen atoms in O-Si 9 H 12 -O d structures from a dangling bond site to a backbond or dimer bond site in the spin-triplet state. Reactant Product Barrier (eV) Structure in Figure 4-12 DB 3 (SiOSiOd) ME 3 (SiOSiO) 0.07 (a) DB 3 (SiSiO2d) ME 3 (SiSiO2) 0.36 (B) DB 3 (SiSiO2d) ME 3 (SiOSiO) 0.25 (C) DB 3 (OSiSiOd) ME 3 (OxSiSiO) 0.23 (D) DB 3 (OSiSiOd) ME 3 (OSiSiO) 0.21 (E) DB 3 (OSiSiOd) ME 3 (SiOSiO) 0.11 (f) The energy barriers for insertion from O-Si 9 H 12 -O d structures into more stable O 2 -Si 9 H 12 isomers are listed in Table 4-5 and the corresponding transition structures are shown in Figure 4-12. The geometric properties of these transition structures are generally similar in that the top of the barrier is reached after only a slight stretching of the Si-Si bond into which the oxygen inserts, and at a fairly large separation between the Si and O atoms that are forming a bond. An interesting observation is that, with one exception, the insertion barriers are lower from the O-Si 9 H 12 -O d structures than from DB 3 (SiSiOd). For example, the barriers for backbond insertion are 0.07, 0.21, 0.23 and 0.36 eV for the two O-atom structures, whereas a barrier of 0.30 eV is predicted for backbond insertion from DB 3 (SiSiOd). Similarly, the barriers for insertion into the dimer bond are 0.11 and 0.25 eV for the O-Si 9 H 12 -O d structures, but is 0.45 eV for the one O-atom cluster. The lower insertion barriers appear to originate from the weakened Si-Si bond that is present in the O-Si 9 H 12 -O d structures. In fact, the lowest barrier is predicted for each O-Si 9 H 12 -O d structure when the oxygen atoms inserts into the elongated Si-Si bond of the cluster, and eliminates this destabilized bond from the cluster. A similar result is predicted for oxygen insertion from DB 3 (OSiOSiOd) (Figure 4-13) to produce ME 3 (OSiOSiO) for which a barrier of only 0.05 eV is predicted.

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97 Coupling the thermodynamic and kinetic information determined to this point, we can begin to refine the preferred reaction pathway (Section 4.3.4.1). The preferred oxidation pathway, based on the lowest energy barriers and the relative energies of the clusters is shown in Figure 4-14. Neglecting significant differences in the preexponential factors, oxygen insertion into a Si-Si backbond is the kinetically preferred pathway from the initial DB 3 (SiSiOd) structure, and will ultimately result in the ME 1 (SiSiO) structure. Oxygen adsorption onto ME 1 (SiSiO) will then produce either DB 3 (OSiSiOd) or DB 3 (SiSiO2d), and the minimum energy insertion pathways from these structures, with barriers of 0.11 and 0.25 eV, respectively, both produce the ME 3 (SiOSiO) structure. Oxygen adsorption onto the ME 3 (SiOSiO) structure will produce DB 3 (OSiOSiOd), which converts to ME 3 (OSiOSiO) after overcoming a barrier of only 0.05 eV. Notice that oxygen adsorption onto ME 3 (SiOSiO) can also generate DB3(SiOSiOd2), which can then convert ultimately to the thermodynamically favorable ME1(SiOSiO2) structure. A barrier for this process was not computed. An alternative route to the favorable ME3(SiOSiO) structure is for DB3(SiSiOd) to overcome a 0.45 eV barrier to generate ME3(SiOSi), followed by a state crossing to ME1(SiOSi) and then oxygen adsorption to produce DB3(SiOSiOd). Finally, the oxygen atom attached at the dangling bond structure must overcome a small 0.07 eV barrier to insert into a backbond and generate the ME3(SiOSiO) structure. Based on the predicted insertion barriers, the bottleneck for the second route to ME3(SiOSiO) is the production of ME1(SiOSi) since the formation of this species competes with the formation of ME1(SiSiO). Unless the preexponential factors differ significantly, the insertion barriers suggest that ME 1 (SiSiO) will be produced preferentially over ME 1 (SiOSi).

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98 Figure 4-12. Structural information of transition state structures for oxygen insertion in O 2 -Si 9 H 12 structures. A) TS 3 (SiOSiOd-SiOSiO). B) TS 3 (SiSiO2d-SiSiO2). C) TS 3 (SiSiO2d-SiOSiO). D) TS 3 (OSiSiOd-OxSiSiO). E) TS 3 (OSiSiOd-OSiSiO). F) TS 3 (OSiSiOd-SiOSiO). Bond lengths are expressed in Oxygen atoms are represented with black balls. Silicon and hydrogen atoms are represented in gray and white balls, respectively. Figure 4-13. Structural information of transition state structure TS 3 (OSiOSiOd-OSiOSiO). Bond lengths are expressed in Oxygen atoms are represented with black balls. Silicon and hydrogen atoms are represented in gray and white balls, respectively.

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99 Figure 4-14. Proposed preferred path for initial steps of oxidation of Si(100)-(2x1) by oxygen atoms. Energy of structures are relative to that of ME 1 (SiSi) plus three O( 3 P) at infinite separation. A) ME 3 (SiSi). B) DB 3 (SiSiOd). C) TS 3 (SiSiOd-SiSiO). D) ME 1 (SiSiO). E) DB 3 (SiOSiOd). F) TS 3 (SiOSiOd-SiOSiO). G) ME 3 (SiOSiO). H) DB 3 (OSiOSiOd). I) TS 3 (OSiOSiOd-OSiOSiO). J) ME 3 (OSiOSiO). Insert: alternative path.

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CHAPTER 5 ATOMIC OXYGEN INSERTION INTO ETHYLENE AND ACETYLENE TERMINATED Si(100) (2x1) 5.1 Introduction Surface functionalization, also known as organic modification, is the process of depositing layers of organic molecules on semiconductor surfaces [36] This process stems from a desire to impart some property of the organic materials to the semiconductor device, especially since the rich chemistry of carbon can be used to provide new capabilities in optical, electronic and mechanical functions to the semiconductor surfaces. For instance, hydrocarbon films may be useful as low-dielectric materials for microelectronics, and would be especially useful if such films can form covalent bonds to the surface that chemically and electrically passivate the surface. Furthermore, covalently bound monolayers may serve as an interface between silicon and other organic materials, particularly if monolayers with a variety of chemical functionality can be produced. The C=C bond in olefins has been found to be very reactive at silicon dangling bond sites, as observed, for example, in early studies of propylene and ethylene chemisorption on the Si(100) surface [ 35 120 121 ]. Based on such results, it has been suggested that the attachment of a variety of bifunctional organic molecules to Si(100) can be achieved by using C=C groups to achieve bonding to the surface while preserving the second bond functionality for further surface reactions [35] These organic molecules might be designed to serve in place of gate oxides in metal-oxide semiconductor field-effect transistor (MOSFET) devices, for example. 100

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101 In MOSFETs, the high wire resistance of smaller metal lines and the crosstalk between closely spaced metal increase the interconnect resistance and capacitance product delay. This requires a low-dielectric constant (low k) material as the interlayer dielectric and low resistance conductors such as copper. Silicon oxycarbide (SiC x O y ), a low k hydrid between organic and inorganic materials, is one of the most favorable candidates for an interlayer dielectric in MOSFETs [39] In addition to its low k value, silicon oxycarbide offers resistance to diffusion of copper and exhibits the desirable material and integration properties of silicon dioxide (SiO 2 ) [39] SiC x O y thin films are generally deposited at high temperature by chemical vapor deposition (CVD) [ 37 38 ]. However, plasma-assisted deposition at low temperature may provide greater control over the properties of silicon oxycarbide films and other oxidized organic layers since thermally activated reactions would be inhibited. For example, thin films of organic oxygenates could be prepared by oxidizing adsorbed hydrocarbon layers using an oxygen plasma. Since plasma-assisted oxidation occurs efficiently at low temperature, it may be possible to introduce specific oxygen-containing chemical functionality into the organic thin film, depending on the reaction selectivity of the active oxygen species toward the organic adsorbate. However, the interactions of gaseous radicals with adsorbate-covered silicon surfaces are relatively poorly understood, mainly because few fundamental studies of radical-surface interactions have been conducted under well-controlled ultrahigh vacuum (UHV) conditions. It is therefore unclear how gaseous atomic oxygen will react with adsorbate-modified Si(100) surfaces. To our knowledge, only a single UHV investigation of the interactions between gaseous O-atoms and organic adsorbates on Si(100) has been reported. In this work,

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102 Auger electron spectroscopy (AES) was used to study the oxidation of acetylene (C 2 H 2 )-, and ethylene (C 2 H 4 )-covered Si(100) by gaseous oxygen atoms [14] Oxygen plasma species were replicated by generating a 50/50 mixture of ground-state O( 3 P) and electronically exited O( 1 D) by the 157 nm photolysis of O 2 From AES measurements, it was found that the O atom adsorption probability for oxygen coverages in the 1-3 monolayer regime is on the order of 0.1 on these olefin-covered surfaces. In addition, the investigators did not observe loss of carbon on either surface and suggested that there first must be extensive oxidation of the near-surface region of silicon before the removal of carbon chemisorbed on Si(100). However, these authors were unable provide chemical state information about the oxidized products so details about the oxidation mechanism remain uncertain. In this work, we used gradient-corrected density functional theory (DFT) to investigate oxygen incorporation into model C 2 H 2 and C 2 H 4 -terminated Si(100) surfaces. We focused on determining the relative energies of the products that result from oxygen atom insertion into the different bonds that are available on the olefin-terminated surfaces, and on elucidating the predominant factors that determine the reaction energetics. We also calculated energy barriers for the oxygen insertion reactions that yield the most energetically favorable products. We find that the formation of siloxane bridges is both thermodynamically and kinetically favored over other insertion pathways on both surfaces. 5.2 Theoretical Approach We used Kohn-Sham density functional theory (DFT) calculations of silicon clusters that represent the local bonding arrangement of the Si(100)-(2x1) surface. We chose to use the Si 9 H 12 cluster (Figure 3-1) which is the smallest structure that

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103 appropriately represents the key structural characteristics of Si(100)-(2x1) (i.e., covalent tetrahedral sp 3 arrangement of Si-Si bonds and a tilted dimer). Truncated Si-Si bonds in this cluster were terminated with hydrogen atoms; which preserve the tetrahedral bonding and have negligible effect on the predicted energies [ 102 115 ]. The clusters were constrained by imposing boundary conditions that mimic the strain that the bulk silicon atoms would impose on the surface dimer under study. This was attained by fixing the hydrogen atoms in their positions. Two types of hydrogen atoms, depending on the Si-Si bond type that they are terminating can be found in a Si 9 H 12 cluster: The bulk atoms that were fixed along tetrahedral directions, and the neighboring dimer atoms, that were constrained in positions that mimic nearest silicon dimer bonds. Third and fourth layer silicon atoms were not directly constrained, while the first and second layer atoms were allowed to completely relax without any constraint, other than the small constraints place on them by the few Si-H bonds that they have. All the chemically active atoms were also allowed to relax completely unconstrained. Kohn-Sham density functional theory (DFT) [ 41 44 ] was used for the electronic structures calculations. We used here the B3LYP hybrid-gradient-corrected method [ 53 116 ] which calculates the exchange correlation term of the electronic energy by means of a linear combination of local, gradient-corrected and Hartree-Fock exact exchange terms with the Becke gradient-corrected term (B88) [49] and the local and gradient-corrected correlation terms of Vosko-Wilk-Nusair (VWN) [48] and Lee-Yang-Parr (LYP) [50] respectively. A mixed approach was used for the basis sets that expand the electronic wave function. A diffuse tripleplus polarization 6-311++G(2d,p) Pople basis set was used to describe the chemically active atoms (i.e.: the surface Si atoms, four Si atoms of

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104 the first subsurface layer, the oxygen atom and the adsorbed olefin), while the remaining subsurface silicon atoms and terminating hydrogen atoms were described with a doubleplus polarization 6-31G(d) Pople basis set. All calculations used an unrestricted approach to calculate the open-shell structures of the spin-triplet states. In all these calculations we did not impose symmetry constraints on the structures, since experimental results [ 122 124 ] indicate that the C-C bond of adsorbed ethylene is twisted by 11.4 with respect to the Si dimer axis, resulting in a 23 meV increase of the binding energy. The quantum chemistry package Gaussian03 [71] was used to run the calculations. 5.3 Results and Discussion A primary goal of this study was to characterize the reaction products that result when gaseous O( 3 P) atoms interact with C 2 H 2 and C 2 H 4 -covered Si(100). Toward this end, we performed energy minimization for various products of O incorporation into the C 2 H 2 and C 2 H 4 -modified surfaces. We specifically considered the relative energies of the products that result when the oxygen atom inserts into a C-C bond, a Si-C bond, a C-H bond or a Si-Si backbond. Since we investigated several structures for both spin-singlet and spin-triplet surfaces, we developed a shorthand notation of capital letters followed by a subscript and a code in parenthesis to label the clusters to facilitate discussion. In this notation the letters A or E designate the surface as either acetyleneor ethylene-covered Si(100), respectively. This letter is followed by a subscript (1 for the spin-singlet and 3 for the spin-triplet states) and a combination of three atomic symbols in parenthesis. These atomic symbols represent the bridge bond that is formed after the oxygen atom inserts into the surface. Four types of oxidized bridges may form: C-O-C, C-O-H, C-O-Si and Si-O-Si in which the oxygen breaks a C-C, C-H, Si-C or Si-Si bond,

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105 respectively; and one case of a single bond formation, in which the oxygen atom adsorbs on a dangling bond. In the latter case, the symbol Od is in parenthesis and it represents that the oxygen atom is attached at dangling bonding site. In the case of the bare clusters or clusters with only olefins adsorbed, the symbols included in parenthesis are SiSi to symbolize that no oxygen atom has reacted with the surface. We ran quantum chemical calculations to analyze the relative energies of the oxidized structures and transitions states to determine the preferred products and details of their formation. The relative energies are defined with respect to the energies of the spin-singlet C 2 H 4 and C 2 H 2 -terminated Si(100) clusters, which we labeled as A 1 (SiSi) (Figure 5-1A) and E 1 (SiSi) (Figure 5-1C), respectively. For oxygen insertion into the ethylene-covered Si(100) structures, we defined the zero of energy as in Equation 5-1a, E 1 (SiSi) + O( 3 P) = 0.00 eV (5-1a) while for oxygen insertion into the C 2 H 2 -terminated clusters, the energy reference was defined in Equation 5-1b. A 1 (SiSi) + O( 3 P) = 0.00 eV (5-1b) Before presenting our results for oxygen atom incorporation into the olefin-modified Si(100) surfaces, we explain some details of the chemistry of formation of A 1 (SiSi) and E 1 (SiSi) and how our results compare to those available in the literature. Given the similarity between a C=C bond and the silicon dimer bond structure with the Si-Si interaction, it has been reported that the formation of A 1 (SiSi) and E 1 (SiSi) is analogous to [2s+2s] cycloaddition reactions [125] In organic systems, cycloadditions are reactions in which two bonded molecules come together to form a new cyclic molecule, losing two bonds and making two new bonds in the process [36]

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106 Specifically, in a [2 s +2 s ] reaction two alkenes come together to form a new four-membered ring, subject to the Woodward-Hoffman selection rules [126] However, while this [2s+2s] cycloaddition model is well accepted for C 2 H 4 chemisorption of C 2 H 2 on Si(100) is still controversial, with some researchers [ 123 127 ] reporting that the unsaturated hydrocarbon molecule is bonded to two surface Si atoms, but that these Si atoms do not necessarily belong to the same dimer. Others [121] even suggest that when C 2 H 2 adsorbs on top of the dimer bond of the Si(100) surface, the silicon dimer bond is cleaved. Figure 5-1. Structural information of optimized clusters for olefin-covered Si(100)-(2x1). Top views include terminating hydrogen atoms to stress adsorbate orientation with respect to the silicon grid. A) A 1 (SiSi). B) A 3 (SiSi). C) E 1 (SiSi). D) E 3 (SiSi). Carbon atoms are in black, silicon atoms in gray and hydrogen atoms in white. Bond lengths are expressed in

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107 Table 5-1. E 1 (SiSi) structure and energy of adsorption compared to those of reported dimerized C 2 H 4 -terminated Si(100) structures. Reference E ads (eV) C-C () C-Si () Si-Si () Si-C-C () This work 1.80 1.57 1.94 2.32 101.1 123 1.89 1.56 1.96 2.37 N/A 125 1.87 1.57 1.953 2.36 101.4 128 1.81 1.52 1.93 2.33 N/A Table 5-2. A 1 (SiSi) structure and energy of adsorption compared to those of reported dimerized C 2 H 2 -terminated Si(100) structures. Reference E ads (eV) C-C () C-Si () Si-Si () Si-C-C () This work 2.58 1.32 1.90 2.32 104.8 123 2.74 1.37 1.91 2.37 N/A 125 2.61 1.353 1.909 2.368 105.5 129 2.81 1.36 1.90 2.36 N/A The energy of adsorption and the main structural characteristics of E 1 (SiSi) and A 1 (SiSi) are compared to theoretical results available in the literature in Tables 5-1 and 5-2, respectively. We calculated the structures of E 1 (SiSi) and A 1 (SiSi) (Figure 5-1), and we predict that in E 1 (SiSi) the ethylene C-C bond stretches upon adsorption from the molecular value of 1.34 to 1.57 as a result of the C-C bond cleavage, and that two identical Si-C bonds (1.90 ) form as well. Also, according to our calculations, the silicon dimer bond is symmetric with respect to the underlying Si atoms and its bond length is 2.32 For A 1 (SiSi), we predict that acetylene forms a dibonded adsorbate complex on the Si(100) surface, retaining a C-C double bond (1.32 ) and decreasing the Si-Si-C bond angle from its optimal value of 109.5 to 104.8. Our results compare reasonably well with the values available in the literature, with the differences attributed to the fact that the other values were obtained using different computational methods. We also optimized the E 3 (SiSi) and A 3 (SiSi) structures, which correspond to the spin-triplet isomers for the C 2 H 4 and C 2 H 2 -covered Si(100), respectively, in which the Si-Si dimer bond is cleaved upon olefin adsorption. These cleaved structures were

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108 investigated to compare their energies to those the spin-singlet counterparts. We find that the energies of olefin adsorption are 0.01 eV for E 3 (SiSi) and 0.83 eV for A 3 (SiSi), which are 1.79 and 1.74 eV less favorable than ther corresponding spin-singlet olefin-terminated structures. These results support the [2s+2s] cycloaddition model for ethylene and acetylene adsorption on the Si(100) surface dimer. Additionally, we observed that the C-C bond of adsorbed C 2 H 4 remains aligned with the silicon dimer bond in E 1 (SiSi) and that the -CH 2 groups remain symmetric with another. In contrast, the C-C bond in the E 3 (SiSi) structure rotates out of the ]011[ direction and the -CH 2 groups also rotate in opposite directions about the C-C bond axis. These structural changes act to relieve strain in the cluster and are possible because the E 3 (SiSi) structure is less rigid than E 1 (SiSi) structure due to the absence of a constrained ring. After confirming that E 1 (SiSi) and A 1 (SiSi) are the minimum energy structures of the adsorbed olefins, we used a bond energy model to estimate the main factors that contribute to the energy of formation of an oxidized cluster. In this model, the calculated energy of an oxidized cluster (E UB3LYP ) is assumed to be given by Equation 5-2. excesscleavagebondLYPUBEEEE 3 (5-2) In Eq. 5-2 E bond is the energy of formation of the oxidized bridge in a representative molecule, E cleavage is the energy required to cleave the dimer bond in the olefin-Si(100) complex, where equals zero or one for spin-singlets and triplets, respectively, and E excess represents the energy contributions to E UB3LYP that are not taken into account through E bond and E cleavage The excess energy is primarily determined by the change in strain energy upon oxygen insertion into the cluster, but it also represents the intrinsic error in this simple model. A key assumption in the bond energy model is that the bonds

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109 being broken and formed are localized and have well-defined energies. The model therefore implies that oxygen insertion modifies other bonds in the cluster only indirectly by altering the geometric structure and hence the strain energy of the cluster. Table 5-3. Calculated energy of formation of the different oxidized bridges that form when an oxygen atom inserts into the C 2 H 2 and C 2 H 4 -terminated Si 9 H 12 clusters. Energy is expressed in eV. Bridge Reaction E bond C-O-C (epoxide) H 2 C=CHSiH 3 +O( 3 P) H 2 COCHSiH 3 -3.54 C-O-C (ether) H 3 C-CH 2 SiH 3 + O( 3 P) H 3 COCH 2 SiH 3 -3.65 C-O-H (alkyl) H 3 C-CH 3 + O( 3 P) H 3 C-CH 2 OH -4.08 C-O-H (vinyl) H 2 C=CH 2 +O( 3 P) H 2 C=CH 2 OH -4.34 Si-O-C H 3 CH 2 C-SiH 3 + O( 3 P) H 3 CH 2 COSiH 3 -5.14 Si-O-Si H 3 Si-SiH 3 + O( 3 P) H 3 SiOSiH 3 -6.68 To estimate the bond energy terms, we calculated the energies of formation of oxidized bridges in small, linear molecules that do not experience the changes in bond strain that occur in the surface clusters upon oxygen insertion. We performed DFT/UB3LYP energy minimization calculations using diffuse tripleplus polarization 6-311++G(2d,p) basis sets for each of the representative molecules before and after an oxygen atom is inserted into a bond to the form the oxidized bridge of interest. The molecular formation reactions and their corresponding E bond values are shown in Table 5-2. Our calculated values compare well with results obtained using values reported by Norman [117] for the crystalline Si-O and Si-Si bonds and by Lide and coworkers [130] for the organic molecule bonds. The E cleavage term is included in the bond energy model because the energies of formation are referenced to the energies of the spin-singlet olefin-surface complexes in which the dimer bond is intact. Thus, the formation of an oxidized cluster of triplet spin may be considered to occur in two steps, namely, cleavage of the dimer bond of the olefin-surface complex, followed by oxygen insertion. The values of E cleavage are 1.74 and 1.79 eV for the ethyleneand acetylene-covered surfaces,

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110 respectively, and represent the difference in energy of the singlet and triplet structures of those reactants. By explicitly including the E cleavage term, the excess energy E excess represents the change in strain resulting from oxygen insertion into either the singlet or triplet reactant. 5.3.1 Relative Energies of Oxidized Ethylene-Covered Si(100) Clusters We obtained the energies of the singlet and triplet oxidized products that result from oxygen insertion into different bonds of the C 2 H 4 -covered surface (Figures 5-4 and 5-5). For each set of products, the trend in relative energies is determined by the strengths of the different oxidized bridges, as given by the bond energy term (Table 5-2). Structures with a C-O-C bridge are the highest energy products, followed by the alcohols, and the incorporation of oxygen into a Si-C bond. The formation of a siloxane-bridge at a silicon backbond results in the minimum energy structure. Additionally, a comparison of Figures 5-4 and 5-5 reveals that the singlet clusters are more favorable than their triplet counterparts in all cases. Thus, the calculations predict that the dimer bond will remain intact during oxygen insertion into any of the available surface bonds. This prediction indicates that any strain relief that may occur by cleaving the dimer bond to accommodate an oxygen atom does not compensate the energy required to cleave the dimer bond. Table 5-4. Penalty energies of spin-singlet O-C 2 H 4 Si 9 H 12 isomers. Contributions of the different energy effects to the relative energies calculated using DFT/UB3LYP (E UB3LYP ). Reference: E 1 (SiSi) + O( 3 P) = 0.00 eV. Energies are given in eV. Column for E cleavage omitted, since it is 0 for all structures. Structure E bond E excess E UB3LYP [Eq. (5-2)] E 1 (COC) -3.65 0.00 -3.65 E 1 (COH) -4.08 -0.07 -4.15 E 1 (SiOC) -5.14 -0.69 -5.83 E 1 (SiOSi) -6.68 0.45 -6.23

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111 Figure 5-2. Relative energies of spin-singlet O-C 2 H 4 -Si 9 H 12 isomers. A) E 1 (COC). B) E 1 (COH). C) E 1 (SiOC). D) E 1 (SiOSi). Reference: E 1 (SiSi) + O( 3 P) = 0.00 eV. Carbon atoms are in black, silicon atoms in gray, oxygen atoms are the large white balls and hydrogen atoms are the small white balls. The bond energy model provides additional insights for understanding the energetic changes that occur upon oxygen insertion and reveals effects that are not evident from chemical intuition only. For example, the formation of ether and alcohol groups produces relatively small changes in the strain energy (Tables 5-4 and 5-5), although these reactions do alter the adsorbate structure (Figure 5-4). For each ether isomer, the CH 2 groups rotate about their respective Si-C bonds and the C-O-C plane is tilted significantly away from the Si-C-C-Si ring. According to the bond energy model that we use in this work, these changes introduce no strain into the dimerized structure E 1 (COC), but cause an 0.18 eV increase in strain for the E 3 (COC) structure. Alcohol formation has only a small effect on the strain as well (Figure 5-4C), which is perhaps more easily anticipated since COH formation occurs on the periphery of the adsorbate molecule and does not affect appreciably the rest of the structure..

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112 Table 5-5. Penalty energies of spin-triplet O-C 2 H 4 Si 9 H 12 isomers. Contributions of the different energy effects to the relative energies calculated using DFT/UB3LYP (E UB3LYP ). Reference: E 1 (SiSi) + O( 3 P) = 0.00 eV. Energies are given in eV. Structure E bond E cleavage E excess E UB3LYP (Eq. 5-2) E 3 (COC) -3.65 1.74 0.19 -1.72 E 3 (COH) -4.08 1.74 0.05 -2.29 E 3 (SiOC) -5.14 1.74 0.09 -3.31 E 3 (SiOSi) -6.68 1.74 0.24 -4.70 Figure 5-3. Relative energies of spin-triplet O-C 2 H 4 -Si 9 H 12 isomers. A) E 3 (COC). B) E 3 (COH). C) E 3 (SiOC). D) E 3 (SiOSi). Reference: E 1 (SiSi) + O( 3 P) = 0.00 eV. Strain effects are more pronounced in the formation of the Si-O-C and Si-O-Si linkages. Specifically, the spin-triplet Si-O-Si and Si-O-C clusters each have smaller E excess values than their spin-singlet isomers (Tables 5-4 and 5-5), which indicates that structures with a cleaved dimer bond are subject to smaller changes in strain during oxygen insertion. This occurs because the triplet clusters are less constrained than the singlets, since they lack a silicon dimer bond, and can therefore more easily adopt geometric configurations that minimize surface strain when oxygen is inserted into the structure. This difference is most evident for the E 3 (SiOC) (Figure 5-4F) and E 1 (SiOC)

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113 (Figure 5-4B) structures. For the triplet cluster, the strain energy increases by only 0.09 eV upon oxygen insertion into the Si-C bond, whereas the formation of E 1 (SiOC) is accompanied by strain relief of 0.69 eV. This substantial decrease in the strain energy can be attributed to the release of ring strain since the formation of E 1 (SiOC) involves the conversion of a four-membered ring to a five-membered ring, and it is well known that threeand four-membered rings are highly strained, while five-membered ring are much less strained [131] The strain energy accompanying oxygen insertion into a Si-Si backbond is also influenced by the presence of the Si-Si dimer bond. In the formation of E 1 (SiOSi), the strain energy increases by 0.45 eV, because the four-membered ring remains intact but is pushed out of its original symmetry plane by the Si-O-Si bridge at the backbond (Figure 5-4). The strain also increases during the formation of E 3 (SiOSi) but only by 0.24 eV, since this structure is better able to accommodate the Si-O-Si bridge. In addition to rotation of the C-C bond out of the original plane of symmetry, a rotation about the C-C axis also occurs, enabling the hydrogen atoms to achieve a staggered conformation in the E 3 (SiOSi) structure (Figure 5-4). Interestingly, the energy of formation of E 1 (SiOSi) is close to that for the incorporation of an oxygen atom into a spin-singlet bare cluster Si 9 H 12 (Table 4-2), which suggests that these structures experience similar amounts of strain. Overall, the binding energy model that we have used appears to adequately capture the dominant contributions to the formation energies of oxidized products on the ethylene-covered Si(100) clusters, despite the initial assumption made regading the localization of the bond formation, and considering the different penalty energies as isolated, independent and additive contributors to the total energy.

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114 Figure 5-4. Structural information of O-C 2 H 4 Si 9 H 12 isomers. Spin-singlet structures: A) E 1 (COC). B) E 1 (COH). C) E 1 (SiOC). D) E 1 (SiOSi). Spin-triplet structures: E) E 3 (COC). F) E 3 (COH). G) E 3 (SiOC). H) E 3 (SiOSi). Oxygen and hydrogen atoms are represented by large and small white balls, respectively. Carbon and silicon atoms are represented by black and gray balls, respectively. Bond lengths are express in

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115 5.3.2 Relative Energies of Oxidized Acetylene-terminated Surfaces The relative energies of the products resulting from oxygen incorporation into the C 2 H 2 -terminated surface (Figures 5-7 and 5-8) follow the same general trend as predicted for the C 2 H 4 -terminated surface. For each type of oxidized bridge, the singlet structure is predicted to be energetically favored over the triplet, indicating that the dimer bond will remain intact during oxygen insertion into any of the bonds on these olefin-terminated surfaces. Also, for each spin state, the relative energies of the products are dictated by the strengths of the oxidized bridges that form; the relative energies of the products are COC > COH > COSi > SiOSi. Similar changes in bond strain are also predicted for oxygen insertion into the C 2 H 2 -terminated surface as found for the C 2 H 4 -terminated surface. For example, upon Si-O-Si formation at the backbond, more strain is introduced into the singlet than the triplet C 2 H 2 -terminated surface (Tables 5-6 and 5-7). Also, oxygen insertion into the Si-C bond results in significantly more strain relief when the dimer bond remains intact than if this bond is cleaved; the excess energies for the formation of A 1 (SiOC) and A 3 (SiOC) are E excess = -0.95 and -0.02 eV, respectively. In fact, the strain relief is so significant for A 1 (SiOC) that its energy is only 0.07 eV higher than that of A 1 (SiOSi), for which the strain increase is 0.52 eV. Table 5-6. Penalty energies of spin-singlet O-C 2 H 2 Si 9 H 12 isomers. Contributions of the different energy effects to the relative energies calculated using DFT/UB3LYP (E UB3LYP ). Reference energy: A 1 (SiSi) + O( 3 P) = 0.00 eV. Energies are given in eV. Column for E cleavage omitted, since it is 0.00 eV for all the spin-singlet structures. Structure E bond E excess E UB3LYP (Eq. 5-2) A 1 (COC) -3.54 0.54 -3.00 A 1 (COH) -4.34 -0.28 -4.62 A 1 (SiOC) -5.14 -0.95 -6.09 A 1 (SiOSi) -6.68 0.52 -6.16

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116 Table 5-7. Penalty energies of spin-triplet O-C 2 H 2 Si 9 H 12 isomers. Contributions of the different energy effects to the relative energies calculated using DFT/UB3LYP (E UB3LYP ). Reference energy: A 1 (SiSi) + O( 3 P) = 0.00 eV. Energies are given in eV. Structure E bond E cleavage E excess E UB3LYP (Eq. 5-2) A 3 (COC) -3.54 1.74 -0.03 -1.83 A 3 (COH) -4.34 1.74 -0.16 -2.76 A 3 (SiOC) -5.14 1.74 -0.02 -3.42 A 3 (SiOSi) -6.68 1.74 0.39 -4.55 While the trends in oxygen insertion chemistry are very similar on the olefin-terminated surfaces investigated, the C-C bond does have a distinct influence on the relative energies and structures of the oxygen insertion products on the C 2 H 2 -terminated surface. Firstly, the change in strain energy is generally greater for oxygen insertion into the acetylene-terminated surface, as may be seen by comparing the excess energies for Si-O-C and Si-O-Si formation on both surfaces (Tables 5-4 to 5-7). Inspection of the corresponding surface structures provides insights for understanding these differences. For example, for each oxidized product on the C 2 H 4 -terminated surface, the C-C bond is predicted to rotate away from the ]011[ direction, and the CH 2 groups also rotate about the Si-C bonds, with these structural changes acting to reduce strain introduced by oxygen insertion (Figure 5-4). In contrast, rotation of and about the C-C double bond of the C 2 H 2 -terminated structures is strongly hindered and, consequently, this bond remains parallel with either the ]011[ direction, for most structures, or with the Si-Si dimer bond. Oxygen insertion into the C-C double bond is a particularly interesting case. This reaction produces an epoxide-like structure, A 3 (COC), on the triplet surface, for which the C-O-C bond is tilted significantly out of the Si-C-C-Si plane. Interestingly, an

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117 Figure 5-5. Structural information of O-C 2 H 2 -Si 9 H 12 isomers. Spin-singlet structures: A) A 1 (COC). B) A 1 (COH). C) A 1 (SiOC). D) A 1 (SiOSi). Spin-triplet structures: E) A 3 (COC). F) A 3 (COH). G) A 3 (SiOC). H) A 3 (SiOSi). Oxygen and hydrogen atoms are represented by large and small white balls, respectively. Carbon and silicon atoms are represented by black and gray balls, respectively. Bond lengths are express in

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118 epoxide does not form when an oxygen atom inserts into the C-C double bond of the singlet C 2 H 2 -terminated surface. Oxygen insertion into the C-C double bond of the singlet cleaves the double bond and produces an C-O-C bridge that lies within a planar, five-membered ring that is parallel with the ]011[ direction (Figure 5-7). The C-O-C bridge of the A 1 (COC) structure involves a interaction in addition to the C-O bonds, and is therefore stronger than an ether-like linkage. Notice that the C-O bond lengths are 1.33 in the A 1 (COC) structure, whereas the C-O bonds are about 1.43 long in the ether linkages of the E 1 (COC) and E 3 (COC) structures. Although the C-O-C linkage of structure A 1 (COC) appears to involve stronger bonding than either an ether or epoxide linkage, its formation also involves cleavage of a C-C double bond. Thus, it is unclear whether the high excess energy of 0.54 eV for A 1 (COC) structure should be viewed as an upper or lower bound of the change in strain energy during A 1 (COC) formation. Figure 5-6. Relative energies of spin-singlet O-C 2 H 2 -Si 9 H 12 structures. A) A 1 (COC). B) A 1 (COH). C) A 1 (SiOC). D) A 1 (SiOSi). Reference energy: A 1 (SiSi) + O( 3 P) = 0.00 eV. Oxygen and silicon atoms are represented by black and gray spheres, respectively. Carbon and hydrogen atoms are represented by large and small white spheres, respectively.

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119 Figure 5-7. Relative energies of spin-triplet O-C 2 H 2 -Si 9 H 12 isomers. A) A 3 (COC). B) A 3 (COH). C) A 3 (SiOC). D) A 3 (SiOSi). Reference energy: A 1 (SiSi) + O( 3 P) = 0.00 eV. Oxygen and silicon atoms are represented by black and gray spheres, respectively. Carbon and hydrogen atoms are represented by large and small white spheres, respectively. 5.3.3 Oxygen Insertion Mechanisms 5.3.3.1 Thermodynamic considerations Our relative energy calculations predict that the oxygen atom has an energetic preference for inserting into a Si-Si backbond over all other bonds that are available on the olefin-covered surfaces. This finding is consistent with an insertion mechanism proposed for oxidation of H-terminated Si(100) by gaseous O-atoms, in which oxygen atoms prefer to insert into Si-Si bonds over abstraction hydrogen atoms [111] Our results also support the conclusions made by Litorja and Buntin [14] that oxidation of acetylene and ethylene-terminated Si(100) proceeds initially by oxidation of the Si atoms. However, this agreement is based only on the prediction that Si-O-Si formation results in the lowest energy reaction products. It is necessary to also consider the energy barriers for oxygen insertion to confirm that the lowest energy products are kinetically preferred.

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120 5.3.3.2 Kinetic considerations According to recent experimental results [107] oxidation of Si(100) by gaseous O( 3 P) atoms is initiated by the adsorption of the O-atom onto a Si dangling bond, followed by oxygen insertion into a Si-Si bond. The precursor structures in this mechanism are spin-triplets since the O( 3 P) atom bonds with only one Si atom of the surface dimer. This interaction leaves one unpaired electron localized on the adsorbed oxygen atom, and the other localized on the opposing Si dimer atom. Similar to our previous study of oxygen insertion into the Si(100) surface, we considered oxygen insertion into the olefin-terminated surfaces as occurring in the triplet spin-state according to the general reaction described in Equation 5-3. C 2 H x -Si 9 H 12 -O d (M s =3) O-C 2 H x -Si 9 H 12 (M s =3) (5-3) In Eq. 5-3 Ms is the spin multiplicity, Od indicates that the oxygen atom is bonded at a Si dangling bond and x = 2 or 4 (Figure 5-8). The heats of O-atom adsorption onto the triplet ethylene and acetylene-terminated surfaces, resulting in E 3 (Od) and A 3 (Od), are found to be 4.86 and 4.95 eV, respectively, and are each slightly higher than the 4.74 eV value found previously for O-atom adsorption onto the bare Si(100) cluster. The Od-Si bond is strengthened on the olefin-terminated surfaces since the adsorbed olefins withdraw charge from the Si atom, resulting in a more polar O-Si bond. We found two energy pathways and the transition state structures for insertion of an oxygen atom into an Si-Si backbond and an Si-C bond of the C 2 H 2 terminated Si(100)-(2x1) surface (Figure 5-9). Oxygen insertion into the silicon backbond to form A 3 (SiOSi) from A 3 (Od) is by far the more kinetically preferred pathway. This reaction has a barrier of 0.31 eV, which is very close to that predicted for the analogous insertion

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121 Figure 5-8. Structural information of clusters involved in oxygen insertion pathways on C 2 H 2 and C 2 H 4 -terminated Si(100) surfaces. Reactants: A) E 3 (Od). B) A 3 (Od). Transition state structures: C) TSE 3 (Od-SiOSi). D) TSE 3 (Od-SiOC). E) TSA 3 (Od-SiOSi). F) TSA 3 (Od-SiOC). Oxygen and hydrogen atoms are represented by large and small white balls, respectively. Carbon and silicon atoms are represented by black and gray balls, respectively. Bond lengths are express in

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122 reaction on the clean Si(100) surface (Chapter 4). The barrier for oxygen insertion into the Si-C bond is predicted to be 1.42 eV, which is significantly higher than that for insertion into the Si-Si bond and therefore suggests that Si-O-C formation does not compete effectively with Si-O-Si formation, at least in the initial stages of oxidation of the C 2 H 2 -terminated surface. The difference between the energy barriers for oxygen insertion into a Si-Si backbond versus into a Si-C bond is even more dramatic for the ethylene-terminated surface. An energy barrier of only 0.10 eV is predicted for oxygen insertion into the silicon backbond of E 3 (Od), which is lower than that computed for the C 2 H 2 -terminated surface (Figure 5-10). In contrast, a barrier of 2.77 eV is predicted for oxygen insertion into the Si-C bond of the ethylene-terminated surface, whereas a lower barrier, but still quite high, was predicted for the same reaction on the C 2 H 2 -terminated surface. Based on the magnitudes of the predicted reaction barriers, our calculations suggest that oxidation of olefin-terminated Si(100)-(2x1) by O( 3 P) atoms will be initiated exclusively by oxygen insertion into Si-Si backbonds, which is consistent with the conclusions reached by Litorja and Buntin [14] Indeed, the barriers for oxygen insertion into Si-C bonds would need to be significantly reduced by the presence of siloxane bridges for Si-O-C bond formation to become a viable reaction pathway at higher oxygen coverages. Additionally, our prediction that the barrier for Si-O-Si formation on the ethylene-terminated surface is lower than that on the acetylene-terminated surface (0.10 vs. 0.31 eV) is also consistent with experimental observations of higher oxidation rates of the C 2 H 4 versus C 2 H 2 -terminated surface by gaseous oxygen atoms [14]

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123 Figure 5-9. Potential energy surfaces for migration of an oxygen atom on an C 2 H 2 covered Si(100)-(2x1) surface. Bond lengths are expressed in and energies in eV. Oxygen and hydrogen atoms represented by large and small white balls, respectively. Carbon and silicon atoms represented by black and gray balls, respectively. Figure 5-10. Potential energy surfaces for migration of an oxygen atom on an C 2 H 4 covered Si(100)-(2x1) surface. Energies expressed in eV. Oxygen and hydrogen atoms represented by large and small white balls, respectively. Carbon and silicon atoms represented by black and gray balls, respectively

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CHAPTER 6 SUMMARY AND CONCLUSIONS 6.1 Nitrogen Atom Abstraction from Si(100)-(2x1) by Gaseous Atomic Oxygen The abstraction of nitrogen atoms from Si(100)-(2x1) by gas-phase O( 3 P) atoms was investigated using DFT for different bonding configurations of nitrogen on the surface. Abstraction in a single elementary step was predicted only for nitrogen bound in a coordinatively saturated configuration, and a barrier of 0.20 eV was determined for this reaction. Despite this relatively low abstraction barrier, oxygen adsorption on a Si dangling bond site appears to be preferred over direct abstraction of coordinatively-saturated nitrogen due to both energetic differences between adsorption and abstraction and the requirement that multiple bonds must break and form during single step abstraction. Abstraction of coordinatively unsaturated nitrogen atoms is predicted to occur by precursor-mediated pathways in which the incident oxygen atom first bonds directly with a surface nitrogen atom to form an adsorbed NO species. A series of elementary steps must then occur for NO to desorb from the surface. Since N-O bond formation releases at least 1.7 eV more energy into the surface than is required to activate the subsequent steps leading to NO desorption, the calculations indicate that gaseous O( 3 P) atoms can efficiently abstract coordinatively-unsaturated nitrogen atoms from Si(100). Although the prediction of facile nitrogen abstraction by gaseous oxygen atoms is consistent with previous observations that nitrogen is depleted from Si surfaces by reaction with oxygen plasma, additional work is needed to clarify the conditions under 124

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125 which atomic nitrogen exists on Si(100) in coordinatively unsaturated configurations, and is therefore susceptible to direct attack by incident oxygen atoms. 6.2 Initial Step of Si(100)-(2x1) Oxidation by Gaseous Atomic Oxygen We studied the incorporation of O( 3 P) atoms onto the Si(100)-(2x1) surface to assess the factors which influence the thermodynamic stability of local structures formed in the in the early stages of Si(100) oxidation by atomic oxygen. Specifically, since we were interested in structures formed during the early stages of oxidation we considered only oxygen atom insertion into the surface dimer bond, one or more backbonds or a dangling bond. Special attention was given to the effects that the spin-state of the surface, the oxidation state of the surface silicon atoms and the strain resulting from oxygen incorporation have on the final energy of the oxidized products. These factors were quantified using a bond energy model that assumes that each one of them contributes additively to the total heat of formation of the products. Transition states for the insertion process were also investigated, to develop a model for the preferred mechanism of the initial steps of Si(100) oxidation by O( 3 P) adsorption. For clusters with one-oxygen atom, the dangling bond structure is the least favorable because it only has one Si-O bond. In all the other more stable one O-atom isomers, two Si-O bonds are formed at the expense of one Si-Si bond, and all the structures have the same suboxide energy penalty. Thus, the differences in the relative energies of these clusters arise from the different spin-states and amounts of strain in them. We found that the spin-singlet isomers are predicted to be more energetically favorable than the spin-triplet isomers. Oxygen adsorption onto the one-atom isomers can result in three distinct dangling bond structures and each of them should form with roughly equal probability. These dangling bond structures have different relative energies

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126 despite their similar structures. These differences are due to delocalization effects associated to the Si-O d bond. The relative energies of the isomers with two and three-adsorbed oxygen atoms are determined by a combination of effects, though the suboxide energy penalty generally has the largest influence. Generally, structures possessing four Si +1 species are less favorable than those with a combination of two Si +1 and one Si +2 species, and the spin-singlet structures are more favorable than their spin-triplet counterparts. It was noticed, though, that the formation of a dangling bond structure with three adsorbed oxygen atoms is 1.28 eV less exothermic than oxygen adsorption on the triplet bare cluster. Based on our kinetic studies we complemented the thermodynamic analysis and propose the following preferred reaction path for oxidation of Si(100)-(2x1) via O( 3 P) adsorption Initially, an oxygen atom adsorbs onto a clean cluster at a dangling bond of the silicon dimer. Then, that oxygen atom inserts into a Si-Si backbond by overcoming an energy barrier of 0.30 eV and ultimately reaches the spin-singlet state, which is more stable than its spin-triplet counterpart by 0.58 eV. A second oxygen atom will adsorb again at a dangling bond site, either with the same or the opposite silicon dimer atom as the first O-atom. Either way, the second O-atom eventually inserts into the dimer bond by overcoming small activation barriers. The reaction barrier for going from having the two oxygen atoms attached at the same silicon dimer is 0.25 eV, and it is only 0.11 eV in the other case. A third oxygen atom adsorbs at the dangling bond of the silicon dimer atom that does not have any backbonded siloxane bridge, eventually inserting into the Si-Si backbond that is on the same side of the dimer as the siloxane-bridge already

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127 present in the structure after overcoming a 0.05 eV activation barrier. Additional oxygen atoms will adsorb following a similar mechanism in which the O-atom first adsorbs at a dangling bond and eventually inserts into a backbond. 6.3 Oxidation of C 2 H 2 and C 2 H 4 -covered Si(100)-(2x1) by Gaseous Atomic Oxygen We used DFT to investigate the reaction energetics of oxygen atom insertion into different bonds of the C 2 H 2 and C 2 H 4 -terminated Si(100) surfaces. Our calculations predict that the Si-Si dimer bond will remain intact when a single oxygen atom inserts into any of the bonds that are available on the olefin-terminated surfaces investigated. In addition, the trend in the relative energies of the oxidized products is found to be determined by the formation energies of the oxidized bridges that form, with changes in strain energy having a secondary influence on the energetics. Specifically, a Si-O-Si bridge at a Si-Si backbond is found to be the most energetically favored product of oxygen insertion, followed by Si-O-C, C-O-H and C-O-C bridge formation. For the C 2 H 2 and C 2 H 4 -terminated surfaces, respectively, the barriers for the insertion of an adsorbed oxygen atom into a Si-Si backbond are predicted to be 0.31 and 0.10 eV, whereas the barriers for oxygen insertion into a Si-C bond are found to be significantly higher at 1.42 and 2.77 eV. This prediction suggests a strong kinetic preference for siloxane formation over Si-O-C formation during the initial stages of oxidation of olefin-terminated Si(100) by gaseous O( 3 P) atoms. The present computational results are consistent with the findings of a recent experimental study [14] which suggest that oxidation of the Si(100) substrate precedes other reaction processes that are stimulated by the interactions between gaseous oxygen atoms and acetylene and ethylene-terminated Si(100).

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APPENDIX A QUANTUM CHEMISTRY SOFTWARE Four computational packages (QCP) were used in this work to properly perform geometry optimization calculations. These programs are Gaussian03 [71] HyperChem 7, Molden, and gOpenMol. Gaussian03 is a program system for ab initio electronic structure calculations created by late John Pople and his collaborators. It is available through the Quantum Theory Project at the University of Florida (QTP), which has all the environment variables set by default. Gaussian03 is the latest in the Gaussian series of electronic structure programs, which have been successfully used for research in established and emerging areas of chemical interest. Based on the basic laws of quantum mechanics, the program predicts the energies, molecular structures, and vibrational frequencies of molecular systems, along with numerous molecular properties derived from these basic computation calculations. Gaussian03 stores all the information relevant to a particular calculation in a binary checkpoint (.chk) file and generates a simplified report of it for the user in an output (.log) file (in text format). To visualize our Gaussia03 computational results and generate initial guesses for geometry optimization or transition state search calculations, we used HyperChem 7 (Figure A-1), which is available on the Windows NT server of QTP. This is a molecular modeling program that is easy to use thanks to its user-friendly Windows interface. The program combines advanced 3D visualization and animation with quantum chemical calculations. All these features made HyperChem 7 our program of choice to perform all our molecular mechanics and semi-empirical geometry optimization calculations, 128

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129 visualize input and output structures, create images of final optimized structures and generate most of our Gaussian03 input files. It was also useful in determine the staring guess structure for transition state calculations. Figure A-1. HyperChem 7 interface. Molden is a software that is mainly used for visualizing molecular densities and orbitals calculated with ab initio packages such as Gaussian03 [71] The program reads all the required information directly from Gaussian03 output (.log) files and displays molecular orbitals, electron density, molecular-atomic density of optimized structures and convergence plots. These capabilities made it the ideal choice to check the day-to-day status of the different optimization calculations that were running at any given time (Figure A-2). Moreover, it also can animate reaction paths and molecular vibrations and, thanks to its advanced Z-matrix editor, it can be used to reorder the Z-matrices (an option that is very useful when creating input files for transition state searches, where several

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130 structures are required, including those of the reactant, the product and, quite often, an initial guess for the transition state). Figure A-2. Molden visualization interface and main command screens. Finally, to plot highest occupied molecular orbitals (HOMO) or any other type of molecular orbitalfrom the information contained in the Gaussian03 checkpoint file we used gOpenMol (Figure A-3), which is a program for the visualization and analysis of molecular structures and their chemical properties. The software was written by Leif Laaksonen and it is available from him on the internet. Since the program can import, display and analyze several different input coordinates and binary trajectory file formats, we also used it to visualize molecular structures available in different file formats. Table A-1 summarizes the information of the different quantum chemistry programs used in this work.

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131 Figure A-3. gOpenMol visualization interface. Main window and command screens. The main screen includes an image of the structure HOMO.

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132 Table A-1. Computational software used in this work Program Platform Applications Gaussian03 UNIX/Linux DFT geometry optimizations DFT transition state searches DFT frequency calculations HyperChem 7 Windows MM geometry optimizations Semi-empirical geometry optimizations Gaussian03 input file generation Generation of initial guess for TS searches Final structure visualizations Final structure image creation Molden Linux Preliminary optimized structure visualization Periodic check of optimization calculations status Creation of transition state movies Z-matrix reordering gOpenMol Linux HOMO visualization Creation of transition state movies Table A-2. File formats used for proper handling of the coordinates systems and results in our geometry optimization calculations. All these files, except .chk, are plain text and can be easily modified by standard text editors. We used EMACS for text editing. File Suffix Format Description Generated by Read by .xyz Cartesian coordinates with no special format Molden gOpenMol HyperChem Babel (from .hin) .zmt Z-Matrix in internal coordinates Molden Babel (from .hin or .xyz) .hin Cartesian coordinates with HyperChem format HyperChem Babel (from .xyz) HyperChem 7 .inp Gaussian03 input Molden Text editor (based on .zmt) Gaussian03 .log Gaussian03 output Gaussian03 Molden gOpenMol Text editor .chk Gaussian03 checkpoint file Gaussian03 Molden (requires use of cubegen and formchk gaussian utilities)

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133 Six different file formats were used in this work (Table A-2). Most of these file formats can be converted into one another by means of the commercial software Babel, or edited using a text editor, such as the commercial UNIX based Emacs. Babel is a program specifically designed to interconvert file formats commonly in molecular modeling. The following sections briefly describe each one of these file types. A.1 File Format .xyz The .xyz file format is the simplest one of all the formats used in this work. It is found in plain text files and consists of four columns: atom identification, and Cartesian x, y and z positions, with each row corresponding to the identification and coordinates of a single atom. The files have a two-line header with the number of atoms in the first line, and a brief description of the file in the second. Even though the order in which the atoms appear in this file format is not important because the coordinates of one atom are completely independent of those of other atoms, the special order shown in Figure A-4 is kept throughout this work. This is all the information that most quantum chemistry programs need to recognize the structure in Cartesian coordinates. A.2 File Format .zmt The order in which the Si 9 H 12+w O x N y C z cluster components are in the .xyz file presented in Figure A-4 is not coincidental. Given that the most effective way to specify Gaussian input files is by using a Z-Matrix format (where the atom positions are specified by bond lengths, bond angles and torsion angles with respect to other atoms in the system [74] ) it is necessary to list first all those atoms that remain fixed during the calculation, followed by the bulk atoms, and the surface and chemically active atoms. In other words, the XYZ matrix described in the .xyz files is constructed in order of decreasing constraints imposed on the atoms.

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134 Figure A-4. Example of a .xyz file describing a Si 9 H 12 O 3 cluster. Molecular geometry expressed in Cartesian coordinates. Insert: The cluster described by the file corresponds. It is important to emphasize that all the files that describe the same structures in a particular geometry optimization calculation keep the same atom order independently of the file format. In this work, the main constraint imposed on the clusters is on all the terminating hydrogen atoms which are fixed in their positions; thus, they must be specified first in the molecule matrix. By doing this, it is guaranteed that all the bond lengths, bond angles and torsion angles that define the hydrogen atoms will remain unchanged during the whole geometry optimization calculation. Following the hydrogen atoms, we list those atoms that are affected the least by the surface reactions (i.e., the third and fourth layer silicon atoms). These atoms will barely move because all of their coordinates are defined relative to those of the hydrogen atoms. The surface silicon atoms and the chemically active atoms are the last to be specified because these atoms are relatively free to move.

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135 Figure A-5. Z-Matrix file describing a Si 9 H 12+x C y cluster. Insert: Top view of Si 9 H 18 C 2 cluster described by this file. Since this file is used in a geometry optimization calculation, the coordinates are specified as variables, whose values are listed below the matrix in the variable section. A Z-Matrix file (Figure A-5) consists of a structure matrix with four columns (corresponding to atom labels, bond lengths, bond angles and torsion angles) followed by a variables section. This variable section shows the values of those coordinates that were specified by a variable name in the structure matrix. Some of these values can also be specified as constants. If that is the case, then a constants section follows immediately

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136 below the variables. Since all the coordinates in a Z-matrix need at least two reference atoms, the first three atom descriptions are incomplete, rendering a non-redundant internal coordinate system that contains only 3N-6 internal coordinates for an N-atom molecule. A.3 HyperChem Files (Format .hin) HyperChem 7 results can be presented in a long and a short file format. Both of them are easily edited using a program like Emacs. HyperChem 7 automatically generates the long file format every time work is saved by the user. This format includes a heading with the quantum chemistry platform used and references for molecule display, which are instructions that the program needs to display the structure the next time it is required by the user. Immediately after the header, the Cartesian coordinate matrix of the structure is presented in the HyperChem matrix format. In addition to the Cartesian coordinates, the format includes the type of atom, the energy associated to it, the order numbers of the atoms to which each atom is bonded and the type of bonding. The file ends with a section displaying all the atoms selected and named by the user. HyperChem short files only have the HyperChem matrix of the structure and are usually generated from .xyz files using Babel, as these files are only mere translations of .xyz files to a format that HyperChem 7 can handle. We used the two HyperChem file formats in two different instances. The .hin long format files are used when exploring for a best initial guess for an optimization calculation (Appendix C). They result from a combination of molecular mechanics and semi-empirical geometry optimization calculations.

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137 Figure A-6. HyperChem 7 files describing a Si 9 H 12 O 3 cluster. A) .hin long format. B) .hin short format. Insert: Side view of the cluster that these files describe. The files are then converted into a Z-matrix format using Babel and used to generate an input file for a UB3LYP/3-21G* calculation in Gaussian03 [71] The .hin short format files are usually obtained from .xyz files that are generated using Molden from Gaussian03 output files and are always used to visualize the final results and

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138 generate the optimized structure images in HyperChem 7 that are presented in the results section of this dissertation. A.4 Gaussian03 Files (Formats .inp, .log and .chk) There are three Gaussian03 files: the input (.inp), checkpoint (.chk), and output (.log) files [71] Contrary to the HyperChem 7 files, the differences among these files go beyond mere formatting. Gaussian03 input files (.inp) (Figure A-7A) are very simple ASCII text files that contain the structure specification (generally in Z-Matrix format) and the Gaussian commands for the simulation that the user wants. The basic structure of an input file includes several sections, which re summarized in Table A-3. Table A-3. Gaussian03 input files sections. An example of an input file is shown in Figure A-1. Section Description Link 0 commands Locates and names scratch files Specifies required memory and number of processors Route section (# lines) Specify desired calculation type, model chemistry and options to keywords. The latter may be specified in any of the following forms: keyword = option keyword(option) keyword=(option1, option2, etc.) keyword(option1, option2, etc.) Title section Brief description of the calculation Molecule specification Specify molecular system to be studied in XYZ or Z-matrix formats Optional additional sections Additional input needed for specific job types Figure A-7(a) is an example of an input file that requests a geometry optimization of a Si 9 H 12 O 3 cluster using the UB3LYP/3-21G* model chemistry. In this job, the route and title sections each consist of a single line. The molecule specification section begins with a line giving the charge and spin multiplicity for the molecule: 0 charge (neutral molecule) and spin multiplicity 3 (triplet) in this case. The charge and spin multiplicity line is followed by lines describing the location of each atom in the molecule (in internal

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139 coordinates in this example). This particular job requests a geometry optimization and the input section including redundant coordinates that follows the molecule specification is used by the Opt(ModRedundant) keyword, as it serves to add constraints in the geometry optimization. The file also includes two Link 0 commands that specify the memory (%Mem) and the number of computer processors (%NProc) that the job requires. Gaussian03 output (.log) files are log ASCII text files that register the optimization calculation main results in user format. A detailed and commented example is presented in Appendix B. These output files includes information such as the energy of the system, the initial structure in several coordinate systems, intermediate structures specification, final structure specification in several coordinate systems, the number of primitive Gaussians, dipole moment, and spin and charge Mulliken analysis, among others. The file is of considerable length but it includes a briefing of the relevant information at the end, so the user can see the main results at a glance (Figure A-7B). Gaussian03 output files are the most important ones in this study, but due to their complexity, the other file formats are required to extract and visualize the information from them. Finally, to generate plots of the highest occupied molecular orbitals (HOMO) using the gOpenMol interface, the Gaussian03 checkpoint file (.chk) is needed. This is a binary file that includes all the information generated by the optimization calculation. The required information is extracted with the Gaussian03 Unix utilities formchk and cubegen [71] Formchk converts a checkpoint file into an ASCII form that visualization programs can read, while cubegen is a standalone utility that generates cubes of electron density and electrostatic potential.

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140 Figure A-7. Gaussian03 file types. A) Sample input file (.inp) for a geometry optimization run. B) View of end section of an output file (.log) showing the most relevant results of the calculation. A schematic of a complete output file is presented in Appendix B

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APPENDIX B EXAMPLE OF GAUSSIAN03 OUTPUT FILE The following is a commented example of a typical geometry optimization Gaussian03 output file. Some explanatory comments were included for the different sections (link outputs) of the file. The first part of the Gaussian output file states in considerable detail the contents of the license agreement. Gaussian03 is no public domain software and University of Florida user must sign an agreement of non-disclosure. Entering Gaussian System, Link 0=g03 Input=OSISIO2d_3_321.inp Output=OSISIO2d_3_321.log Initial command: /usr/local/Gaussian/g03/l1.exe /storage/Gau-21781.inp -scrdir=/storage/ Entering Link 1 = /usr/local/Gaussian/g03/l1.exe PID= 21782. Copyright (c) 1988,1990,1992,1993,1995,1998,2003, Gaussian, Inc. All Rights Reserved. This is the Gaussian(R) 03 program. It is based on the the Gaussian(R) 98 system (copyright 1998, Gaussian, Inc.), the Gaussian(R) 94 system (copyright 1995, Gaussian, Inc.), the Gaussian 92(TM) system (copyright 1992, Gaussian, Inc.), the Gaussian 90(TM) system (copyright 1990, Gaussian, Inc.), the Gaussian 88(TM) system (copyright 1988, Gaussian, Inc.), the Gaussian 86(TM) system (copyright 1986, Carnegie Mellon University), and the Gaussian 82(TM) system (copyright 1983, Carnegie Mellon University). Gaussian is a federally registered trademark of Gaussian, Inc. This software contains proprietary and confidential information, including trade secrets, belonging to Gaussian, Inc. This software is provided under written license and may be used, copied, transmitted, or stored only in accord with that written license. The following legend is applicable only to US Government contracts under DFARS: RESTRICTED RIGHTS LEGEND Use, duplication or disclosure by the US Government is subject to restrictions as set forth in subparagraph (c)(1)(ii) of the Rights in Technical Data and Computer Software clause at DFARS 252.227-7013. Gaussian, Inc. Carnegie Office Park, Building 6, Pittsburgh, PA 15106 USA 141

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142 The following legend is applicable only to US Government contracts under FAR: RESTRICTED RIGHTS LEGEND Use, reproduction and disclosure by the US Government is subject to restrictions as set forth in subparagraph (c) of the Commercial Computer Software Restricted Rights clause at FAR 52.227-19. Gaussian, Inc. Carnegie Office Park, Building 6, Pittsburgh, PA 15106 USA --------------------------------------------------------------Warning -This program may not be used in any manner that competes with the business of Gaussian, Inc. or will provide assistance to any competitor of Gaussian, Inc. The licensee of this program is prohibited from giving any competitor of Gaussian, Inc. access to this program. By using this program, the user acknowledges that Gaussian, Inc. is engaged in the business of creating and licensing software in the field of computational chemistry and represents and warrants to the licensee that it is not a competitor of Gaussian, Inc. and that it will not use this program in any manner prohibited above. --------------------------------------------------------------Cite this work as: Gaussian 03, Revision B.04, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Pittsburgh PA, 2003. Actual program output specific to a certain calculation starts with a statement of the date, program version, Gaussian revision (here B.4), and system software (here LINUX). Subsequently the keywords used in the input file are repeated together with other general settings such as the amount of main memory needed for the calculations (here 250 MB), and the location of a binary checkpoint file for storage of important results (here OSISIO2d_3_321.chk). The OPT keyword used here specifies geometry optimization and the quantum mechanical method used is UB3LYP. The U in UB3LYP stands for unrestricted, an approach that is used any time one is dealing with open-shell systems such as the spin-triplet of spin-double cluster in this work.

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143 ********************************************* Gaussian 03: x86-Linux-G03RevB.04 2-Jun-2003 19-Oct-2004 ********************************************* %Mem=250MB %Nproc=2 Will use up to 2 processors via shared memory. --------------------------------------#ub3lyp/3-21G* Opt(ModRedundant) Nosymm --------------------------------------The keywords are transformed by Gaussian into a sequence of subroutine calls termed links. The links are given together with the corresponding options set for each link in a proprietary format. 1/14=-1,18=120,26=3,38=1/1,3; 2/9=110,15=1,17=6,18=5,40=1/2; 3/5=5,7=1,11=2,16=1,25=1,30=1,74=-5/1,2,3; 4/7=2/1; 5/5=2,38=5/2; 6/7=2,8=2,9=2,10=2,28=1/1; 7/30=1/1,2,3,16; 1/14=-1,18=20/3(1); 99//99; 2/9=110,15=1/2; 3/5=5,7=1,11=2,16=1,25=1,30=1,74=-5/1,2,3; 4/5=5,7=2,16=3/1; 5/5=2,38=5/2; 7/30=1/1,2,3,16; 1/14=-1,18=20/3(-5); 2/9=110,15=1/2; 6/7=2,8=2,9=2,10=2,19=2,28=1/1; 99/9=1/99; In link101 the program reads in or retrieves from the checkpoint file the structure of the system together with other parameters and prints the structure (in a slightly modified format) together with overall charge and spin multiplicity and the comments supplied in the input file. It is good practice to include the name of the input file in the comments of the job. It is also convenient to keep a log of all the unsuccessful attemps that one might have had to do in order to accomplish a successful initial geometry or a successful geometry optimization. The system chosen here for optimization is a Si 9 H 12 O 3 cluster in its triplet electronic ground-state with one of the oxygen atoms bonded at a dangling bond site.

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144 -----------------OSISIO2d_3_321.inp -----------------Symbolic Z-matrix: Charge = 0 Multiplicity = 3 H H 1 r2 H 1 r3 2 a3 H 1 r4 2 a4 3 d4 0 H 3 r5 1 a5 2 d5 0 H 5 r6 3 a6 1 d6 0 H 6 r7 5 a7 3 d7 0 H 7 r8 6 a8 5 d8 0 H 4 r9 1 a9 2 d9 0 H 9 r10 4 a10 1 d10 0 H 5 r11 3 a11 1 d11 0 H 11 r12 5 a12 3 d12 0 Si 2 r13 1 a13 3 d13 0 Si 3 r14 1 a14 2 d14 0 Si 4 r15 1 a15 2 d15 0 Si 6 r16 5 a16 3 d16 0 Si 8 r17 7 a17 6 d17 0 Si 9 r18 4 a18 1 d18 0 Si 12 r19 11 a19 5 d19 0 Si 19 r20 12 a20 11 d20 0 Si 18 r21 9 a21 4 d21 0 O 20 r22 19 a22 12 d22 0 O 20 r23 19 a23 12 d23 0 O 21 r24 18 a24 9 d24 0 O 22 r25 18 a25 9 d25 0 Variables: r2 2.4392 r3 3.6038 a3 70.9 r4 3.6142 a4 70.29 d4 225.98 r5 3.9791 a5 68.81 d5 221.74 r6 2.4695 a6 62.12 d6 151.1 r7 2.7071 a7 133.13 d7 313.47 r8 2.4716 a8 137.71 d8 8.98 [...] The following ModRedundant input section has been read: X 1 B X 2 B X 3 B X 4 B X 5 B X 6 B X 7 B X 8 B X 9 B X 10 B X 11 B X 12 B X F

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145 In link103 the program selects a coordinate system in which the geometry optimization will be performed. The string GradGradGrad... delimits the output from the Berny optimization procedures. On the first, initialization pass, the program prints a table giving the initial values of the variables to be optimized. Notice that for optimizations in redundant internal coordinates, all coordinates in use are displayed in the table and not merely those present in the molecule specification section. Also, observe that the manner in which the initial second derivative are provided is indicated under the heading Derivative Info (in this case the second derivatives are frozen for all the coordinates of the constrained hydrogen atoms, while they are estimated for all the other coordinates). GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Berny optimization. Initialization pass. ---------------------------! Initial Parameters ! (Angstroms and Degrees) --------------------------------------------------! Name Definition Value Derivative Info. -------------------------------------------------------------------------------! X1 R(1,-1) 0.0 Frozen ! Y1 R(1,-2) 0.0 Frozen ! Z1 R(1,-3) 0.0 Frozen ! X2 R(2,-1) 0.0 Frozen ! Y2 R(2,-2) 0.0 Frozen ! Z2 R(2,-3) 2.4392 Frozen ! X3 R(3,-1) 3.4054 Frozen ! Y3 R(3,-2) 0.0 Frozen ! Z3 R(3,-3) 1.1792 Frozen ! X4 R(4,-1) -2.3644 Frozen ! Y4 R(4,-2) 2.4467 Frozen ! Z4 R(4,-3) 1.2189 Frozen ! X5 R(5,-1) 2.9521 Frozen ! Y5 R(5,-2) 2.47 Frozen ! Z5 R(5,-3) -1.9074 Frozen ! X6 R(6,-1) 4.7063 Frozen ! Y6 R(6,-2) 3.0002 Frozen ! Z6 R(6,-3) -0.2519 Frozen ! X7 R(7,-1) 4.7401 Frozen . D80 D(22,20,21,18) -9.5614 estimate D2E/DX2 ! D81 D(22,20,21,24) -144.608 estimate D2E/DX2 ! D82 D(23,20,21,17) 6.2737 estimate D2E/DX2 ! D83 D(23,20,21,18) -113.1856 estimate D2E/DX2 ! D84 D(23,20,21,24) 111.7678 estimate D2E/DX2 ! D85 D(21,20,22,19) -26.2057 estimate D2E/DX2 ! D86 D(23,20,22,19) 81.7555 estimate D2E/DX2 ! D87 D(21,20,23,16) 50.02 estimate D2E/DX2 ! D88 D(22,20,23,16) -62.7957 estimate D2E/DX2 -------------------------------------------------------------------------------Trust Radius=3.00D-01 FncErr=1.00D-07 GrdErr=1.00D-06 Number of steps in this run= 144 maximum allowed number of steps= 144.

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146 Link202 determines, among others, the symmetry of the system, decides on the symmetry properties that will be used in the actual quantum mechanical calculations and rotates the molecule such that the center of mass is located in the origin of the cartesian coordinate system, the principal axis (if present) points along the z-axis, and the principal plane of symmetry is located in the yz-plane. The resulting orientation is printed as standard orientation, which is then assumed for all information regarding the wavefunction and first and second derivatives of the energy with respect to structural parameters. GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Input orientation: --------------------------------------------------------------------Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z --------------------------------------------------------------------1 1 0 0.000000 0.000000 0.000000 2 1 0 0.000000 0.000000 2.439200 3 1 0 3.405407 0.000000 1.179228 4 1 0 -2.364395 2.446693 1.218924 5 1 0 2.952147 2.469978 -1.907356 6 1 0 4.706251 3.000198 -0.251935 7 1 0 4.740128 3.058559 2.454324 8 1 0 3.254646 2.398272 4.316089 9 1 0 -0.435065 3.862356 4.363306 . 20 14 0 2.346874 5.611148 0.481206 21 14 0 2.013479 4.776043 2.699130 22 8 0 0.991805 5.409444 -0.396979 23 8 0 3.242276 4.465825 -0.239658 24 8 0 2.521716 5.336089 3.523604 --------------------------------------------------------------------Distance matrix (angstroms): 1 2 3 4 5 1 H 0.000000 2 H 2.439200 0.000000 3 H 3.603800 3.631023 0.000000 4 H 3.614200 3.614657 6.267255 0.000000 5 H 4.295809 5.805903 3.979100 6.167640 0.000000 . 21 22 23 24 21 Si 0.000000 22 O 3.321281 0.000000 23 O 3.200415 2.445360 0.000000 24 O 1.118800 4.209154 3.929212 0.000000 Symmetry turned off by external request. Stoichiometry H12O3Si9(3) Framework group C1[X(H12O3Si9)] Deg. of freedom 66 Full point group C1 NOp 1 Rotational constants (GHZ): 0.3685978 0.3641897 0.3294472

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147 Link 301 loads all components necessary for the actual quantum mechanical part of the calculation. Standard basis: 3-21G* (6D, 7F) Integral buffers will be 262144 words long. Raffenetti 2 integral format. Two-electron integral symmetry is turned off. 222 basis functions, 378 primitive gaussians, 222 cartesian basis functions 82 alpha electrons 80 beta electrons nuclear repulsion energy 1911.7608705331 Hartrees. NAtoms= 24 NActive= 24 NUniq= 24 SFac= 1.00D+00 NAtFMM= 60 Big=F One-electron integrals computed using PRISM. NBasis= 222 RedAO= T NBF= 222 NBsUse= 222 1.00D-06 NBFU= 222 Before the actual energy calculation is performed a guess for the wavefunction is obtained using the Harris method. Alternatively, a guess could also be read from the checkpoint or input file. Harris functional with IExCor= 402 diagonalized for initial guess. ExpMin= 9.33D-02 ExpMax= 9.11D+02 ExpMxC= 9.11D+02 IAcc=2 IRadAn= 4 AccDes= 0.00D+00 HarFok: IExCor= 402 AccDes= 0.00D+00 IRadAn= 4 IDoV=1 ScaDFX= 1.000000 1.000000 1.000000 1.000000 of initial guess= 2.0000 The UB3LYP energy of the system is calculated with link 502. The heat of formation of the system is given relative to the corresponding elements in their standard states at 298K and 1bar pressure. The energy is given in atomic units (a.u., Hartree). Requested convergence on RMS density matrix=1.00D-08 within 128 cycles. Requested convergence on MAX density matrix=1.00D-06. Requested convergence on energy=1.00D-06. No special actions if energy rises. Integral accuracy reduced to 1.0D-05 until final iterations. EnCoef did 1 forward-backward iterations Initial convergence to 1.0D-05 achieved. Increase integral accuracy. SCF Done: E(UB+HF-LYP) = -2823.75111938 A.U. after 24 cycles Convg = 0.4891D-08 -V/T = 2.0092 S**2 = 2.0088 Annihilation of the first spin contaminant: S**2 before annihilation 2.0088, after 2.0000 Selected information on the optimized wavefunction is printed along with a Mulliken population analysis in link601.

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148 ********************************************************************** Population analysis using the SCF density. ********************************************************************** Alpha occ. eigenvalues --65.77906 -65.77859 -65.77530 -65.77283 -65.76357 Alpha occ. eigenvalues --65.75541 -65.75457 -65.75179 -65.68641 -19.03988 Alpha occ. eigenvalues --19.03842 -18.98261 -5.30490 -5.29808 -5.29518 . Alpha virt. eigenvalues -2.00237 2.12797 2.18049 2.27291 2.39592 Alpha virt. eigenvalues -2.40934 2.43628 2.54564 2.56050 2.57939 Alpha virt. eigenvalues -2.63137 2.68744 3.11002 3.20990 3.23101 Beta occ. eigenvalues --65.77858 -65.77692 -65.77518 -65.77049 -65.76320 Beta occ. eigenvalues --65.75539 -65.75457 -65.75085 -65.68568 -19.03825 Beta occ. eigenvalues --19.03691 -18.97581 -5.30251 -5.29795 -5.29504 . Beta virt. eigenvalues -2.27530 2.39890 2.41167 2.44024 2.54778 Beta virt. eigenvalues -2.56315 2.58295 2.63355 2.69171 3.11300 Beta virt. eigenvalues -3.22159 3.23421 Condensed to atoms (all electrons): 1 2 3 4 5 6 1 H 0.692854 -0.010127 -0.000067 -0.000078 0.000018 0.000000 2 H -0.010127 0.694888 -0.000106 -0.000094 0.000000 0.000000 3 H -0.000067 -0.000106 0.671425 0.000000 0.000150 -0.000063 . 22 O 0.369732 0.285670 -0.009719 8.021088 -0.020436 0.000005 23 O 0.001505 0.283892 -0.010110 -0.020436 8.042494 0.000014 24 O -0.000007 -0.024351 0.750376 0.000005 0.000014 7.309498 Mulliken atomic charges: 1 1 H -0.021871 2 H -0.020478 3 H 0.004839 . 22 O -0.586526 23 O -0.591986 24 O 0.010565 Sum of Mulliken charges= 0.00000 Atomic charges with hydrogens summed into heavy atoms: 1 1 H 0.000000 2 H 0.000000 3 H 0.000000 . 22 O -0.586526 23 O -0.591986 24 O 0.010565 Sum of Mulliken charges= 0.00000 Atomic-Atomic Spin Densities. 1 2 3 4 5 6 1 H -0.001263 -0.000016 -0.000023 0.000021 0.000000 0.000000 2 H -0.000016 0.000754 -0.000009 -0.000018 0.000000 0.000000 3 H -0.000023 -0.000009 0.043209 0.000000 -0.000045 0.000051 .

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149 22 O 0.002464 -0.028064 -0.001103 0.072536 0.000189 0.000001 23 O 0.000386 -0.024576 -0.003074 0.000189 0.065566 0.000003 24 O 0.000033 0.004674 -0.022670 0.000001 0.000003 0.320169 Mulliken atomic spin densities: 1 1 H -0.000556 2 H 0.000480 3 H 0.031374 . 22 O 0.045404 23 O 0.041377 24 O 0.258965 Sum of Mulliken spin densities= 2.00000 Electronic spatial extent (au): = 13816.3675 Charge= 0.0000 electrons In link 716 the forces acting on the nuclei in the current structure are calculated. Based on this information and extrapolated second derivatives, the geometry of the system is then optimized. Dipole moment (field-independent basis, Debye): X= -1.1949 Y= -3.6322 Z= -0.3149 Tot= 3.8366 Quadrupole moment (field-independent basis, Debye-Ang): XX= -144.4356 YY= -170.6914 ZZ= -145.9753 XY= -14.6396 XZ= -2.1017 YZ= -6.7256 Traceless Quadrupole moment (field-independent basis, Debye-Ang): XX= 9.2651 YY= -16.9906 ZZ= 7.7255 XY= -14.6396 XZ= -2.1017 YZ= -6.7256 Octapole moment (field-independent basis, Debye-Ang**2): XXX= -629.2157 YYY= -1642.3646 ZZZ= -559.1831 XYY= -300.4322 XXY= -527.3560 XXZ= -181.7506 XZZ= -221.5762 YZZ= -529.1159 YYZ= -230.2137 XYZ= -29.5500 Hexadecapole moment (field-independent basis, Debye-Ang**3): XXXX= -4298.3983 YYYY=-13438.7596 ZZZZ= -3339.2862 XXXY= -2229.0299 XXXZ= -767.6104 YYYX= -2646.7614 YYYZ= -2166.1740 ZZZX= -867.5263 ZZZY= -2035.6036 XXYY= -3041.4153 XXZZ= -1276.1823 YYZZ= -2888.0082 XXYZ= -690.3738 YYXZ= -481.6349 ZZXY= -850.6601 N-N= 1.911760870533D+03 E-N=-1.052403736265D+04 KE= 2.798035909220D+03 Isotropic Fermi Contact Couplings Atom a.u. MegaHertz Gauss 10(-4) cm-1 1 H(1) -0.00012 -0.26082 -0.09307 -0.08700 2 H(1) 0.00016 0.36726 0.13105 0.12251 3 H(1) 0.00964 21.55067 7.68981 7.18853 . 22 O(17) 0.05858 -17.75587 -6.33574 -5.92272 23 O(17) 0.05212 -15.79840 -5.63726 -5.26978 24 O(17) 0.05180 -15.70185 -5.60281 -5.23757 -------------------------------------------------------Center ---Spin Dipole Couplings ---3XX-RR 3YY-RR 3ZZ-RR -------------------------------------------------------1 Atom -0.000051 0.000947 -0.000895 2 Atom 0.000187 0.001831 -0.002018 3 Atom -0.002328 0.003961 -0.001633 . 22 Atom -0.071605 0.069397 0.002207 23 Atom -0.049814 0.063602 -0.013788 24 Atom 0.053542 0.290091 -0.343633

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150 -------------------------------------------------------XY XZ YZ -------------------------------------------------------1 Atom 0.001181 0.000838 0.001238 2 Atom 0.002070 -0.000158 0.000002 3 Atom -0.002765 0.001309 0.002136 . 22 Atom 0.053535 -0.029846 -0.049940 23 Atom 0.057857 -0.041620 -0.103531 24 Atom -0.632678 -0.099642 0.034107 ---------------------------------------------------------------------------------------------------------------------------------------Anisotropic Spin Dipole Couplings in Principal Axis System --------------------------------------------------------------------------------Atom a.u. MegaHertz Gauss 10(-4) cm-1 Axes Baa -0.0016 -0.839 -0.299 -0.280 -0.2430 -0.3336 0.9108 1 H(1) Bbb -0.0008 -0.445 -0.159 -0.148 0.8282 -0.5602 0.0158 Bcc 0.0024 1.284 0.458 0.428 0.5050 0.7582 0.4124 Baa -0.0020 -1.089 -0.389 -0.363 0.1406 -0.0758 0.9872 2 H(1) Bbb -0.0012 -0.639 -0.228 -0.213 0.8151 -0.5571 -0.1589 Bcc 0.0032 1.728 0.616 0.576 0.5620 0.8270 -0.0166 Baa -0.0047 -2.528 -0.902 -0.843 0.7352 0.3727 -0.5662 3 H(1) Bbb -0.0006 -0.334 -0.119 -0.112 0.6104 -0.0008 0.7921 Bcc 0.0054 2.862 1.021 0.955 -0.2948 0.9279 0.2281 Baa -0.0020 -1.061 -0.379 -0.354 -0.1667 0.7020 -0.6924 . 22 O(17) Bbb -0.0244 1.763 0.629 0.588 -0.0151 0.4767 0.8789 Bcc 0.1160 -8.394 -2.995 -2.800 0.3105 0.8378 -0.4491 Baa -0.0857 6.204 2.214 2.069 0.0932 0.5426 0.8348 23 O(17) Bbb -0.0739 5.350 1.909 1.785 0.9421 -0.3193 0.1023 Bcc 0.1597 -11.554 -4.123 -3.854 0.3221 0.7769 -0.5410 Baa -0.4929 35.664 12.726 11.896 0.7338 0.5774 0.3580 24 O(17) Bbb -0.3295 23.845 8.508 7.954 -0.2294 -0.2854 0.9305 Bcc 0.8224 -59.508 -21.234 -19.850 -0.6395 0.7650 0.0770 --------------------------------------------------------------------------------------------------------------------------------------------------Center Atomic Forces (Hartrees/Bohr) Number Number X Y Z ------------------------------------------------------------------1 1 0.001146096 0.000418453 0.002297002 2 1 0.000438619 0.000309862 -0.002321601 3 1 -0.000917556 0.012633446 -0.000046790 . 22 8 -0.004878416 -0.006357674 -0.002443661 23 8 0.000940840 -0.011682918 -0.002452035 24 8 0.774765502 0.846384629 1.262745525 ------------------------------------------------------------------Cartesian Forces: Max 1.262745525 RMS 0.282001930 Each subsequent step of the optimization is delimited by lines like these: GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Berny optimization. Internal Forces: Max 0.021671785 RMS 0.018595738 Search for a local minimum. Step number 1 out of a maximum of 20 All quantities printed in internal units (Hartrees-Bohrs-Radians) Swaping is turned off. Second derivative matrix not updated -first step. lowest eigenvalue of the Hessian is 0.1414 RFO step: Lambda= 8.34629057D-04. Linear search not attempted -option 19 set. Iteration 1 RMS(Cart)= 0.03412487 RMS(Int)= 0.00111569

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151 Iteration 2 RMS(Cart)= 0.00126040 RMS(Int)= 0.00000053 Iteration 3 RMS(Cart)= 0.00000036 RMS(Int)= 0.00000000 Variable Old X -DE/DX Delta X Delta X Delta X New X (Linear) (Quad) (Total) R1 1.86973 -0.02167 0.00000 -0.04367 -0.04367 1.82605 R2 1.86973 -0.02167 0.00000 -0.04367 -0.04367 1.82605 A1 1.74579 0.00990 0.00000 0.06169 0.06169 1.80748 Item Value Threshold Converged? Maximum Force 0.021672 0.000450 NO RMS Force 0.018596 0.000300 NO Maximum Displacement 0.038954 0.001800 NO RMS Displacement 0.033876 0.001200 NO Predicted change in Energy=-1.250480D-03 Link103 decides whether the optimization converged or not. Its decision is based on four convergence criteria called maximum force, RMS force (meaning average force), maximum displacement and RMS displacement. For each criterion, the actual value is printed together with the convergence limit. As soon as the actual value falls below the convergence limit, the converged-label changes from NO to YES. Here the optimization is far from convergence as none of the four criteria are fulfilled. Gaussian then jumps back to link103 to set up for the next iteration of energy and gradient calculation. GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Input orientation: --------------------------------------------------------------------Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z --------------------------------------------------------------------1 1 0 0.000000 0.000000 0.000000 2 1 0 0.000000 0.000000 2.439200 . 23 8 0 2.579160 3.976724 0.097904 24 8 0 2.745370 5.492015 4.068503 --------------------------------------------------------------------Distance matrix (angstroms): 1 2 3 4 5 1 H 0.000000 2 H 2.439200 0.000000 3 H 3.603800 3.631023 0.000000 . 22 O 3.369816 0.000000 23 O 2.854058 2.380927 0.000000 24 O 1.624705 4.742915 4.253162 0.000000 Symmetry turned off by external request. Stoichiometry H12O3Si9(3) Framework group C1[X(H12O3Si9)] Deg. of freedom 66 Full point group C1 NOp 1 Rotational constants (GHZ): 0.3712325 0.3571113 0.3219723 Standard basis: 3-21G* (6D, 7F) Integral buffers will be 262144 words long. Raffenetti 2 integral format. Two-electron integral symmetry is turned off.

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152 222 basis functions, 378 primitive gaussians, 222 cartesian basis functions 82 alpha electrons 80 beta electrons nuclear repulsion energy 1893.1559509431 Hartrees. NAtoms= 24 NActive= 24 NUniq= 24 SFac= 1.00D+00 NAtFMM= 60 Big=F One-electron integrals computed using PRISM. NBasis= 222 RedAO= T NBF= 222 NBsUse= 222 1.00D-06 NBFU= 222 Initial guess read from the read-write file: of initial guess= 2.0134 Requested convergence on RMS density matrix=1.00D-08 within 128 cycles. Requested convergence on MAX density matrix=1.00D-06. Requested convergence on energy=1.00D-06. No special actions if energy rises. SCF Done: E(UB+HF-LYP) = -2824.26068137 A.U. after 13 cycles Convg = 0.3699D-08 -V/T = 2.0106 S**2 = 2.0134 Annihilation of the first spin contaminant: S**2 before annihilation 2.0134, after 2.0001 ------------------------------------------------------------------Center Atomic Forces (Hartrees/Bohr) Number Number X Y Z ------------------------------------------------------------------1 1 0.000016299 0.000621727 0.002496590 2 1 -0.000687029 0.002187895 -0.003350990 3 1 0.003233696 -0.006125911 0.005663815 . 22 8 0.000005178 0.000020516 -0.000005388 23 8 0.000022974 -0.000030844 -0.000030268 24 8 0.000003770 0.000015339 -0.000017250 ------------------------------------------------------------------Cartesian Forces: Max 0.044119745 RMS 0.006837095 GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Berny optimization. Internal Forces: Max 0.029278539 RMS 0.003303414 Search for a local minimum. Step number 50 out of a maximum of 144 All quantities printed in internal units (Hartrees-Bohrs-Radians) Update second derivatives using D2CorX and points 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Trust test= 1.30D+00 RLast= 5.15D-03 DXMaxT set to 1.00D+00 Eigenvalues --0.00089 0.00372 0.00972 0.01314 0.03277 Eigenvalues --1000.000001000.000001000.00000 RFO step: Lambda=-3.01087941D-08. Quartic linear search produced a step of 0.42139. Iteration 1 RMS(Cart)= 0.00012700 RMS(Int)= 0.00000008 Iteration 2 RMS(Cart)= 0.00000004 RMS(Int)= 0.00000005 Variable Old X -DE/DX Delta X Delta X Delta X New X (Linear) (Quad) (Total) X1 0.00000 0.00121 0.00000 0.00000 0.00000 0.00000 Y1 0.00000 0.00152 0.00000 0.00000 0.00000 0.00000 Z1 0.00000 0.00168 0.00000 0.00000 0.00000 0.00000 X2 0.00000 -0.00008 0.00000 0.00000 0.00000 0.00000 D87 -2.10410 0.00012 -0.00053 0.00090 0.00037 -2.10373 D88 2.23848 -0.00026 -0.00070 0.00095 0.00026 2.23874 Item Value Threshold Converged? Maximum Force 0.000025 0.000450 YES RMS Force 0.000004 0.000300 YES Maximum Displacement 0.001274 0.001800 YES RMS Displacement 0.000127 0.001200 YES Predicted change in Energy=-2.550445D-08 Optimization completed. -Stationary point found.

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153 After several steps all four convergence criteria are fulfilled. The structure and the wavefunction information for the converged structure are printed. This information includes the redundant internal coordinate definitions, which are given in the second column of the table. The numbers in parentheses refer to the atoms within the molecule specification. For example, the variable R1, defined as R(2,1), specifies the bond length between atoms 1 and 2. The energy for the optimized structure will be found in the output from the final optimization step, which precedes this table in the output file. ---------------------------! Optimized Parameters ! (Angstroms and Degrees) --------------------------------------------------! Name Definition Value Derivative Info. -------------------------------------------------------------------------------! X1 R(1,-1) 0.0 -DE/DX = 0.0012 ! Y1 R(1,-2) 0.0 -DE/DX = 0.0015 ! Z1 R(1,-3) 0.0 -DE/DX = 0.0017 . D85 D(21,20,22,19) -51.1358 -DE/DX = -0.0006 ! D86 D(23,20,22,19) 36.8774 -DE/DX = 0.0003 ! D87 D(21,20,23,16) -120.5561 -DE/DX = 0.0001 ! D88 D(22,20,23,16) 128.2555 -DE/DX = -0.0003 -------------------------------------------------------------------------------GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Input orientation: --------------------------------------------------------------------Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z --------------------------------------------------------------------1 1 0 0.000000 0.000000 0.000000 2 1 0 0.000000 0.000000 2.439200 3 1 0 3.405407 0.000000 1.179228 . 22 8 0 0.974438 5.683461 -0.327222 23 8 0 2.579160 3.976724 0.097904 24 8 0 2.745370 5.492015 4.068503 --------------------------------------------------------------------Distance matrix (angstroms): 1 2 3 4 5 1 H 0.000000 2 H 2.439200 0.000000 3 H 3.603800 3.631023 0.000000 . 24 O 5.399743 3.010877 3.449808 5.529620 3.580723 21 22 23 24 21 Si 0.000000 22 O 3.369816 0.000000 23 O 2.854058 2.380927 0.000000 24 O 1.624705 4.742915 4.253162 0.000000

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154 Symmetry turned off by external request. Stoichiometry H12O3Si9(3) Framework group C1[X(H12O3Si9)] Deg. of freedom 66 Full point group C1 NOp 1 Rotational constants (GHZ): 0.3712325 0.3571113 0.3219723 ********************************************************************** Population analysis using the SCF density. ********************************************************************** Alpha occ. eigenvalues --65.79592 -65.78218 -65.77961 -65.77065 -65.76617 Alpha occ. eigenvalues --65.75838 -65.75411 -65.75241 -65.74816 -19.05796 . Beta virt. eigenvalues -2.56999 2.58775 2.64011 2.70122 3.04247 Beta virt. eigenvalues -3.07188 3.23086 Condensed to atoms (all electrons): 1 2 3 4 5 6 1 H 0.696297 -0.010010 -0.000082 -0.000066 0.000013 0.000000 2 H -0.010010 0.692841 -0.000224 -0.000049 0.000000 0.000000 3 H -0.000082 -0.000224 0.708613 0.000000 0.000229 0.000320 . 22 O 0.346186 0.321110 -0.012542 8.006027 -0.025423 0.000000 23 O 0.038052 0.197931 0.010408 -0.025423 8.156629 0.000004 24 O 0.000088 -0.019458 0.475573 0.000000 0.000004 7.869431 Mulliken atomic charges: 1 1 H -0.024166 2 H -0.020714 3 H -0.014334 . 22 O -0.573538 23 O -0.660483 24 O -0.310216 Sum of Mulliken charges= 0.00000 Atomic charges with hydrogens summed into heavy atoms: 1 1 H 0.000000 2 H 0.000000 3 H 0.000000 . 22 O -0.573538 23 O -0.660483 24 O -0.310216 Sum of Mulliken charges= 0.00000 Atomic-Atomic Spin Densities. 1 2 3 4 5 6 1 H -0.000801 -0.000002 -0.000012 0.000008 0.000001 0.000000 2 H -0.000002 0.000023 0.000009 -0.000003 0.000000 0.000000 3 H -0.000012 0.000009 0.027041 0.000000 -0.000016 0.000090 . 22 O -0.002501 -0.020613 -0.000375 0.058907 0.000493 0.000000 23 O 0.003803 -0.027737 -0.007816 0.000493 0.097992 0.000002 24 O 0.000026 0.002442 -0.036721 0.000000 0.000002 0.787974 Mulliken atomic spin densities: 1 1 H -0.000326 2 H 0.000163 3 H 0.015724 .

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155 22 O 0.035698 23 O 0.069940 24 O 0.737440 Sum of Mulliken spin densities= 2.00000 Electronic spatial extent (au): = 13848.9089 Charge= 0.0000 electrons Dipole moment (field-independent basis, Debye): X= -0.4917 Y= -2.9114 Z= -1.8324 Tot= 3.4750 Quadrupole moment (field-independent basis, Debye-Ang): XX= -142.4744 YY= -166.3022 ZZ= -151.8657 XY= -8.5875 XZ= -8.5891 YZ= -11.8887 Traceless Quadrupole moment (field-independent basis, Debye-Ang): XX= 11.0731 YY= -12.7548 ZZ= 1.6818 XY= -8.5875 XZ= -8.5891 YZ= -11.8887 Octapole moment (field-independent basis, Debye-Ang**2): XXX= -619.3850 YYY= -1616.0681 ZZZ= -613.8510 XYY= -264.6661 XXY= -511.9235 XXZ= -215.3116 XZZ= -243.8888 YZZ= -552.9745 YYZ= -254.8585 XYZ= -52.3854 Hexadecapole moment (field-independent basis, Debye-Ang**3): XXXX= -4196.8582 YYYY=-13371.7449 ZZZZ= -3684.9219 XXXY= -2153.0753 XXXZ= -953.5072 YYYX= -2418.0039 YYYZ= -2329.4342 ZZZX= -1058.3133 ZZZY= -2310.3032 XXYY= -2963.2311 XXZZ= -1377.2064 YYZZ= -3035.5007 XXYZ= -831.3117 YYXZ= -596.1096 ZZXY= -936.1710 N-N= 1.893155950943D+03 E-N=-1.048456408998D+04 KE= 2.794521444675D+03 Isotropic Fermi Contact Couplings Atom a.u. MegaHertz Gauss 10(-4) cm-1 1 H(1) -0.00005 -0.11278 -0.04024 -0.03762 2 H(1) 0.00006 0.13201 0.04711 0.04404 3 H(1) 0.00471 10.52060 3.75401 3.50930 . 22 O(17) 0.05724 -17.34952 -6.19074 -5.78718 23 O(17) 0.01157 -3.50666 -1.25126 -1.16970 24 O(17) 0.09758 -29.57505 -10.55311 -9.86517 -------------------------------------------------------Center ---Spin Dipole Couplings ---3XX-RR 3YY-RR 3ZZ-RR -------------------------------------------------------1 Atom -0.000200 0.000943 -0.000743 2 Atom -0.000145 0.001681 -0.001536 3 Atom -0.001908 0.003063 -0.001155 . 22 Atom 0.056015 -0.058041 0.002025 23 Atom -0.027896 -0.023651 0.051547 24 Atom 0.069880 0.863590 -0.933469 -------------------------------------------------------XY XZ YZ -------------------------------------------------------1 Atom 0.001059 0.000555 0.000992 2 Atom 0.001623 -0.000070 0.000079 3 Atom -0.001901 0.000789 0.001445 . 22 Atom 0.060087 -0.086903 -0.039493 23 Atom 0.114981 -0.135505 -0.147065 24 Atom -1.543051 -0.379540 0.271562 ---------------------------------------------------------------------------------------------------------------------------------------Anisotropic Spin Dipole Couplings in Principal Axis System --------------------------------------------------------------------------------Atom a.u. MegaHertz Gauss 10(-4) cm-1 Axes Baa -0.0012 -0.645 -0.230 -0.215 -0.1257 -0.3636 0.9230

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156 1 H(1) Bbb -0.0008 -0.443 -0.158 -0.148 0.8732 -0.4821 -0.0710 Bcc 0.0020 1.088 0.388 0.363 0.4708 0.7971 0.3781 . Baa -1.2095 87.520 31.229 29.194 0.7136 0.4622 0.5265 24 O(17) Bbb -0.9160 66.279 23.650 22.108 -0.3428 -0.4250 0.8378 Bcc 2.1255 -153.799 -54.879 -51.302 -0.6110 0.7783 0.1449 --------------------------------------------------------------------------------When a Z-matrix was used for the initial molecule specification, this output will be followed by an expression of the optimized structure in that format, whenever possible. Final structure in terms of initial Z-matrix: H H,1,r2 H,1,r3,2,a3 . O,20,r22,19,a22,12,d22,0 O,20,r23,19,a23,12,d23,0 O,21,r24,18,a24,9,d24,0 Variables: r2=2.4392 r3=3.6038 a3=70.9 . r24=1.62470532 a24=117.48711308 d24=-32.67810492 At the very end of each Gaussian calculation, an archive entry in a very compact format is printed to summarize the results. This archive entry is frequently used as supplemental material in publications of theoretical results. 1\1\GINC-THOR\FOpt\UB3LYP\3-21G*\H12O3Si9(3)\PHERRERA\21-Oct-2004\0\\# UB3LYP/3-21G* OPT(MODREDUNDANT) NOSYMM\\OSISIO2d_3_321.inp\\0,3\H,0.,0 .,0.\H,0.,0.,2.4392\H,3.4054068896,0.,1.1792278643\H,-2.3643946794,2.4 634634,4.0085326199,-1.8676148497\H,-1.1265747899,5.5302583028,-0.0337 0145066,4.0685026139\\Version=x86-Linux-G03RevB.04\HF=-2824.2606814\S2 =2.013446\S2-1=0.\S2A=2.000087\RMSD=3.699e-09\RMSF=6.837e-03\Dipole=-0 .1934417,-1.1454323,-0.7209343\PG=C01 [X(H12O3Si9)]\\@ From a database of citations, Gaussian prints one entry together with some timing information. WAR ES EIN GOTT DER DIESE ZEICHEN SCHRIEB? LUDWIG BOLTZMANN, QUOTING GOETHE, ABOUT MAXWELL'S EQUATIONS. Job cpu time: 0 days 22 hours 14 minutes 51.5 seconds. File lengths (MBytes): RWF= 69 Int= 0 D2E= 0 Chk= 11 Scr= 1 Normal termination of Gaussian 03 at Thu Oct 21 07:39:55 2004.

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APPENDIX C CALCULATION PROTOCOL FOR INITIAL GEOMETRY FOR GEOMETRY OPTIMIZATIONS It is not practical to start a geometry optimization calculation using directly the UB3LYP/6-311++G(2d,p) model chemistry, since it would be computationally very demanding and time consuming. Instead, a computational protocol (summarized in Table C-1) that uses molecular mechanics and semi-empirical methods followed by DFT/UB3LYP calculations with small basis sets is used to obtain initial structures that are very close to the one that corresponds to the optimized geometry of the clusters. Table C-1. Calculation protocol used to determine the best initial geometry for the geometry optimization, transition state search and frequency calculations. Step Description Software required Input file Output file 1 Draw initial sketch of structure Options: modify a stored calculation result or start from scratch HyperChem 7 None or .hin .hin 2 Molecular mechanics optimization HyperChem 7 .hin .hin 3 AM1 [132] semi-empirical optimization HyperChem 7 .hin .hin 4 Generate internal coordinates matrix Babel .hin .zmt 5 Generate the Gaussian03 input file from .zmt file Text editor .zmt .inp 6 Density functional theory optimization using the UB3LYP/3-21G* model chemistry Gaussian03 .inp .log The calculation protocol starts with HyperChem 7 where a new structure is built using the sorfaware, or one already available in the calculation database is modified. When the initial draft of the structure is finished, the hydrogen atoms are labeled with their respective numbers in the molecule matrix to impose constraints. These atoms are 157

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158 fixed at their positions by modifying their force constants from the default value of 7 to an arbitrary value of 5000, a. change necessary in HyperChem 7 to guarantee that the hydrogen remain fixed during calculations. Once the boundary conditions and constraints are specified the structure is ready to undergo the geometry optimization process. The molecular mechanics (MM+) calculation is run first, and once it is done it is immediately followed by an AM1 optimization [132] The resulting HyperChem 7 output file (in .hin long format) is changed with the Emacs text file editor into a .hin short file, converted into a .zmt file using Babel and modified again to insert an input section to generate the Gaussian03 input file (Figure A-7A). This file is used by Gaussian03 [71] to run the UB3LYP/3-21G* calculation and the Gaussian log file is read using Molden. In this visualization program, the user has the option to generate a .xyz file of the final structure. This .xyz file is transformed into a .zmt and a .hin short file using Babel. While the .zmt file is used to generate a Gaussian03 input file for the actual 6-311++G(2d,p) calculation, the .hin file is used to generate an image of the initial structure.

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APPENDIX D CALCULATION PROTOCOL FOR INITIAL GEOMETRY FOR TRANSITION STATE SEARCHES We used Gaussian03 options OPT(QST2) and OPT(QST3) to search for transition state structures [71] (Here OPT stands for Optimization calculation). Both methods search for a transition structure using the synchronous transit-guided quasi-Newton (STQN) method [ 74 82 ] and require at least the reactant and product structures in the input file. OPT(QST3) requires a third structure, corresponding to an initial guess for the transition state. It is imperative that the atoms in all the structures are labeled exactly in the same order. All the transition state calculations are run with the UB3LYP/3-21G* level of theory to speed the convergence process. We specified OPT(NoEigen) so the simulation did not stop after two iterations. This option overrides the default option OPT(Eigen), which searches for the number of negative eigenvalues after each iteration. Given the structural configuration of the Si 9 H 12 clusters, and the constraints imposed on them, the initial iterations of this STQN method always result in more than one negative eigenvalue. OPT(CalcAll) is used to calculate the Hessian matrix after each iteration. We do not update the Hessian (as we do with the geometry optimization calculations because of computation economy) because it is required that the description of the force fields be as accurate as possible so the transition state structure can be located more precisely [71] OPT(QST2) is generally straight forward. Having optimized structures for the reactant and the product of the reaction of interest, it is simple to combine the two in a Gaussian03 input (.inp) file and run the calculation. 159

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160 Figure D-1. Input file for a transition state search that uses the OPT(QST2) option. In this case, there was not need to reorder the atoms in the structures.

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161 Figure D-2. Diagram of the structures of reactant and product used to search for a transition state structure using OPT(QST2) or OPT(QST3) in Gaussian03. Since the atom order is different for the two structures, one of them must be reordered using Molden. Insert: Molden commands used to change the Z-matrix order for the structures. Sometimes, however, the bonds being broken (or formed) must be increased by approximately 1.5 times so both the reactant and product initial structures are closer to the estimated transition state geometry (Figure D-1). In some cases, the order of the atoms (especially that of the adsorbed atoms) differs considerably between the reactant and the product. In such cases, the program Molden is used to visualize both structures

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162 and reorder one of them so its atom sequence matches that of the other structure, by using the powerful Z-matrix command of this visualization software. Notice that the latter the file depicted in Figure D-3 is analogous to the sample file presented in Figure A-5 and that the only difference is the notation used to specify the variables. One may refer to the notation used to name the variables in Figure A-5 as the Babel notation, while the one used in Figure D-3 would be the Molden notation. Finally, the result is a Gaussian03 input file that has the reactant (product) structure expressed in Babel notation and the product (reactant) expressed in Molden Notation (Figure D-4). Figure D-3. Z-matrix (.zmt) file generated by Molden. This file is completely analogous to the Z-matrix presented in Figure A-5. The only difference the two files is the notation used for the variables. While the Babel notation used in Figure A-5 is based on the number of bond length, bond angle or dihedral angel, the Molden notation is based on the atomic symbol of the atoms involved in the bond length, bond angle or dihedral angle.

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163 Figure D-4. Gaussian03 input file for a transition state search where the product structure had to be reordered using Molden. This file is analogous to the Gaussian03 input file presented in Figure D-1.

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164 The OPT(QST3) calculations are performed when the OPT(QST2) initialization does not converge. OPT(QST2) has initial convergence problems when the reactant and product structures differ significantly. In this case, an intermediate structure is generated using HyperChem 7. The reactant and product structures are merged in HyperChem 7, where either one of them is labeled as the reactant and the other, as the product. Once the structures are labeled, the option Reaction Map is used to match the atom numbers in both structures. The user must match pairs of atoms individually, making sure that their positions in space have meaning for a reaction path following algorithm. Figure D-5. Process used to generate an initial guess for the transition state search using OPT(QST3) in Gaussian03. HyperChem 7 is the visualization software used. A) Reactant structure is opened and merged with the product. B) Each structure is selected and named as either REACTANT or PRODUCT. C) A Reaction Map is created by mapping the two structures. The factor lambda tells Hyperchem 7 how close the structure will be to that of the reactant (lambda = 0) or the product (lambda = 1). D) The atoms are matched one by one. At the end of the process, Hyperchem 7 generates the TS structure.

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165 As in the case of OPT(QST2), the order of the atoms must be exactly the same in both structures. After the reaction map is completed, HyperChem automatically generates the structure of the transition state. The end structure must be saved in a different .hin file, that will be transformed into a .zmt file with Babel. The typical Gaussian03 input file that uses OPT(QST3) is similar to that which uses OPT(QST2), but with a third structure following that of the product. One must be aware that the same constrains must be imposed on the same atoms in both the reactant and product. These constraints, however, are not accepted for the transition state structure in the Gaussian03 calculation. Nonetheless, since Gaussian03 [71] would not run the calculation until some constraint are specified for the transition state structure (because OPT(ModRedundant) is specified), we added a fixed atom constraint for all our transtition state structures. The atom chosen was the first one specified in the Z-Matrix of the Gaussian03 input file, since by keeping it fixed, we do not alter the calculation.

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BIOGRAPHICAL SKETCH Paulo Emilio Herrera-Morales was born on April, 13, 1971, in Guatemala City, Guatemala. He is the son of Victor Manuel Herrera and Emma Morales de Herrera. After graduating from high school in 1988, he pursued his studies at Universidad de San Carlos de Guatemala, and earned a bachelors degree in chemical engineering in 1994. After graduating, he worked in industry in Guatemala. From 1995 to 1999, he was a faculty member in the Universidad Rafael Landvar College of Engineering in Guatemala. Paulo Emilio was very involved in academic and extracurricular activities in Universidad de San Carlos and Universidad Rafael Landvar, becoming the President of Chemical Engineering Students in 1992, and Faculty Advisor of the Fifth Latin-American Chemical Engineering Student Meeting Executive Board in 1999. In 1999, he received an overseas assistantship from the University of Florida (UF), and he began his Ph.D. degree in surface science technology in the Chemical Engineering Department at the UF under the supervision of Dr. Jason F. Weaver. He spent four semesters in Gainesville taking classes before he began his research in the Surface Science Laboratory, at the Nuclear Sciences Building. He worked as a graduate assistant and as a teaching assistant while at UF. Paulo Emilo is engaged to Sonia Mara Garca, whom he plans to marry, once he finishes this Ph.D. 173


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QUANTUM CHEMISTRY CALCULATIONS OF THE REACTIONS OF GASEOUS
OXYGEN ATOMS WITH CLEAN AND ADSORBATE-TERMINATED Si(100)-(2x1)














By

PAULO EMILIO HERRERA-MORALES


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Paulo Emilio Herrera-Morales
































To my late grandmother, Catalina Caal Prado.















ACKNOWLEDGMENTS

I wish to express my sincere appreciation to Dr. Jason F. Weaver (my supervisory

committee chair) for all his encouragement and advice throughout my Ph.D. program. I

would like to extend my deepest gratitude to the members of my advisory committee (Dr.

Gar B. Hoflund, Dr. Vaneica Y. Young, and Dr. Fan Ren) for their assistance, technical

support, and guidance. I also thank Dr. David Micha, Dr. Erick Deumens, and Dr. Adrian

Roitberg (from the Quantum Theory Project) and Dr. Helena Hagelin-Weaver (from the

Chemical Engineering Department) for their valuable support with quantum chemistry

matters. I would like to thank my colleagues at Universidad Rafael Landivar in

Guatemala, for their support and encouragement.

I would like to express my sincere appreciation and gratitude to my parents, Victor

Manuel Herrera and Emma Morales de Herrera; and to my sister, Servanda Virginia

Herrera, for supporting me and my parents for the duration of this work. Special thanks

go to my fiancee Sonia Maria Garcia, who supported me more than once. She possesses

admirable patience.

I would like to thank the Quantum Theory Project of the University of Florida for

the use of its computational facilities; and Brad Shumbera, for his excellent work in

setting up the computer network for our research laboratory. I also want to recognize the

other Weaver Research Group members, Alex A. Gerrard, Jau Jiun Chen, Sunil

Devarajan and Heywood Kan, for their help with different issues through the years.
















TABLE OF CONTENTS


page

A C K N O W L ED G M EN T S ................................................ ........................................... iv

LIST OF TABLES .............................................. viii

LIST O F FIG U RE S .............. ......................... ........................... ....................... .. .. .... .x

A B S T R A C T .................................................................................................... ........ .. x iii

CHAPTER

1 IN T R O D U C T IO N .................................................................. .. ... .... ............... 1

1.1 Research Objective ..................... .. ........ ..... ..........................
1.2 T he Si(100)-(2x l) Surface .................... ................................. ................. ............... 1
1.3 Chemistry of Atomic Oxygen on Si(100)-(2xl).......................... ...............3......
1.4 Important Silicon-based Materials for Microelectronics Industry.......................4...

2 THEORETICAL BACKGROUND OF QUANTUM CHEMISTRY
C A L C U L A T IO N S ......................................................................... .. . .. ...............8

2 .1 Introdu action ...................... .... ... ..................... ..................... ............ .. .8
2.2 Energy Functionals and Early Density Functional Theory (DFT) .................... 10
2.3 Hohenberg-Kohn-Sham Density Functional Theory (KS-DFT) .......................14
2.3.1 Hohenberg-Kohn Lemmas and Exchange-Correlation Functional
D e fin itio n .............. .... ........ ..................................................................... 1 5
2.3.2 Kohn-Sham Self-consistent Field Method .............................................17
2.3.3 Exchange and Correlation Functional Approximations ..............................18
2.3.3.1 Local density approximation (LDA)........................................ 19
2.3.3.2 D ensity gradient corrections...................................... ................ 20
2 .3.3.3 H ybrid m ethods ......................................................... ................ 22
2.3.3.4 E m pirical D FT m ethods ................ .......... .......... ...................... 24
2.3.4 Kohn-Sham Density Functional Theory Computational Chemistry ...........25
2.4 B asis Sets ................. ........................................ ............... 27
2.5 Geometry Optimization and Transition State Search Calculations...................31
2.5.1 Geometry Optimization of Energy Minima ..........................................33
2.5.2 T transition State Searches ...................................................... ................ 38









3 NITROGEN ATOM ABSTRACTION FROM Si(100)-(2xl)...............................41

3 .1 In tro d u ctio n ........................................................................................................... 4 1
3.2 C om putational A pproach.................................... ....................... ................ 43
3 .3 R e su lts...................... ... .. .............................. ........................................... .. 4 5
3.3.1 Bonding Configurations of a Nitrogen Atom on Si(100)-(2xl)..............45
3.3.2 Nitrogen Abstraction by a Gas-Phase Oxygen Atom...............................46
3.3.3 Abstraction of N Adsorbed at the Dangling Bond [R(ad)] .....................48
3.3.4 Abstraction of the Nitrogen Bonded Across the Dimer [R(db)] .............51
3.3.5 Abstraction of the Nitrogen Bonded at a Backbond [R(bb)]...................54
3.3.6 Abstraction from the N=Si3 Structure [R(sat)]..................................... 59
3 .4 D iscu ssio n ............................................................................................................. 6 3

4 CHEMISTRY ON Si(100)-(2xl) DURING EARLY STAGES OF OXIDATION
W ITH O (3P) .................................................................................. . .................66

4 .1 In tro d u ctio n ...........................................................................................................6 6
4 .2 T heoretical A approach .......................................... ......................... ................ 69
4 .3 R esu lts an d D iscu ssion ....................................................................... ............... 70
4.3.1 Structures with One Adsorbed Oxygen Atom (Oi-Si9H12) .....................77
4.3.2 Structures with Two Adsorbed Oxygen Atoms (02-Si9H12)...................80
4.3.3 Structures with Three Adsorbed Oxygen Atoms (03-Si9H12) .................86
4 .3 .4 O x y g en In sertio n ........................................................................................9 1
4.3.4.1 Therm odynam ic considerations ................................ ................ 91
4.3.4.2 K inetic considerations ............................................... ................ 94

5 ATOMIC OXYGEN INSERTION INTO ETHYLENE- AND
ACETYLENE-TERMINATED Si(100)-(2xl).............................................100

5 .1 In tro du ctio n ......................................................................................................... 10 0
5.2 Theoretical A approach ................. ............................................................ 102
5.3 R results and D iscu ssion ........................................................... ...................... 104
5.3.1 Relative Energies of Oxidized Ethylene-Covered Si(100) Clusters.........110
5.3.2 Relative Energies of Oxidized Acetylene-terminated Surfaces .............115
5.3.3 Oxygen Insertion M echanism s....... .......... ....................................... 119
5.3.3.1 Therm dynamic considerations ........................ ................... 119
5.3.3.2 K inetic considerations ....... ........... ....................................... 120

6 SUMMARY AND CONCLUSIONS.......... ..........................124

6.1 Nitrogen Atom Abstraction from Si(100)-(2xl) by Gaseous Atomic Oxygen ..124
6.2 Initial Step of Si(100)-(2xl) Oxidation by Gaseous Atomic Oxygen..............125
6.3 Oxidation of C2H2- and C2H4-covered Si(100)-(2xl) by Gaseous Atomic
O x y g e n ............................................................................................................ . 12 7









APPENDIX

A QUANTUM CHEMISTRY SOFTWARE ............................................................ 128

A I1 F ile F orm at .xyz .... ... ......................................... ....................... . .......... 133
A .2 F ile F orm at .zm t ............................................. .. ..................... ... ......... ............ 33
A.3 HyperChem Files (Format .hin)................ ............................................ 136
A.4 Gaussian03 Files (Formats .inp, .log and .chk) ...................... ................... 138

B EXAMPLE OF GAUSSIAN03 OUTPUT FILE ........................... ..................... 141

C CALCULATION PROTOCOL FOR INITIAL GEOMETRY FOR GEOMETRY
OPTIMIZATIONS ............................. .......... ........ ............... 157

D CALCULATION PROTOCOL FOR INITIAL GEOMETRY FOR TRANSITION
STATE SEARCHES ................................... .............................. 159

LIST O F R EFEREN CE S ... ................................................................... ............... 166

BIOGRAPH ICAL SKETCH .................. .............................................................. 173















LIST OF TABLES


Table page

1-1 Selected properties of silicon dioxide ................................................... ...............6...

2-1 Approximations made to correct Thomas-Fermi-DFT before the development of
Kohn-Sham DFT ...................................... ........ ................... 13

2-2 Variables used to express the approximate exchange-correlation functionals
E x c [p (r )] ............................................................................................................... ... 1 8

4-1 Suboxide penalty energies for various silicon oxidation states...............................75

4-2 Penalty energies of O i-Si9H 12 isom ers ................................................ ................ 78

4-3 Penalty energies of 0 2-Si9H 12 isom ers ................................................ ................ 81

4-4 Penalty energies of 0 3-Si9H 12 isom ers .....................................................................87

4-5 Energy barriers for migration of oxygen atoms in O-Si9H12-Od structures from a
dangling bond site to a backbond or dimer bond site in the spin-triplet state..........96

5-1 Ei(SiSi) structure and energy of adsorption compared to those of reported
dimerized C2H4-terminated Si(100) structures.......................... ...................107

5-2 Ai(SiSi) structure and energy of adsorption compared to those of reported
dimerized C2H2-terminated Si(100) structures.......................... .................. 107

5-3 Calculated energy of formation of the different oxidized bridges that form when an
oxygen atom inserts into the C2H2- and C2H4-terminated Si9H12 clusters ...........109

5-4 Penalty energies of spin-singlet O-C2H4-Si9H12 isomers................................. 110

5-5 Penalty energies of spin-triplet O-C2H4-Si9H12 isomers ..................................... 111

5-6 Penalty energies of spin-singlet O-C2H2-Si9H12 isomers..................................115

5-7 Penalty energies of spin-triplet O-C2H2-Si9H12 isomers ..................................116

A-i Quantun chemistry programs used in this work......................... ...................132









A-2 File formats used for proper handling of the coordinates systems and results in our
geometry optimization calculations ....... ....... ...................... 132

A-3 Gaussian03 input files sections ....... ........ .......... ...................... 138

C-1 Calculation protocol used to determine the best initial geometry for the geometry
optimization, transition state search and frequency calculations .........................157















LIST OF FIGURES


Figure page

1-1 Si(100) -2x1 surface show ing the dim er row s. ............................... ..................... 2

1-2 Metal-oxide-semiconductor field-effect transistor (MOSFET). ............................. 6

2-1 Flowchart of the Kohn-Sham SCF procedure ............................... ..................... 26

2-2 Tw o-dim ensional potential energy surface ......................................... ................ 31

2-3 Flowcharts for a quasi-Newton algorithm for geometry optimization.................. 36

3-1 The Si9H12 cluster used in our UB3LYP calculations.........................................47

3-2 Structural inform ation of N -Si9H 12 clusters .................................. ..................... 48

3-3 Reaction pathway for the nitrogen abstraction from R(ad)................................ 49

3-4 Critical point structures of the nitrogen abstraction from R(ad)........................... 50

3-5 Reaction pathway of nitrogen abstraction from R(db) .............................................52

3-6 Critical point structures of the nitrogen abstraction from R(db)...........................53

3-7 Reaction pathways for the abstraction of the nitrogen atom inserted in a Si-Si
b a c k b o n d ................................................................................................................. 5 6

3-8 Molecular precursors formed after O-chemisorption onto R(bb) .........................57

3-9 Structures formed during nitrogen abstraction from MP(bbt) .............................. 58

3-10 Structures involved in the nitrogen abstraction from MP(bbs)............................. 59

3-11 Reaction pathways for N-abstraction from (and O-atom adsorption on) R(sat)......61

3-12 Structures formed during the direct nitrogen abstraction from R(sat) ..................62

3-13 Structures involved in the O-atom chemisorption on R(sat)............................... 62

4-1 Structural information and highest occupied molecular orbital plot of clean Si9H12
clu sters .................................................................................................... ........ .. 7 4









4-2 R elative energies of O 1-Si9H 12 isom ers............................................... ................ 77

4-3 Structural characteristics of Si9H12 clusters with one oxygen atom......................79

4-4 R elative energies of 0 2-Si9H 12 isom ers............................................... ................ 83

4-5 Structural information of dangling bond isomers with two oxygen atoms adsorbed
o n S i(10 0 ) .............................................................................. .. ............... 8 4

4-6 Structural inform ation of 02-Si9H 12 isom ers....................................... ................ 85

4-7 Relative energies of optimized structures for three oxygen atom incorporation into
S i( 1 0 0 ) .................................................................................................................. ... 8 8

4-8 Structural information of the dangling bond 03-Si9H12 isomer DB3(OSiOSiOd) ...89

4-9 Structural characteristics 03-Si9H 12 isom ers ....................................... ................ 90

4-10 Qualitative representation of a minimum energy path for oxygen atom
incorporation on Si(100)-(2xl) based on the relative energy of the structures........ 93

4-11 Structural information of transition state structures for oxygen migration in
0 1-Si9H 12 structures.. .................................................................................... 95

4-12 Structural information of transition state structures for oxygen insertion in
02-Si9H 12 structures.. .................................................................................... 98

4-13 Structural information of transition state structure TS3(OSiOSiOd-OSiOSiO) ......98

4-14 Proposed preferred path for initial steps of oxidation of Si(100)-(2xl) by oxygen
ato m s ...................................................................................................... ....... .. 9 9

5-1 Structural information of optimized clusters for olefin-covered Si(100)-(2xl).....106

5-2 Relative energies of spin-singlet O-C2H4-Si9H12 isomers....................................111

5-3 Relative energies of spin-triplet O-C2H4-Si9H12 isomers..................................112

5-4 Structural information of O-C2H4- Si9H12 isomers ..................... ...................114

5-5 Structural information of O-C2H2-Si9H12 isomers ...................... ...................117

5-6 Relative energies of spin-singlet O-C2H2-Si9H12 structures.................................118

5-7 Relative energies of spin-triplet O-C2H2-Si9H12 isomers..................................119

5-8 Structural information of clusters involved in oxygen insertion pathways on C2H2-
and C2H4-terminated Si(100) surfaces ........................................121









5-9 Potential energy surfaces for insertion of an oxygen atom on an C2H2 covered
Si(100)-(2x l) surface ............. ............... ................................................ 123

5-10 Potential energy surfaces for insertion of an oxygen atom on an C2H4 covered
Si(100)-(2x l) surface ............. ............... ................................................ 123

A -i H yperChem 7 interface .................. ............................................................ 129

A-2 Molden visualization interface and main command screens................................130

A-3 gOpenM ol visualization interface ....... ...... ...... ..................... 131

A-4 Example of a .xyz file describing a Si9H1203 cluster.................. ...................134

A-5 Z-M atrix file describing a Si9H12+xCy cluster ....... ... ................. ................... 135

A-6 HyperChem 7 files describing a Si9H1203 cluster....................... ...................137

A-7 Gaussian03 file types ............................................................. 140

D-1 Input file for a transition state search that uses the OPT(QST2) option ............. 160

D-2 Diagram of the structures of reactant and product used to search for a transition
state structure using OPT(QST2) or OPT(QST3) in Gaussian03 ........................161

D-3 Z-matrix (.zmt) file generated by M olden ........................................ ............... 162

D-4 Gaussian03 input file for a transition state search where the product structure had to
be reordered using M olden.................................... ...................... ............... 163

D-5 Process used to generate an initial guess for the transition state search using
OPT(Q ST3) in G aussian03 .......................................................... 164















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

QUANTUM CHEMISTRY CALCULATIONS OF THE REACTIONS OF GASEOUS
OXYGEN ATOMS WITH CLEAN AND ADSORBATE-TERMINATED Si(100)-(2xl)

By

Paulo Emilio Herrera-Morales

May 2005

Chair: Jason F. Weaver
Major Department: Chemical Engineering

Reactions of gas-phase radicals at solid surfaces are fundamental to the

plasma-assisted processing of semiconductor materials. In addition to adsorbing

efficiently, radicals incident from the gas-phase can also stimulate several types of

elementary processes before thermally accommodating to the surface. Reactions that

occur under such conditions may be classified as non-thermal; and examples include

atom insertion, direct-atom abstraction and collision-induced reaction and desorption.

Indeed, non-thermal surface reactions play a critical role in determining the enhanced

surface reactivity afforded by plasma processing. Advancing the fundamental

understanding of radical-surface reactions is therefore of considerable importance to

improving control in plasma-assisted materials processing. Our study used quantum

chemical calculations to investigate the interactions of gas-phase oxygen atoms [0(3P)]

with clean and adsorbate-modified Si(100)-(2xl) surfaces. We carefully studied reaction

pathways for the oxygen insertion, adsorbate abstraction and adsorbate migration of these









species on reaction with 0(3P) using density functional theory (DFT) and silicon clusters.

Knowledge of these reaction pathways helps determine the viability of certain reactions

on Si(100)-(2xl) by gas-phase oxygen atoms in plasma-assisted material processing in

general.

Another goal of our study was to better understand the chemistry of certain

ultrathin silicon-based films. Silicon dioxide has been at the center of the

microelectronics industry given its multiple electrical properties. However, as the size of

the devices keeps shrinking according to Moore's law predictions of number of

transistors, the demands imposed on this material necessitate a detailed understanding of

its chemistry at the molecular level urging researchers to look for several alternatives.

Materials such as silicon oxynitride (SiOxNy) and silicon oxycarbide (SiOxCy) have been

suggested as possible silicon dioxide substitutes as gate dielectric and interface dielectric,

respectively. Silicon oxynitride can be obtained by either inserting nitrogen atoms into

silicon dioxide or oxygen atoms into a silicon nitride film, resulting in ultrathin layers

that enhance properties of pure silicon dioxide as a gate dielectric. Silicon oxycarbide is

one of the products that evolved as an alternative for silicon dioxide as a result of the

recent trend of organic functionalization of the silicon surfaces (a process by which

hydrocarbon molecules are chemisorbed on top of the silicon surfaces to combine the

semiconductor properties of silicon with the organic functionality of carbon). Thus we

used quantum chemistry calculations to study the initial steps of oxidation of the clean

Si(100)-(2xl) surface, of the abstraction of nitrogen atoms, and for insertion of gaseous

oxygen atoms on acetylene- and ethylene-terminated Si(100).














CHAPTER 1
INTRODUCTION

1.1 Research Objective

Our main objective was to advance fundamental understanding of the reactions of

gaseous oxygen atoms with clean and adsorbate-modified Si(100)-(2xl) surfaces,

motivated by an interest in gaining insights into the possible reaction pathways during the

initial steps of oxidation. Our work was also motivated by the potential benefits of

fabricating ultrathin silicon-based films and by the need to precisely control the

properties of such films. Density functional theory (DFT) quantum chemistry

calculations were run to investigate the possible reactions. This chapter provides a brief

introduction to the Si(100)-(2xl) surface, an overview of the current state of knowledge

of the oxidation process of this surface, and a brief description of the industrially

important silicon dioxide, silicon oxynitride, and silicon oxycarbide, semiconductor

materials whose production can be deeply impacted by knowledge of oxygen-atom

chemistry at the molecular level. Density functional theory and the background of the

quantum chemistry calculations are explained in detail in Chapter 2.

1.2 The Si(100)-(2xl) Surface

Because of its numerous industrial applications and model character for

semiconductor surface science, the industrially important Si(100) surface has been widely

studied, dating back to the early work of Schlier and Farnsworth [1]. It is known that

when first formed, the surface reconstructs by forming silicon dimmers, and that the

surface adopts a (2xl) orientation (where 2x1 designates the new periodicity of the









surface atoms). The surface forms these dimers so it can reduce the number of dangling

bonds per surface atom, from two in the bulk-terminated (lxl) surface to one in the

reconstructed (2xl), and also because their formation lowers the surface energy by about

1.0 eV [2] (Figure 1-1).













Figure 1-1. Si(100) -(2xl) surface showing the dimer rows.

It is generally accepted [3-8] that the dimers are asymmetric (buckled) and consist

of an sp2-like bonded down atom, that moves closer to the plane of its three nearest

neighbors; and an up atom that moves away from the plane of its neighbors, and

possesses an s-like dangling bond. This rehybridization process from the bulk sp3 to the

sp2-like configuration of the silicon dimer atoms is accompanied by a charge transfer

from the down to the up atom. To minimize this electrostatic effect (and to relieve local

stress) the direction of buckling alternates within the dimer rows. The silicon dimer bonds

in Si(100)-(2xl) are similar to a C=C double bond [9]. The organic double bond consists

of two types of interactions: a a bond with symmetry around the axis connecting the two

atoms, and a sr bond with a nodal plane along the axis. Similar interactions exist in the

Si-Si dimer bonds, with the difference that the ar interaction is sufficiently weak that the

dimer is not held in a symmetric configuration, thus adopting the buckled asymmetry that

constitutes its more recognizable structural feature [10] (Figure 1-1).









1.3 Chemistry of Atomic Oxygen on Si(100)-(2xl)

Among the most reliable sources of gas-phase radicals such as ground-state atomic

oxygen [0(3P)] are plasmas, which typically contain a variety of highly reactive, unstable

species (radicals and charged particles) that, in addition to adsorbing efficiently, can also

stimulate several types of elementary processes before thermally accommodating to the

surface. These elementary processes include atom insertion, direct-atom abstraction, and

collision-induced reaction and desorption. It is generally accepted that atom abstraction

can occur in several ways, ranging from the Eley-Rideal mechanism (ER) of direct

abstraction (in which an atom is abstracted directly from the surface in a single collision

with an incident species, without reaching thermal equilibrium with the surface) [ 11]; and

the Langmuir-Hinshelwood mechanism (LH) of collision-induced reaction and

desorption (in which the incident species reaches thermal equilibrium with the surface

before reacting with the adsorbate and eventually evolving into the gas-phase) [12]. An

intermediate reaction mechanism for atom abstraction is the the hot atom mechanism

(HA) in which the incident species experiences multiple collisions with the surface, but

does not fully thermalize before the reactive encounter [13].

To study the aforementioned reaction mechanisms, it is necessary to have a

convenient source for all types of radicals. Unfortunately, such a source exists only for

hydrogen radicals, which can be produced efficiently by thermal dissociation of

molecular H2 over a hot filament [14]. Thus, abstraction reactions on silicon have been

widely investigated only for hydrogen atoms [15-16]; and several kinetic models for the

H/Si(100) system have been proposed, most of them favoring the ER mechanism [17-18],

the hot atom precursor scenario [19-21], or both [22-25]. Other systems, however (such

as those of reactions of oxygen atoms on metal and semiconductor surfaces) have not









been studied as often, despite their industrial and technological importance.

Nevertheless, a few studies on oxidation of silicon surfaces by gaseous oxygen atoms

have been reported. Engel et al. [26-31] performed a detailed kinetic study of oxidation

of clean Si(100) by gaseous oxygen-atoms and reported that O-atom oxidation is facile

compared with that caused by molecular oxygen, and that several monolayers of oxide

can be formed efficiently via the direct insertion of O atoms into near-surface bonds.

Orellana et al. [32] studying the effect of the spin-state of the surface on the oxidation of

Si(100) by 02, found that the oxidation proceeds with a high probability along the triplet

potential energy surface (PES), and concluded that the spin state of the surface may be an

important factor in determining the quality of the Si/Si02 interface.

Despite much study, the reaction mechanism for abstracting hydrogen atoms from

Si(100)-(2xl) is still unclear. So the oxygen-atom chemistry on Si(100) is far from

understood. Thus, we used quantum chemical calculations to investigate interactions of

the O-atoms with clean Si(100), spin-doublet N-covered Si(100), and spin-singlet and

spin-triplet C2H2- and C2H4-terminated Si(100) surfaces. We focused on determining the

reaction pathways leading to oxygen insertion or adsorbate direct abstraction from these

surfaces, and on the effects that the spin-state has on the processes. This information

may be important for the microscopic understanding of the initial steps of the silicon

oxidation process by gaseous oxygen atoms. It may also be relevant for developing

improved techniques for the fabrication of ultrathin layers of silicon-based industrial

materials.

1.4 Important Silicon-based Materials for Microelectronics Industry

Semiconductors are extremely useful for electronic purposes, because they can

carry an electric current by electron propagation or hole propagation, and because this









current is generally unidirectional, and the amount of current may be influenced by an

external agent [33]. The most widely used insulator is silicon dioxide (SiO2), which is

also one of the most commonly encountered substances in daily life. Modern integrated

circuit industry was made possible by the unique properties of silicon dioxide. It is the

only native oxide of silicon that is stable in water and at elevated temperatures, is an

excellent electrical insulator, is a mask to common diffusing species, and is capable of

forming a nearly perfect electrical interface with its substrate (Table 1-1). SiO2 has

strong, directional covalent bonds, and has a well-defined local structure in which four

oxygen atoms are arrayed at the corners of a tetrahedron around a central silicon atom.

The Si-O-Si bond angles are essentially the tetrahedral angle (109.40), and the Si-O

distance is 1.61 A with very little variation. It is these siloxane-bridge bonds between

silicon atoms that give SiO2 many of its unique properties [33].

The Si/SiO2 interface is one of the key components of the commonly used

metal-oxide-semiconductor field-effect transistor (MOSFET), the building block of

integrated circuits (Figure 1-2). This type of transistor continues to be the predominant

device in ultralarge scale integrated circuits (USIC) because it is simple to scale down

[34]. MOSFETs consists of a source and a drain (regions of doped silicon), a gate

dielectric (an extremely thin silicon dioxide layer), and a gate electrode (a layer of

polycrystalline silicon), rendered conductive by heavy doping, to bias the gate dielectric.

Other MOSFET elements include thin-metal interconnect layers (made of Cu) to connect

the transistor electrically to other parts of the circuit, and SiO2 dielectric layers to provide

electrical isolation between the Cu interconnects and other devices. Basically, MOSFETs

are switches that allow current to flow from the source to the drain only when the gate

electrode supplies the appropriate bias voltage through the gate dielectric [34].









Table 1-1. Selected properties of silicon dioxide.

Oxide native to silicon
Low interfacial defect density (-1010eV-cm-2)
Melting point = 1713C
Energy gap = 9 eV
Resistivity 1015 cm
Dielectric strength ~ 107 V/cm
Dielectric constant relative to air (K) =3.9



Field
Oxide Gate
/ C, /



Silicon



Source Drain
Gate
dielectric
Figure 1-2. Metal-oxide-semiconductor field-effect transistor (MOSFET).

Silicon dioxide has a relatively low-dielectric constant (K = 3.9) [33]. Thus, since

high gate dielectric capacitance is necessary to produce the required drive currents from

submicron devices, and because capacitance is inversely proportional to gate dielectric

thickness, the SiO2 layers have been scaled to extremely thin dimensions where the

Si/Si02 interface becomes a more critical (and limiting) part of the gate dielectric. As a

result, 1.0 nm Si02 layers are commonly found in modern materials. Such a thin silicon

dioxide layer is mostly interface, and it contains about five layers of Si atoms (at least

two of which reside at the interface). This has given rise to several electrical and

performance problems (including impurity penetration through the dioxide, and a lack of

convenient insulating properties) that make it necessary to look for alternate gate

dielectric materials. Over the past several years, ultrathin silicon oxynitride (SiOxNy)









films have been incorporated into metal-oxide-semiconductor (MOS) devices as an

alternative for pure SiO2 [33]. Incorporating nitrogen into SiO2 is a relatively simple

method for fabricating silicon oxynitride films, since the nitrogen atoms tend to aggregate

at the Si/SiO2 interface, and results in dielectric layers that enhance resistance to gate

current leakage and inhibit boron penetration into the dielectric [34].

In recent years, surface functionalization also has been intensively investigated

[35-36]. This is the process of depositing layers of organic molecules on semiconductor

surfaces to impart some property of the organic materials to the semiconductor device

[35-36]. The organic molecules might be designed to serve in place of gate oxides in

metal-oxide semiconductor field-effect transistor (MOSFET) devices, where the higher

wire resistance of smaller metal lines and the crosstalk of closely spaced metal increase

the interconnect resistance and capacitance product delay. This requires a low-dielectric

constant (low k) material as the interlayer dielectric, and low-resistance conductors (such

as aluminum). Until recently, silicon dioxide was the material of choice, but the decrease

of the device dimensions and the resulting change of aluminum for copper have

necessitated the search for alternative materials. Silicon oxycarbide (SiCxOy), a low k

hydrid between organic and inorganic materials obtained by organic functionalization, is

one of the most favorable candidates for interlayer dielectric in MOSFETs [37-39].















CHAPTER 2
THEORETICAL BACKGROUND OF QUANTUM CHEMISTRY CALCULATIONS

2.1 Introduction

From a chemistry and physics viewpoint three types of systems exist [40]: Systems

that possess a very small number of particles where the particles do not interact and that

can be studied classically and exactly, systems that have many particles and that require

statistical methods to give a nearly exact solution of the properties under study, and

systems where only a few particles exist, but where the particles interact considerably

with one another. The latter cannot be solved exactly (neither classically nor by

statistical modeling) so a different strategy is needed. Electron interactions with other

electrons and nuclei within a molecule are an example of these complicated systems for

which classical and statistical approaches are insufficient. If one considers that the

molecule has N electrons with electronic coordinates ri, the total electronic energy of the

system is obtained by solving the Schrodinger equation




where H is the Hamiltonian operator, expressed as in Equation 2-1.

1 n n n 1
H=- V + V(r) + (2-1)
2 I I j r, rj

The three terms on the right-hand side of Equation 2-1 correspond to the kinetic energy,

the classical Coulomb electrostatic potential between nuclei and electrons, and the

electron-electron interactions, respectively. From these, the electron-electron interaction

term highly complicates the calculation, given that it describes the highly coupled









motions of the electrons. If one attempts the classical approach, one can ignore the

electron-electron interactions completely. This decouples the electronic motions and

factorizes the problem into N completely independent problems, one for each electron.

However, this approach loses essential elements of the physics of the system. If the

statistical approach is attempted there is no need for intuitive simplifications, but a large

supercomputer is needed to solve the Schrodinger equation directly. This requires due

care in each calculation step, and has the drawback that the result is difficult to interpret

and gives little physical insight. Thus, a molecular system is one of those systems for

which neither classical nor statistical methods are sufficient. As indicated, these systems

require a different approach, and a hybrid between the classical and statistical methods

seems the most reasonable. The electron-electron interaction term in Equation 2-1 is

replaced by a term that can be solved statistically or computationally, while the first two

terms are solved classically. The available methods give a loss of few of the finer details

of the system, but they factorize the original problem into N independent one-electron

problems that can be solved iteratively (given that usually the one-electron equations

depend on the energy, while the energy depends on the one-electron equations) [40].

The two more-widely used iterative computational chemistry methods are

Hartree-Fock (HF) and Kohn-Sham density functional theory (Kohn-Sham DFT) [41-44].

Both methods are based on the self-consistent-field (SCF) concept and share many

conceptual and computational features. However, they are essentially different because

while HF is based on single-electron orbitals {t (r)} and many electron wavefunctions

(') constructed from them, Kohn-Sham DFT is based on the electron density p(r) and the

fictitious single-particle orbitals {(p, (r)} [42-45].









This chapter presents the most-common energy functionals (Section 2.2), followed

by a detailed description of the Kohn-Sham SCF scheme and its implementation in

Gaussian03 (Section 2.3), an introduction to the basis sets (Section 2.4) and an

explanation of the geometry optimization schemes used in our study (Section 2.5). Since

our main focus is on DFT calculations, we present only the main characteristics of the

Hartree-Fock method, and do not discuss the more complex and exact post-Hartree-Fock

methods (such as CI and MP2).

2.2 Energy Functionals and Early Density Functional Theory (DFT)

It is useful to express the total electronic energy of a molecular system partitioned

in energy functionals, as follows

E = E +E+E + Ex + Ec (2-2)

given that the computer programs available find the approximate solutions to the

Schrodinger equation by calculating each one of the functionals of Equation 2-2

separately. A functional is a mathematical device that maps objects onto numbers, and

can be viewed as a function whose argument is also a function. Terms on the right-hand

side correspond to the kinetic energy of the electrons, the Coulomb energy of electrons

caused by their attraction to the nuclei, the Coulomb energy that the electrons would have

in their own field if they moved independently and if each electron repelled itself, a

correction for electron exchange, and a correction for electron correlation, respectively.

The last two terms can be combined as

Exc = Ex + Ec

rendering a correction for the false assumptions involved in Ej(i.e., that electrons do not

perturb one another at close approach and that their motions are independent). The

exchange energy Exis the electron stabilization resulting from the Fermi correlation (i.e.,









the dependence of motion arising from the Pauli Principle); while Ec (the correlation

energy) arises from the correlation between the motions of electrons with different spins.

These five terms have entirely different magnitudes, with ET, EV and Ej constituting most

of E. The trend is as follows: ET > Ev > Ej >> Ex >> Ec. Additionally (as seen later), the

expressions for Ev and Ej are common for all SCF methods used in quantum chemistry;

while the formulas for ET, Ex and Ec are the ones that differentiate those methods [40,

45].

The Hartree-Fock approximation (the first SCF method developed) is based on

orbital functionals, which are well-defined procedures that take the orbitals of a system

and return an energy. The early orbital functionals that were suggested are the Hartree

kinetic functional, the self-interaction-correction functional and the Fock exchange

functional. In 1928, Hartree [40, 45] presented a model in which the ith electron in an

atom moves completely independent of the other in an orbital y, (r). Since all electrons

are represented this way, the system is then a collection of uncorrelated electrons, and the

total kinetic energy of the atom ET is the sum of the kinetic energies of the electrons.

H "
E7H Y f V, (r)V 2, (rr (2-3)


Molecular orbital (MO) theory originated when Hartree's model was extended to

molecules by delocalizing the { V., (r) } over several atoms. However, the model neither

gives the exact ET (given that the electrons do not move independently) nor excludes the

self-interactions of electrons (i.e., the exchange energy), and an additional

self-interaction-correction functional is required.

A correction functional was presented in 1930 by Fock [40, 45], who indicated that

the Hartree wavefunction violates the Pauli Exclusion Principle because it is not properly









antisymmetric. He then solved the problem by rendering the wavefunction antisymmetric

as follows

EF30 = -l Idrdr2 (2-4).
2 2 ri r2

At this point in the history of quantum chemistry, it was realized that it is

cumbersome to work with wavefunctions, given that they have essentially no physical

interpretation and have units of probability density to the one-half power. An alternative

method was suggested, based on an observable that should allow the construction of the

electronic Hamiltonian. Since the Hamiltonian depends on the position and atomic

number of nuclei and the total number of electrons N, the best observable for the

alternative method is the electron density p, given that the total number of electrons Nis

obtained from

N = p(r)dr (2-5)

and, since the nuclei correspond to point charges, the nuclei positions are local maxima of

p. Only the relationship between the atomic number ZA and the electron density was not

clear, but it was found that ZA is related to the spherically averaged density. With this

solved, Eq. 2-2 for the total electronic energy can be expressed in terms of density

functional

E[p(r)] = E, [p(r)] + E, [p(r)] + Ej [p(r)] + Ex [p(r)] (2-6)

The simplest approximation for the total electron energy was the approach of

Thomas-Fermi [46-47], who assumed that the system had classical behavior, and then

reduced Equation 2-6 to Equation 2-7.

E[p(r)] = E, [p(r)] + E, [p(r)] + E, [p(r)] (2-7)









In Equation 2-7, the Coulomb interaction terms in the right-hand side are given by

Equation 2-8 and 2-9.

nuclei Z
E,[p(r)] J= p(rr (2-8)


and


E [r)] p(r pr dr2 (2-9)
2 r, r2

E, [p(r)] and Ej [p(r)] are extremely useful for modern SCF methods, whether

based on HF or DFT theories. Equation 2-8 is exact and is used in all the available SCF

methods. Equation 2-9 is an approximation that considers that the electrons move

independently and that each electron experiences the field caused by all the electrons

(including itself).

Thomas and Fermi [46-47] also derived a kinetic functional based on the uniform

electron gas, orjellium, which is an idealized system in the limit of N and V oo of a

system of N electrons in a cubical box of volume V, that has a uniformly distributed

positive charge that renders the system neutral. Thomas and Fermi chose jellium because

although it is a many-electron system, it is completely defined by its electron density p

[46-47]. The functional is as shown in Equation 2-10.

3 6 2 2/3 5/3(r r
E,4p(r)] = (60f a (2-10)

The Thomas-Fermi kinetic energy functional is the only density functional that has

an elegant mathematical derivation, but unfortunately, it is not accurate enough to be

chemically useful. Also, it was the first DFT functional as such, because it showed that

non-electrostatic energy terms can be expressed in terms of the electron density.









Table 2-1. Approximations made to correct Thomas-Fermi-DFT before the development
of Kohn-Sham DFT
Functional Author Year Comments
Hole function: Slater 1930 The hole function h(ri; r2)
[ 1 p( f(r,)p ()dr2 Wigner 1933 is centered on electron
E, [p(r)] = 2 r, -r21 and 1 and a function of
+ p(rl)h(rl; r2)r1dr2- Seitz electron 2 coordinates.
2 |r, -_r2
Hole exchange energy ofjellium: Bloch and 1930 When combined with the
3 1/ 3 3 Dirac work of Thomas and
EXD3[p(r)]= p43 (r)dr Fermi, is known as the
2 4;rj Thomas-Fermi-Dirac
approximation.
Slater exchange hole function: Slater 1951 Completely ignores
9(31/3 correlation effects. The
Efs[p(r)]= J43 (r)dr exchange hole function
0 8 at any position is
calculated as a sphere
of constant potential
with a radius that
depends on the
magnitude of the
electron density at that
position.
Xa: Gaspar 1954 a= 4 works better than
9a 3 )1/3 P ( r Slater 1971 both 1 and 2/3.
Ex[P(r)]=-- (r)dr and
Wood
with a = 1 for a = 1 for Exs51 and
2/3 for ExD3o

However, these Thomas-Fermi functionals (known as Thomas-Fermi DFT) are no

longer widely used, because they predict that molecules are unstable with respect to the

dissociation of their constituent atoms. Table 2-1 shows the approximations made to

correct for problems with Thomas-Fermi-DFT before Kohn-Sham density functional

theory was developed; of these approximations, Xa is the only one that still is used in

certain scientific endeavours.

2.3 Hohenberg-Kohn-Sham Density Functional Theory (KS-DFT)

The early DFT model presented in Section 2.2 did not have widespread use because

it resulted in fairly large errors in molecular calculations, and because the theories were








not rigorously founded. This changed in 1964, when Hohenberg and Kohn (HK) [41]

presented two key theorems to the scientific community. Before we further describe the

HK theorems, we must clarify that we have limited this description to the simplest

systems, that is, N non-relativistic, interacting electrons in a non-magnetic state with

Hamiltonian

E[p(r)]= ET [p(r)] + E [op(r)] + Ej [p(r)] (2-11)


where ET[p(r)]- Ey[p(r)] Zv(rj) and Ej[p(r)] 1
2 2i.jri -rj

For mathematical reasons, we considered a broad class of Hamiltonians with electrons

moving in an arbitrary external potential v(r), besides the physically relevant Coulomb

potentials due to point nuclei.

2.3.1 Hohenberg-Kohn Lemmas and Exchange-Correlation Functional Definition

The starting point of KS-DFT is the rigorous, simple-existence lemma of

Hohenberg and Kohn (HK) [41] who considered that the specification of the ground-state

density, p(r), determines the external potential v(r) uniquely (to within an additive

constant C). Since p(r) also determines N by integration,

N[p(r)] = pc (r)dr = N,

it determines the full Hamiltonian H (and, implicitly, all the properties determined by H).

Hohenberg and Kohn [41, 45] proved that the ground-state density determines the

external potential via reduction adabsurdum. They assumed that there were two

potentials, va(r) and vb(r), with ground-state wave functions Yo,a and TO,b, respectively.

They also assumed that both wave functions give rise to the same density p (r). Unless

[vb(r) va(r) = constant] To,a and TO,b cannot be equal since they satisfy different








Schrodinger equations. If one denotes the Hamiltonian and ground-state energies

associated with Yo,a and YO,b by Ha and Hb and Eo,a and Eo,b, then one has by the minimal

property of the ground-state,

Eo,b =
so that

EO,a
By interchanging a and b, HK found in exactly the same way

EO,b
Finally, adding Equations 2-12 and 2-13 leads to the inconsistency

Eo,a + Eo,b < Eo,a + Eo,b. (2-14)

With the help of the existence lemma, HK also showed (in a second lemma) that

the density obeys a variational (or minimal) principle. For a given v(r), they defined the

following energy functional of p(r)

Ev(r) [P(r)] = f v(r)p(r)dr + F[p(r)], (2-15)

where F[p(r)] ('P[p(r)IET + Ej 'P[p(r)]). (2-16)

One should notice that F is a functional of p(r), since the wave function itself is a

functional of p(r). The variational principle is then given by the expression

Ev(r) [p(r)]> Ev(r) [o (r)] E (2-17)

where po(r) and E are the density and energy of the ground-state. The equality in Eq.

2-17 holds only if po(r) = p(r); in other words, the variational principle states that any

calculated energy will be higher than that of the ground-state. Then, Hohenberg and

Kohn [41, 45] extracted from F[p(r)] its largest and elementary contributions








by writing Equation 2-18,

F[p(r)]= ET [(r)]+ -I p(r ) drdr' + EXC [(r)] (2-18)
2 r -r'

where E,[p(r)] is the kinetic energy of a non-interacting system of electrons with density

p(r), and the next term is the classical expression for the interaction energy. Equation

2-18 is the KS-DFT definition of Exc[p(r)].

2.3.2 Kohn-Sham Self-consistent Field Method

If Exc is ignored in Eq. 2-18, the physical content of the theory becomes identical

to that of the Hartree approximation [44]. Kohn and Sham (KS) [42] noticed that and

transformed the Euler-Lagrange equation associated with the stationarity of Ev[p(r)] into

a new set of self-consistent equations, the Kohn-Sham (KS) equations.


-1 V + v(r)+ f (dr'+Vxc(r)- -j (r)= 0, (2-19a)

N
N[p(r)] = (r2 (2-19b)
j=1

and VXC = (r) (2-19c).

These equations are analogous to Hartree-Fock equations, although they also

include correlation effects. They must be solved self-consistently, like the Hartree-Fock

equations, calculating Vxc in each cycle from Eq. 2-19c, with an appropriate

approximation for Exc[p(r)] (Section 2.3.3). Despite the appearance of simple,

single-particle orbitals, the KS equations are in principle exact provided that the exact

Exc is used in Eq. 2-19c; that is, the only error in the theory is due to approximations of









Exc[p(r)]. Once Equations 2-19 are self-consistently solved, the ground-state energy is

obtained by


E- 2- r-r r
E = rcj 2r -r drdr'- f VXC (r)p(r)dr + Exc [o(r)] (2-20)

1

where e, and p are the self-consistent quantities. The individual eigenfunctions and

eigenvalues, p, and e,, of the Equations 2-19 have no strict physical significance.

2.3.3 Exchange and Correlation Functional Approximations

In principle, the KS-DFT exchange-correlation functional accounts for the classical and

quantum mechanical electron-electron repulsion and corrects for the difference in kinetic

energy between the fictitious non-interacting system and the real electron system.

Table 2-2. Variables used to express the approximate exchange-correlation functionals
Exc[p(r)]
Variable Equation Comments
Energy density (cxc) Ex [p(r)]= p(r),c [p(r)]dr Function of the electron
density, given by the
sum of individual
exchange and correlation
contributions in the
system.
Effective radius [rs(r)] r 1/3 Used when the electron
r (r)= density is expressed for
4;Tp(r) exactly one electron.
This electron would be
contained within a
perfect sphere that has
the same density
throughout its center.
Normalized spin (p(r)- p (r) Used to express spin
polarization () (r) r= p(r) densities at any position
Sa p n in open-shell systems. By
where: a corresponds to convention, the spin
spin-up- and fs to spin-down densities are given by
electrons. 1
P (r)= P(r)(+1)
2
and
p (r)= p(r)- pa(r).








However, there is no known systematic way to achieve an arbitrarily high level of

accuracy for this functional. Therefore, an approximate form for the correct

exchange-correlation functional must be selected so that the Kohn-Sham differential

equations for the orbitals can be obtained by minimization of the total energy in Eq. 2-20.

These Ex [,o(r)] approximations can be broadly classified in four major groups:

the local density approximation functionals, the gradient-corrected functionals, the hybrid

functionals and semi-empirical DFT. Moreover, all these approximate functionals can be

expressed in terms of the three variables summarized in Table 2-2.

2.3.3.1 Local density approximation (LDA)

The designation local density approximation originally referred to any DFT

functional where the energy density at some position r was computed exclusively from

the value of the electron density at that particular position (i.e., the local value of p).

However, since the only functionals that follow this definition are those derived based on

the uniform electron gas electron density, generally the name LDA is applied only to

exchange and correlations functionals derived from jellium.

In the special case when one needs to calculate the exchange energy density for a

spin polarized system, LDA can be modified with Equation 2-21.


ex [p(r), = [01=(r)]+ e [p(r)]- [p(r)] (1+ )42( 1- 4/3 -2 (2-21)


9Q (31/3
where c4 p(r)= 9a 3 3 [r],
810

and s\x [p(r)] is analogous to so [p(r)]. It describes a uniform electron gas composed of

electrons with the same spin. This expression is the local spin density approximation

(LSDA) [40, 45].








Since there has been little success in finding analytical expressions for c [p(r)],

most of the work available on correlation energy density calculations is approximated by
numerical techniques. One of the most widely used functionals is that of Vosko, Wilk
and Nusair (VWN) [48]. After Ceperly and Alder used Monte Carlo techniques in 1980
to compute the total energy for jellium of various densities and found their correlation
energy by subtracting the exchange, VWN developed a spin polarized correlation local
density functional analogous to the exchange functional presented in Eq. 2-21 by fitting
these computational results. The VWN proposed expression is Equation 2-22.

r 2b f(4c- b2)12
In + 2b tan 1 (4c -b 2
A rs +b(r)/2 c 4c-b2/2 2r/2 +b (2-22)
fc(r < >:2bln| -- -) (2-22)
C 2 bxo r/2- 2 2(b+2xo) tan 1(4c- b2 1/2
x2 +bxo +c r+br +C (4c-b22 /2 2 -/2 +b-

where i = 0 or 1 (analogous to E [p0(r)] and Ex [p(r)]), and with different empirical

constant sets for gc [p(r)] and cc [p(r)]. This correlation functional, known

asEcjv[p(r)], is exact for jellium but not for molecular systems and it is a good example

of how many empirically optimized constants are required in DFT to approximate the

unknown exactExc [p(r)]. Furthermore, the semi-empirical flavor of functionals such

as Ecv [p(r)] is the main inspiration for the development of the semi-empirical DFT

methods (Section 2.3.3.4).
2.3.3.2 Density gradient corrections
Because the LDA exchange functionals were derived from a HF density matrix
constructed with the plane wave orbitals of the uniform electron gas, they typically
underestimate the exchange energy by roughly 10 to 15 %, and, consequently, need a









correction for the non-uniformity of atomic and molecular electron densities. An

improvement would be the generation of exchange and correlation functionals that

depend on the local value of the electron density, and on the extent to which the density is

locally changing (i.e., the density gradient). These types of functionals are called

gradient-corrected functionals, and the methodology by which they are formed is known

as the generalized gradient approximation (GGA). Most of these gradient-corrected

functionals are constructed by adding a correction to an LDA functional energy density

expression, that is, Equation 2-23.

'c [p(r), Vp(r)] = WLD [p(r)] + A /cx(r) (2-23)

where x(r) is the reduced density gradient given by Eq. 2-24,


x(r)= Vp( (2-24),


and X/C indicates that this could be either an exchange or a correlation gradient-corrected

density functional). The reduced density gradient has small values in bonding regions,

larger values in core regions and very large values in Rydberg regions of molecules.

The most widely used GGA exchange functional is the one developed by Becke in

1988 [49], which combines LSDA exchange with a gradient correction, and yields

accurate exchange energies for atoms and correct exchange energy density in Rydberg

regions. In its simplest form, the functional is given by Equation 2-25,

2
E88 = ED30 p )4/3 X dr (2-25)
x 'x VJ ) 1+ 6/fxt sinh x,


where the semi-empirical parameter fl has a value of 0.0042, obtained by fitting to the

ExF30 (Hartree-Fock) exchange energies of six noble gas atoms (He to Rn). Alternative

GGA exchange functionals are do not include empirically optimized parameters.








With respect to correlation, E'j N[p(r)] predicts correlation energies that are too

large, making it a poor starting point for a gradient-corrected functional. Instead, the most
popular GGA correlation functional developed by Lee, Yang and Parr (LYP) [50]
computes full correlation energies. LYP abandoned jellium in favor of the He atom, the
simplest system with a non-vanishing correlation energy. Their approach was based on
earlier work of Colle and Salvetti [51], and it was improved for spin-compensated and
spin-polarized versions by Miehlich et al. [52]. The expression reported by Miehlich et
al. for this gradient-corrected correlation functional, denoted as a second-order gradient
expansion is Equation 2-26.

Ej [p,Vp]= -1a 4 )
1 1 Vv1 i+dp 226

-ab {Cp [211/3CF((a)8/3 +(P)8/3)+ 47 7 2 (2-26)

2 18 9 [p P
2P2Vp2 2 (pa)2)Vp2+(2p2 (-P)2)V] 2}


where = exp( 1)p 11-/3, 1 3 + dp 1 CF (3 2)23 and the
1+ dp-13 1d+dp-1'3 10

parameters a, b, c and d were obtained from the work of Colle and Salvetti [51]. The
values for the parameters are a = 0.04918, b = 0.132, c = 0.2533 and d= 0.349. ECL [p(r)]

yields better results than EwN'[p(r)] and is very important in DFT despite some

theoretical deficiencies.
2.3.3.3 Hybrid methods
Given that the basis of the Kohn-Sham approximation is the non-interactive
reference system, it was suggested that one can control the conversion from a non-









interacting to a real interacting system. With the Hollmann-Feynman theorem, the DFT

exchange-correlation energy can be computed as shown in Equation 2-27.

Exc = (Aj (A) Y (A))dA (2-27)

where 2 is the extent of interelectronic interaction, going from 0 (non-interaction system)

to 1 (real system). Development of hybrid DFT techniques resulted in methods that are

significantly more efficient than ab initio methods of comparable accuracy. The relatively

good accuracy and computational efficiency of hybrid methods indicate that further

improvements might lead to methods that are both highly accurate and relatively

efficient. All these methods include Hartree-Fock exchange because a small exact-

exchange component is a necessary constituent of any exchange-correlation

approximation aiming for accurate molecular energetic. The methods use the

conventional mixing method introduced by Becke [53], which expands the functional

with respect to the electron density and its gradient according to Eq. 2-23 and adds

adjustable coeffiencts, c,

yB, (2-28)


an approach that is appealing because its application to a new molecule requires only a

single Kohn-Sham calculation. Becke developed the B3LYP exchange-correlation

functional [53], which is defined by Equation 2-29.

EB3LY E28 +Ev +Ej +(I-c1)E 0 +c1E3 +cAE88 +(1-c3)EWN +c3EL (2-29)

where the coefficients cl, c2 and c3 are semi-empirical coefficients determined by a fit to

experimental data. This method cannot be derived rigorously and, at least in its current

status, is basically empirical. Nonetheless, it is widely used because its ability to predict









the atomization energies of normal systems is very close to their exact value. Kang and

Musgrave [54] proposed a hybrid DFT method in which the exchange functional is a mix

of Slater exchange and exact exchange. The KMLYP energy functional is given by Eq.

2-30.

ELYP EH28 + EV + E +(1- a)Ef30 +aE3 + (I b)E vwN + bEL (2-30)

where a= 0.557 and b = 0.448 and the subscripts are the same as in Eq. 2-29.

2.3.3.4 Empirical DFT methods

These methods increased the emphasis on empirical parameterization, arguing that

by modestly increasing parameterization it may be possible to obtain good functionals.

Additionally, empirical DFT methods (EDF) looked for an exchange-correlation

functional that was optimized for a relatively small basis set and questioned the need for

including Hartree-Fock exchange to obtain good agreement with experiment. Hybrid

methods suppose that a small exact-exchange component is a necessary constituent of

any exchange-correlation approximation aiming for accurate molecular energetic, but

this introduces non-local effects and consequent computational complications that EDF

methods tried to eliminate.

Empirical density functional theory 1 (EDF1) [55] developed a DFT functional by

linearly combining several existing functionals. It was optimized to yield accurate

thermochemistry when used with the 6-3 1+G* Pople basis set (Section 2.4). Among

other variational schemes, it considered linear combinations of different functionals or

linear combinations of the same functionals with different parameters. The component

functionals used are those of Becke [49] and Lee, Yang and Parr [50], discussed above.

EDF2 [56] was developed following the method used in EDF1 [55], in which a

DFT functional is obtained by linearly combining several existing orbital and density









functionals, with a special focus on giving accurate harmonic frequencies when used with

the cc-pVTZ [57] basis sets. The expression EDF2 is as shown in Equation 2-31.

EXF2= blEF3 []+ b2ED3 []+b3E"88 [,]+b4EDFo[p]+ bE"N []+b6E [po]++bEDF l[p] (2-31)

where b, = 0.1695, b2 = 0.2811, b3 = 0.6227, b4 = -0.0551, b5 = 0.3029, b6 = 0.5998, and

b7 = -0.0053.

2.3.4 Kohn-Sham Density Functional Theory Computational Chemistry

Figure 2-2 summarizes the steps involved in a KS-DFT geometry optimization. For

the most part, they are the same as those used in Hartree-Fock theory. First, a basis set is

chosen to construct the KS orbitals (Eq. 2-19) and then an initial estimate of the

molecular geometry. After that, the overlap integrals and the kinetic energy and nuclear

attraction integrals are computed. The latter two kinds of integrals are called one-electron

integrals in HF theory to distinguish them from the two-electron Coulomb and exchange

integrals. (However, in KS theory all integrals can in some sense be regarded as

one-electron integrals since every one reflects the interactions of each one electron with

external potentials). To evaluate the remaining integrals, one must guess an initial

density, and this density can be constructed as a matrix entirely equivalent to the P HF

density matrix, which describes the degree to which individual basis functions contribute

to the many-electron wave function, and thus, how important the Coulomb and exchange

integrals should be.

The elements of P are then computed as indicated in Equation 2-32.

occupied
P,'0 = 2 aia7 (2-32)


where the coefficients aA, and a, specify the contribution of the basis functions to the

molecular orbital i and the factor of two appears because this is a restricted calculation,









meaning that it is considering only single wave functions in which all orbitals are doubly

occupied [45]. Pople and Nesbet [58] presented the equation for P for radicals and

excited states, which is known as unrestricted Hartree-Fock and which has one wave

function for each electron.




/Choose basis set


Figure 2-1. Flowchart of the Kohn-Sham SCF procedure









Once the guess density is determined one can construct Vxc (Eq. 2-19c) and

evaluate the remaining integrals in each KS matrix element. New orbitals are determined

from solution of the secular equation, the density is determined from those orbitals, and it

is compared to the density from the preceding iteration. Once convergence of the SCF is

achieved, the energy is computed by plugging the final density into Eq. 2-20. At this

point, the calculation proceeds according to the geometry optimization algorithm (i.e., if

the geometry does not correspond to the optimal point, a new structure will be found and

the KS SCF process will be run again, until the optimum is reached [45]). The geometry

optimization algorithm is described in Section 2.5, but first, we present a brief

introduction to the basis sets used in this work.

2.4 Basis Sets

As discussed, Equations 2-19 are formally similar to the HF SCF ones, except that

the exchange nonlocal one-electron operator is replaced by a local exchange-correlation

operator depending on the total electron density. Thus, the KS equations can be

efficiently calculated and the single electron wave functions can be represented by

several basis functions such as Slater-type orbitals. A basis set is then, a set of

mathematical basis functions from which the wave function is constructed in a quantum

chemistry calculation.

Slater-type orbitals have a number of attractive features primarily associated with the

degree to which they closely resemble hydrogenic atomic orbitals. They suffer, however,

from a fairly significant limitation. There is no analytical solution available for the

general four-index integral given en Equation 2-33 when the basis functions are STOs.

The requirement that such integrals be solved by numerical methods severely limits their

utility in molecular systems of any significant size.










1 2

Alternatives to the use of STOs have been proposed. One of them indicates that for

there to be an analytical solution of the general four-index integral (Eq. 2-33) formed

from such functions is required that the radial decay of the STOs be changed from e-' to

e-r2. That is, the AO-like functions are chosen to have the form of a Gaussian function.

The general functional form of a normalized Gaussian-type orbital (GTO) in

atom-centered Cartesian coordinates is as shown in Equation 2-34.


(x,y,z;a,i,j,k) 2a3/4 (8a)i+j+ki!j!k! /2 yj zk e (x2 + +z2) (2-34)
KL) (2i) (2j) (2k)!

Smalls GTO sets can be used for a wider range of chemical problems but involve

some loss of flexibility in the resulting molecular orbitals. The simplest level of basis is

minimal and corresponds to one basis function per atomic orbital. The next level is

split-valence or valence-multiple-Cin which two basis functions are used for each valence

atomic orbital. This second level is known to give a better description of the relative

energies and of some geometrical features of molecules. Further improvement of a basis

set requires addition of functions of higher angular quantum numbers (polarization

functions) [59].

The basis functions are generally contracted (i.e., each basis function is a linear

combination of a number of primitive Gaussian functions). The contracted GTOs used in

DFT to represent the electronic orbital wave functions, often were originally developed

for Hartree-Fock calculations. A considerable increase in computational efficiency can

be achieved if the exponents of the Gaussian primitives are shared between different

basis functions [60]. At the split-valence level, this has been exploited by sharing









primitive exponents among s and p functions for the valence functions. In particular, a

series of basis was defined and designated K-LMG by Pople and coworkers [61-68]

where K, L, and M are integers. Such a basis for a first-row element (Li to Ne) consists

of an s-type inner-shell function with K Gaussians, an inner set of valence s- and p-type

functions with L Gaussians, and another outer sp set with M Gaussians. Both valence

sets have shared exponents. For hydrogen, only two s-type valence functions (with L and

M Gaussians) are used. Among this basis sets are 3-21G [67-68], 4-31G [64, 69], and

6-31G [63, 65-66]. The original 4-31G and 6-31G split-valence basis sets were obtained

by optimizing all Gaussian exponents and contraction coefficients to give the lowest

spin-unrestricted Hartree-Fock energy for the atomic ground-state. The Pople basis sets

have seen widespread use among the scientific community [45].

A problem with the calculations based on K-LMG basis sets is that s and p

functions centered on the atoms do not provide sufficient mathematical flexibility to

adequately describe the wave function for some geometries. This occurs because the

molecular orbitals require more mathematical flexibility than do the atoms. Because of

the utility of AO-like GTOs, this flexibility is almost always added in the form of basis

functions corresponding to one quantum number higher angular momentum than the

valence orbitals. Thus, for a first-row atom the most useful polarization functions are d

GTOs, and for hydrogen, p GTOs. A variety of molecular properties prove to be

sensitive to the presence of polarization functions; for example, d functions on

second-row atoms are absolutely required to make reasonable predictions for the

geometries of molecules including such atoms in formally hypervalent bonding situations

(e.g., siliconates). Because the total number of functions begins to grow rather quickly

with the addition of polarization functions, early calculations typically made use of only a









single set. Pople and coworkers introduced first a simple nomenclature scheme to

indicate the presence of the polarization functions, the "*" (star). Thus, 6-31G* implies a

set of d functions added to polarize the p functions in 6-31G. A second star implies p

functions on H and He (e.g., 6-311G**) [45]. However, realizing the tendency to use

more than one set of polarization functions in modern calculations, their basis set

nomenclature now typically includes an explicit enumeration of those functions instead

of the star nomenclature. Thus, 6-31G(d) is preferred over 6-31G* because the former

generalizes to allow names such as, for example, 6-31G(3d2fg,2pd), which implies heavy

atoms polarized by three sets of d functions, two sets of f functions, and a set of g

functions, and hydrogen atoms by two sets of p functions and one of d.

Finally, we must indicate that the highest energy MOs of anions and highly excited

electronic states tend to be much more spatially diffuse than the molecular orbitals

described so far. When a basis set does not have the flexibility necessary to allow a

weakly bound electron to localize far from the remaining density, significant errors in

energies and other molecular properties can occur. To address this limitation, standard

basis sets are often augmented with diffuse basis functions. In the Pople family of basis

sets, the presence of diffuse functions is indicated by a "+" in the basis set name. Thus,

6-3 l+G(d) indicates that heavy atoms have been augmented with an additional one s and

one set of p functions having small exponents. A second plus indicates the presence of

diffuse s functions on H [e.g., 6-31 l1++G(2d,p)]. For the Pople basis sets, the exponents

for diffuse functions were variationally optimized on the anionic one-heavy-atom

hydrides and are the same for 3-21G, 6-31G and 6-311G [45]. In this work we use the

3-21G*, the double-p plus polarization 6-31G(d), and the diffuse triple-p plus polarization

6-31 l1++G(2d,p) Pople basis sets.









2.5 Geometry Optimization and Transition State Search Calculations

Before we discuss how to perform geometry optimizations of molecular systems,

we must understand the concept of a potential energy surface (PES) since a potential

energy surface describes the energy of a particular molecule as a function of its geometry,

and a geometry optimization is a process to find minima on a potential energy surface.

Finding the total PES is a very complicated task, mainly because molecules have many

atoms and many coordinates that describe their geometry. A simplified three-dimensional

PES can be represented as a topographic surface with valleys and saddle points (Figure

2-2).

saddle point
Global maximum (transition state)


local minum =- ;:J mnumu
(reactant) 'Tr Te :tAble product)

Local minimum
(alternate product)





Figure 2-2. Two-dimensional potential energy surface

Generally, calculation of most potential energy surfaces is based on the

Born-Oppenheimer approximation which specifies that since the electrons are much

lighter than the nuclei, the electronic part of the wave function readjusts almost

immediately to any nuclear motion. Thus, potential energy surfaces can be obtained by

calculating the total electronic energy for all the possible fixed nuclear positions that the

molecule may adopt, an approach that is satisfactory for ground-state systems. On a PES,

a reaction path is the movement from a valley of reactants to a valley of products, and a









reaction mechanism is the path across the PES. Then, the important points in the

calculated PES are the minima and the saddle points, with the equilibrium geometries

corresponding to minima and the saddle points to the highest points on a reaction path

that requires the least energy to get from the reactants to the products. Both PES minima

and saddle points are called stationary points (or critical points), because for them the

gradient (first derivative) is zero, and in classical mechanics, negatives of the first

derivatives of the potential energy surface are the forces on the atoms in the molecule, as

expressed by Eq. 2-34.

F = -VEV (2-34)

An energy minimum must satisfy two conditions [70]:

its first derivative (i.e., gradient of the forces) must be zero (critical point).
its second derivative matrix must be positive definite (i.e., all eigenvalues of the
Hessian must be positive or all of the vibrational frequencies must be real).

Because of the difficulty of finding a complete PES, most quantum chemistry

geometry optimization methods are focused on finding equilibrium structures and

transition states directly. In this work, we performed geometry optimizations of

Si9H12+wOxNyCz (w,x,y,z = 0,1,2, etc.) clusters using the unrestricted hybrid density

functional method B3LYP combined with the diffuse triple-L plus polarization

6-31 l1++G(2d,p) basis set to expand the molecular orbitals of chemically active atoms

(the surface Si atoms, four Si atoms of the first subsurface layer, the carbon atoms, the

nitrogen atoms, and the oxygen atoms), and the double-t plus polarization 6-31G(d) basis

set to describe the remaining subsurface silicon atoms and terminating hydrogen atoms

(Section 2.4). Usually, finding equilibrium geometries and transition structures requires

application of unconstrained optimization on the PES. However, we imposed constraints









on the hydrogen atoms that terminate broken Si-Si bonds in the clusters, to mimic the

constraints that the rest of the solid should impose on the surface dimer under study.

2.5.1 Geometry Optimization of Energy Minima

There are three types of geometry optimization algorithms: Methods that use only

the energy, methods that use the first derivatives of the potential energy surface with

respect to geometric parameters (i.e., gradient methods), and methods that require second

derivatives (i.e., Newton or Newton-Raphson methods). The methods that only use the

energy are the most widely applicable, but the slowest to converge, while the second

derivative methods converge very fast but, given that analytic second derivatives are not

readily available and more costly than the gradient methods. Thus, the gradient

algorithms are the preferred methods, and (if analytical gradients are not available) it is

usually efficient to calculate them numerically. In this work we performed quantum

chemistry calculations with Gaussian03 [71], which uses the gradient method known as

the Berny algorithm [72-73] (Figure 2-3) as the standard method for calculations of both

geometry optimizations to a local minimum and transition state searches.

Gradient methods, such as the Berny algorithm, approximate the potential energy

surface as a quadratic function and calculate the energy with the expressions


E(r)= Eo + gAr + 1-Ar'HAr (2-35a)
2

g = go + HAr (2-35b)

where Ar = r ro, g is the gradient vector, and H is the Hessian matrix. Since for an

energy minimum g = 0, the minimum structure can be obtained by solving linear

equations Eq. 2-36a and Eq. 2-36b.

g = go + HAr = 0 (2-36a)









Ar = -H g0 (2-36b)

known as the Newton step. These first derivatives can be found analytically or

numerically. The rate of convergence of the Berny algorithm depends predominantly on

six factors:

* Initial geometry of the structure to be optimized
* Coordinate system chosen.
* Initial guess for the Hessian matrix.
* Hessian matrix updating method.
* Step size control.

The initial geometry of the structure is a critical issue. It cannot be just any

structure that resembles the molecule under study as it would imply that the optimization

using a large basis set would require numerous unnecessary steps. Generally, molecular

mechanics and semi-empirical methods are used to refine a raw structure and obtain a

much better initial guess. In this work, we also ran a DFT/UB3LYP geometry

optimization with the 3-21G* Pople basis set after the semi-empirical calculations to get

as close as possible to the final minimum energy structure.

Early geometry optimization used non-redundant internal coordinates (e.g., the Z

matrix internal coordinates), but later it was shown that Cartesian coordinates and a

combination of Cartesian and internal coordinates had some advantages for particular

systems [74]. Cartesian coordinates are the simplest and give an unambiguous

representation of the structure, but they are also strongly coupled (i.e., to change a bond

length one must change the x, y, and z positions of several atoms). Alternatively, internal

coordinates (based on bond lengths, valence angles and dihedral angles) avoid problems

with rigid body rotation s and translations. In internal coordinates, the coupling is much

smaller than with Cartesian coordinates, and as a result the Hessian is more diagonal.









(Appendix C reviews the protocol used to determine the initial coordinates for our

calculations).

The rate of convergence of a geometry optimization depends also on the initial

guess of the nuclear Hessian since a closer guess to the actual Hessian will translate into a

faster convergence. Computationally inexpensive empirical methods to generate initial

nuclear Hessian have been proposed by Schlegel [75] and Fischer and Alml6f [76], who

also compared the effect of different coordinate systems on the geometry optimization.

An initial empirical estimate of a diagonal Hessian can be quite satisfactory for

redundant internal coordinates and can be readily transformed to other coordinate

systems. However, Baker [77-78] showed that minimizations with Cartesian and

redundant internal coordinates were similar if a good initial estimate of the Hessian is

used (i.e., a molecular mechanics or semi-empirical Hessian instead of a unit Hessian).

The Hessian from an optimization or a frequency calculation previously performed at a

lower level of theory is often a very good initial guess for an optimization at a higher

level. Some difficult optimizations may require an accurate initial Hessian computed

analytically or numerically at the same level of theory as that used for the optimization.

The default in Gaussian03 estimates bond stretching force constants from empirical rules,

and obtains average angle-bending and torsion constants from vibrational spectra or

theoretical calculations, using smaller and less expensive basis sets to calculate the force

constants. Finally, since we are interested in constrained structures (given the rigidity

that the silicon bulk would impose on the surface dimer chemistry) it is important to

know that any components of the gradient vector corresponding to frozen variables are

set to zero or projected out, thereby eliminating their direct contribution to the next

optimization step.










choose coordinate syste "

//Choose coordinate system/


Figure 2-3. Flowcharts for a quasi-Newton algorithm for geometry optimization

Proper update of the Hessian is essential for efficient optimizations, as the quality

of the Hessian at each point is critical for the success of the optimization. However, the

calculation of the exact Hessian at each point would make the process lengthy and costly.

Instead, the Hessian is adjusted in the quadratic approximation represented by Eq. 2-36









so that it fits the gradient gi at the current point ri and the gradient gi-1 at the previous

point. This leads to Equation 2-37.

H,Ar = Ag (2-37)

In Eq. 2-37 Ar = r, r, and Ag = g, g,- There are numerous methods to update the

Hessian, for example that of Broyden, Fletcher, Goldfab and Shanno (BFGS) [70], which

can be written as expressed in Eq. 2-38.

H, = H1 + AgAg'/Ar'Ag H,-ArAr'H-1 /Ar'H,-1Ar (2-38)

Eq. 2-38 is symmetric, positive definite, and minimizes the norm of the change in the

Hessian. Schlegel et al. [74] indicate that most modem updating algorithms give similar

results.

Once there is a new dependable Hessian, a Newton step is taken on the model

quadratic surface. The Hessian must be positive definite (i.e., all of its eigenvalues must

be positive, for the step to be in the downhill direction). If the structure is far from the

minimum (e.g., large gradients) or the potential energy surface is very flat (one or more

small eigenvalues of the Hessian), then a simple Newton step may be too large, taking the

molecule beyond the region where the model quadratic surface is valid. In this case, a

shorter step must be taken. One can limit the step to be no larger than a trust radius

which can be adjusted as the calculation proceeds, depending on whether the change in

the predicted energy compares well with the actual calculated energy difference. In the

Bemy algorithm of Gaussian03 the trust radius is updated using the method of Fletcher

[79].









2.5.2 Transition State Searches

A transition structure is the highest point on the reaction path that requires the least

energy to get from the reactant A to the product B. In other words, it is a stationary point

that is an energy maximum in one direction and a minimum in all others. For a point to

be considered to be a transition state structure the first derivatives must be zero and the

energy must be a maximum along the reaction path connecting the valley of reactants

with the valley of products on the potential energy surface. The transition state structure

must be a critical point of index one (i.e., one of the eigenvalues of the Hessian matrix

must be negative and all others must be positive). Energy, structure and vibrational

information for transition state structures are obtained from the transition state theory. As

the case for minima, transition structures can be found by geometry optimization,

although some modifications must be incorporated into the procedure.

Initial estimates for transition structures are more difficult to obtain than for

equilibrium structures since transition state geometries vary much more than equilibrium

geometries. Moreover, molecular mechanics generally cannot handle transition states

involving the making and breaking of bonds. As a result, when searching for transition

states, many quasi-Newton methods need to start fairly close to the quadratic region of

the transition state. Several techniques have been proposed to search for the initial

structure of the transition state, including synchronous transit [80] and eigenvector

following [81]. Gaussian03 combines synchronous transit and quasi-Newton methods to

find transition states [74, 80, 82]. This method requires two or three structures to start the

optimization (one in the reactant valley, the second in the product valley, and an optional

third structure as an initial guess for the TS geometry). If there is no third structure, the

initial guess is obtained by interpolating between the reactants and products in redundant









internal coordinates. The first few steps search for a maximum along an arc of circle

connecting the reactant-like with the product-like structure, and a minimum in all other

directions. The initial guess of the points along the reaction path is obtained by

interpolating between two input structures; the energy and gradient are calculated at each

point and an empirical estimate of the Hessian is obtained also at each point. The highest

energy point on the path is chosen to be optimized to the closest TS, dividing the path

into two downhill segments. In the remaining optimization steps, a quasi Newton based

eigenvector following optimization is guided by the tangent to the arc of circle passing

through the presumed TS and minima according to the implementation developed by

Baker in 1986 [81].

As in the geometry optimizations, the use of a certain coordinate system can

improve the transition state search considerably. Ayala and Schlegel [73] showed that

the combined used of redundant internal coordinates and the tangent to the quadratic

synchronous transit (QST) path improves considerably the transition state optimization,

mainly because a better Hessian is obtained in the first few steps and improves the search

direction. These redundant internal coordinates are based on the work of Pulay et

al.[83-84], who defined a natural internal coordinate system that minimizes the number

of redundancies by using local pseudosymmetry coordinates about each atom and special

coordinates for ring structures, and Peng et al. [74], who reduced the number of special

cases of these coordinates by using a simpler set of internal coordinates composed of all

bond lengths, bond angles and dihedral angles and applied it successfully for transitions

state searches. Peng et al. [74] defined their coordinates as follows: First, the

interatomic distances are examined to determine which atoms are bonded. Then, a bond

angle bend coordinate is assigned for any two atoms bonded to the same third atom.









Special attention must be given to linear bond angles; if the bond angle is greater than

-1750, then two orthogonal linear angle bend coordinates are generated. Finally, a

dihedral angle coordinate is assigned for each pair of atoms bonded to opposite ends of a

bond. If one or both of the bond angles involved in a dihedral angle is linear, then the

dihedral is omitted. In addition to the redundant internal coordinates generated

automatically, extra stretch, bend and dihedral angle coordinates can be specified in the

input. For regular transition state optimizations starting from one structure, the bonds

being made or broken need to be specified in the input.

Some difficult transition state searches may require an accurate initial Hessian

computed analytically or numerically at the same level of theory as that used for the

optimization instead of an updated one. Furthermore, Hessian update methods are

suitable for finding minima for small and medium-sized molecules, but for difficult cases,

such as some of the transition state structures investigated in this work, the Hessian has to

be recalculated every few steps (or even every step) instead of being updated. This is

equivalent to a Newton or Newton-Raphson algorithm. This is a very expensive method,

and by default, Gaussian03 avoids it and updates the Hessian; however, its use has the

advantage of obtaining an excellent description of the Hessian at each point of the

calculation and also, since a vibrational frequency analysis is automatically done at the

converged structure there is no need for an additional frequency job to determine the

number of positive eigenvalues or zero point vibrational energies. The protocol that we

used for transition state searches is presented in Appendix D.














CHAPTER 3
NITROGEN ATOM ABSTRACTION FROM Si(100)-(2xl)

3.1 Introduction

The reactions of gas-phase radicals at solid surfaces are fundamental to the

plasma-assisted processing of semiconductor materials. In addition to adsorbing

efficiently, radicals incident from the gas-phase can also stimulate several types of

elementary processes before thermally accommodating to the surface, including

direct-atom abstraction and collision-induced reaction and desorption. Direct-atom

abstraction can occur by an Eley-Rideal mechanism in which an atom is abstracted from

the surface in a single collision with an incident species [11] or by a hot atom mechanism

in which the incident species experiences multiple collisions with the surface but does not

fully thermalize before the reactive encounter [13]. Indeed, non-thermal surface

reactions such as these play a critical role in determining the enhanced surface reactivity

afforded by plasma processing. Advancing the fundamental understanding of

radical-surface reactions is therefore of considerable importance to improving control in

plasma-assisted materials processing in addition to being of scientific interest.

In the present study, we have used quantum chemical calculations to investigate the

interactions of an oxygen atom with nitrogen atoms incorporated into the Si(100)-(2xl)

surface, focusing on pathways that lead to direct nitrogen abstraction from the surface by

the formation of gaseous NO. This investigation is motivated firstly by an interest in

determining the viability of direct nitrogen abstraction from Si(100) by gas-phase oxygen

atoms, and to gain insights into the possible pathways for this surface reaction. It is









further motivated by the potential benefits that may be realized by incorporating ultrathin

silicon oxynitride films into metal-oxide-semiconductor (MOS) devices and the need to

precisely control the properties of such films. Over the past several years, ultrathin

silicon oxynitride (SiOxNy) films have been incorporated into MOS devices as an

alternative for SiO2, which is no longer suitable as an insulator for MOS devices because

of the decreasing dimensions of microelectronic devices imposed by Moore's law [33,

85]. Incorporating nitrogen into SiO2 is a relatively simple method for fabricating silicon

oxynitride films, and results in dielectric layers that enhance resistance to gate current

leakage and inhibit boron penetration into the dielectric [86]. However, as the sizes of

semiconductor devices continue to decrease, it is becoming critical to develop processing

methods that afford control of film properties and composition at the monolayer level.

Low-temperature remote plasma processing offers distinct advantages over other

ultrathin silicon oxynitride film preparation methods [33]. For example, researchers

[87-88] have reported that the concentration profiles of nitrogen and oxygen atoms within

SiOxNy films grown by remote plasma processing on Si(100) can vary significantly with

both the feed gas composition and processing protocol that is used. Of particular interest

is the observation that the exposure of Si(100) to an N20 or N2/02 plasma produces a film

with a highly heterogeneous composition profile in which the nitrogen atoms accumulate

at the film-Si interface and the oxygen atoms remain close to the film-vacuum interface.

It was suggested that this composition profile results, at least in part, by chemical

reactions in which gaseous oxygen atoms efficiently scavenge nitrogen atoms located

closest to the film-vacuum interface. In a separate study, Watanabe and Tatsumi [89]

also observed a decrease in surface nitrogen concentration after exposing a nitrided

Si(100) surface to an oxygen plasma, and asserted that nitrogen atoms are removed from









the surface through reactions with gaseous oxygen atoms. While thermally activated

reactions between accommodated oxygen and nitrogen atoms could have taken place at

the surface temperature of 750 C used in the work of Watanabe and Tatsumi, the other

investigations cited were conducted with the substrate held at 300 C, which is well

below the temperature for appreciable thermal decomposition of silicon oxide [30] and

nitride surfaces [90-91]. It is therefore likely that the oxygen-induced removal of

nitrogen from the surfaces of these films occurs by non-thermal processes such as

Eley-Rideal abstraction. In the present study, we find that direct nitrogen abstraction by

gaseous oxygen atoms is energetically favorable when nitrogen is bound at the Si(100)

surface in coordinatively unsaturated configurations, but less so when the nitrogen is

triply coordinated with surrounding silicon atoms.

3.2 Computational Approach

Quantum chemical calculations of oxygen induced abstraction of a nitrogen atom

from the Si(100)-(2x1) surface were performed using density functional theory (DFT)

and cluster models of the surface. Cluster energies were computed using the unrestricted,

hybrid three-parameter Becke method (UB3LYP), which combines the gradient-corrected

exchange functional of Becke [49, 53] with the Lee-Yang-Parr (LYP) correlation

functional [50]. An unrestricted approach was used for all calculations because the

reactants, products and transition states are open shell structures. Geometry optimization

methods based on the Berny algorithm were used for calculating local minima [73-74],

while transition state structures were obtained using a multiple step procedure. In this

approach, an initial transition structure is obtained using the quadratic synchronous

transit-guided quasi-Newton method (QSTN) [74, 82]. A frequency analysis is then

performed to determine the normal mode of vibration that has the largest imaginary









frequency. In the final step, this normal mode is selected to search for a transition

structure using an eigenvector following method. This extensive approach was found to

be more robust than the standard search routines available in the Gaussian03 package for

identifying transition structures for the open shell systems we investigated. After

convergence, a final frequency analysis was performed to confirm that the local minima

and transition structures were zero or first order saddle points, respectively [92].

Figure 3-1 shows the Si9H12 one-dimer cluster that was used to model the

Si(100)-(2xl) surface in these calculations. This cluster model has been widely used to

study chemical reactions on Si(100)-(2xl) using DFT [10, 93-101], mainly because it is

the smallest structure that adequately represents the main structural characteristics of the

Si(100)-(2xl) surface such as the tilted silicon dimer bond and well-oriented sp3 covalent

bonds. DFT calculations using the one-dimer cluster have been found to accurately

predict bond energies and barriers for several reactions on Si(100)-(2xl) [10, 99] for

which non-local electronic effects [98] are of minor importance. Hydrogen atoms are

used to terminate the bonds of the Si cluster (i.e., hydrogen atoms are used as substituents

for the bulk silicon atoms that are removed by truncation of Si-Si bonds at the exterior of

the cluster). These H atoms preserve the tetrahedral sp3 bonding environment of

subsurface Si atoms, mimic strain that the bulk silicon atoms would impose on the

boundary of the cluster, and generally have a negligible effect on the quantum chemistry

calculation itself [99, 102]. Geometric constraints are imposed on the hydrogen atoms to

improve the simulation of bulk strain effects as follows: All H atoms in place of bulk Si

atoms are held fixed in their ideal tetrahedral configurations, while those replacing silicon

atoms in neighboring dimers are fixed in positions that mimic the buckled dimer structure

[96, 99]. The third and fourth layer Si atoms are not directly constrained, although their









displacements are hindered by the constraints imposed on their neighbors and terminating

hydrogen atoms; the fourth layer Si atom is practically held fixed in space due to the

limited displacement of the third layer silicon atoms. Finally, all chemically active

atoms, which include the silicon dimer atoms, the second layer Si atoms, and the nitrogen

and oxygen atoms, are allowed to fully relax during the geometry optimizations.

To reduce the computational expense of the calculations, a mixed basis set was

used to expand the electronic wave function. A diffuse triple-p plus polarization

6-31 l1++G(2d,p) basis set was used to describe the chemically active atoms while the

remaining subsurface silicon atoms and terminating hydrogen atoms are described with a

double-p plus polarization 6-31G(d) basis set [92]. All the structures investigated have a

spin doublet multiplicity and the calculations were done using the Gaussian03 program

[71].

3.3 Results

3.3.1 Bonding Configurations of a Nitrogen Atom on Si(100)-(2xl)

We investigated the pathways for nitrogen abstraction from four different

configurations of a nitrogen atom bonded on the Si(100)-(2x1) surface to explore the

influence of the local surface bonding environment on these reactions. The N-Si(100)

structures that we examined were recently predicted by Widjaja et al. [98] to be possible

equilibrium structures resulting from incorporating a single nitrogen atom into the silicon

surface. To benchmark our calculations, we optimized the N-Si(100) structures using the

same computational procedure as used by those authors.

The structures and the corresponding energies of formation that we predicted are

shown in Figure 3-2, where the zero of energy is taken to be the energy of the nitrogen

atom and silicon cluster at infinite separation. Each structure has a spin multiplicity of









two. The lowest energy structure, which has an energy of formation of 2.69 eV, is

obtained when the nitrogen atom bonds to a single dimer atom [R(ad)]. The remaining

structures are the N atom bonded with both dimer atoms in an epoxide-like structure

[R(db)], the N atom inserted into a Si-Si backbond [R(bb)] and finally the N atom bonded

with one dimer atom and two second layer Si atoms [R(sat)]. The geometrical properties

of these structures are in excellent agreement with those reported by Widjaja et al. [98],

and the energies of structures R(ad), R(db) and R(sat) differ by less than 0.11 eV from

their results. However, the energy of formation of structure R(bb) is found to be higher

by 0.38 eV from the prediction of Widjaja et al. The differences in the energies predicted

in these studies most likely arise from slight differences in the way geometric constraints

are imposed in the calculations. It is noted that the energy of structure R(bb) is

particularly sensitive to the positions of the terminating hydrogen atoms. Despite this

difference, the geometries and trends in the relative energies of the N-Si(100) structures

are very close to those reported in the study of Widjaja et al. [98]. Finally, each of the

structures (Figure 3-2) is predicted to be a local minimum on the doublet N-Si(100)

potential energy surface, and the lowest barrier for interconversion between the structures

is 0.48 eV [98]. These characteristics suggest that at moderate surface temperature each

structure could exist in appreciable concentrations during the initial stages of nitridation.

3.3.2 Nitrogen Abstraction by a Gas-Phase Oxygen Atom

We investigated reactions between species only in their respective electronic

ground-states. All of the reactions occur on a doublet potential energy surface and









First-layer dimerr)


Second-layer


Third-layer
[100]

Fourth-layer

[110] [110]

Figure 3-1. The Si9H12 cluster used in our UB3LYP calculations. Si atoms are
represented by dark spheres and H atoms are shown as light-colored spheres.

may be represented by the general equation,

0(3P) + N-Si(100) (Ms = 2) -- NO(21) + Si(100) (Ms= 1),

where Ms is the initial spin multiplicity of the surface cluster. For each of the N-Si(100)

structures investigated, nitrogen abstraction by a gas-phase O-atom was found to be

highly exothermic. This conclusion is reached by considering that the bond energy of

NO in its doublet ground-state is about 6.5 eV, whereas the energies of formation of the

N-Si(100) structures range from about 2.7 to 4.7 eV (Figure 3-2). We initially explored

direct pathways for abstraction in which the N-O bond forms and the Si-N bonds break in

a single elementary step, as depicted by the reaction equation shown above. Extensive

searches revealed a transition structure for only one single-step abstraction process,

namely, direct abstraction from the R(sat) structure. For structures R(ad), R(db) and

R(bb), the abstraction pathways exhibit energy wells between the reactants and products

due to the formation and interconversion of adsorbed NO species. These molecular

precursor structures are produced when the 0(3P) atom attaches directly to a nitrogen

dangling bond, yielding an NO species that is bound to the surface.














A C











B D





Figure 3-2. Structural information of N-Si9H12 clusters. A) R(ad). B) R(db). C) R(bb).
D) R(sat). All distances are in A. Terminating hydrogen atoms have been
excluded for clarity. Si atoms are shown as light-colored spheres and N
atoms as darker spheres.

3.3.3 Abstraction of N Adsorbed at the Dangling Bond [R(ad)]

Figure 3-3 shows the predicted pathway by which an oxygen atom abstracts a

nitrogen atom that is adsorbed on a single Si atom of the surface dimer [R(ad)]. The

reaction involves the initial formation of molecular precursor MP(ad) (Figure 3-4B),

which then decomposes to produce a gaseous NO molecule and the bare Si cluster,

structure P (Figure 3-4C). The production of MP(ad) is highly exothermic (4.84 eV),

since this reaction involves the formation of a strong N-O double bond without cleavage

of other bonds in the cluster. The subsequent decomposition of MP(ad) via the formation

of a gaseous NO molecule is endothermic by 0.91 eV. A transition structure could not be









found along the path from structure MP(ad) to P so only the thermochemical barrier must

be overcome in this step. Although the molecular precursor lies at a lower energy than

the abstraction product, a minimum of 3.93 eV of energy would need to be dissipated to

the solid for the molecular precursor to stabilize. Moreover, since the NO species

couples to the solid mainly through the Si-N bond, it is likely that the Si-N bond

0
ReAfrence;
S0 + + :



























-7.53
MP(ad)
Figure 3-3. Reaction pathway for the nitrogen abstraction from R(ad). Energiesarein
units of eV and the zero of energy is taken to be the ground-state nitrogen
atom, the 0(3P) atom and the singlet Si cluster at infinite separation.
atom, the 0(3P) atom and the singlet Si cluster at infinite separation.








will become highly excited and break during N-O bond formation since only 0.91 eV are

needed to detach the NO molecule from the structure. This result therefore suggests that

nitrogen abstraction from structure R(ad) will be highly facile, occurring effectively in a

single atom-surface collision.


0


110.81


Figure 3-4. Critical point structures of the nitrogen abstraction from R(ad). A) R(ad) B)
MP(ad). C) P (final product). All distances are in A and the terminating
hydrogen atoms have been excluded for clarity.

The geometric and electronic changes that occur during the reaction shown in

Figure 3-3 provide additional insights for understanding this reaction. In reactant R(ad),

two unpaired alpha electrons are localized on the N atom, and one beta electron resides

mainly on the triply coordinated Si atom of the surface dimer. Since the 0(3P) atom also









has two unpaired electrons, its interaction with the adsorbed N atom results in the highly

exothermic formation of an NO double bond. The N-O bond in structure MP(ad) is close

in length to that of the N-O bond of an isolated NO molecule (1.20 vs. 1.15 A,

respectively, Figure 3-4). The predominant change in the relative positions of the Si and

N atoms that occurs during the first reaction step is elongation of the Si-N bond from

1.75 to 1.91 A, which indicates a weakening of the Si-N bond when the N-O double bond

of MP(ad) is formed. In the final reaction step, the molecular precursor decomposes by

cleavage of the Si-N bond, with one electron transferring to a ;r* orbital of the NO

molecule and the other remaining at the surface. This remaining electron experiences a

weak ;r-interaction with the initially unpaired electron on the opposing dimer atom since

the bare Si surface in these calculations is taken to be the singlet, buckled dimer structure

that is thought by many to be the ground-state of the Si(100)-(2xl) surface [103-104].

3.3.4 Abstraction of the Nitrogen Bonded Across the Dimer [R(db)]

Shown in Figure 3-5 is the pathway predicted for the abstraction of a nitrogen atom

bonded across the surface dimer [R(db), Figure 3-6(A)]. The first step in this pathway is

N-O bond formation, resulting in structure MP(db). The exothermicity of this reaction is

3 eV, which is quite significant but is still lower by 1.84 eV than N-O bond formation in

the analogous reaction R(ab) -- MP(ab) (Figure 3-3). Formation of structure MP(db) is

followed by NO migration from the bridging site to the dangling bond site from which

NO desorbs. The migration path is predicted to have an energy barrier of 0.48 eV, and

the final desorption step presents a thermochemical barrier of 0.91 eV as discussed. Each

of these barriers is significantly less than the energy released in the initial formation of

the N-O bond. Thus, unless energy is efficiently dissipated away from the initial

collision zone, NO desorption can occur rapidly by this pathway as well.






52




0 0 + + -.joev


O Oxygen
.'. Nitrogen




-4.66 O
R(db)









TS(db-ad) -7.52
-7.66 MP(ad)
MP(db)
Figure 3-5. Reaction pathway of nitrogen abstraction from R(db). Energies expressed are
in eV.

The geometric changes that the cluster undergoes during these reactions steps are

illustrated in Figure 3-6. N-O bond formation to produce MP(db) causes each Si-N bond

to stretch from 1.68 to 1.80 A, and the Si-Si dimer bond to contract from 2.55 to 2.37 A,

which is indicative of weaker Si-N bonds and a stronger Si dimer bond. Also, the N-O

bond in MP(db) is longer than that in MP(ad) (1.27 vs. 1.20 A), suggesting a weaker N-O

bond in the former. These changes may be understood by considering the corresponding

changes in the alpha spin densities on the surface atoms. In the initial reactant R(db), an

unpaired electron is distributed mainly between the Si atoms of the epoxide-like ring.

The alpha spin density on these Si atoms decreases to zero upon formation of molecular

precursor MP(db), and an unpaired electron is distributed almost evenly between the N











126.0e


A .C











.,, ,.4
B ,- D


1'-.




Figure 3-6. Critical point structures of the nitrogen abstraction from R(db). Structural
information and relative energy. A) R(db). B) MP(db). C) TS(db-ad). D)
MP(ad). All distances are in A. The terminating hydrogen atoms were
excluded for clarity. The final product is shown in Figure 3-4(C).

and 0 atoms in the MP(db) structure. These changes indicate that NO bond formation

leading to MP(db) involves a pairing between the lone electron on the reactant R(db) and

an unpaired electron on the incident 0(3P) atom. The second lone electron on the 0(3P)

atom is transferred to a 7r*-like orbital between the N and 0 atoms. The resulting N-O

bond effectively has a bond order of 1.5 and is therefore weaker and longer than the NO

double bond of MP(ad).

The migration of the NO species from the bridging to the dangling bond position is

activated by 0.48 eV, since it involves the simultaneous cleavage of a Si-N bond and the









formation of an NO double bond. In this reaction, an electron from the initial Si-N bond

is transferred to the NO species to form the double bond, while the second electron

remains localized on the Si atom. The transition structure TS(db-ad) for this reaction is

shown in Figure 3-6(C). To reach TS(db-ad) from MP(db), the NO species tilts away

from one of the Si dimer atoms, causing the Si-N bond to stretch to 2.44 A. At

TS(db-ad), the NO bond contracted to 1.21 A, and the remaining Si-N bond is elongated

from 1.68 to 1.85 A. On the path from TS(db-ad) to MP(ad), the structure adopts a more

stable configuration as the NO species tilts farther away from the opposing Si dimer

atom, and the N-O bond contracts to it final value of 1.20 A.

3.3.5 Abstraction of the Nitrogen Bonded at a Backbond [R(bb)]

We found two pathways for nitrogen abstraction from the backbonded position

[structure R(bb)] (Figure 3-7). For each path, the oxygen atom is predicted to first bond

directly with the nitrogen atom of structure R(bb) to generate molecular precursor

MP(bbs) (Figure 3-8B) in which the N-O bond is nearly parallel to the bisector of the

Si-N-Si bond angle. In one pathway, the 0 atom tilts toward the Si atom to form an Si-O

bond resulting in structure MP(bbt) which can then decompose by sequential N-Si bond

cleavage, forming MP(ad) and then a gaseous NO molecule. The alternative pathway

involves direct N-Si bond cleavage, causing the symmetric structure MP(bbs) to convert

directly to MP(ad) from which the NO molecule detaches (Figure 3-7, inset). The

reaction to produce MP(bbs) from the nitrogen backbonded structure is exothermic by

2.68 eV, and only a small energy barrier of 0.03 eV must be overcome for MP(bbs) to

transform into the more stable molecular precursor MP(bbt). An energy barrier of

0.38 eV must be overcome for MP(bbt) to transform to MP(ad), while a barrier of

0.55 eV is predicted for the direct conversion of MP(bbs) to the MP(ad) structure. The









overall reaction is exothermic by 2.25 eV, and the energy released during the initial N-O

bond formation is much greater than the barriers that must be surmounted for the NO

molecule to desorb from the cluster.

The general bonding characteristics of the MP(bbs) structure (Figure 3-8B) are

similar to those of the MP(db) structure (Figure 3-6B) discussed above. In particular,

formation of the N-O bond in MP(bbs) involves a pairing between lone electrons on the

N and 0 atoms, with the second lone electron of the 0 atom transferring to a z*-like

orbital localized along the N-O bond. Despite these similarities, however, structure

MP(bbs) is less energetically favorable than MP(db) by 0.61 eV. A comparison of the

structures (Figures 3-6B and 3-8B) suggests that MP(bbs) is more strained since its

formation is accompanied by an increase of the Si-N-Si bond angle and elongation of the

Si-N bonds. The N-O bond in MP(bbs) is also longer than in MP(db), which suggests

that additional strain would be imposed if the N-O bond were to achieve its optimum

length.

One of the pathways for N abstraction involves the conversion of structure

MP(bbs) to MP(bbt) (Figure 3-8D), which is exothermic by 0.25 eV and presents a

barrier of only 0.03 eV (Figure 3-7). Although the energy barrier is quite small for this

reaction, vibrational analysis indicates that structures MP(bbs) and TS(bbs-bbt) do

correspond to zero and first order saddle points, respectively. The exothermicity for this

reaction step is also relatively small mainly because unfavorable structural changes offset

the energy gained in forming the strong Si-O bond.

The predominant structural changes include stretching of the N-O bond from 1.32

to 1.51 A and an increase of the Si-N-Si bond angle from 132.60 to 164.50. Elongation of

the N-O bond indicates a weakening of this bond, and the substantial increase in the









Alternative path 3-1....r


o 0 TS(bbs-ad)'t at -D
e-0-- -7.05 "
dQ. .0 MP(bbs)

-7.53
-4.37 ______,______
R(bb) C O
0--Q


T r-6 62
-7.05 -7.02 r-6 92> P
MP() TS(bsbbt TS(b d

P(bbt) 53
MP(a d)




0 + + :1 A.0 eoov

0 Oxygen
Nitrogen

Figure 3-7. Reaction pathways for the abstraction of the nitrogen atom inserted in a Si-Si
backbond. Main panel: 0 + R(bb) MP(bbs)- TS(bbs-bbt) --MP(bbt) --
TS(bbt-ad) -- MP(ad) -- P. Top inset: 0 + R(bb) -- MP(bbs)- TS(bbs-ad)
-- MP(ad) -- P. Energies are in units of eV.

Si-N-Si bond angle imparts strain on the Si-Si bonds in underlying layers. Energy is also

required for this reaction because an electron must be removed from a ;r-like bond along

the Si surface dimer for the Si-O bond to form. The remaining steps in this abstraction

pathway include migration of the NO species of structure MP(bbt) to the dangling bond

position, and then NO desorption from MP(ad). The migration reaction MP(bbt) ->









MP(ad) is exothermic by 0.23 eV and has an energy barrier of 0.38 eV (Figure 3-7). The

geometry of the transition structure TS(bbt-ad) (Figure 3-9B) reveals that the NO species

tilts away from the Si dimer atom, causing the Si-O bond to break and the N-O bond to

strengthen and contract early along the reaction path. Concurrently, the Si-N-Si bond

angle decreases to allow the Si-Si backbond to begin forming and the lower Si-N bond to

break. Formation of the Si-Si backbond and the N-O bond is completed on the MP(ad)

side of the barrier.

In the second abstraction pathway for N(bb), the symmetric MP(bbs) structure

converts directly to MP(ad) by the path shown in the inset of Figure 3-7. This reaction

step is exothermic by 0.48 eV and has an energy barrier of 0.55 eV. The corresponding

transition structure TS(bbs-ad) for this path is shown in Figure 3-10(B). The bond


0
1058,

A 103.20










B D





Figure 3-8. Molecular precursors formed after O-chemisorption onto R(bb). Structural
information of the structures involved in the reaction pathway. A) R(bb). B)
MP(bbs). C) TS(bbs-bbt). D) MP(bbt). All distances are in A. The
terminating hydrogen atoms were excluded for clarity.









lengths in TS(bbs-ad) are nearly identical to those in TS(bbt-ad) (Figure 3-9B), which is

not surprising since the reactions that proceed through these transition states have a

common product. Nevertheless, the energies of these transition structures, referenced to

the same initial state, differ by 0.42 eV. The most striking structural difference is that in

TS(bbt-ad) the NO bond is nearly parallel with the plane defined by the Si-N-Si ring,

whereas the NO bond is significantly tilted out of this plane in TS(bbs-ad). This suggests

that overlap of the ,r-like orbitals in the NO species with those in the Si-N-Si ring is

enhanced in the planar configuration of TS(bbt-ad), thereby lowering the energy of the

structure relative to that of TS(bbs-ad).


Figure 3-9. Structures formed during nitrogen abstraction from MP(bbt). A) MP(bbt). B)
TS(bbt-ad). C) MP(ad). D) P. All distances are in A.
















A





















B) TS(bbs-ad). C) MP(ad). D) P. All distances are in A

3.3.6 Abstraction from the N=Si3 Structure [R(sat)]

Figure 3-11 shows two possible pathways for N abstraction from the N Si3

structure [R(sat), Figure 3-12(A)] by an incoming 0(3P) atom, and one pathway for the

adsorption of the 0 atom at a Si dangling bond site. The first abstraction pathway is a

direct process wherein the incident 0 atom removes the N atom from the surface in a

single step. This reaction is the only single-step abstraction process that was identified in

the study. Direct N abstraction from the N Si3 structure is predicted to be exothermic by

2.57 eV and presents a relatively small energy barrier of 0.20 eV. The N-0 separation at

the transition structure is 1.23 A (TS(sat-P) (Figure 3-12B), which is only 0.12 A longer









than the NO bond of the isolated NO molecule. In addition, each second-layer Si-N bond

has broken and two Si-Si backbonds have formed on the path to the transition structure.

This indicates that direct abstraction from N-Si3 is a concerted process in which three

Si-N bonds break while two Si-Si backbonds and a N-O bond simultaneously form

during a single gas-surface collision. This reaction would require the upper Si dimer

atom to move toward the second layer Si atoms by a considerable distance, and the N

atom to simultaneously move out of the plane defined by these three Si atoms as the 0

atom approaches the cluster and the N-O bond begins to form. The other possible

pathway for N abstraction from the N-Si3 structure is also a concerted process in which

the interaction of the O-atom with the cluster produces molecular precursor MP(bbs)

(Figure 3-8B), which then decomposes by the pathways shown in Figure 3-7. However,

neither an intermediate structure nor a transition structure could be found for the reaction

0(3P) + R(sat) "- MP(bbs). Thus, for this reaction to occur, an N-O bond and a Si-Si

backbond must form, and a Si-N bond must break during a single collision of the O-atom

with the cluster.

Although neither of these abstraction pathways is energetically prohibitive, they

each involve the formation and scission of multiple bonds in a single gas-surface

collision. Moreover, these reactions require that the O-atom come in close proximity to

the dangling bond of the Si dimer atom where it could adsorb. Indeed, adsorption of the

O-atom at the dangling bond site (Figure 3-11) is predicted to be barrierless and highly

exothermic (6.35 eV), and is therefore concluded that to be the predominant outcome of

the interaction between an 0(3P) atom and the N-Si3 structure. As may be seen in Figure

3-12, adsorption of an 0(3P) atom at the Si dangling bond causes the local N-Si3

configuration to change negligibly, but it does result in contraction of the Si-Si dimer






61






. -3.85 "
-4 05 >TS(sat-P)
-4.05',
R(sa t)

I N







S /'TS(bbs-ad)', -6.62
-7.05 \ c p
MIP (bbs)

-7.53
1MP(ad)



Q + + J>. 6J, UOeV
0

*',Q Oxygen
C Nitrogen
-10.41
O(ad)-R(sat)
Figure 3-11. Reaction pathways for N-abstraction from (and O-atom adsorption on)
R(sat). Energies are in units of eV and the zero of energy is taken to be the
ground-state nitrogen atom, the 0(3P) atom and the singlet Si cluster at
infinite separation.

bond. This contraction indicates that O-Si formation alleviates repulsive interactions

between the two unpaired electrons that are present on the silicon dimer atoms of the

N=Si3 structure. In fact, we find that the energy of O-atom adsorption at the Si dangling

bond site of R(sat) is 1.9 eV higher than that for 0-adsorption on the clean Si(100)-(2xl)

surface. This indicates that significant electronic repulsion exists along the dimer of the

R(sat) structure, and that O-atom adsorption is highly favorable on this surface.
























IC.S*


Figure 3-12.


Structures formed during the direct nitrogen abstraction from R(sat). A)
0(3P) attacks the nitrogen atom of R(sat). B) Transition structure
[TS(sat-P)]. C) NO molecule desorbed into the gas phase and bare Si
surface (P). All distances are in A. The terminating hydrogen atoms were
excluded for clarity.


O


04 8,


Figure 3-13. Structures involved in the O-atom chemisorption on R(sat). Structural
information and relative energy. A) 0(3P) atom approaching structure
R(sat) and B) O-atom adsorbed at the Si dangling bond site [O(ad) -R(sat)]
All distances are in A. The terminating hydrogen atoms were excluded for
clarity.









3.4 Discussion

Nitrogen abstraction by a gas-phase 0(3P) atom is highly exothermic for each of

the N-Si(100) structures investigated in this study. However, abstraction is predicted to

occur in a single step only for the reaction of 0(3P) with the coordinatively saturated N-

atom of the N=Si3 structure, and (in this case) single-step abstraction appears to be much

less probable than 0 adsorption due to the energetic differences in these reactions and

because multiple bonds must break and form for single-step abstraction to occur. For

each of the coordinatively unsaturated N-Si(100) structures, N abstraction is predicted to

occur by a precursor-mediated pathway that is initiated by the formation of an N-O bond

and the release of between 2.7 and 4.8 eV into the surface. Since the subsequent

elementary steps leading to NO desorption have barriers that are less than 1.0 eV, NO

bond formation provides the system with excess energy that could readily promote local

bond rearrangements and ultimately NO desorption. Indeed, nitrogen abstraction by such

a pathway would effectively be an Eley-Rideal process since the NO product would

evolve into the gas-phase within no more than a few vibrational periods after the initial

gas-surface collision. Alternatively, the energy released during NO bond formation could

be dissipated away from the surface bonds, allowing the adsorbed NO species to

equilibrate in one of the configurations predicted. However, considering the significant

amount of energy (-2 to 4 eV) that would need to be channeled away from the initial

collision zone for a precursor to stabilize, it is reasonable to expect NO desorption to be

the more likely outcome of initial N-O bond formation in these systems. Nevertheless,

calculations of the dynamics for abstraction, and particularly the efficiency of energy

dissipation to the solid, are needed to explore the propensity for 0(3P) atoms to abstract

nitrogen from Si(100) by the pathways predicted here.









Prior experimental studies have provided indirect evidence for nitrogen abstraction

from Si surfaces by gaseous atomic oxygen. For example, exposure of Si(100) to

plasmas containing both 0 and N atoms results in a surface that is depleted of nitrogen

[87, 88, 105]. Similarly, treating nitrided Si(100) with an oxygen plasma has been

reported to lower the surface nitrogen concentration [89, 106]. We recently conducted

reactive scattering experiments in UHV to directly examine the abstraction of nitrogen

from Si(100) by an atomic oxygen beam, but we did not observe the evolution of gaseous

NO using mass spectrometry or nitrogen depletion from the surface [107]. However, the

absence of measurable nitrogen abstraction in our experiments can be attributed to the

bonding state of nitrogen that was investigated. In that work, nitrogen was incorporated

into Si(100) by thermally decomposing NH3 on the surface at a substrate temperature of

900 K; adsorption at this high surface temperature is necessary to ensure the complete

desorption of hydrogen. Above about 700 K, however, nitrogen has been found to

diffuse into the subsurface region of Si(100) [108] and is apparently inaccessible for

direct abstraction by gaseous O-atoms. Our experimental observations therefore suggest

that the N-Si(100) structures considered in the present computational study are not stable

at temperatures greater than about 700 K. In fact, recent quantum chemical

investigations predict that a single nitrogen atom does have an energetic preference to

bond in the subsurface of Si(100) [109-110].

Nevertheless, the N-Si(100) structures investigated in the present study, which

were reported originally by Widjaja et al. [98], may be stable at surface temperatures

lower than 700 K since, to our knowledge, barriers for the migration of nitrogen into the

Si(100) subsurface have not been reported, and may be high enough to enable nitrogen

atoms to stabilize at the surface at low to moderate temperatures. Hence, nitrogen









abstraction by gaseous O-atoms may indeed occur during plasma-enhanced

oxynitridation of Si(100) if the surface temperature is maintained sufficiently low.

Another possibility is that the barriers for nitrogen diffusion to the subsurface are small,

which would imply that the N-Si(100) structures that we investigated in this

computational study (Figure 3-2) do not exist in appreciable concentrations at low to

moderate surface temperature. In this case, a mechanism other than direct abstraction by

gaseous oxygen atoms would be needed to explain observations of nitrogen depletion at

the Si(100) surface by reaction with oxygen plasmas. Experiments to directly investigate

nitrogen abstraction by atomic oxygen will require a method for adsorbing N atoms onto

Si(100) at low surface temperature, which should be possible using an active nitrogen

source such as gaseous N-atoms or through non-thermal activation of nitrogen-containing

adsorbates.














CHAPTER 4
CHEMISTRY ON Si(100)-(2x1) DURING EARLY STAGES OF OXIDATION WITH
O(3P)

4.1 Introduction

Oxygen adsorption on Si(100)-(2xl) is an important process because it corresponds

to the initial stages of the formation of the Si/Si02 interface, a system of high

technological relevance in microelectronic devices, especially in the metal-oxide-

semiconductor field-effect transistor (MOSFET) gate dielectrics. Because of the

demands imposed by Moore's law, the thickness of the silicon dioxide gate dielectrics

grown on Si(100) is crucial for the future of the microelectronic industry. As these layers

approach the sub nanometer region (it is now possible to fabricate devices with Si02 gate

thickness of 1.3 nm [34], which correspond to about ten silicon atoms across the Si/Si02

interface) it is important to understand the detailed mechanism of initial oxidation of

silicon surfaces. This small thickness is a limit for the use of Si02 gate dielectrics given

that the concentration of undesirable silicon suboxides (Si Si2+ and Si3+) in the Si/Si02

interface becomes increasingly greater. Because of this, aggressive research for

alternative gate dielectric materials is taking place, but even for these new gate high-K

dielectric materials, knowledge of the initial steps of Si02 formation is critical given that

many of them still involve one or two monolayers of Si02 deposited over the gate

channel region.

Several methods to fabricate silicon dioxide have been used successfully over the

year [33-34]. However, because of the continuous scaling of the devices, these









traditional methods are becoming increasingly less efficient. Furthermore, as the size of

the silicon dioxide layers approach molecular and atomic dimensions, it would be ideal to

have a fabrication method based on molecular chemistry and low temperature, so the

product obtained by the manufacturing process remains in a given configuration and is

not affected by thermal changes of the system. One of these methods is plasma

processing of silicon substrates. The interactions of radical species such as oxygen

atoms, which can be present in large fractions in plasma environment, with silicon

surfaces are relatively poorly understood. Engel and coworkers [26-31] reported detailed

kinetics studies of the 0 atom oxidation of clean silicon using a plasma-based atomic

beam source. Also, Yasuda et al. [111] studied the 0 atom oxidation of

hydrogen-terminated silicon using a hot-filament source. These studies revealed that 0

atom oxidation is favorable compared with that due to 02, and that several monolayers of

oxide can be formed efficiently via the direct insertion of 0 atoms into near-surface

bonds.

We explore the structures formed during the initial steps of 0(3P) incorporation on

clean Si(100)-(2xl) using small Si9H12 cluster models and gradient-corrected density

functional theory (DFT). From the time when early studies of asymmetric dimer

descriptions [112] and initial adsorption of hydrogen on Si(100) [94] were made, the

Si9H12 cluster has been proven a very useful tool for performing theoretical studies of

local chemistry on Si(100)-(2xl). Recent comprehensive oxidation studies which

combine Si9H12 clusters with gradient-corrected density functional theory (DFT)

techniques yielded accurate energetic and vibrational frequencies for several oxidation

products, although none of them has treated oxidation by 0(3P) specifically. Most of

previous work has focused on the water-induced oxidation of the clean Si(100)-(2x 1)









surface [34, 95, 96, 104, 113, 114], and some on oxidation by molecular oxygen [100].

What has been found so far using the DFT/cluster approach can be summarized as

follows: Initially, water adsorbs dissociatively on top of single silicon dimer and forms H

and OH fragments which are most stable when they are bonded with the hydroxyl

oriented away from the surface dimer bond [95, 104]. Thermodynamic [96] and

mechanistic studies [113-114] revealed a strong tendency for 0 to agglomerate on the

dimers of Si(100) and predicted that the three- and five-oxygen agglomerated structures

were the most stable. Formation of epoxide-like rings upon dehydrogenation of the

surface in the event of oxygen agglomeration was also reported. This tendency for 0 to

agglomerate was found to be thermodynamically driven since it is energetically more

favorable to have one dimer with n oxygens, for n=2-5, and (n-1) oxygen-free dimers,

than it is to have n dimers each with one oxygen.

In this work, we performed DFT/cluster calculations to determine minimum energy

geometries to elucidate the relative thermodynamic energies of the possible oxidation

products that can form by insertion of up to three 0(3P) atoms, and we use this result to

predict a minimum energy reaction path. These oxidized Si9H12 clusters were classified

in three different sets of isomers (Oi-Si9H12, 02-Si9H12 or 03-Si9H12) depending on the

number of oxygen atoms adsorbed on the surface. The energy of these isomers was then

compared to one another to determine how it was affected by the formation of the two

Si-O bonds in place of one Si-Si bond, by the oxidation state of the surface silicon

atoms, by the spin-state of the surface (spin-singlet vs. spin-triplet) and by strain effects

(all quantified by using a bond energy model that assumes that they are independent and

additive). Finally, we investigated transition states for the insertion process, to develop a

model for the preferred mechanism of the initial steps of Si(100) oxidation by 0(3P).









4.2 Theoretical Approach

Our theoretical approach was based on Kohn-Sham density functional theory

(DFT) calculations of clusters of silicon that represent the local bonding arrangement of

the surface. The silicon cluster used was the Si9H12 (Figure 3-1) which is the smallest

structure that appropriately represents the main structural characteristics of the

Si(100)-(2xl) surface (i.e., covalent tetrahedral sp3 arrangement of Si-Si bonds and a

tilted dimer). This cluster has twelve H atoms to terminate all dangling bonds resulting

from truncation of Si-Si bonds at the exterior of the cluster; these terminating hydrogen

atoms preserve the tetrahedral bonding and have a negligible effect on the predicted

energies of the different clusters [102, 115]. The clusters were constrained by imposing

boundary conditions that mimic the strain that the bulk silicon would impose on the

surface dimer under study. In particular, the hydrogen atoms were fixed in their positions

along the directions of truncated Si-Si bonds that they terminate. Two types of hydrogen

atoms can be found in this cluster. The bulk hydrogen atoms, fixed along tetrahedral

directions, and the neighboring dimer hydrogen atoms, constrained in positions that

mimic nearest silicon dimers. The third and fourth layer silicon atoms were not directly

constrained, but the displacement of these atoms is minimal because they are surrounded

by fixed H atoms. All the chemically active atoms including the first and second layer

silicon atoms were allowed to relax completely unconstrained.

Kohn-Sham density functional theory [41-44] is used for the electronic structures

calculations. Specifically, we use the B3LYP hybrid-gradient-corrected method [53, 116]

which calculates the exchange correlation term by means of a linear combination of local,

gradient-corrected and exact Hartree-Fock exchange terms with the Becke

gradient-corrected term (B88) [49], and the local and gradient-corrected correlations









terms of Vosko-Wilk-Nusair (VWN) [48] and Lee-Yang-Parr (LYP) [50], respectively.

The electronic wave function was expanded using mixed Gaussian basis sets.

Diffuse triple-(plus polarization 6-31 l1++G(2d,p) was used for the oxygen atoms and

first and second layer silicon atoms, while subsurface silicon and terminating H atoms

were expanded with a double-(plus polarization 6-3 1G(d) basis set. This approach, used

successfully in similar studies [97-100], focuses basis functions on the chemically active

portion of the cluster and accurately describes orbitals involved in the reaction while

minimizing the computational expense. All calculations were run using an unrestricted

approach to calculate the open-shell structures of the spin-triplet systems. Comparison

between restricted and unrestricted calculation results for closed shell systems (i.e.,

spin-singlet state clusters) showed negligible differences. The commercial program

Gaussian03 [71] was used to run all the calculations.

4.3 Results and Discussion

A primary goal of this study was to assess the factors which influence the

thermodynamic stability of local structures formed in the early stages of Si(100)

oxidation. Toward this end, we performed energy minimization for each possible isomer

in which one, two or three oxygen atoms are inserted into the first and second layer Si-Si

bonds of the Si9H12 cluster. Since these calculations generated many structures, a

shorthand notation of capital letters followed by a subscript and a code in parenthesis is

used for labeling the clusters to facilitate discussion. The first capital letters -which are

DB, ME or TS- stand for dangling bond, minimum energy, and transition state structure,

respectively. Subscripts 1 or 3 appear after the capital letter to indicate whether the

structure is a spin singlet or triplet. Finally, the code in parenthesis is based on the

notation used by Stefanov and Raghavachari [96] in which SiSi corresponds to a non-









oxidized cluster, SiOSi is a cluster with the 0 atom inserted into the dimer bond, SiSiO2

is a structure with two oxygen atoms inserted into the two backbonds of the same dimer

silicon atom, OSiSiO has two oxygen atoms inserted in a backbond pair at the same side

of the dimer Si-Si bond, OxSiSiO has the two oxygen atoms in opposite-side backbonds,

and SiSiOd is a cluster with an oxygen atom bonded to a single silicon atom of the dimer.

Several factors determine the heat of formation of an oxidized cluster, including the

Si oxidation states, bond strain and the spin state of the cluster. We investigated clusters

in both singlet and triplet spin states since the lowest energy structures of the clean

Si(100) correspond to these spin states. The spin has two primary effects on the energy

of the cluster. Firstly, a singlet cluster has one extra Si-Si bond compared with a triplet

cluster, which tends to lower the energy of the structure. However, the formation of this

extra bond also alters the geometric structure of a cluster and hence the strain energy.

The strain is typically higher in a singlet cluster than in the analogous triplet structure for

the one-dimer model. To directly compare the energies of singlet and triplet clusters, the

energy of an oxidized cluster is defined with respect to a common reference state, which

was taken to be the energy of the appropriate number of isolated oxygen atoms plus that

of the clean (unoxidized) cluster in its singlet ground-state, [structure MEi(SiSi)].

The singlet bare cluster was chosen as the reference state because DFT predicts that

this structure minimizes the energy of the clean Si(100) surface. The dangling bonds on

the Si dimer atoms are unpaired in the spin-triplet state of the bare cluster, [ME3(SiSi),

Figure 4-1(a)], but they pair to form a weak ar bond along the dimer in the spin-singlet

state, [MEi(SiSi), Figure 4-1(B)]. This extra bond in MEi(SiSi) shortens and tilts the

silicon dimer bond along the [110] direction, resulting in an asymmetric structure that is









0.37 eV more stable than its symmetric ME3(SiSi) counterpart. The highest occupied

molecular orbitals (HOMO) of both MEi(SiSi) and ME3(SiSi) were also calculated

(Figure 4-1). The 0.37 eV energy difference, labeled as AEspin, reflects the energy gained

by the 7c interaction, which has been reported to be 1.24 eV [117]. It also reflects the

difference in strain energies between the singlet and triplet bare clusters. Since the initial

surface is taken to be the singlet bare cluster, the production of an oxidized cluster in the

triplet spin state may be considered to occur in two steps, namely, conversion of the bare

cluster from the singlet to the triplet ground-state, and then insertion of oxygen atoms

into the triplet bare cluster. Thus, when comparing the energies of the oxidized clusters,

it is important to recognize that the singlet to triplet conversion step contributes an energy

penalty of 0.37 eV to the heat of formation of a triplet cluster. This energy is taken into

account in our analysis of other factors that determine the cluster energies as discussed

below.

We quantified factors which contribute to the energy of an oxidized cluster by

invoking a bond energy model, which has previously been shown to provide a reasonable

representation of the heats of silicon suboxide formation [118]. In the bond energy

model, bond strain and distinct Si-O bond strengths for Si+1, Si+2 and Si+3 oxidation states

are treated as separable effects that contribute additively to the total heat of formation. In

cases for which each oxygen atom in the cluster is inserted into a Si-Si bond, the bond

energy model yields the following expression for the energy of an oxidized cluster,

AEUB3LP = N(2so s) + AEsuboxide + 5(AEsp + AEtran (4-1)

where AEUB3LYP is the energy of the oxidized cluster calculated by DFT, Nis the total

number of oxygen atoms in the cluster, s,o is the Si-O bond energy in stoichiometric ft









quartz crystal and s,, is the Si-Si bond energy in bulk silicon. In addition, AEuboxde is

the total suboxide penalty energy, AEpn is the 0.37 eV energy difference between the

triplet and singlet bare clusters, where 8 is zero for the spin-singlet and one for the

spin-triplet clusters, and AEexess is the excess energy that is mainly related to changes in

strain that result from oxygen insertion into the bare cluster. The suboxide energy

penalty is defined by the Equation 4-2.

AEsuboxide ZNs (Aj) (4-2)
x

where Nsgi is the number of silicon atoms bonded to x oxygen atoms (with x = 1, 2 or

3), and Ax is the energy penalty of a Si atom in the +x oxidation state.

The concept of suboxide penalty energies was introduced in the bond energy model

of Hamman [119] to take into account the effective increase in Si-O bond strength as the

Si oxidation state increases. This effect originates mainly from the greater amount of

ionic character in Si-O bonds involving Si atoms in higher oxidation states, but quantum

resonance has also been suggested to enhance the bond strengths [119]. To quantify the

differences in suboxide Si-O bond strengths, penalty energies were computed by Hamann

[119] and Bongiorno and Pasquarello [118] for each Si suboxide state, under the

assumption that the penalty energies can simply be added to the (2esio-esisi) term to

compute the heats of formation of suboxide structures. Since the quantity esio is defined

as the Si-O bond energy in crystalline SiO2, the penalty energies have positive values so

that their contribution reduces the energy change in forming Si-O suboxide bonds. Table

4-1 shows the values of the suboxide penalty energies determined in the separate studies

of Hamann [119] and Bongiorno and Pasquarello [118].









Relative stability


[110]
L [iT0]


Structural information and highest occupied molecular orbital plot of clean
Si9H12 clusters. A) Symmetric spin-triplet surface ME3(SiSi). B) asymmetric
spin-singlet surface MEi(SiSi). MEi(SiSi) is calculated to be more stable
than ME3(SiSi) by 0.37 eV. Silicon and hydrogen are represented by grey
and white balls, respectively. Bond lengths are expressed in A.


Figure 4-1.









Table 4-1. Suboxide penalty energies for various silicon oxidation states. The
superscripts denote the number of oxygen atoms bonded to a particular
silicon atom. Energies are given in eV.
Oxidation state Energy penalty Ref 118 Ref 119
Si+1 A1 0.50 0.47
Si+2 A2 0.51 0.51
Si+3 A3 0.22 0.24

The penalty energies determined in these studies by Hamann [119] and Bongiorno

and Pasquarello [118] are in good agreement with one another and are significant in value

(Table 4-1), which suggests that maximizing the Si oxidation states is an important

driving force that determines the types of local structures that form on Si(100) during

initial oxidation.

We used the bond energy model given in Eq. 4-2 to quantify changes in strain

energy resulting from the insertion of oxygen atoms into Si-Si bonds of the bare cluster.

We chose to use an average of the suboxide penalty energies reported by Hamann [119]

and Bongiorno and Pasquarello [118] in our analysis. This approach appears justified

considering the close agreement in the penalty energies determined in those

investigations. In addition, the values of Eso and Es,s were taken to be -4.35 and -

2.02 eV, respectively, as reported for the Si-O bond energy in crystalline Si02 (/f quartz,

the lowest-energy form), and for the Si-Si bond energy in bulk silicon [96, 117].

Additionally, for spin-singlet structures where the oxygen atom inserts into the silicon

dimer, the appropriate value of s,s, in Eq. 4-2 corresponds to the energy for Si-Si bondd

cleavage (-1.24 eV) [117]. These choices introduce some uncertainty in the absolute

strain energies that are calculated because Eso and Es,s are experimentally determined

values and are, therefore, not subject to the systematic errors that may affect the energies

predicted by DFT.









We also investigated clusters in which one of the oxygen atoms adsorbs on a single

Si atom of the surface dimer and forms only one Si-O bond, which we label as Si-Od.

These structures can serve as precursors to the formation of Si-O-Si linkages which form

when the adsorbed 0 atom inserts into Si-Si bonds near the surface. In fact, a recent

experimental study by Gerrard et al. [107] shows that gaseous 0(3P) atoms initially

adsorb at surface dangling bond sites on Si(100), forming Si-Od bonds, before

incorporating into the solid. Hence, investigating these dangling bond structures is

essential for developing a mechanistic understanding of Si(100) oxidation by gaseous

oxygen atoms. For the dangling bond structures, the following variation of Eq. 4-2 was

used to quantify strain energies,

AUB3LYP = (N- X2sSiO SiSi) + -SiOd + Esuboxide + 8(AEspin)+ Aexcess (4-3)

where all quantities have the same definitions as given above except for eSiOd which is the

energy of the Si-Od bonds. We find that Si-Od bonds have a more delocalized character

than the Si-O bonds in Si-O-Si linkages due to the lone electron from the oxygen atom.

One consequence of this delocalization is that the bond energy model with the parameters

defined above, yields excessive values for the relative strain energies for the dangling

bond structures (Section 4.3.2). In virtue of this, the value of eSiOd has been defined by

the difference in energy between the DB3(SiSiOd) and MEi(SiSi) structures.

Finally, it is important to point out that charge transfer involved in the

incorporation of adsorbed species on Si(100)-(2xl) can exhibit nonlocal effects (i.e.,

charge transfer to a neighboring dimer on the surface) which may limit the overall

usefulness of the Ox-Si9H12 clusters in estimating heats of formation. However, Widjaja

and Musgrave [100] reported that the electron density in the similar system of 02









adsorbed on Si(100)-(2xl) remains localized within the one-dimer environment. This

observation supports the use of the small Si9H12 cluster for our investigation.

4.3.1 Structures with One Adsorbed Oxygen Atom (Oi-Si9H12)

Since we are interested in structures formed during the early stages of oxidation

of the Si(100)-(2xl) surface, we considered only oxygen atom insertion into the surface

dimer bond, one or more backbonds or a dangling bond. Optimized total energies of the

different oxidized clusters are used to evaluate their stability relative to that of the clean

clusters. For now, we postpone discussing the mechanistic aspects (i.e., reaction barriers)

of these reactions and focus exclusively on the relative thermodynamic stabilities of the

different isomers.

-3 -

-3.5 -








-5.5-559
--5.81 -

-6.5 -

Structure
Figure 4-2. Relative energies of Oi-Si9H12 isomers. A) DB3(SiSiOd). B) ME3(SiSiO).
C) ME3(SiOSi). D) MEi(SiOSi). E) MEi(SiSiO). Energy reference:
MEi(SiSi) + 0(3P) = 0.00 eV. Oxygen and silicon atoms are represented by
black and gray balls, respectively.

For the minimum energy analysis, the energies of the Oi-Si9H12 isomers are

referenced to the reaction

0(3P) + Si9H12 (Ms = 1) -- Oi-Si9H12 (Ms = 1 or 3) (4-4)









where Ms is the spin multiplicity. Table 4-2 shows the different energies that contribute

to the calculated UB3LYP energies of the O1-Si9H12 clusters, in order of ascendant

stability. The DB3(SiSiOd) structure (Figure 4-3A) is the least favorable cluster because

it only has one new Si-O bond. According to the bond energy model for dangling bond

structures (Eq. 4-3) and the definition of the ESiOd energy, there is not any stress

associated to DB3(SiSiOd). This value is not surprising, since the only difference in the

geometric characteristics of the DB3(SiSiOd) and ME3(SiSi) structures (Figure 4-1A) is

an 0.02 A elongation of each of the backbonds that are located on the oxygen side of the

dimer.

Table 4-2. Penalty energies of O1-Si9H12 isomers. Contributions of the different factors
to the calculated DFT/UB3LYP energies, according to the bond energy
model. Energies are given in eV. For all structures: N= 1. Isomers are listed
in order of increasing thermodynamic stability. Energy reference: MEi(SiSi)
+ 20(3P) = 0.00 eV.
Structure N(2esio-esisi)a ESiOd AEsuboxide 6(AEspin) MAexcess AEUB3LYP
(Eq. 4-3)
Dangling Bond Isomers
DB3(SiSiOd) 0.00 -4.74 0.00 0.37 0.00 -4.37
Minimum Energy Isomers
ME3(SiSiO) -6.68 N/A 1.00 0.37 -0.28 -5.59
ME3(SiOSi) -6.68 N/A 1.00 0.37 -0.50 -5.80
MEI(SiOSi)b -7.46 N/A 1.00 0.00 0.56 -5.90
MEi(SiSiO) -6.68 N/A 1.00 0.00 -0.49 -6.17
a: corresponds to (N-1) (2esio-esisi) for dangling bond clusters.
b: uses a value of -1.24 eV for the esisi, which corresponds to the Si-Si xZ bond

In the remaining O1-Si9H12 structures, two Si-O bonds are formed at the expense of

one Si-Si bond, so each structure has the same suboxide energy penalty. Thus,

differences in the relative energies arise from the spin penalty energy and the different

amounts of strain in the structures. Firstly, the spin-singlet isomers MEi(SiOSi) (Figure

4-3D) and MEi(SiSiO) (Figure 4-3E) are predicted to be the more energetically favorable

structures, indicating that strain relief in the triplet structures is insufficient to





















































Spin-triplet


2.34.


,. =97.50
. =42.8
S=39.70


[100]


[110] [11iTb]


Spin-singlet


Structural characteristics of Si9H12 clusters with one oxygen atom. A)
DB3(SiSiOd). B) ME3(SiOSi). C) ME3(SiSiO). D) MEi(SiOSi). E)
MEi(SiSiO). Oxygen and silicon atoms are represented by black and gray
balls, respectively. Bond lengths are expressed in A. The crystalographic
coordinate origin gives the orientation of the clusters. The surface is along
the [100] plane.


Figure 4-3.









compensate the spin penalty energy. Structure MEi(SiOSi) is more strained, and

therefore less favorable than MEi(SiSiO) by 0.27 eV, since formation of the epoxide-like

ring stretches and weakens the Si-Si dimer bond. This result agrees with observations

made by Weldon et al. [113-114], who were the first researchers to report the

epoxide-like ring structures and suggested that they were the thermodynamically favored

products after oxygen agglomeration (i.e., after three or more oxygen atoms absorb on the

Si(100) surface). For the triplet structures, ME3(SiOSi) is more favorable than

ME3(SiSiO) by 0.22 eV since oxygen insertion breaks the dimer bond and thereby

relieves the strain that the dimer bond imposes on the bare cluster. Notice in Figure

4-3(B) that the silicon dimer atoms are separated by 3.00 A in the ME3(SiOSi) structure.

Finally, the bond energy model predicts negative strain energy (i.e. strain relief, for

almost all of the O1-Si9H12 structures, suggesting that each structure possesses less strain

than the corresponding bare cluster). The only exception is the MEi(SiOSi) structure, in

which the predominant structural change is a significant elongation of the silicon dimer

bond to a length of 2.57 A, that increases the strain in the cluster by 0.56 eV.

Nevertheless, the silicon dimer bond is not cleaved in this structure given that the

formation of the Si-O bond only disrupts the weak Si-Si interaction along the dimer.

This formation does not break the stronger Si-Si o- bond, although it elongates the bond

considerably.

4.3.2 Structures with Two Adsorbed Oxygen Atoms (02-Si9H12)

Figure 4-4 shows the relative thermodynamic stability of all the isomers with two

oxygen atoms (02-Si9H12), and the various contributions to these relative stabilities are

presented in Table 4-3, in order of ascendant stability. The energies of the isomers are

referenced to the reaction expressed in Eq. 4-5.









20(3P) + Si(100) (Ms = 1) -- 02-Si(100) (Ms = 1 or 3) (4-5)

where, as before, Ms is the spin multiplicity. As in the case of the Oi-Si9H12 clusters, the

dangling bond structures are the least favorable structures, because they only have three

Si-O bonds instead of the four that form on all the other isomers given that one of the

oxygen atoms bonds to a silicon dangling bond. An interesting structural effect is

predicted by the DFT/UB3LYP calculations for the dangling bond structures which

significantly influences the heats of Si-Od bond formation. We find that Si-Od bond Si-Si

bond lengths and angles relative to the reactant ME3(SiSi). In contrast, formation of an

Si-Od bond on the ME3(SiOSi) and ME3(SiSiO) structures causes one Si-Si bond length

to increase, thereby weakening the structure. In particular, with respect to the reactant

structures, the dimer Si-Si bond is stretched by about 0.07 A in DB3(OSiSiOd) and

DB3(SiSiO2d), and one of the Si-Si backbonds on the Od side of the dimer is

Table 4-3. Penalty energies of 02-Si9H12 isomers. Contributions of the different factors
to the calculated DFT/UB3LYP energies, according to the bond energy
model. Energies are given in eV. For all structures: N= 2. Isomers are listed
in order of increasing thermodynamic stability. Energy reference: MEi(SiSi)
+ 20(3P) = 0.00 eV
AEUB3LYP
Structure N(2esio-esisi)a ESiOd AEsuboxide 6( AEspin) AEexcess (Eq. 4-3)
Dangling Bond Isomers
DB3(OSiSiOd) -6.68 -4.74 1.00 0.37 0.23 -9.82
DB3(SiSiO2d) -6.68 -4.74 1.00 0.37 -0.27 -10.32
DB3(SiOSiOd) -6.68 -4.74 1.00 0.37 -0.38 -10.43
Minimum Energy Isomers
MEi(OSiSiO) -13.37 N/A 2.00 0.00 0.33 -11.04
ME3(OxSiSiO) -13.37 N/A 2.00 0.37 -0.26 -11.25
MEi(OxSiSiO) -13.37 N/A 2.00 0.00 0.06 -11.31
ME3(SiSiO2) -13.37 N/A 1.51 0.37 0.05 -11.44
ME3(OSiSiO) -13.37 N/A 2.00 0.37 -0.70 -11.69
ME3(SiOSiO) -13.37 N/A 1.51 0.37 -0.57 -12.05
MEi(SiOSiO)b -14.15 N/A 1.51 0.00 0.48 -12.15
MEi(SiSiO2) -13.37 N/A 1.51 0.00 -0.30 -12.16
a: corresponds to (N-1) (2esio-esisi) for dangling bond clusters.
b: uses a value of -1.24 eV for the esisi, which corresponds to the Si-Si 7r bond.









stretched by 0.10 A in the DB3(SiOSiOd) structure. As expected, this Si-Si bond

elongation lowers the heat of Si-Od bond formation. For example, the heat of the reaction

ME3(SiSi) + 0(3P) = VME3(SiSiOd) is 0.50 eV greater than the heat of the reaction

ME3(SiSiO) + 0(3P) = ME3(OSiSiOd). This energy difference may be attributed almost

entirely to the observed Si-Si bond weakening since the Si-Od bonds in both structures

involve a Si+1 species, and the only appreciable structural difference between reactants

and products is the longer Si-Si dimer bond in ME3(OSiSiOd). It is tempting to conclude

that Si-Si bond weakening occurs in the O2-Si9H12 structures because the Si atoms in

these bonds have higher partial positive charges than in ME3(SiSiOd). For example, the

dimer bond in ME3(SiSiOd) involves an Sio and Si+1 species, whereas in VIME3(OSiSiOd)

the dimer Si atoms are both nominally in the +1 oxidation state. However, similar Si-Si

bond stretching is not observed in 02-Si9H12 structures in which each oxygen atom is

present in an Si-O-Si linkage (Figure 4-6). Hence, delocalization of the unpaired electron

of the Od atom must play an important role in weakening Si-Si bonds in the O2-Si9H12

dangling bond structures.

The remaining 02-Si9H12 isomers result from the formation of Si-O bonds in

siloxane bridges or epoxide-like rings in place of the dimer bond or backbonds (Figure

4-6). The relative energies of these structures are determined by a combination of effects,

though the suboxide energy penalty generally has the largest influence. The overall trend

is that the structures possessing four Si+1 species are less favorable than those with a

combination of two Si+ and one Si+2 species, and that the spin-singlet structures are more

favorable than their spin-triplet counterparts. Indeed, three of the four isomers possessing

a Si+2 atom are the most energetically favorable among the two 0-atom isomers and, in

order of increasing stability, the most favorable structures are ME3(SiOSiO) <









MEi(SiOSiO) < MEi(SiSiO2). The ME3(SiSiO2) structure falls out of this trend as it is

0.25 eV less favorable than the ME3(OSiSiO) structure which has four Si+1 species and,

consequently, should be less stable. After accounting for the suboxide penalties, the bond

energy model suggests a substantial 0.75 eV difference in the excess energies of these

two structures, which probably arises from a combination of strain effects associated to

the bonding site of the oxygen atom. The results suggest that forming two

siloxane-bridges with two different silicon dimer atoms is the favored product.

-9.1 A

-9.6 C
-9.82 .
-10.0.33

-10.6 -10.43 E F G


-11.25 -1131 J K
-11.6 -1144 --

-12.1 --
-12.05 -12.15 -12.16
-12.6 -
Structure
Figure 4-4. Relative energies of 02-Si9H12 isomers. A) DB3(OSiSiOd). B)
DB3(SiSiO2d). C) DB3(SiOSiOd). D) MEi(OSiSiO). E) ME3(OxSiSiO).
F) MEi(OxSiSiO). G) ME3(SiSiO2). H) ME3(OSiSiO). I) ME3(SiOSiO).
J) MEi(SiOSiO). K) MEi(SiSiO2). Energy reference: MEi(SiSi) + 20(3p) =
0.00 eV. Oxygen and silicon atoms are represented by black and gray
spheres, respectively.

Another substantial effect of strain is the observation that structure ME3(OSiSiO)

(Figure 4-6A) is significantly lower in energy than structure MEi(OSiSiO) (Figure 4-6E),

despite the spin penalty energy that applies to the triplet structure. After taking into

account the spin penalty energy, the bond energy model suggests that MEi(OSiSiO) has

1.02 eV more strain than ME3(OSiSiO). In both of these structures, the formation of two













































p =108.5o

Side view Top view

Figure 4-5. Structural information of dangling bond isomers with two oxygen atoms
adsorbed on Si(100). A) DB3(OSiSiOd). B) DB3(SiSiO2d). C)
DB3(SiOSiOd). Bond lengths are expressed in A. Oxygen atoms are
represented with black balls. Silicon and hydrogen atoms are represented in
gray and white balls, respectively.
















































Side view Top view Side view Top view
Spin-triplet Spin-singlet
Figure 4-6. Structural information of 02-Si9H12 isomers. Structures are listed by
increasing relative stability of the spin-singlet isomers. A) ME3(OSiSiO).
B) ME3(OxSiSiO). C) ME3(SiOSiO). D) ME3(SiSiO2). E) MEi(OSiSiO).
F) MEi(OxSiSiO). G) MEi(SiOSiO). H) MEi(SiSiO2). Bond lengths are
expressed in A. Oxygen atoms are represented with black balls. Silicon and
hydrogen atoms are represented in gray and white balls, respectively.









siloxane bridges on the same side of the silicon dimer bond shifts the dimer out of its

original symmetry plane, without changing its orientation along the [110] direction.

However, these structures experience markedly different strain effects due to the

different bonding interactions along their respective Si-Si dimers. In ME3(OSiSiO), the

formation of the two siloxane bridges alleviates strain in the remaining Si-Si backbonds,

which each shorten by 0.04 A. Aside from being shifted out of plane, the dimer bond

remains largely unaltered by the formation of ME3(OSiSiO). These structural changes

are predicted to relieve about 0.70 eV of strain from the cluster, and result in considerable

stabilization of the structure, making it the most energetically favorable of all the

structures with four Si+1 species. In contrast, the formation of the siloxane bridges

significantly weakens the w interaction across the silicon dimer in MEi(OSiSiO), causing

the dimer bond to stretch by 0.16 A, resulting in an 0.33 eV increase in strain relative to

the singlet bare cluster, according to the bond energy analysis.

4.3.3 Structures with Three Adsorbed Oxygen Atoms (O3-Si9H12)

Figure 4-7 shows relative energies of isomers with three oxygen atoms and table

4-4 shows the different contributions to the relative energies of the structures, as

determined from the bond energy model.

As in the previous two cases discussed above, the energies of the isomers are

defined with respect to the reaction described by Equation 4-6.

Si9H12(Ms=l) + 30(3P) -- 03-Si9H12 (Ms=1 or 3) (4-6)

We calculated the relative energies and structural characteristics of the 03-Si9H12

isomers (Figures 4-8 and Figure 4-9). We found that the relative energy of the structures

is determined by a combination of effects, including strain and suboxide formation.