First-Principles Simulations in Multiple Dimensions

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First-Principles Simulations in Multiple Dimensions Nano-Particles, Surface and Bulk
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english
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Wu, Yuning
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Doctorate ( Ph.D.)
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University of Florida
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Physics
Committee Chair:
Cheng, Hai Ping
Committee Co-Chair:
Biswas, Amlan
Committee Members:
Hirschfeld, Peter J
Monkhorst, Hendrik J
Chen, Youping

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dft -- nano-scale
Physics -- Dissertations, Academic -- UF
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Physics thesis, Ph.D.
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Abstract:
Systems in different dimensions can exhibit different and nearlyunique properties, such as catalytic behavior in zero-dimensional (0D) metalclusters, adsorption patterns of molecules on two-dimensional (2D) noble metalsurfaces, and mechanical and magnetic properties in three-dimensional (3D)crystals. Accurate description of quantum effects and electron correlations is imperativeto correctly study these systems. Density functional theory (DFT) is a reliableand efficient method to determine the electronic structure across many methodsand systems. We have studiedthe silver cluster cations and the adsorption of N2 and O2molecules onto them. We have interpretedthe mutually cooperative co-adsorption of oxygen and nitrogen as a result of theN2-induced increase in charge transfer from Agn+ cations to O2. To understand the role of chlorine in the fragmentation ofAg nanostructures, we have studied the diffusion of Agn and AgnClm(n = 1 to 4) clusters on an Ag(111) surface, and the interaction strength of(Ag55)2 dimers with and without chloridization. Bondweakening and enhanced mobility are two important mechanisms underlyingcorrosion and fragmentation processes. We have demonstrated the concept of molecularmagnetocapacitance, in which the quantum part of the capacitance becomesspin-dependent. The nano-magnet Mn3O(sao)3(O2CMe)(H2O)(py)3shows a 6% difference in capacitance. For 2D systems, wehave investigated the structure, energetics, electronic and magnetic structuresof Fen-doped C60 monolayers supported by h-BN monolayer covered Ni(111) surfaces.The binding energy, charge transfer (from Fen to C60),and magnetic moment all increase monotonically as functions of n. The electron charge transfer is fromthe spin minority population. We have also studied the adsorption of C60molecules on Au(111) surface defects. The adsorption energy of the strongbonding configuration is much higher than the weak bonding configuration. We haveinvestigated the effect of epitaxial strain on a BiMnO3 thin.Anti-ferromagnetism starts to emerge under about 2% epitaxial strain. The AFMorder can break inversion symmetry and induce electric polarization, a result whichis comparable to experiment. Finally, we have studied the structure, energetics, elastic tensors andmechanical properties of four crystalline forms of Ta2O5with exact stoichiometry as well as a model amorphous structure.
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by Yuning Wu.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Cheng, Hai Ping.
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Co-adviser: Biswas, Amlan.
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1 FIRST PRINCIPLES SIMULATIO NS IN MULTIPLE DIMENSIONS: NANO PARTICLES, SURFACE AND BULK By YUNING WU 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 2012

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2 2012 Yuning Wu

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3 To my wife and my parents

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4 ACKNOWLEDGMENTS First I would like to thank my adv isor, Profe ssor Hai Ping Cheng, who has taught me so much. I benefit enormously from her professional guidance and rigorous scientific attitude. She has always been watchful and helpful for me, not only as my advisor, but also as a friend. I will take her diligence, strictness and enthusiasm in science as my role model for the rest of my life. I am very grateful to all my collaborators, including Dr. XiaoGuang Zhang, Professor Amlan Biswas, Dr. Schmidt Martin, Dr. Nouari Ke bali, and Dr. Catherine Brchig nac for their valuable discussions and suggestions. It is great pleasure to collaborate with them. I would also like to thank my committee members, including Professor s Henk Monkhorst, Peter H irschf eld, Amlan Biswas, and Youping Chen, as well as Professor Samuel Trickey Their effort s to improve the quality of this thesis are highly appreciated. I also want to thank previous and current group members, especially Dr. Chao Cao. Discussion with him has always been very beneficial, valuable and pleasant I h ave learned a lot also from Dr. Yan Wang. In addi tion, my gratitude also goes to my friends in Gainesville for their delightful companionship. They are Xue and family, Yue and family, Bo and family, Yan and family, Xingyuan and family, Yilin and family, Mengxing and family, and Jin. Finally, I want to thank my parents and my wife, for their endless love and support. I thank the US/DOE grant DEFG 0202ER 45995, NSF/PHY 0855292 and NSF / DMR0804407, as well as the UF/HPC and DOE/NERSC for c omputing resources.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 8 LIST OF FIGURES ........................................................................................................ 10 LIST OF ABBREVIATIONS ........................................................................................... 13 ABSTRACT ................................................................................................................... 14 CHAPTER 1 BACKGROUND ...................................................................................................... 16 2 THEORE TICAL AND COMPUTATIO NAL METHODS ............................................ 20 2.1 Density Functional Theory ................................................................................ 20 2.1.1 Fundamental ........................................................................................... 20 2.1.2 KohnSham Procedure ............................................................................ 21 2.2 ExchangeCorrelation Functionals .................................................................... 24 2.2.1 Local Density Approximation (LDA) ......................................................... 24 2.2.2 Generalized Gradient Approximation (GGA) ........................................... 25 2.2.3 Hybrid Functionals ................................................................................... 26 2.3 Plane Wave Basis Set ...................................................................................... 27 2.4 Pseudopotential ................................................................................................ 28 2.4.1 Norm Conserving Pseudopotential .......................................................... 2 8 2.4.2 Ultrasoft Pseudopotential and Projector Augmented Wave (PAW) Method .......................................................................................................... 30 2.5 Modern Theory of Polarization .......................................................................... 34 2.5.1 Conventional Definition ............................................................................ 34 2.5.2 Fundamental of Polarization Theory ........................................................ 35 2.5.3 Berry Phase Description .......................................................................... 36 2.5.4 The Quantum of Polarization ................................................................... 38 2.5.5 The Connection to Wannier Orbitals ....................................................... 39 3 CO ADSORPTION OF N2 AND O2 ON SILVER CLUSTERS ................................. 40 3.1 Methods and Computational Details ................................................................. 42 3.2 Results and Discussion ..................................................................................... 43 3. 2.1 Structure of Pristine Agn Clusters ............................................................ 43 3.2.2 Physisorption of N2 Molecules ................................................................. 46 3.2.3 Chemisorption of O2 Molecules ............................................................... 50 3.2.4 (N2)nO2 Co adsorption on Ag4 + ............................................................... 52

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6 3.3 Summary and Conclusion ................................................................................. 53 4 ENHANCE OF AG CLUSTE R MOBILITY ON AG SUR FACE BY CHLORIDIZATION ................................................................................................. 57 4.1 Methods and Computational Details ................................................................. 59 4.2 Results and Discussion ..................................................................................... 60 4.2.1 AgnClm Clusters on Ag(111) Surfaces ..................................................... 60 4.2.1.1 Ag Monomer and AgCl ................................................................. 61 4.2.1.2 Ag dimer and Ag2Cl ..................................................................... 62 4. 2.1.3 Ag Trimer and Ag3Cl3 ................................................................... 65 4.2.1.4 Ag Tetramer and Ag4Cl4 ............................................................... 67 4.2.1.5 Formation Energies ...................................................................... 69 4. 2.2 (Ag55)2 dumbbell structure ....................................................................... 70 4.3 Summary and Concluion ................................................................................... 72 5 MOLECULAR MAGNETOCAP ACITANCE ............................................................. 74 5.1 Method and Computational Details ................................................................... 75 5.2 Results and Discussion ..................................................................................... 76 5.3 Summary and Conclusion ................................................................................. 85 6 FIRST PRINCIPLES STUDIES O F C60 MONOLAYERS ON METAL SURFACES ............................................................................................................ 87 6.1 Fedoped monolayer C60 on h BN/Ni(111) Surface .......................................... 87 6.1.1 Method and Computational Details ......................................................... 89 6.1.2 Results and Discussions ......................................................................... 92 6.1.2.1 Structure ...................................................................................... 92 6.1.2.2 Electronic and Magnetic Properties ............................................. 99 6.1.3 Summary and Conclusions .................................................................... 111 6.2 C60 on defected Au(111) surface .................................................................... 113 6.2.1 Method and Computational Details ....................................................... 114 6.2.2 Results and Discussions ....................................................................... 114 7 SYMMETRY BREAKING INDUCED BY EPITAXIAL STRAIN IN BIMNO3 THIN FILM ...................................................................................................................... 118 7.1 Method and Computational Details ................................................................. 119 7.2 Results and Discussions ................................................................................. 119 7.3 Summary and Conclusions ............................................................................. 125 8 FIRST PRINCIPLES STUDIES O F TA2O5 POLYMORPHS ................................. 126 8.1 Method and Computational Details ................................................................. 128 8.2 Results and Discussions ................................................................................. 132 8.2.1 Structures .............................................................................................. 132 8.2.1.1 and L Ta2O5 ......................................................................... 132

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7 8.2.1.3 77atom LTa2O5 ........................................................................ 137 8.2. 1.4 Model amorphous Ta2O5 ............................................................ 138 8.2.2 Energetics and electronic structure ....................................................... 139 8.2.3 Elastic Moduli ........................................................................................ 141 8.2.4 Technical remarks: VCA and direct approach ....................................... 142 8.3 Summary and Conclusion ............................................................................... 147 9 CONCLUSIONS ................................................................................................... 148 LIST OF REFERENCES ............................................................................................. 152 BIOGRAPHICAL SKETCH .......................................................................................... 166

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8 LIST OF TABLES Table page 3 1 Total energy (TE), binding energy per atom (BE/ n ), and ionization potential (IP) obtained by using HSE (2nd and 3rd columns) and PBE (4th and 5th columns), respectively. ....................................................................................... 45 3 2 Total energy (TE) and adsorption energy per N2 molecule (AE/m) obtained by using HSE (2nd and 3rd columns) and PBE (4th and 5th columns), respectively. ........................................................................................................ 47 3 3 Adsorption energy and charge transfer to O2 molecules adsorbed on Agn + cations. The lowest energy states are written in italic. ........................................ 52 4 1 The energies E and formation energies Ef (in meV) of Agn and AgnClm clusters on an Ag(111) surface. The ground state energy of each cluster is set to zero. .......................................................................................................... 69 4 2 Diffusion barriers Ed of Agn and AgnClm clusters on an Ag(111) surface. All energies are in meV. .......................................................................................... 70 4 3 Distance to the center of cluster (in ) of each layer (labeled by LCl/O, L 1, L 2, and Lctrons. ........................................................................ 71 4 4 Binding energy (in eV) between two Ag55based clusters. .................................. 71 5 1 Energies of neutral, cation, and anion of both HS and LS states. Adding/removing one spin up/down electron are all considered. The energy of neutral LS st ate (ground state) is set to be 0. ................................................. 78 5 2 Ionic potential (IP), electron affinity (EA), capacitance (C), and charging energy (Ec) of both HS and LS states. ................................................................ 79 5 3 Energies of neutral, cation and anion of both AFM and FM states. Adding/removing one spin up/down electron are all considered. The energy of neutral AFM s tate (ground state) is set to be 0. .............................................. 84 5 4 The ionization potential, electron affinity, charging energy, and capacitance of the Fe C60Fe system for both AFM and FM states. ....................................... 85 5 5 Energies, magnetizations and estimated switch fields (from AFM to FM) of FenC60Fen systems. .......................................................................................... 85 6 1 Binding energy of Fe atoms at various sites in eV. ............................................. 94 6 2 Charge transfer and magnetic moments as computed by the Bader method ... 103

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9 6 3 Spin populations on the individual orbitals of Fe and C60, spin up, spin down, m magnetic moment. .............................................................................. 111 6 4 Structure and energies: dz is the average inter average intraxy the average lateral displacement of atoms surrounding a vacancy (see Figure 611) .................... 117 8 1 The optimized lattice constants, a, b, c Numbers in parenthesis are calcu lated with symmetry constraints. All lengths are in unit of .................... 135 8 2 Comparison of bond lengths and angles of symmetric and optimized phase structure. All lengths are in unit of and angles in degree. .............................. 136 8 3 Comparison of bond lengths and angles of symmetric and optimized phase structure. All lengths are in unit of and angles in degree. .............................. 137 8 4 Numbers of oxygen and tantalum atoms, and cohesive energy. Values in parentheses are calculated with symmetry constraints. Al l lengths are in unit of .................................................................................................................. 141 8 5 Elastic constants, Bulk moduli, Shear moduli, Youngs moduli, and Poissons ratio of different phases of Ta2O5. Yx, Yy and Yz are Youngs modulus along a, b and c axis. ................................................................................................. 144 8 6 Bulk moduli, Youngs moduli, and Poissons ratio of different phases of Ta2O5. Y are Youngs modulus along c axis, calculated via direct method for validating VCA. ................................................................................................. 146

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10 LIST OF FIGURES Figure page 3 1 (a) Intensity histograms of Ag4 +(N2)m and ( b) The mean number of N2 adsorbed on Ag4 + and on Ag4 +O2 as a function of nitrogen pressure. (c) The mean number of oxygen molecules on all Ag4+(N2)m clusters ........................... 41 3 2 Structure of Agn ( n =5 7) neutral clusters and cations. The structures (a) in the left column are the most stable ions. (a d) indicates decreasing stability of the ions. .................................................................................................................... 46 3 3 N2adsorption on Agn + ( n =3, 4, 6, 7). Both PBE and HSE06 XC were used. Results are listed in Table 32. ........................................................................... 48 3 4 O2adsorption on Agn + ( n =1 6) cations. The adsorption energies and the Bader charge transfer are listed in Table 33. .................................................... 51 3 5 The lowest energy states of Ag4O2 +(N2)m and of Ag4 +(N2)m with m =2 5, respectively. ........................................................................................................ 54 4 1 Ag adatoms and AgCl clusters or separately adsorbed adatoms on an Ag(111) surface. ................................................................................................. 62 4 2 Ag dimer and Ag2Cl clusters adsorbed on Ag(111) surface. .............................. 64 4 3 Ag trimers and Ag3Cl3 clusters adsorbed on an Ag(111) surface. ...................... 66 4 4 Ag tetramers and Ag4Cl4 clusters adsorbed on an Ag(111) surface .................. 68 4 5 (a) Ag55Ag55 dumbbell structure. (b) Optimized Ag55Cl19 structure. (c) Optimize d (Ag55Cl19)2 dumbbell structure. (d) Optimized (Ag55O19)2 dumbbell structure. ............................................................................................................ 73 5 1 Optimized structure of [Mn3] molecule. Panel (a) is top view and panel (b) is side view. ............................................................................................................ 77 5 2 Isosurfaces of charge difference (a) between neutral molecule and cation of HS, (b) between anion and neutral molecule of HS, (c) between neutral molecule and cation of LS, (d) between anion and neutral molecule of LS. ....... 81 5 3 Isosurfaces of charge density of (a) highspin neutral state HOMO, (b) highspin anion state HOMO, (c) low spin neutral state HOMO and (d) low spin anion state HOMO. ............................................................................................. 82 5 4 C60 with 2 Fe atoms attached. C atoms are in yellow and Fe atoms are in red. ..................................................................................................................... 84

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11 6 1 Fe1doped C60 monolayer configurations on an h BN/Ni (111) surface. Panel (a) (d): Structure of the configurations according to binding energy. (a) is the most stable and (d) the least. Left column is side view and right is top view. ..... 93 6 2 Same as Figure 61 for Fe2doped C60 monolayer on an h BN/Ni (111) surface. ............................................................................................................... 95 6 3 Fe3doped C60 monolayer on an h BN/Ni (111) surface. Similar to Figure 61 with (a) being the most stable on and (c) the l east. ............................................ 97 6 4 Same as Figure 63 for Fe4doped C60 monolayer on an h BN/Ni (111) surface. ............................................................................................................... 98 6 5 Two isomers of a Fe15doped C60 monolayer on an h BN/Ni (111) surface. Panel (a) is more stable then panel (b). ............................................................ 100 6 6 Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and a Fe atom for FeC60/ h BN/Ni(111) Panel (c) and (d) are iso surfaces of charge density difference and spin density ........... 102 6 7 Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and two Fe atoms for Fe2C60/ h BN/Ni(111). Panel (e) (f) present the same information as panel (a) (d) for the AFM state. 105 6 8 Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and three Fe atoms for Fe3C60/ h BN/Ni(111) ...... 108 6 9 Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and four Fe atoms for Fe4C60/ h BN/Ni(111) ........ 110 6 10 Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and fifteen Fe atoms for Fe15C60/ h BN/Ni(111) ... 112 6 11 C60 molecules on vacancy sites of Au(111) surface. Panel (a) is side view of C60 on a 7atom pit. Panel (b), (c) and (d) are top view of C60 on 1, 3 and 7 atom pits, respectively .................................................................................. 116 7 1 (a) Monoclinic 40atom unit cell of BMO. (b) 20 atom primitive cell of BMO. Bi atoms are in grey. Mn atoms are large blue (dark) spheres. O atoms are small red (dark) spheres. .................................................................................. 120 7 2 The energies per primitive cell for FM, AFM_2, and Ferri_2 magnetic configurations. (a) is the LDA+U result and (b) is from the HSE06 hybrid functional. Only the most stable AFM and Ferri orders are shown. .................. 122 7 3 (a) The atomic displacements of a primitive cell under 2.2% strain. (b) Isosurface of spindensity of the states within 2 eV below Fermi surface. (c) The calc ulated polarization corresponding to the strain. ................................... 1 24

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12 8 1 Panel (a): Structure of phase Ta2O5. Panel (b): Structure of phase Ta2O5. Panel (c): Structure of tetragonal phase Ta2O5. Panel (d): Structure of low temperature phase 77 atom Ta2O5. .................................................................. 134 8 2 Panel (a): Structure of amorphous Ta2O5. Panel (b): pair correlation function of TaTa in a 2 2 2 super cell. Panel (c): Pair correlation function of TaO in a 2 2 2 super cell. ................................................................................... 139 8 3 Panel (a): DOS of Ta2O5. Panel (b): DOS of Ta2O5. Panel (c): DOS of tetragonal H Ta2O5. Panel (d): DOS of tetragonal H Ta2O5 with VCA. Panel (e): DOS of low temperature 77 atoms. ............................................................ 143

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13 LIST OF ABBREVIATIONS B3LYP Hybrid functional of Becke threeparameter exchange, and LeeYangParr correlation DFT Density functional Theory DOS Density of states GGA Generalized gradient approximation HOMO Highest occupied molecular orbital HSE Hybrid functional of Heyd, Scuseria and Ernzerhof LDA Local density approximation LUMO Lowest unoccupied molecular orbital NEB Nudged elastic band PBE Generalized gr adient approximation of Perdew, Burke and Ernzerhof PDOS Projected density of states PW91 Generalized gradient approximation of Perdew and Wang SD Single determinant SMM Single molecule magnets VCA Virtual crystal approximation

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14 Abstract of Dissertation Pr esented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FIRST PRINCIPLES SIMULATIO NS IN MULTIPLE DIMENSIONS: NANO PARTICLES, SURFACE AND BULK By Yuning Wu August 2012 Chair: HaiPing Cheng Cochair: Amlan Biswas Major: Physics Systems in different dimensions can exhibit different and nearly unique properties, such as catalytic behavior in zerodimensional (0D) metal clusters, adsorption patterns of molecules on two dimensional ( 2D ) noble metal surfaces, and mechanical and magnetic properties in threedimensional ( 3D ) crystals. Accurate description of quantum eff ects and electron correlations is imperative to correctly study th e se systems. D ensity functional theory (DFT) is a reliable and efficient method to determine the electronic structure across many methods and systems. We have studied the silver cluster cations and the adsorption of N2 and O2 molecules ont o them We have interpret ed the mutually cooperative coadsorption of oxygen and nitrogen as a result of the N2induced increase in charge transfer from Agn + cations to O2. To understand the role of chlorine in the fragmentation of Ag nanostructures, we have studied the diffusion of Agn and AgnClm (n = 1 to 4) clusters on an Ag(111) surface, and the interaction strength of (Ag55)2 dimers with and without chloridization. Bond

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15 weakening and enhanced mobility are two important mechanisms underlying corrosion and fragmentation processes. W e have demonstrate d the concept of molecular magnetocapacitance, in which the quantum part of the capacitance becomes spindependent T he nanomagnet [Mn3O(sao)3(O2CMe)(H2O)(py)3] shows a 6% difference in capacitance. For 2D systems, w e have investigated the structure, energetics, electronic and magnetic structures of Fendoped C60 monolayers supported by h BN monolayer covered Ni(111) surfaces. The binding energy, charge transfer (from Fen to C60), and magnetic moment all increase monotonically as functions of n The electron charge transfer is from the spin minority population. We have also studied the adsorption of C60 molecules on Au(111) surface defects The adsorption energy of the strong bonding configuration is much higher than the weak bonding configuration. We have investigated the effect of epitaxial strain on a BiMnO3 thin. Anti ferromagnetism starts to emerge under about 2% epitaxial strain. The AFM order can break inversion symmetry and induce electric polarization, a result which is comparable to experiment Finally, we have studied the structure, energetics, elastic tensors and mechanical properties of four crystalline forms of Ta2O5 with exact stoichiometry as well as a model amorphous structure.

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16 CHAPTER 1 BACKGROUND I mprovement of experiment al technology, especially the nanoscale technology developed in the last three decades, has broadened the scope of physics and material science enormously from traditional 3D and 2D systems to so called 0D and 1D systems. A typical example is the existence of carbon in 0D to 3D form s. 0D is C60, 1D is carbon nanotube, 2D is graphe n e and 3D is graphite. All are all highlights of scientific research in recent decades. A variety of interesting properties are obse r ved, depending on the dimension and size of the system. For example, zerodimensional nano clusters can present catalytic behavior.19 In recent years huge amount s of research have been focused on the 2D graphene, due to its rare zerogap band structure (Dirac cone).1014 Furthermore, some specific types of 1D nanowire s, including carbon nanotubes1520 and nanoribbo n s,2125 can present semi conductor propert ies Utilizing such pro perties for new functional materials and devices has become the cuttingedge and challen ging task for condensed matter physicist s and material scientists. In principle, most physical properties of interest are determined by the positions of the atomic nucl ei and the electron distribution They appear together in the complete wave function i Ir R where IR are the coordinates of atomic nuclei and ir are the electron coordinates. The complete wave function is the solution of the Schr dinger equation with the Hamiltonian j i j i i I i I I i i J I I I J I I I Ir r r R Z R R Z Z M H 1 2 1 2 1 2 1 2 1, 2 2 ( 1 1 ) Here Hartree atomic units ( 1 e me ) are adopted. IM and IZ are the mass and charge of the atomic nucleus I Solving for the wave function can be simplified by

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17 introducing the Born Oppenheimer approximation, which exploits the much faster velocity of electronic motion than ionic motion (or equi valently, the mass of a nucleus being much larger than the electron mass). The BornOppenheimer approximation allows the separation of the complete wave function into nuclear and electron parts: I n i e i IR rr R ( 1 2 ) i er can be obtained by solving the many electron Schrdinger equation with the Halmitonian j i j i i I i I I i i er r r R Z H 1 2 1 2 1, 2 ( 1 3 ) This separation leaves aside the nuclear kinetic energy (term in Equation 1 1) and nuclear Coulomb repulsions (second term in Equation 1 1). The nuclear motion then can be determined from the Hamiltonian with the nuclear terms (first two terms in Equation 1 1), plus the electronic energy as a function of IR derived from the electronic Hamiltonian. Solving for the electronic wave function is still not an easy task. The third term in Equation 1 3 which introduces many body electronelectron correlation, makes the problem so difficult that generally it is not solvable analytically. A many body Schrdinger equation can be solved approximately by two categories of approaches The first category, quantum chemistry methods,26 is mostly based on the HartreeFock theory, which approximates the wave function as a Slater determinant. HartreeFock theory captures exact electron exchange, but omits electron correlation. It can be improved by including more determinant s in the wave function, such as i n configuration interaction (CI) and many body perturbation theory (MBPT). The coupled cluster (CC)

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18 method is not based conceptually on S later determinant, but in practice a single Slater determinant often is the reference function. The accuracy of CI or C C can be improved arbitrarily in principle, however, the high time consumption limits computational application to systems of tens to hundreds of atoms. D ensity functional theory27 (DFT) is the second category. It is based on the Hohenberg Kohn theorem, which states that the ground state energy is a unique functional of the electron density. The KohnSham self cons iste nt procedure,28 proposed in 1965, makes the HohenbergKohn theorem a practical tool to solve for the ground state energy of a many body system. In KohnSham DFT, the interacting many electron system is treated as a set of noninteracting electrons with an effective potential. KohnSham DFT has achieved great success, because with reasonable approximate functionals, it can obtain good accuracy with relatively low computational effort. With the help of super computer s, pseudopotential s and efficient algorithm s, DFT can treat system s as large as 1000 electrons. The major approximation is the exchangecorrelation functional. The simplest level is the local density approximation29, 30 (LDA). Other exchangecorrelation functionals such as GGA,31 LDA+U, GGA+U,23, 3234 and hybrid functional s,3537 have been devised proposed to improve upon LDA in treating large classes of systems. To proceed, the remaining chapters are organized as following. C hapter 2 provides a general review of DFT, pseudo potential s, and the modern theory of polarization. In C hapter 3 DFT study of co adsorption of N2 and O2 on silver cluster cations is presented. C hapter 4 discusses enhancement of Ag cluster mobility on Ag(111) surfaces by chloridization. The concept of molecular magnetocapacitance is

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19 elaborated in C hapter 5. Studies of C60 monolayer on defected Au(111) and hBNcovered Ni(111) surface are presented in C hapter 6. T he strain effect of the multiferroic BiMnO3 thin film is discussed in C hapter 7. Finally the structural, energetic, mechanical properties of Ta2O5 polymophs are shown on C hapter 8, and the work as a whole is summarized in C hapter 9.

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20 CHAPTER 2 THEORETICAL AND COMPUTATIONAL METHODS 2.1 Density Functional Theory 2.1.1 Fundamental As explained in Chapter 1, the Born Oppenheimer approximation allows the separation of the electronic many body Hamiltonian from the total Hamiltonian. The electronic Hamiltonian ( Equation 1 3) can be rewritten as intV V T Hext ( 2 1 ) where the kinetic energy T is i iT22 1 ( 2 2 ) the external potential extV from the nuclei electron interaction is i I i I I extr R Z V, ( 2 3 ) and the internal electronelectron Coulomb interaction is j i j ir r V 1 2 1int ( 2 4 ) For all N electron systems, T and intV are the same if they are taken as operators. Therefore, the ground state wave function ir0 is decided by extV. The corresponding ground state electron density is given by N i N ir r r r dr r n2 2 32 0 0 0 0... ( 2 5 ) The HohenbergKohn theorem27 states that, for an N electron system, (1) The external potential ext V is completely and uniquely determined by the corresponding ground state electron density r n up to an additive constant.

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21 (2) For all V representable densities r n the energy corresponding to r n is no less than the ground state energy. The ground state energy can be obtained by the variational minimization of a functional of r n Here a density is V representable if it derives from an eigenstate of a Hamiltonian with a certain external potential extV Thus, the ground state expectation value of the Hamiltonian can be reformed as a functional of r n n V nV n T n Eext int ( 2 6 ) The required V representability can be weakened to N represen tability according to Levy and Liebs constrained search formulation.38, 39 The N represeatability, which is much weaker than V represent ability, only requires that the total integral of the density is N, and the proper differentiability of n In their formulation, the variational minimization is done in two steps, first of which is searching among the wave functions yi elding a specific density n and the second step is varying the n It can be presented as } ] [ min { minint 0 dr r n r V V T Eext n n ( 2 7 ) 2.1.2 KohnSham P rocedure Although the ground state energy is well proven to be a functional of the charge density, the HohenbergKohn theorem is not practical for real systems because the functional cannot be defined, or specifically, the kinetic energy and the nonclassical part of the inter electronic interact ions cannot be written as a functional of the charge density. T o take advantage of the theorem, a self consistent procedure is proposed by Kohn and Sham .28 It maps the many body interacting electrons to fictitious non-

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22 interacting electrons The ground state charge density is assumed to be reproducible by the noninteracting electrons. In Equation 2 6, int V can be separated into two parts, a classical part and a nonclassical part. The classical part is 2 1 r r rn r n n Vh ( 2 8 ) which is also called the Hart r ee energy The nonclassical part (denoted as xcV ), which is called the exchangecorrelation energy cannot in general be explicitly written as a density functional Thus Equation 2 6 can be rearranged into n V n V n V n T n Exc h ext ( 2 9 ) The essence of the KohnSham procedure is the introduction of the kinetic energy of the assumed noninteracting electrons ( n T0 ) and the difference between n T0 and n T is combined with the unkown to from Then we can rewrite Equation 2 9 into n V n V n V n T n Exc h ext 0 ( 2 10) We can thus perfor m the variational minimization of this functional with a Lagrange multiplier to constrain the number of electrons to be N: 00 N dr r n n V n V n V n Txc h ext ( 2 11) Then we have r n n V n V n V r n n Txc h ext 0 ( 2 12)

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23 We define dr r r r n r n n V r vh h ( 2 13) and r n n V vxc xc ( 2 14) We can thus define a total onebody potential r v r v r v r vxc h ext eff ( 2 15) Note that Equation 2 12 is exactly the same equation as for a system of noninteracting electrons in an external potential r veff Therefore, the wave functions of the noninteracting electrons can be obtained by r r r vi i i eff 22 1 ( 2 16) The total kinetic energy of noninteracting electrons is calculated from N i i ir r n T1 2 02 1 ( 2 17) and the charge density is constructed as N i ir r n1 2 (2 18) Equations 2 13 to 21 8 form a self consistent loop, which defines a practical way to minimize the functional ( Equation 2 10) A guess of the charge density can form a set of noninteracting orbitals, constructing a new charge density for the next iteration. The

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24 functional is minimized once the input and the output charge density are the same up to a desired level of accuracy 2.2 ExchangeC orrelation F unctionals Although the KohnSham recipe of DFT is exact in principle, the XC functional is nearly impossible to derive explicitly The difficulty comes from the fact that the XC potential at position r depends on the value of the XC energy ( n vxc in our notation) not only at this point, but also its variation nearby. We can expand the XC energy in terms of the gradients of the charge density, ,... r n r n r n v r n vxc xc (2 19) Normally, only the local density r n and its gradient r n (if necessary) are considered in the expansion, because higher order gradients in the expansion are known to misbehave. Commonly used XC functionals include local density approximation29, 30 (LDA), generalized gradient approximation (GGA) and hybrid functionals. We summarize these next. 2.2.1 Local D ensity A pproximation (LDA) The simplest way to construct the XC functional is to evaluate the XC energy density locally at every point, i.e., the XC energy density at point r is assumed to be the same as the XC potential at this point in a homogeneous electron gas with density r n Thus the total XC energy ( n Vxc in our previous notation) is dr r n r n r n Exc LDA xchom (2 20) where nxc is the XC energy per electron in a homogeneous electron gas of density n which is already calculated to great accuracy. Due to the assumption that electrons at

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25 a point feel the XC potential of a homogeneous electrons gas, one would expect LDA to be good for systems close to the h omogeneous electron gas, and poor in very inhom ogeneous systems. Although unphysical in its strict locality, LDA works surprisingly successfully. Several parameterizations of LDA have been proposed and they all work quite well. The LDA helps to give useful predictions on electron densities, atomic pos itions, etc but it can fail in predicting band gaps. 2.2.2 Generalized G radient A pproximation (GGA) The LDA can be improved by introducing the dependence of spatial variation of the charge density into the XC energy. Many forms of GGA have been proposed to do this. They generally are of the from a form40 dr r n r n F r n r n dr r n r n r n Exc x xc GGA xc ,hom (2 21) where hom x is the exchange energy of homogeneous electron gas and xcFis a dimensionless function. In more careful notation, xcF is separated into two parts, the exchange part xF and the correlation part cF Both of them can be expressed as a function of n and s where n is the charge density and 3 4 3 1 23 2 n n s (2 22) The correlation part typically contributes much less than the exchange. Different GGA recipes, using different forms for xcF have been proposed. The m ost commonly used GGA functionals include PW91,41 B88,42 PBE,31 etc. GGA

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26 functions cover the nonlocal effect to some degree, and hence they show better performance in systems with high density gradient than LDA. 2.2.3 Hybrid F unctionals Hybrid functionals are a group of XC functionals which combine the exact exchange from orbital dependent singledeterminant (SD) exchange and conventional DFT functionals.43 Hybrid functionals introduce a certain fraction of SD exchange (normally 20% to 25%), which has an error canceling effect on the self interaction error from conventional functionals. Hybrid functionals generally work better in ener getic calculations, such as molecular properties and structural calculations,44 than LDA or GGA. The general form of the hybrid functionals45 is LDA GGA xc LDA GGA x SD x hybrid xcE E E E/ / (2 23) where SD xE is the exact HF exchange, and the hybrid coefficient often is fitted according to atomic and molecular data. Many hybrid funtional s have been proposed, including B3LYP,46 PBE0,47 HSE,36 etc. B3LYP has the form of c c Becke x x DFA x HF x LDA xc LYP B xcE a E a E E a EE 0 3 (2 24) in which 0a xa and ca vary according to the elements and molecules. The PBE0 functional contains 25% of exact exchange, combined with 75% of PBEGGA exchange. The correlation is totally represented by PBE G GA. It has the following form: PBE c PBE x HF xc PBExcE E E E 4 3 4 10 (2 25)

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27 The PBE0 functional yields obvious improvement s in certain problems over LDA/GGA, however, its self consistent SD calculation is very time consuming. The HSE03/HSE06 hybrid functionals were invented to reducing the computing cost without significant los s of accuracy. The idea is to separate the SD exchange in to longrange (LR) and short range (SR) term s In HSE03/HSE06, the LR term is replaced by the PBE LR exchange. The resulting expression is thus, PBE c LR PBE x SR PBE x SR x HSE xcE E E E E ,43 4 1 (2 26) In other words, the hybridization of exact exchange and explicit DFT exchange only happens in short range. HSE03/HSE06 give very similar results to PBE0, but greatly reduces the computational effort. The rangeseparation is determined by the parameter A value of 0.3 means HSE03, while 0.2 is used in HSE06. is also related to a characteristic di stance, / 2 where the SR term goes negligible. 2.3 Plane W ave B asis S et There exist several ways of representing the onebody wave functions or orbitals in the Kohn Sham equations including plane waves, atomic orbital s, and real space grids. The plane wave basis set is commonly used, especially for periodic systems, due to its computational efficiency in the fast Fourier transform (FFT) and easy convergence control. In a plane wave basis a Bloch function can be expanded as G r G k i G k n k ne c r, (2 2 7 ) where G are reciprocal lattice vectors. The number of plane waves is controlled by the kinetic energy cutoff cutE by

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28 cutE G k 22 1 (2 28) The kinetic energy is used as the criteria because it contributes the most to the total energy. As the cutoff energy and the number of plane waves increase, the convergence can be obtained systematically for all forms of wave functions. Compared to other basis sets, the number of necessary plane waves to reach convergence is normally large. The FFT somewhat reduces the efficiency of parallelization because it is nonlocal and hard to be parallelized. 2. 4 Pseudopotential The large number of the plane waves needed for the convergence is mostly due to the high kinetic energy of the core electrons and the associated nuclear electron cusp in density The idea of pseudopotential s is to replace the real potential from nuclei and core electrons by a smooth effecti ve core potential or pseudopotential and focus on valence electrons. The rapid oscillations of the valence electron wave function near the nuclei are replaced by the nodefree pseudo wave functions which are forced to coincide with the real wave functions outside the core region. 2.4.1 Norm Conserving Pseudopotential The pseudopotential and pseudo wave function are obtained by solution of the all electron scalar relativistic radial atomic Schrdinger equation with the KS DFT potential 0 ) ( ) ( ) 1 ( 1 ) ( ) ( 2 ) ( 2 12 2 2 r u r V r l l r dr d r r r dV r M dr d r Ml l (2 29) in which

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29 )) ( ( 2 1 ) (2r V r Ml (2 30) and is the fine structure constant. The pseudo wave function ) ( ~ r ul is constructed according to the following conditions : (1) The eigenvalue of ) ( ~ r ul must match the eigenvalue of the true wave function ) ( r ul (2) ) ( ~ r ul must coincide with ) ( r ul beyond a chosen core radius clr (3) ) ( ~ r ul and ) ( r ul include the same amount of charge inside the core radius. (4) The first and second derivatives at clr are identical for ) ( ~ r ul and ) ( r ul After ) ( ~ r ul is chosen, the pseudopotential lV ~ then can be obtained by inversion of the nonrelativistic radial Schrodinger equation, 0 ) ( ~ ~ ) ( ~ ) 1 ( 2 12 2 2 r u r V r l l dr dl l (2 31) and thus ) ( ~ ) ( ~ 2 1 ) 1 ( ~ )( ~2 2 2r u dr d r u r l l r Vl l (2 32) An accurate and practical pseudopotential requires a nonlocal form becuase different angular momentum states (partial waves) have different scattering properties. ) ( ~ r Vl can be used to build the nonlocal part of the pseudopotential in the way proposed by Kleinmen and Bylander,48 m l lm l lm l lm lm l local KBV V V V V,~ ~ ~ ~ ~ (2 33)

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30 in which local l lV V V ~ ~ ~ (2 34) lm lmY r u ) ( ~ ~ (2 35) and localV ~ is an arbitrary local potential. Many recipes for generating the norm conserving pseudopotential have been proposed, depending on different ways of defining pseudo wave functions in the core region. The Troullier and Martin recipe49, 50 uses a polynomial up to 12th order of r A linear combination of spherical Bessel functions was also proposed by Rappe and Rabe.51 2.4.2 Ultrasoft P seudopotential and P rojector A ugmented W ave (PAW) M ethod Despite the success of norm conserving pseudopotential s for various elements, such as Si and Al, they still need expensive kinetic energy cutoff for the elements with localized valence states. Norm conservation requires that the core radius cannot be chosen too much beyond the maximum of the true wave function, so the pse udo wave function under the norm conserving condition is not significantly smoother than the true wave function. Vanderbilt52 and Bl chl53 proposed the ultrasoft and projector augumented wave (PAW) pseudopotential respectively, to resolve this problem. T he ultrasoft pseudopotential avoids the norm conserving condition by introducing a nonlocal overlap operator, j i j i ijQ S,1 (2 36) where ijQ is the matrix based on true wave functions and pseudo wave functions inside the core radius clr

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31 cl clr j i r j i ijQ ~ ~ (2 37) and i are projector functions defined as the dual of i~ The norm conserving condition is 0 ijQ which is generalized to cl clr j i r j iS ~ ~ (2 38) This indicates that and ~ have the same amplitude beyond the core radius. Define the matrix ijB as j i ijB ~ (2 39) in which i loc i iV T ~ ) ( (2 40) and locV is the chosen local part of the pseudopotential, then the nonlocal part can be determined as, j i j i ij i ij NLQ B V,) ( (2 41) In such a way, the pseudo wave function i~ is the solution of the modified Schrodinger equation, 0 ~ ) ( ~ ) ( i i NL loc i iS V V T S H (2 42) The elimination of the norm conserving condition allows smoother pseudo wave functions and thus a lower energy cutoff. Similar to the ultrasoft pseudopotential, the projector augmented wave method makes use of projectors and auxiliary localized functions. The calculations of physical quantities, such as energy, involve not only smooth functions over all space, but also

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32 loc alized contributions inside the core regions of ions. The PAW method keeps and uses true wave functions from all electron KS calculation, which is different from the ultrasoft pseudopotential. The basic idea of the PAW method is shown as following. More details can be found in Bl chls original literature.53 The all electron wave function and the smooth auxiliary wave function (or pseudo wave function) ~ are related by a linear transformation n nT ~ (2 43) The transformation should modify ~ only inside augmentation spheres for each atom, so the operator T can be written as, R RT T 1 (2 44) where R means the atomic sites, and RT only acts in the augmentation sphere of site R For each site, the auxiliary atomic wave function i~ can be defined from the all electron partial waves i by i R i iT T ~ ) 1 ( ~ for R i (2 45) Then a set of projector functions ip ~ is constructed inside the augmentation sphere, following ij j ip ~ ~ (2 46) and 1 ~ ~ i i ip (2 47) Inside the augmentation sphere of site R the auxiliary wave function can be expanded by auxiliary atomic wave functions as

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33 i R i n i np ~ ~ ~ ~ (2 48) Plug in Equation 2 45 and we can get R i n i i i n Rp T ~ ~ ~ ~ (2 49) so the transformation is R i R i i ip T ~ ~ 1 (2 5 0 ) In other words, the transformation operator is completely defined as a combination of smooth auxiliary partial waves, nonsmooth all electron partial waves, and the projector functions. By applying the operator onto auxiliary wave function, we get R R i R i n i i n i i n np p ~ ~ ~ ~ ~ ~ (2 51) The expectation value of any onebody operator O is coreN i i c i c R R j i R ij j i j i n n n nO D O O O f O1 ~ ~ ~ ~ (2 52) where n n j i n n R ijp p f D ~ ~ ~ ~ (2 53) and is the core wave function. The PAW method is closely related to, however somewhat better than the ultrasoft pseudopotential. The ultrasoft pseudopotential is the same as the linearization of the onsite compensation energy calculation (inside the augmentation sphere) in PAW scheme. The PAW method is more trustable than the ultrasoft pseudopotential especially in magnetic systems.

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34 2. 5 Modern T heory of P olarization The macroscopic electric polarization, defined as electric dipole per unit volume, is an old but a very important concept regarding dielectric media. Ferroelectric materials, which exhibit spontaneous macroscopic polarization, are of great interest and application, especially if magnetic properties coexist (multiferroics). Despite the importance and significance of this polarization, a mature physical understanding at the firstprinciple s level was not achieved until the 1990s. In this section, the modern theory of polarization is reviewed briefly 2.5.1 C onventional D efinition s The conventional definition of electric polarization is based on the Clausius Mossoti model,54 in which polarizable units are assumed. The macroscopic electric polarization is thus the accumulation of the dipole moments contributed from the polarizable units, divided by the volume of the w hole system. In crystalline systems, the unit cell can be identified as the polarizable unit and the calculation of polarization is within the unit cell, dr r r V Pcell cell cell1 (2 54) The key point of the validity of the CM model is that the induced charge can be clearly divided and attributed to the polarizable units. The CM model may work in ionic crystals, in which charge is quite localized due to the ionic bonds. However, in systems with covalent character, the CM model is not valid, because the delocalized charge distribution makes the identification of polarizable units ambiguous. Other attempts to define polarization through the charge distribution have proven to be failures Exa mples include the polarization in a macroscopic volume and the cell

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35 average of the microscopic polarization. Macroscopic polarization can be illustrated by the following equation, dr r r V Psample sample sample1 (2 55) The sample volume is a macroscopic but finite volume, which contains the bulk and surface. This model proves unhelpful because it is very hard to distinguish th e surface from the bulk region and because this model has little connection to the bulk property based on periodic boundary conditions. Thus, i t is not a useful definition. The second attempt to model polarization is through a cell average of the microscopic polarization r r Pmicro (2 56) is also not meaningful, because any divergencefree contribution can be added to r Pmicro without violating equation (2 56) Overall, a more sophisticated definition of polarization is needed, especially for crystalline systems with periodic boundary condition. 2.5.2 F undamental of Polarization Theory In practice, i t has been realized that the change of polarization P is more physica l l y meaningful than the absolute value of P .55 The experimental measurements either of induced polarization or spontaneous polarization, are based on the change of polarization.56 Because of the fundamental equation t j dt t dP (2 57) in which t j is the current density, the change of polarization can be written as

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36 dt t j P t P Pt 00 (2 58) This implies the change of the polarization can be achieved by accumulation through an adiabatic process. Not e that t j is a bulk property. By introducing an adiabatic time ,55 the last equation can be rewritten to d d dP P 1 0 (2 59) where 0 or 1 refer s to the initial or final state, respectively The initial and final states can be the states before and after the application of electric field, strains, etc. It is convenient, especially when studying the spontaneous polarization, to set the centrosymmetric geometry as the reference point, i.e. 0, and thus the polarized geometry can be set to 1. B oth nuclei and electrons contribute to polarization. The ionic contribution is rather straightforward. In terms of the electronic contribution, it is quite interesting to realize that the change of polarization is essentially the accumulation of the adiab atic electronic current, which is closely related to the phase of the wave function in the quantum mechanic al picture. The phase of the wave function vanishes during the calculation of the charge density, explaining why the description of polarization by c harge density is insufficient for bulk system. 2.5.3 Berry P hase D escription57, 58 In a crystalline system with periodic boundary condition, the Bloch theorem tells us that the eigenfunctions have the Bloch wave form r u e rnk r ik nk where r unk has periodicity of the lattice vectors and follows from nk nk nk ku E u H The Hamiltonian in k space is

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37 V m k p Hk 22 (2 60) with V the Fourier transform of the effective oneelectron potential. Considering the adiabatic process depending on the parameter the variation of the wave function can be obtained through a first order perturbation, n m mk mk nk nk mk nkE E dt d i (2 61) where is the derivative with respect to This variation can be utilized to construct the current contributed from the nth band, n m mk nk nk mk mk nk e n nc c E E p dk dt d m e i dt dP j 23 (2 62) in which c c indicates the complex conjugate. / dt dP can be converted to /d dP by multiplying by /d dt Combined with applying a perturbation to the Hamiltonian in k space, the following form of /d dP can be obtained, 23c c u u dk ie dt dPnk nk k n (2 63) Therefore, after an integration with respect to and summation over all occupied bands, the electronic polarization is n nk k nk elu u dk e P Im 23 (2 64) Here is the index for all occupied states, including up and down states. With the ionic contribution combined, the total polar ization including ionic and electronic contributions, is formula ted as n nk k nk i i i el ionu u dk e R Z e P P P Im 23 (2 65)

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38 where iZ and iR are the atomic numbers and coordinates of the ion with index i. Equation 2 65 is the core statement of the modern theory of polarization. The term nk k nku u i is the gauge potential in Berry phase theory,59, 60 and its integral over a closed manifold (the Brillouin zone in the polarization case) is the Berry phase. The change of polarization is independent of the path the system undergoes during the adiabatic process. To study the ferroelectric material, the starting point is commonly set as the centrosymmetric geometry which gives zero polarization. 2.5.4 The Q uantum of P olarization For simplicity, an 1D case is initially considered. Due to the periodic boundary conditions, the periodic Bloch function nku can be shifted by a phase factor k ie in which m a 2 0 / 2 and m is an integer. Thus the polarization contributed from the th band will have an additional term e m This implies that the polarization is only welldefined by a series of branches with modulus e Extended to the 3D case the polarization is only well defined by a series of many branches with modulus / eR where i i iR m R and iR are the lattice vectors of the primitive cell. The formula to compute the change of the polarization is generalized to 0 1: P P P mod eR (2 66) The : means that the P should be one of the group of values on the right hand side, i.e. only a branch. All values corresponding to the points along the evolving path should be in the same branch. In practice, several intermediate points normally need to be included in order to stay along the same branch.

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39 2.5.5 The C onnection to Wannier O rbitals The W annier orbitals provide another view of the Berry phase polarization. The W annier orbitals are defined as dk e wnk R ik nR 32 (2 67) which is similar to the Fourier transform of Bloch wave. Interestingly, the center of a Wannier orbital nR nRw r w turns out to be R P e rn nR (2 68) The Berry phase polarization thus can be rec onsidered in a Wannier orbital picture. T he contribution to the Berry phase polarization from band n is equivalent to placing an electron sitting at the center of the corresponding Wannier orbital

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40 CHAPTER 3 CO ADSORPTION OF N2 AND O2 ON SILVER CLUSTERS1 Metal clusters were a subject of intense research interest in the 1980s because of the fascinating sizedependence of their physical properties .6264 In the lat e 1990s, the unexpected catalytic behavior of charged small noble metal particles 13, 6, 65 67 brought such clusters into the spotlight and continues to stimulate curr ent research interest. Charging of metal clusters is typically achieved by surface charge transfer if the clusters are supported or by ionization and electron attachment in free clusters. To understand the chemical reactivity of small particles, adsorption and reaction of small molecules on clusters are therefore crucial. Several experimental groups have been working on the subject using free clusters in ion beams.6874 It was noticed that oxygen is molecularly chemisorbed to silver clusters at low temperature but forms silver oxide under heating.7274 Furthermore a cooperative effect in the co adsorption of O2, and N2 was found (as shown in Figure 3 1) .74 In contrast to silver, adsorption of small molecules on gold clusters has been extensively studied theoretically.7579 However, there is no theoretical work addressing this enhancement in coadsorption, although the same phenomenon but less pronounced was reported on gold in an experimental paper by Lang et al .71 The stronger chemical activity of Ag over Au makes the silver clusters interesting but also adds complexity to, in particular, consideration of the metal oxygen interaction. Structures of pristine metal clusters are critical for theoretical investigations of molecular adsorption. It has been noticed that small gold and silver clusters assume a flat geometry; a 2D to 3D structure transition occurs within the size range from 5 to 1 This wo rk has been published in Journal of Chemical Physics.61

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41 Figure 3 1 (a) Intensity histograms of Ag4 +(N2)m over m at different nitrogen pressures. (b) The mean number of N2 adsorbed on Ag4+ and on Ag4+ O2 is plotted in comparison as a function of nitrogen pressure p The presence of chemisorbed O2 enforces the physisorption of N2. (c) The mean number of oxygen molecules on all Ag4+(N2)m clusters depending on the mean number of nitrogen on Ag4+. 12 atoms, depending on material and charge state. Anions tend to stay in a 2D geometry at a size slightly larger than the neutral state and the cation, and the gold cluster remains in 2D at a size slightly larger than the silver cluster For example, a 2D model for Ag7 is presented by Xiang et al. to study the anion state when silver clusters are surrounded by organic ligands.80 Nevertheless, in 2002, Kappes group reported ion mobility measurements of Agn + cation clusters.81 It was shown that cation silver cluster s transform from the plana r structure to 3D at 4 5 atoms. Quantum chemistry calculations performed by the same group at the MP2 level of approximation support this conclus ion.81 Density functional theory (DFT)28 with generalized gradient approximations (GGA), specifically the Perdew Burke Ernzerhof (PBE) functional31, were used to study oxidiz ation of silver clusters, but a planar

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42 geometry was found to be the ground state of the pristine Ag6 + and Ag7 +. Fernndez and coworkers used the same functional, however they predicted that the 2D to 3D transition occurs from Ag5 + to Ag6 +.82 This discrepancy raises a question regarding the validity of DFT in determining the structure of small noble metal clusters. In a more recent paper by Trub lars group, a number of new functionals have been constructed and applied to study gold anion clusters.65 It is evident that the PB E functional fails to handle such subtle structural transformations. Trub lars group also pointed out the significance of electronic structure of single atoms (specifically, the transition energy from d10s1 to d9s2).65 A revisit of structure of silver clusters is thus desirable. Structures of pristine clusters are important for understanding adsorption and reaction. This chapter presents calculations of structures of small silver cluster cations using hybrid functional s. This approach predicts to a 2D to 3D transition at 4 to 5 silver atoms. With such a calculation, we go further and study the molecular adsorption of oxygen and nitrogen on different sites of the silver clusters, and coadsorption in order to compare with experimental data. The rest of this chapter is organized as follows: Section 3. 1 discusses the theoretical approach, and section 3. 2 shows the results on pristine silver clusters, on the N2 physisorption, on the O2chemisorption and on the O2(N2)m co adsorption. The chapter ends with summ a ry in section 3. 3 3.1 Methods and Computational Details All systems have been investigated by density functional theory (DFT)28 with the spin unrestricted GGA31 in the PBE f orm We used planewave basis in conjunction with the PAW53, 83 pseudopotential as implemented in the VASP package.84, 85 For issues concerning 2D vs 3D structures and possible corrections to the adsorption energy, the Heyd Scuseria Ernze rhof (HSE06) hybrid exchangecorrelation functional36 also was

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43 used to confirm and verify our results. The unit cell typically was 25 25 25 ( ) with only the gamma point in the first Brillouin zone. Tests have been performed to guarrantee that clusters are isolated from neighboring images for both neutral and charged systems. For charged clusters, a jellium background with dipole corrections86 was used to screen out longrange Coulomb interactions properly To reach the energy convergence of charged clusters when using HSE06, unit cells of some systems were enlarged to 30 30 30 ( ). A 500 eV cut off energy was used to truncate the planewave basis and ensure precision better than 1 meV/atom in the total energy. Structural optimizations were performed using PBE energy functional with an energy convergence of 105 eV/cell and atomic force convergence of 0.02 eV/, respectively. Finally, the Bader method87 was used for charge analysis. The calculated nitrogen adsorption energy is defined as m nN Ag total adE E E E2 (3 1) and the coadsorption energy as, mN nN O Ag total adE E E E2 2 (3 2) where is the total energy of cation with adsorpants, m represents the number of N2 molecules, and nAgE and 2O AgnE are total energies of Agn + and AgnO2 + cations without adsorpants, respectively. 3.2 Results and Discussion 3.2.1 Structure of P ristine Agn C lusters T o understand N2 adsorption, it is necessary to understand structures of pure silver cations. We therefore have revisited structures of Agn + ( n =5 7) using the

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44 computationally expensive HSE06 hybrid functional as implemented in the VASP package. Table 31 lists binding energies of both neutral clusters and cations. For neutral clusters, PBE and HSE06 are in agreement qualitatively. For n =5, t he pl anar bowtie is most stable (Figure 3 2 c5), followed by the bi pyramid (Figure 3 2 b5) and the twisted bowtie (Figure 3 2 a5, 3D). When ionized, the most stable and the least stable isomers swap shapes and the bi pyramid remains as the second stable isomer It is interesting to note that for both ionized and neutral clusters the energy difference between the first and second configuration is much smaller than between the second and the third. The two functionals, PBE and HSE06, predict the same geometry for cations. For n =6, PBE and HSE06 also are in agreement with HSE06 for neutral clusters. The neutral cluster still prefers the 2D triangle structure (Figure 3 2 c6). There are also two 3D isomers including a rhombic wedge (Figure 3 2 a6) and a pyramid (Figure 3 2 d6). Upon ionization, PBE and HSE06 predict different ground state geometr ies The planar structure (Figure 3 2 b6) is more stable than the others in the PBE calculations. When computed by HSE06, the rhombic wedge becomes the most stable shape. The planar (Figure 3 2 b6 and c6) geometries shift to second and third position in terms of stability. Differing from the pentamers, the energy differences among neutral hexamers are much larger than among the cations. The heptamer, n =7 was analyzed using the same method. For the neutral cluster, HSE06 and PBE predictions again agree. The most stable geometry is the bi pyramid (Figure 32 a7) followed by the cap (Fi gure 32 c7) and the hexagonal planar (Figure 32 b7). The 2D 3D transition of neutral Agn is in agreement with previous theoretical

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45 Table 31 Total energy (TE), binding energy per atom (BE/ n ), and ionization potential (IP) obtained by using HSE (2nd and 3rd columns) and PBE (4th and 5th columns), respectively. q nAg (n, q) Figur e 3 2 TE HSE BE/n TE PBE BE/n (5,0) a5 0.62 0.99 0.24 1.11 (5,0) b5 0.08 1.02 0.08 1.14 (5,0) c5 0 1.03 0 1.15 (6,0) a6 0.96 1.12 0.90 1.259 (6,0) b6 0.30 1.23 0.22 1.37 (6,0) c6 0 1.28 0 1.409 (6,0) d6 0.44 1.21 0.37 1.348 (7,0) a7 0 1.123 0 1.434 (7,0) b7 0.35 1.258 0.40 1.377 (7,0) c7 0.22 1.275 0.27 1.395 IP IP (5,1) a5 0 5.11 0 5.36 (5,1) b5 0.16 5.54 0.03 5.62 (5,1) c5 0.24 5.73 0.48 5.98 (6,1) a6 0 5.72 0.02 6.22 (6,1) b6 0.09 5.59 0 5.88 (6,1) c6 0.20 6.44 0.15 6.76 (6,1) d6 0.37 6.82 0.18 7.01 (7,1) a7 0 5.60 0.10 5.78 (7,1) b7 0.002 5.26 0 5.38 (7,1) c7 0.114 5.50 0.23 5.78 Note: All energies are in units of eV and shifted to zero for the lowest energy state. The letter a5c7 denotes clusters size n =5 7 and various isomer structures shown in Figure 3 2; the superscript q indicate the charge state. Numbers in italic refer to the lowest isomer. studies. Like all above mentioned cases, HSE06 and PBE predict similar trends and order of magnitude energy differences among all heptamers. However, the situation is very different for the cations. According to HSE06, the bi pyramid(Figure a7) and the planar configurations tie to be most stable, and the capshaped (Figure 32 c7) isomer

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46 is only slightly less stable (by 0.12 eV). Meanwhile, PBE predicts the planar (Figure 32 b7) form to be most stable, followed by the pyramid and the cap. Figure 3 2 Structure of Agn ( n =5 7) neutral clusters and cations. The structures (a) in the left column are the most stable ions. (ad) indicates decreasing stability of the ions. Italic indicates the most stable neutral structures. The energetics and IP potential are listed in Table 31 for two different DFT approximations. So, the hybrid functional seems to correct a slight PBE bias for the 2D structure when treating noble metal clusters. It also predicts systematically smaller binding energies and smaller ionization potentials (IP). Therefore, PBE is still a very good approximation for studying the trends in clusters. As a reference point, PBE and HSE06 predict the ionization energy of a single Ag atom as 7.93 and 7.59 eV, r espectively, co mpared to the ex erimental value of 7.57 eV.88 3.2.2 Physisorption of N2 M olecules After clarification of the structure s of pristine clusters, we can discuss results of N2 adsorption. Figure 3 3 depicts adsorption geometr ies for a few selected clusters to illustrate the physical picture. We first focus on trimer and tetramer cations since they are directly related to the ex perimental results.61

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47 Table 3 2 Total energy (TE) and adsorption energy per N2 molecule (AE/m) obtained by using HSE (2nd and 3rd columns) and PBE (4th and 5th columns ), respectively. mN Agn 2 (n,m) Figure 3 4 TE HSE AE/m TE PBE AE/m (3,1) a 0 0.36 0 0.43 (4,1) b 0 0.30 0 0.40 (4,1) c 0.12 0.18 0.12 0.28 (6,1) d 0 0.31 0 0.45 (6,1) e 0.21 0.10 0.25 0.20 (6,1) f 0.23 0.09 0.25 0.20 (7,1) g 0 0.35 0 0.43 (7,1) h 0.21 0.14 0.27 0.16 (7,6) i 0 0.19 0 0.22 (7,6) j 0.26 0.14 0.35 0.17 (7,6) k 0.44 0.12 0.31 0.16 Note: All energies are in units of eV and shifted to zero for the lowest energy state. The letters a h refers to various isomer structures and the superscript q to charge states. Numbers in italics refer to the lowest isomer. The first part of Table 32 (rows with m =1) shows our calculated total energies and adsorption energies of Ag cations relative to states of lowest energies. For n =3, the problem is straightforward. There is only one geometric arrangement (Figure 3 3 a). The adsorption energy is 0.43 eV from PBE and 0.36 eV from HSE06 and the same degree of reduction applies to all clusters. The energy gain by adding the second N2 t o a trimer cation is 0.39 eV. For n =4, there are two inequivalent sites as shown in Figure 3 3 bc. Our calculations (via PBE) show that N2 prefers an obtuse corner over an acute one by 0.12 eV, or 0.40 vs. 0.28 eV. This is different from O2 physisorption on Ag4 +, in which an acute corner is preferred, although it is not the ground state (Table 32, Figure 6gh). When three or four corners are all occupied by three or four N2 molecules, the total

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48 energy gain is 1.02 and 1.24 eV, respectively. Note that when the corners are all occupied, the total adsorption energy is 110 meV less than merely counting the number of sites and adding up single bond energies. In contrast with the coadsorption of oxygen and nitrogen, nitrogen molecules on different sites of the same cluster do not enhance but reduce the binding of each other. Whereas oxygen and nitrogen act cooperatively nitrogen molecules act mutually competitively For the hexamer, we have studied three inequivalent N2 adsorption sites in the 3D cluster as shown in Figure 3 3 df, in which the three sites are ordered from high to low adsorption energy. The adsorption energies are 0.45 eV, 0.20 eV, 0.20 eV for PBE, and 0.31 eV, 0.10 eV, 0.09 eV for HSE06, respectively. Adsorption for the geometry i n Figure 3 3 d is about 0.2 eV stronger than the other two geometries by both functionals. Fig ure 3 3 N 2 adsorption on Ag n + ( n =3, 4, 6, 7). Both PBE and HSE06 XC were used R esults are listed in Table 32 The physisorption to sites at obtuse corners with high coordination number is found to be stronger than to those at acute ones.

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49 Next we present our calculations on the silver heptamer cation, with its interesting and controversial 2D vs. 3D feature. For single N2 adsorption on the ground state heptamer of bipyramid geometry, there are two inequivalent sites, the top site and the side site (Figure 3 3 gh). We notice that N2 favors the top site of a cation. This preference is consistent with what is observed in the Ag tetramer and hexamer cation, in which an obtuse corner is preferred. Similar to Ag4 +, we also observe the competitive effect when multiple N2 molecules are adsorbed on Ag7 +. The total binding of N2 molecules on top and bottom sites is 20 meV less than twice the binding of one molecule o n one of the two sites. The total binding of four N2 molecules in the middle plane in Figure 3 3 i is 130 meV less than four times the binding of one N2 in the plane (with top and bottom sites occupied). Magic numbers have been observed at two physisorbed nitrogen molecules for Ag7 + as well as for Ag6 + in experiments .61 This is consistent with the calculated most stable 3D isomers and with the nitrogen preference for obtuse corners, since only those isomers have two sites on obtuse corners with highcoordinated Agatoms. The second part of Table 32 (rows with m =6) lists energies of Ag7 with six nitrogen molecules. The number m =6 is chosen to compare the 3D bipyramid (Figure 3 3 i j) with the planar geometry (Figure 3 3 k). For 6 N2 on a bipyramid, there is a vacant site. Adsorption on the cation gives rise to 0.10.2 eV/molecule of adsorption energy. Adsorbed with six nitrogen molecules, the total energy of an Ag7 + bipyramid is clearly more stable than for the 2D structure, low er by 0.44 eV for HSE06 and 0.31 eV for PBE than planar cations. This difference indicates that N2 molecules can stabilize the 3D structure and hence exert a nonnegligible effect.

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50 3.2.3 Chemisorption of O2 M olecules Fig ure 3 4 depicts O2 adsorption on Agn + for n =1 6. For n =1 3, there is one stable configuration which is physisorbed in nature. The oxygen molecule lies in the same plane as the silver ions. The adsorption energies are 0.45, 0.26, and 0.23 eV, respectively, and charge transfers are essentiall y zero (Table 33). The adsorption energy is computed according to Equation 3 1 but replacing N2 by O2 and setting m =1. For comparison, we have performed a calculation on a trimer using the HSE06 functional. The calculated adsorption energy is 0.16 eV, 0.07 eV lower than the PBE treatment. For n >3, we have obtained multiple isomers. The most interesting case is Ag4 +, since the lowest energy state upon O2 adsorption corresponds to a 3D structure (Figure 3 4 d) which is different from a pristine Ag4 + planar rhombus. The next three isomers with 2D geometry are shown in Figure 3 4 f h. The isomer in Figure 3 4 f has charge transfer and adsorption energy both lower than the ground state cluster, and the clusters in Figure 3 4 g h follow exactly the same pattern. We therefore can conclude that among isomers of a given size cluster, adsorption state energy and charge transfer are correlated and that measurements of one signal the other. The substantial difference between trimers and tetramers can be explained simply by the number of electrons in the two cations In Ag3 + the two s electrons are spinpaired but in Ag4 + there is a spin unpaired electron. The calculated O2 bond lengths are 1.23 in free space, 1.24 in Ag3 +O2, and 1.31 in Ag4 +O2, respectively. Calculations on O2adsorbed Ag5 + (Figure 3 4 i m) and Ag6 + (Figure 3 4 nq) cations also indicate interesting physical processes in these small clusters. Similar to trimers and tetramers, respectively, pentamers and hexamers have even and odd number of electrons. It is expected that an O2 binds to a pentamer more weakly than to

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51 a haxamer as shown in Table 33. Nevertheless, there is a sizable charge transfer in pentamers, which is not the case in trimers. This difference reflects the fact that the size does matter when dealing with clusters. The adsorption energy of a pentamer increases by 40 meV from the value of a trimer, and the O2 bond length is 1.26 or 2% longer than that of a trimer. Interestingly, and contrary to Ag4 +O2, the O2adsorbed Ag5 + is a planer. Figure 3 4. O 2 adsorption on Ag n + ( n =1 6) cations. Small spheres represent O atoms and large ones Ag. For n =4 6, isosurfaces of charge difference (at the value of 0.02 e / 3 ), are plotted for the ground state cluster. The blue (darker) color

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52 represents electron accumulation and yellow (lighter) deficiency. The adsorption energies and the Bader charge transfer are listed in Table 33. The hexamer shows the strongest adsorption energy and the hexamer ground state also has the largest charge transfer among all the small clusters. This trend agrees with experimental observation.61 The bond length of the oxygen molecule is 1.33 8% larger than that of a free oxygen molecule, clearly indicating a chemisorption state. Without repeating the tedious effort we made to understand Ag4 +O2, we have determined that enhanced coadsorption should occur in the hexamer and other large clusters based on the same mechanisms as in the tetramer. Table 3 3 Adsorption energy and charge transfer to O2 molecules adsorbed on Agn + cations. The lowest energy states are written in italic. n adsorption energies and charge transfers of isomers (a) (h) (Figure 3 4) 1 (a) 0.45, none 2 (b) 0.26, none 3 (c) 0.23, none 4 (d) 0.62, 0.41 (f) 0.35, 0.33 (g) 0.29, 0.18 (h) 0.15, 0.08 5 (i) 0.27, 0.20 (k) 0.18, none (l) 0.18, none (m) 0.12, none 6 (n) 0.81, 0.52 (p) 0.51, 0.39 (q) 0.33, 0.24 Note: To put the charge state into perspective, our test calculation shows that the Bader analysis gives a 0.8 e charge transfer for a NaCl unit. All energies are in eV and charge transfer in e. 3.2.4 (N2)nO2 C o adsorption on Ag4 + With clues obtained from experimental data, we have carried out an extensive search for the cluster structure and adsorption configurations. Figures 1bc show the cooperati ve effect in coadsorption of N2 and O2 for the example of Ag4 +. Fig ure 3 5 ad depict the lowest energy adsorption states of Ag4O2 + with 2 5 N2 molecules and Figure 3 5 eh show Ag4 + with 2 5 N2. Compared to Ead ( Equation 3 1), the calculated Eadco ( Equation 3 4) for m =2 5 are 48, 35, 5, and 14 meV higher, indicating a small but clear enhancement. Compared to cation Ag4O2 + without N2

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53 adsorption, the charge transfer from Ag4 + to O2 increases by 0.01, 0.06, 0.12, and 0.12 electron, respectively. The enhancement in both energy and charge transfer is accompanied by an O2 bond length increase of 0.00, 0.01, 0.02 and 0.02 respectively. In addition, our calculated binding energies of N2 molecules on Ag4 + are in good agreement with previous theoretical and experimental studies.88, 89 One very subtle point needs to be mentioned: the first two N2, on the opposite side from the O2 (Figure 3 5 a), contribute adsorption energy merely through electrostatic energy, but the two N2 on the side of O2 (near the oxygen molecule, Figure 3 5 c) affect charge transfer to the molecule and thus the O2 bond length. Therefore, the fifth N2 adsorption gains more energy compared to the fifth N2 in the cluster without an O2. The observed enhancement in nitrogenoxygen coadsorption involves complex mechanisms. We have also performed calculations of O2(N2)2 co adsorption on Ag3 + clusters. An 18 meV enhancement in adsorption energy has been found due to the existence of an O2 molecule, but there is no charge transfer even with the two N2 molecules Our calculated results agree qualitatively with the experimental trend that oxygen molecules can increase the average number of adsorbed N2 molecules and vice versa; the N2 induces further charge transfer from Ag4 + core to the O2 molecule. 3.3 Summary and Conclusion Our comparative studies on two exchangecorrelation functionals, PBE and HSE06, show that PBE is able to capture most of the trends in adsorption processes. The only failed predictions are the planar structures for Ag6 + and Ag7 +. Calculated with a hybrid functional they are found to be 3D in agreement with the experimentally deduced

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54 nitrogen adsorption abundances and with earlier ion mobility experiments.61 The HSE06 functional provides, additionally, a better ionization potential. Fig ure 3 5 The lowest energy states of Ag4O2 + (N2)m and of Ag4 + (N2)m with m =2 5, respecti vely. For adsorption of nitrogen molecules, polarization dominates the process resulting in exclusively physisorption states. We found that the physisorption to sites at obtuse corners is stronger than for those at acute ones. This is opposite to findings of earlier experiments on neutral clusters. In charged clusters different sites differ in the distance to the charge center. Typically, sites on obtuse corners are closer to the charge center then acute ones. This effect enhances the adsorption on obtuse corners and is therefore opposite to the one described in the literature. For the same reason the nitrogen physisorption tends to decrease with increasing cluster size, as seen in experiment as in our calculations. Since the relative distances of sites to the charge center in the small silver cation clusters vary strongly, their impact on the bindi ng of nitrogen is more important than the effect of field enhancement on acute corners that was noticed in earlier studies on neutral and bigger clusters. Based on this, experimentally observed abundances in the number of adsorbed nitrogen molecules

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55 were used to confirm the 2D 3D transition, as predicted based on hybrid functional results As we have seen in the Ag4 + and Ag7 + cases, physisorption of nitrogen is a competitive effect. The binding of nitrogen to any site is reduced by the presence of other nit rogen molecules on other sites. For oxygen molecules, a transition from physisorption to molecular chemisorption occurs with n >3, which is accompanied by charge transfer from cations to the O2 and O2 bond elongation. For a given size of cluster, the bind ing energy increases as charge transfer increases among various isomers. The amount of charge transfer displayed a pattern which coincides with evenodd number of s electrons in the cation clusters. We have used Ag4 +O2 as an example to show that nitrogen and oxygen coadsorption is a cooperative effect. The enhancement mechanism is, however, neither simple nor straightforward. First, the N2 adsorption energy of some sites far from O2 increases when there is an O2 adsorbed on a Agn + without inducing extra charge transfer to O2. S econd, when sites near O2 are also occupied, the charge transfer to O2 and the O2 bond length both increase but the adsorption energy stay s comparable to pure N2 adsorption. T hird, an N2induced extra charge transfer results in enhanc ed binding for the second adsorption shell. It is important to keep in mind that the energetic information does not include effects caused by kinetics and dynamics during evaporation process es as occur in experiments where vibrational frequencies are also important. Nevertheless, energetics and charge transfer capture the essential physics underlying the adsorption state and adsorption process.

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5 6 Our investigations on silver should also shed light on coadsorption in gold clusters and nanoparticles. The physisorption (N2) induced strengthening of chemisoption (O2) is of great general interest in nanocatalytic processes.

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57 CHAPTER 4 ENHANCE OF AG CLUSTE R MOBILITY ON AG SUR FACE BY CHLORIDIZATI ON The interactions between nanostructures and chemical additives in ambient e nvironment can lead to many in teresting phenomena. These interactions are of fundamental importance to nanostructure resistance to aging and consequently to system stability. Extensive experimental and theoretical work has been done on relevant phenomena such as metal surface coarsening, chemi sorptioninduced surface reconstruction, nanocluster fragmentation, etc.9097 Chalcogens (mainly, sulfur and oxygen), commonly used as additives, have been found to enhance the coarsening of Ag and Cu surfaces. Shen et al .96 have observed accelerated coarsening of an Ag(111) surface if the sul fur coverage exceeds the critical value of 0.01 monolayer in experiment. They have proposed several surface units (in the form of AgnSm) to explain the phenomenon, supported by DFT calculations.28 Similar investigations also have been done on Cu(111) surfaces by Feibelman,93 who showed that the presence of sulfur significantly reduces the formation energy of Agn or Cun clusters (n> 2) when sulfur is added, and thus enhances the material mass transport significantly The use of preformed clusters as primary buil ding blocks is a major tool i n the building of new architec tures such as dendritic structures at nanoscale.98 It has been observed that the deposition of silver clusters with a diameter of a few nanometer s leads to silver fractal islands. The fractal relaxation can be activated by using a surfactant, such as molybdenum oxide molecules, carried by silver clusters in a subsequent deposition.99 The surfactant increases the surface diffusion processes that lead to fractal pearling fragmentation.99 More recently, an experimental study100 on

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58 silver fractal aging shows sulfur and chlorine ar e the main impurities to play a role in fractal fragmentation. One possible explanation is that these additive molecules act as surfactants, which increase the surface diffusion of fractal arms thereby leading to their fragmentation. This chapter attempts to address the role of chlorine in the observed phenomena related to Ag nanostruc tures using first principles DFT calculations. We have studied the effectiveness of mass transport and bond weakening, two fundamental aspects critical to corrosion and fragm entation, which are intertwined in the observed fractal degradation process. The enhancement of the mobility or the mass transport rate of Ag clusters has been demonstrated by Ag cluster diffusion on Ag(111) surfaces before and after chloridization. T he decrease of interaction strength has been examined using a model that consists of two Ag55 clusters (details are discussed below ). The problem of Agn and AgnClm diffusion on sil ver surfaces is a n interesting topic in the field of metal surfaces.101106 It is thus relevant to mention related theoretical work on transition and noble metal surfaces. A systematic study of surface structure, diffusion path and barriers of Aln (n = 1 to 5) clusters on Al(111) surfaces was carried by Chang et al. using DFT.102 In the same paper, by use of the empirical embedded atom method (EAM), they calculate d the diffusion barriers of a series of transition metals and noble metal clusters on ( 111) sur faces. Diffusion of Cun clusters on Cu(111) and Ag(111) surfaces also was studied using DFT.104, 105 To our knowl edge, the diffusion behavior of Agn clusters and AgnClm clusters on Ag(111) surfaces has not been investigated at the firstprinciples level. Following the practice of previous work ,93, 96 our calculations and

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59 analysis have been focused on the comparison between the mobility of Agn and AgnClm clusters on Ag(111) surfaces. The rest of this chapter is organized as follows: Section 4.1 describes our models and the first principle calculation methods; Section 4.2 presents our results on Agn and AgnClm (n = 1 to 4) clusters on Ag(111) surface and the (Ag55)2 dumbbell structures Section 4. 3 concludes the chapt er. 4 .1 Methods and Computational Details All systems were investigated by DFT calculations with the spinunrestricted PBE31 form of the GGA functional. We used the planewave basis set i n conjunction with the PAW53, 83 potential as implemented in the VASP package.84, 85 A 500 eV cut off energy was used to truncate the planewave expansion. For the first part of the investigation, a sevenlayer slab with the two bottom layers fixed was employed to simulate the Ag(111) surface, on top of which Agn and AgnClm cluster adsorption and diffusion was investigated. The choice of (111) surface orientation enables the study of both fcc and hcp adsorption sites. The thickness of the vacuum between the surface clusters and the neighboring metal surface was estimated to be larger than 15 A ( 3 2 3 2 ) R30 surface unit cell and a (3 3 1) Monkhorst Pack kpoint mesh107 was used. We define the formation energy Ef as )], ( ) ( [( )] ( ) ( [ )] ( ) ( [ ) (m n ad m n ad ad m n fCl Ag E Cl Ag E Cl E Cl E m Ag E Ag E n Cl Ag E ( 4 1 ) where Ead is the surface adsorption energy, given by ), ( ) ( ) ( ) ( surface E X E surface X E X Ead ( 4 2)

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60 where X can be a single Ag or Cl atom, or an AgnClm cluster. The formation energy describes the energy gain for assembling an AgnClm cluster on an Ag(111) surface from separately adsorbed atoms. It affects the relative population of a given size of AgnClm clust er on a surface. Low formation energy correlates to a high population. Note that our definition of formation energy differs from that in studies of surface coarsening,93, 96 which used the difference between the adsorption energy of an Ag adatom on a pure surface and on the edge of an Ag island. Since the difference is a constant per Ag atom, our definition of formation energy provides a good criterion for evaluating the relative population of different sizes of clusters. For the second part of our investigation, we used an Ag55 dimer model, in which each Ag55 is a Mackay i cosahedron. We used a 30 20 20 box and consider point. The binding energy of the dimer Eb is defined as ) ( 2 ) (55 2 55 m m bCl Ag ECl Ag E E ( 4 3) Structural optimizations were performed using an energy convergence of 10 5 eV/cell and atomic f orce convergence of 0.02 eV/ respectively. For ener gy barrier eval uations, we performed nudged elastic band108 (NEB) calculations. Five or sev en images, including the start ing and final images, were considered in the calculation, depending on the distance between starting and final images. The normal force was converged within 0.03 eV / Finally, the Bader method87 was used for charge analysis. 4 .2 Results and Discussion 4.2.1 AgnClm C lusters on Ag(111) S urfaces The first part of our inv estigation addresses issues related to mass transpor t on Ag surfaces. We therefore studied adsorption of a number of Agn and AgnClm (n = 1 to 4)

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61 clusters on Ag(111) surfaces. The impor tant quantities are diffusion barriers of chloridized clus ters on surfaces compared to pure Agn clust e rs. To avoid unnecessary complication only clusters with stoichiom etry m= n were our calculations, an Ag2Cl2 cluster is not stable on an Ag(111) surface, but Ag2Cl is. Figures 4 1 to 44 show the different surface states of Agn and AgnClm that are discussed in detail below and Tables 4 1 and 4 2 summarize the calculated energies of surface states, formation energies and dif fusion energy barriers. 4.2.1.1 Ag M onomer and AgCl There exist two locally stable sites for a single Ag adatom adsorption, fcc and hcp sites ( Figures 4 1a and b, respectively). The fcc site is more stable than the hcp site by only 4 meV. The best diffusion path between fcc and hcp sites is the bri dge crossing, with a 59 meV energy barrier. This barrier is about 20 meV higher than the value calculated using EAM. For AgCl, we have found t hat AgCl dissociated adsorption ( Fig 41f), in which the AgCl bond is completely broken, is energetically more favorable than the configuration in which the Ag and Cl atoms are bonded, or AgCl cluster adsorption (Figure 4 1c). In the bonded configuration, AgCl with Ag at an fcc site (Figure 4 1c) is 60 meV higher than the dissociated state (Figure 4 1 f ), while AgCl with Ag at an hcp site (Figure 4 1 d) is 72 meV higher. The inter mediate stage (Figure 4 1 e ), in which the Ag and Cl are halfway dissociated (wit h an Al Cl distance somewhere between 1c and 1f), is 66meV higher than the ground state. The diffusion bar rier of AgCl as a cluster from Figure 4 1c to Figure 4 1d is 46meV, which is 13meV lower than the barrier for dif fusion of a single Ag. This 22% difference is significant, since the mobility is an exponential function of the energy barrier. Furthermore, although the AgCl cluster is not the most stable structure on an Ag(111) surface, the dissociation barrier from Figure 4 1c to

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62 Figure 4 1e is 72meV, whi ch is 26 meV higher than the diffusion barrier. This means that once the Ag and Cl atoms form a cluster, the prob ability of diffusion as a c luster is higher than the probability of separation. The dissociation barrier from the configuration of Figure 4 1e to that of Figure 4 1f is 30 meV. Fig ure 4 1. Ag adatoms and AgCl clusters or separately adsorbed adatoms on an Ag(111) surface: (a): an adatom on an fcc site. (b): an adatom on an hcp site. (c): an AgCl cluster with Ag at an fcc site. (d): an AgCl cluster with Ag at an hcp site. (e): an intermediate stage of AgCl separation. (f): Ag and Cl separated and adsorbed. Ag adatoms are in blue (dark) and Cl adatoms are in green (small, dark). The top layer of the Ag(111) surface is in li ght grey, while the second and third layer are in yellow (grey) and red (dark), respectively. Hexagons are used to illustrate the same positions in (a) to (f). 4.2.1.2 Ag dimer and Ag2Cl For Ag2 clusters on an Ag(111) surface, three different configurations were studied. Figures 4 2a to 2c show fcc fcc (FF), fcc hcp (FH) and hcphcp (HH) adsorption geometries, respectively. Here, fcc fcc refers to the arrangement with both Ag atoms

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63 located at fcc sites and fcc hcp refers to one Ag at fcc and one at hcp, etc The ground state FF, is more stable than FH by 31 meV. The HH structure is 10 meV hi gher than the ground state. We investigated both local rotation and longrange intercell translation, as was done in reference15. The local rotation can be illustrated by Figure 4 Figure 4 Figure 4 2c. The energy barrier between configurations of Figure 4 2a and Figure 4 2b is 62meV, while that between configurations of Figure 4 2b and in Figure 4 2c is 55meV. However, this rotational motion does not allow the Ag dimer to diffuse into the next 7Ag hexagon. Longrange intercell translation requires the center of mass of the two Ag atoms to move in a given direction continuo usly from one cell to the next Figure 4 Figure 4 2d. Figure 4 2 i shows that the cluster moves along a zig zag path. The calculated diffusion energy barrier of long range motion is 123 meV. As mentioned previously, an Ag2Cl2 cluster is not a stable structure when pla ced on the Ag(111) surface, al though this cluster is stable in vacuum. No ener gy barrier is found for one Cl atom to drift away from the cluster and find a stable adsorption site on the surface. The one Cl atom that stays in the cluster is located above the center of mass of the Ag dimer and forms an Ag2Cl cluster. The surface struc tures of Ag2Cl are shown in Figures 4 2 eh As was found for the Ag dimer, the most favorable state is FF. The energies of FH and HH structures are 6 and 13 meV higher than the ground state, respectively. Our calculations have found a 61 meV energy barrier dur ing the local rotation shown in Figure 4 2d Figure 4 2e while a 62meV bar rier has been obtained for the rotation shown in Figure 4 Figure 4 2f Compared to results for the Ag2 dimer, the former is practically the same, but the latter is 7 meV higher. Thus, the local rotation dynamics is suppressed slightly by chloridization. In contrast, the intercell

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64 translation is enhanced. W e found an 8 meV decrease in the translational barrier height for Ag2Cl compared to a pure silver dimer. Figure 4 2. Ag dimer and Ag2Cl clusters adsorbed on Ag(111) surface: (a): Both Ag atoms are at fcc sites (FF); (b): One Ag atom is at an fcc site and the other is at a hcp site (FH); (c): Both Ag atoms are at hcp sites (HH); (d): Both Ag atoms in (a) diffuse together (h): Different adsorption configurations of Ag2Cl, with the same Ag pat terns as (a) (d), respectively. The Cl atom stays above the middle of the two Ag atoms. Panel (i) shows the translational motion of an Ag2Cl cluster. The skeleton grid represents the top

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65 Ag layer and the arrows point to the direction of the movement of Ag (and chlorine) atoms in each step. 4.2.1.3 Ag T rimer and Ag3Cl3 Four different surface configurations of Ag trimers exist on the Ag(111) surface, fcct (Figure 4 3a), fcch (Figure 4 3b), hcpt (Figure 4 3c) and hcph (Figure 4 3d). Here, fcc or hcp refer to the adsorption site of Ag atoms, and t or h mean top or hollow, labeling the location of the center of mass of the cluster. Our calculations reveal that fcct and fcch configurations are nearly degenerate in energy, and so are the hcpt and hcph configurations. The fcch structure is the ground state, 14meV lower than the hcpt or hcph configurations The energy barrier for the longrange diffusion from fcch to hc pt is 178 meV, and the energy barrier between fcct and hcph is equally high. In an Ag3Cl3 cluster in vacuum, Cl atoms prefer to bind at the edge sites of a triangle formed by the three Ag atoms On a n Ag(111) surface, Cl atoms remain at side sites, but som ewhat buckled above the Ag3 plane. Compared to an Ag trimer, an Ag3Cl3 cluster is more stable at the fcct (Figure 4 3e) and the hcpt (Figure 4 3h) sites. These two sites are energetically equivalent, and they are 102 meV lower than the hcph (Figure 4 3f) site and 99meV lower than the fcch (Figure 4 3g) site. Unlike an Ag trimer, in which the positions of Ag atoms determine the ground state, the position of the center of mass is the critical factor to the cluster ground state. The calculated diffusion barri er is 159 meV for moving from fcct to hcph and 158 meV for moving from fcch to hcpt, which is about a 20meV reduction from the diffusion barriers of a pure Ag trimer on the surface. The minimal energy diffusion path is again a zigzag path (Figure 4 3i).

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66 Figure 43. Ag trimers and Ag3Cl3 clusters adsorbed on an Ag(111) surface: (a): All three Ag atoms are at fcc sites and the center of mass is at the ontop site (fcct); (b): Ag atoms are at hcp sites and the center of mass is at the hollow site (fcch); (c): Ag atoms are at fcc sites and t he center of mass is at the hollow site (fcch) ; (d): Ag atoms are at hcp sites and the center of mass is at the ontop site (hcpt). (e) (h): Adsorption configurations of Ag3Cl3 with same Ag patterns as (a) (d) respectively. Similar to Figure 4 2i panel ( i) shows the translational motion of Ag3Cl3 cluster.

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67 4.2.1.4 Ag T etramer and Ag4Cl4 The most stable Ag tetramer on the surface is a com pact fcc structure with all Ag atoms sitti ng at fcc sites, shown in Figure 4 4a. We have denoted this configuration as 4F. This structure has an energy about 100 meV lower than the 4H geometry, in which all Ag atoms are located at hcp sites (shown in Figure 4 4c). The diffusion between 4F and 4H can be realized by moving four Ag atoms together, or by a twoat a time motion which we label manuevering through an intermediate 2F2H state, shown in Figure 4 4b. Along the first path, the calculated energy barrier is 225meV. The second path was studied by Chang102 to explain the surface diffusion barrier drop from trimer to tetramer for Irn/Ir(111) and Nin/Ni(111) systems, in which this motion has lower barriers than the corresponding trimmers. T he same mechanism treated by DFT also explained the Aln/Al(111) system. However, in the case of Agn/Ag(111), the 2F2H structure is 207meV higher than the 4F state and the barrier through the second path is 218meV, just slightly lower than the first path and significantly higher than the EAM result. Chloridized Ag tetramers behave quite differently from the pure metal ones. The ground state of Ag4Cl4 is the 2F2H structure Figure 4 4d, which is 127 m eV lower than the 4F structure Figure 4 4e. Figure 4 4f also shows a different 2F2H state, which can be obtained by shifting the Ag atoms on the left side upward in Figure 4 4e. Figure 4 4g illustrates the diffusion path of the Ag4Cl4. The first three steps correspond to Figure 4 4d, 4e and 4f. As shown in the path, only two of the four Ag atoms move at each step, finally achieving a step in the direction of diffusion. This complicated maneuvering path has a 153 meV bar rier, which is much lower than the barrier of translational motion for the Ag tetramer.

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68 Figure 44. Ag tetramers and Ag4Cl4 clusters adsorbed on an Ag(111) surface: (a): Ag atoms are at fcc sites (4F); (b): Two Ag atoms are at fcc sites and the other two are at hcp sites (2F2H); (c): Ag atoms are at hcp sites (4H); (d): 2F2H structure of Ag4Cl4; (e): 4F structure of Ag4Cl4; and (f): 2F2H structure of Ag4Cl4, with the cluster shifted from (d). Panel ( g) shows the diffusion path of Ag4Cl4. Again, the skeleton grid represents the top Ag layer. The arrows depict the movement of Ag atoms in each step. Chlorine atoms just follow

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69 Table 41. The energies E and formation energies Ef (in meV) of Agn and AgnClm clusters on an Ag(111) surface. The ground state energy of each cluster is set to zero. cluster Adsorption site E (meV) E f (meV) Ag fcc (4 1a) 0 0 hcp (4 1b) 4 4 Ag2 FF (4 2a) 0 114 FH (4 2b) 3 2 83 HH (4 2c) 10 104 HH (4 2d) 10 104 Ag3 fcc t (4 3a) 3 491 hcp h (4 3b) 14 480 fcc h (4 3c) 0 494 hcp t (4 3d) 14 480 Ag4 4F (4 4a) 0 919 2H2F (4 4b) 207 712 4H (4 4c) 100 819 AgCl fcc (4 1c) 60 3 hcp (4 1d) 72 8 mid_stage (4 1e) 66 3 separate (4 1f) 0 63 Ag2Cl FF (4 2e) 0 551 FH (4 2f) 6 557 HH (4 2g) 13 564 HH (4 2h) 13 564 Ag3Cl3 fcc t (4 3e) 0 843 hcp h (4 3f) 102 741 fcc h (4 3g) 99 744 hcp t (4 3h) 0 843 Ag4Cl4 2H2F (4 4d) 0 1392 4F (4 4e) 127 1265 2H2F (4 4f) 2 1390 4.2.1.5 Formation E nergies Formation energies of AgnClm are listed in Table 4 1 Only the ground stat es are discussed except for n= 1, since the ground state is dissociated adsorption, which is not the focus of this chapter. For n=m=1, the forma which means that adsorption of an AgCl cluster is almost comparable to that of a single Ag adatom. For n=2, m=1, the formation energy is 551 meV, which is much higher than

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70 ans that the cluster is energet ically higher than the fragmented state while adsorbed on the surface. The ground states of both Ag3Cl3 and Ag4Cl4 have negative formation energies, 281 meV and 348 meV per AgCl unit. Both values are significantly lower than the corresponding Ag clusters without chlorine. Based on the format ion energies, one can predict that silver surface coarsening can be accelerated by chlorine. One can also expect that Ag3Cl3 and Ag4Cl4 have higher population than AgCl and Ag2Cl clusters on the surface, and thus are good candidates as basic surface diffus ion units. Table 42. Diffusion barriers Ed of Agn and AgnClm clusters on an Ag(111) surface. All energies are in meV. cluster transition E d clusters transition E d Ag 1a 1b 59 AgCl 1c 1d 46 1c 1e 72 Ag 2 2a 2b 62 Ag 2 Cl 2e 2f 61 2b 2c 55 2f 2g 62 2a 2d 123 2e 2h 115 Ag 3 3a 3b 178 Ag 3 Cl 3 3e 3f 159 3c 3d 178 3g 3h 158 Ag 4 4a 4b 216 Ag 4 Cl 4 4d 4e 153 4b 4c 218 4e 4f 149 4a 4c 225 4.2.2 (Ag55)2 dumbbell structure In the second part of our investigation we studied the Ag55 Mackay icosahedron and Ag55Ag55 dumbbell dimer structure (Figure 4 5a), and we calculated the binding energies of (Ag55)2 itself and with Cl or O adsorption (see Table 43 ). In the dumbbell structure, two Ag55 clusters ar e connected through two parallel facets, one of which is rotated 60 degrees with respect to the other one, mimicking fcc stacking. Our model

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71 structure with Cl adsorption is (Ag55Cl19)2, with one Cl atom on each facet except for those in contact with the other clus ter. Table 43. Distance to the center of cluster (in ) of each layer (labeled by LCl/O, L 1, L 2, and L3) and change in charge (in e) in each layer. + means gaining Layer_Cl/O L_1 L_2 L_3 distance Ag 55 5.54 4.84 2.80 Ag 55 Cl 19 6.65 5.65 4.95 2.83 Ag 55 O 19 5.78 5.53 5.17 2.82 c harge Ag 55 Cl 19 +9.99 2.59 8.24 +0.80 Ag 55 O 19 +17.40 2.28 15.46 +0.37 Table 44. Binding energy (in eV) between two Ag55based clusters dimer Binding energy Ag 55 Ag 55 4.61 Ag 55 Cl 19 Ag 55 Cl 19 3.82 Ag 55 O 19 Ag 55 O 19 5.29 To construct an (Ag55)2 dumbbell structure with Cl adsorption, we started with Cl adsorption on an Ag55 icosahedron. For single Cl atom adsorption on Ag55, we found that the center hollow site of a facet has is the most stable site. The calculated the second most stable. Then, 19 Cl atoms were placed at the center hollow sites of 19 facets, with one facet left for binding with a second chloridized cluster to later form Ag55Cl19Ag55Cl19. After optimization, some Cl atoms have diffused on facets and have deviated from the center hollow sites (Figure 4 5b). Meanwhile, the structure of Ag55 is significantly influenced by C l atoms. The first part of Table III shows the aver age distance from each layer to the center of the Ag55 cluster. On average, with Cl atoms the surface and the second Ag layers are both 0.11 further away from the center than values without (2.0% and 2.3% expansion, respectively): the Ag55 cluster expands under chloridization. For comparison, an oxidized Ag55 with 19 O atoms, one on each

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72 facet, also was studied. Our calculations show that oxygen atoms prefer to stay firmly at the center hollow sites, and they are closer to the center by 0.87 than Cl atoms. Though the second layer expands by 0.33 (6.8%), the distance between the surface layer and the center of the cluster remains nearly unchanged. Charge (electron) analy sis (see the second part of Table 4 3 ) shows that the second Ag layer contributes a much higher percentage of the total charge transferred to O than Cl atoms, which indicates the strong interaction between oxygen atoms and Ag atoms in the layer. Based on the optimized Ag55Cl19, we then constructed the chloridized dumbbell structure, in the same way as the (Ag55)2 construction described above. The optimization of the (Ag55)2 geometry leads to a binding energy of 4.61eV, but the binding energy of (Ag55Cl19)2 is 3.82 eV (reduced by 17% ). C hloridization clearly weak ens the bonding strength between the two Ag55 clusters in the dimer. For comparison, we calculated the binding energy of the oxidized dimer. Interestingly, the binding energy is 5.29 eV, higher than for both the pure silver cluster dimer and the chloridized cluster dimer, indicating a bond strengthening due to oxidation. Therefore, oxygen chemisorption on silver dendritic nanostruc tures does not enhance the fragmentation, but chlorine does. 4.3 Summary and Concluion To understand t he enhancement of Ag nanofractal fragmentation by chlorine pollution, this chap t er has presented a detailed density functional study of (Ag55)2 dumbbell structures with and without chloridization, as well as surface diffusion of Agn and AgnClm (n = 1 to 4) clusters on Ag(111) surfaces. For a single Ag55 cluster, surface adsorption of Cl atoms tends to loosen the first two layers on the surface. We have shown that the binding energy between two Ag55 icosa hedrons in the dumbbell structure

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73 is reduced by 17% with surface coverage of chlorine (19 Cl atoms on each Ag55). We have also demonstrated that AgnClm clusters are generally more mobile than Agn clust ers on Ag(111) surfaces for n= 1 4. The formation energies of AgnClm imply that Ag3Cl3 and Ag4Cl4 are good candidates as basic units for Ag surface diffusion in addition to the monomer A g Cl; the energy barrier calculations indicate that AgCl and Ag4Cl4 have barriers substantially lower than their corresponding pure silver clusters, and Ag3Cl3 has a barrier slightly lower than Ag3. Finally, our investigations also have uncovered diffusion paths of the clusters. Figure 45 (a) Ag55Ag55 dumbbell structure. (b) Optimized Ag55Cl19 structure. Green (small, dark) balls are Cl atoms. (c) Optimized (Ag55Cl19)2 dumbbell structure. (d) Opti mized (Ag55O19)2 dumbbell structure. Red (small, dark) balls are O atoms. Our theoretical investigation suggests that the enhancement of Agn surface mobility by chloridization explains the acceleration of fragmentation of silver nanofractals and the effects of those impurities in the aging process under ambient air.

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74 CHAPTER 5 MOLECULAR MAGNETOCAP ACITANCE Quantum mechanical effects can change t he capacitance of a mesoscopic capacitor by a contribution due to the density of states .109 O ne quantum consequence is magnetocapacitance due to the asymmetry in the capacitance tensor elements under field reversal .110 On the nanoscale, quantum capacitance of a molecule may depend on the charge density distributions of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). For single molecule nanomagnets (SMM) and other magnetic nanosystems whose HOMO and LUMO are determined by their magnetic states, it is natural to expect that their self capacitances are in turn dependent on their magnetic states. Such an effect, if proven to exist, will provide a much simpler path to achieve magnetocapacitance in nanoscale materials. By merely switching the magnetic state of a molecular nanomagnet, one can change its capacitance. In this chapter, we demonstrate the concept of molecular magnetocapacitance based on first principles calculations of single molecular nanomagnets. Molecular nanomagnets are stable at room temperature and can be crystallized or used in singlemolecule tunneling juncti ons.111115 A rich array of magnetic states or spin states has been probed and t unneling transport through a Mn12 single electron transistor has been studied using DFT .116 Our model system is a single molecular nanomagnet [Mn3O(sao )3(O2CMe)(H2O)(py)3] that contains three MnIII ions, the key for its magnetic properties, three pyridine ligands, one carboxylate group and a water molecule. For simplicity, we abbreviate the molecular formula as [Mn3]. The SMM can be in an S =6 highspin (HS) state or in an S =2 low spin (LS) state depending on the relative spin

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75 orientations of the three MnIII ions. The LS state is observed as the ground state in experiment. The system also can be viewed as a zerodimensional quantum dot. 5.1 Method and Computational Details Following the classical definition, we relate potential energy change upon charging and discharging to the capacitance of the system, but with a full quantum description of electrons coupled with molecular configurations. For nanodots, the capacitance can be obtained by the ratio of the charge variation to the chemical potential variation. The important quantity is the charging energy (sometimes also called capacitive energy) which is the difference between the ionization potential (IP) and the electron affinity (EA),117, 118 = ( ) = ( ) ( ) (5 1) where N is the number of electrons in the system, IP and EA are the least energy needed to subtract an electron from, and the most energy released to attach an electron to a system of N electrons, respectively. With this important least most energy principle in mind, we carefully examine the physical properties of [Mn3]. The basic procedure consists of four steps as, 1) optimize the molecular configuration and obtain the electronic structure and magnetic pattern, 2) add and/or subtract an electron of spinup and/or spin down followed by optimization again, 3) extract according to the least most energy principle, and 4) calculate magnetic quantum conductance. As detailed in C hapter 2, we performed DFT28 calculations to investigate the ionization potential and electron affinity of the [Mn3] SMM system. We used the spinpolarized PBE31 exchangecorrelatio n functional in the PAW53, 83 as implemented in the VASP package.84, 85 The [Mn3] molecule was placed in a 35 by 35 by 35 unit cell

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76 for isolation from neighboring molecules for both neutral and charged systems T hus only the point was used for the first Brillouin zone.107 The planewave energy cutoff was 500 eV. Thresholds for self consistent calculation and structur e optimization wer e set as 105 eV and 0.02 eV/, respectively. The polarizability wa s calculated by linear interpolation of the induced dipole moment and applied electric field. The dipole moment was calculated with the same criteria as self consistent calculation with dipole correction. 5 .2 Results and Discussion Configuration optimization is essential to nanosystems that contain 102103 atoms because of the strong interplay between structure and properties. All calculations should be performed using same theoretical treatment for maximal error cancellations. Figure 5 1 shows the optimized structure of a [Mn3] molecule. It can be seen that three pyridine ligands are attached to MnIII ions above the [MnIII]3plane. Below the [MnIII]3 plane, one carboxylate group is shared by Mn2 and Mn3, while a water molecule is attached to Mn1. The rest of the atoms of the molecule lie almost in the [MnIII]3plane. The largest deviation from the plane is the position of the middle oxygen (O1) atom119, 120 (0.39 above the plane, in good agreement with the experimental value of 0.33 ), followed by the position of one side oxygen (O2) atom (0.33 below the plane). The deviations of all other atoms range from 0 to 0.20 Both the HS and LS states show very similar structure s after optimization. There are three distinct low spin configurations, 1=(down, up, up), 2=(up, down, up) and 3=(up, up, down), but the second and the third ones are equivalent due to the system symmetry. The second LS configuration, or LS2, is predicted to be more stable

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77 than the first LS type by 21 meV. Our calculations also show that LS2 is energetically more stable than HS state by 37 meV, in good agreement with previous calculations.61 (a) (b) Figure 5 1. Optimized structure of [Mn3] molecule. Panel (a) is top view and panel (b) is side view. Mn atoms are in purple. O atoms are in red. Blue spheres are N atoms. Grey and bluegreen sticks stand for C and H atoms. Next, in step 2, we proceed to examine energy changes in various initial and final states of [Mn3] upon adding or removing an electron. We consider only the most stable LS state. Anions (cations) were prepared by adding (removing) a spinup or a spindown electron such that we created ions of all possible different spin states. Table 51 shows the energies of the neutral molecule, cation, and anion of both HS and LS states. We denote anion_up and anion_down as gaining a spin up and a spindown electron, res pectively. Similarly, cation_up and cation_down refer to losing a spinup and spindown electron, respectively. Structural optimizations were performed for all states The relaxation energies, defined as the energy difference before and after structural relaxation for a charged system from the neutral structure are 33, 57, 78 and 54 meV for anion_up, anion_down, cation_up and cation_down in the HS state, while those of LS

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78 states are 88, 86, 79 and 73 meV. As shown, the HS state prefers to adsorb a spinup electron over a spindown electron by 75 meV, and favors losing a spinup electron rather than a spindown one by 592 meV. In contrast to the HS state, the LS state, which is the ground state, prefers to gain a spindown electron over a spinup electron by 45 meV. However, it prefers to give away a spinup electron than a spindown one by 80 meV. Table 51. Energies of neutral, cation, and anion of both HS and LS states. Adding/removing one spin up/down electron are all considered. The energy of neutral LS state (ground state) is set to be 0. High spin state Low spin state Energy (eV) Magnetization B ) Energy (eV) Magnetization B ) neutral 0.037 12 0 4 anion_up 1.627 13 1.544 5 anion_down 1.552 11 1.589 3 cation_up 5.677 11 5.825 3 cation_down 6.269 13 5.905 5 In s tep 3, we followed the least most energy principle and select the most stable anion and cation states for calculations of ionization potential and electron affinity. Table 5 2 lists the Ionization potential ( IP ), electron affinity (EA), capacitance ( C ) and charging energy ( Ec) of both HS and LS states (Step 4 followed immediately once the right IP and EA were identified). Charging energy and capacitance were calculated according to Equation 5 1 It can been seen that the HS state is 175 meV lower in IP and 75 meV higher in EA than the LS state, resulting in a capacitance of the HS state that is 6% (or 0.2471020 F) higher than in the LS state and a charging energy that is 260 meV lower than those of the LS state.

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79 Table 52. Ionic potential (IP), electron affinity (EA), capacitance (C), and charging energy (Ec) of both HS and LS states. High spin state Low spin state I P(eV) 5.640 5.825 3% EA (eV) 1.664 1.589 4% C (10 20 F) 4.029 3.782 6% E c (eV) 3.976 4.236 6% The difference in Ec between highspin and low spin state constitutes the physical foundation for the concept of quantum magnetocapacitance. Without a magnetic field, the molecule stays in the LS ground state, which has a high charging energy. The system can be switched into the HS state, which has a lower charging energy, by applying a sufficiently high magnetic field, resulting in a change in the quantum capacitance of the molecule or a quantum magnetocapacitance. We estimate the magnitude of the switching magnetic field via M g E BB / where E is the energy difference between LS and HS states (37 meV), M is the magnetic moment difference between HS and LS states, B =0.058 meV/T, and the g factor is equal to 2. With these values, the switching magnetic field is approximately 40 T at 0 K. It is important to understand the microscopic origin of the charging energy difference between HS and LS states. We thus calculated the spatial distribution of total charge difference between the neutral and the charged [Mn3] for both anions and cations in the HS and LS states. Figure 5 2, panels (a) and (b) depict the difference between the neutral molecule and the anion in the HS st ate, and panel (c) and (d) show those in the LS state. By comparing Figure 5 2 with electron orbitals Figure 5 3, we fo u nd that the charge density difference is mainly from the highest occupied electron orbitals (HOMOs). Note that the electron in the HOMO of the neutral molecule is the electron lost in the ionization process and the electron in the HOMO of the anion is the

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80 electron gained when attaching an electron. Panels (a) and (c) (corresponding to HOMOs of the HS and LS neutral atoms, respectively) show significant difference between the HS and the LS cations, especially at the Mn2 site. Drastically different distributions of the lost electron between the HS and the LS states lead to a relatively large difference in IP (175 meV). Meanwhile, panels (b) and (d) (correspond to the HOMOs of the HS and the LS anions, respectively) display some similarities, especially on all three Mn atoms, which explains the relatively small difference in EA (75 meV). The main difference is that in panel (b), the center oxyg en atom has more charge than the one in panel (d). The mechanism of quantum magnetocapacitance is therefore clear T he charging process in a magnetic system depends on the magnetic state of the system and also on the spin of the incoming and out going electrons. The capacitance can be controlled by the external magnetic field, by changing the spin configuration of a quantum dot. We stress here that the proposed controllable magnetic quantum capacitance is fundamentally different from tuning the quantum capacitance by utilizing Landau levels 121. There, the system itself is nonmagnetic and thus the capacitance is not spin dependent. As the size of a sys tem is reduced, it becomes harder and harder to utilize Landau levels. To generate one magnetic flux quantum through a quantum dot of 2 2 nm2 (the size of our molecule) in cross sectional area, such as the one in our study (in the xy plane), the requir ed magnetic field is 500 Tesla. The switching field for our model molecule of about 40 T does not allow even one electron in each Landau level, and the capacitance cannot be modulated through Landau levels under such a field.

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81 ( a ) (b) ( c) ( d ) Fig ure 5 2. Isosurfaces of charge difference of (a) between neutral molecule and cation of highspin state, (b) between anion and neutral molecule of high spin state, (c) between neutral molecule and cation of low spin state and (d) between anion and neutral molecule of low spin state. Isovalue is 0.015 e/ 3. The type of molecular magnetocapacitance presented here is best exploited through the Coulomb blockade effect. Recently it has been proposed122, 123 that a small spin dependence of the charging energy of a quantum dot can lead to a giant Coulomb blockade magnetoresistance effect. Molecular magnets and magnetic nanostructures that demonstrate magnetocapacitance are the perfect candidates for realizing this effect.

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82 (a) (b) ( c) ( d ) Figure 53. Isosurfaces of charge density of (a) highspin neutral state HOMO, (b) highspin anion state HOMO, (c) low spin neutral state HOMO and (d) low spin anion state HOMO. Isovalue is 0.015 e/ 3. The concept of the quantum self capacitance should be distinguished from the polarizability of a molecule, even though both are related to the concept of a capacitance at some level. The polarizability is only a factor affecting the mutual capacitance betw een the source and the drain if such molecules are used as a dielectric medium. The charging energy, on the other hand, is the essential quantity in the Coulomb blockade effect whereby an electron is injected onto the molecule. The pertinent capacitance in the latter case is the self capacitance of the molecule (or more precisely the mutual capacitance between the molecule and an electrode). We have

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83 shown that the quantum self capacitance has a strong spindependence. It is natural to further ask whether the molecular polarizability also has a similar spindependence. We have performed dielectric constant and polarizability calculations of the molecule in the HS and the LS states within the linear response regime. T hat is, we assume a linear dependence between the dipole moment and applied electric field. By comparing the calculated response tensor elements, we have found that the maximum difference between the LS and HS states is less than 0.5%. The sharp contrast between the energy calculations and the polarizibility (or dielectric constant) calculations highlights the different physics represented by these two quantities. The polarizability reflects how all electrons collectively respond to an external field, whereas the quantum capacitance is mainly determ ined by only the HOMO and LUMO orbitals. Therefore, a molecule may have different self capacitances in two spin states but a spinindependent polarizability. It is clear then that when the molecule is used as a dielectric medium its magnetic moment does not affect the polarizability. When it is used as a quantum dot for Coulomb blockade a strong spindependence in the current should appear. The capacitances for these two applications are entirely different. Finally, the proposed magnetic quantum capacitor can be realized by a nanostructure other than SMM For example, a system that consists of two Fe particles separated by a C60 molecule (Figure 5 4) can have an AFM ground state. Our calculations show that the energy difference between the AFM state and the FM state is a function of Fen particle size. Table 53 and 54 present the energetic and capacitance information of FeC60Fe, corresponding to Table 51 and 52 for [ Mn3] molecule. The

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84 AFM state is 321 meV lower than the FM state in charging energy, indicating a 6 percent higher capacitance for the AFM state. Figure 54. C60 with 2 Fe atoms attached. C atoms are in yellow and Fe atoms are in red. Table 5 3 Energies of neutral cation and anion of both AFM and FM states. Adding/removing one spin up/down electron are all considered. The energy of neutral AFM state (ground state) is set to be 0. AFM FM Energy (eV) Magnetization B ) Energy (eV) Magnetization B ) neutral 0.0 0.269 0.083 6.022 anion 2.558 1.033 2.641 6.573 cation 5.402 0.023 5.498 5.300 Table 55 lists the energy differences between AFM and FM states, as well as the switching fields, for different values of n. The estimated switching field is 124.4 T for the FeC60Fe system. By enlarging the attached Fe clusters, the magnetic moment difference between AFM and FM states increases and the energy difference decreases. This leads to a drop of the switching field. The switching field of Fe15C60Fe15 is estimated as 1.2 T. Beyond a certain size, the ground state of the system transits from the AFM state to the FM state. Our findings on SMM s and FenC60Fen suggest that a relentless search for candidate systems can be very fruitful. Future synthesis of SMM

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85 guided by the energy principle may hold the key for realizing quantum dot s with capacitance that is tunable using a magnetic field under 1 T. Table 54 The ionization potential, electron affinity, charging energy and capacitance of the FeC60Fe system for both AFM and FM states. AFM FM IP (eV) 5.402 5.415 1% EA (eV) 2.558 2.724 13% C (10 20 F) 5.633 5.954 6% E c (eV) 2.844 2.691 6% Table 55 Energies, magnetizations and estimated switch fields (from AFM to FM) of FenC60Fen systems. The switching field decreases as the number of Fe atoms (or the magnetic moment) increases. Fe50C60Fe50 turns out to have a FM ground state. AFM FM Switching Field (T) Energy (eV) Magnetization B ) Energy (eV) Magnetization B ) Fe C 60 Fe 0.0 0.27 0.083 6.02 124.4 Fe 10 C 60 Fe 10 0.0 0.00 0.022 53.23 3.6 Fe 15 C 60 Fe 15 0.0 0.00 0.012 86.73 1.2 Fe 50 C 60 Fe 50 0.014 0.91 0.0 281.34 N/A 5 .3 Summary and Conclusion In this chapter we have demonstrated the concept of molecular magnetocapacitance. The quantum part of the capacitance becomes spindependent, and is tunable by an external magnetic field. Such molecular magnetocapacitance can be realized using single molecule nanomagnets and/or other nanostructures that have antiferromagnetic ground states. As a proof of principle, first principles calculation of the nanomagnet [Mn3O(sao)3(O2CMe)(H2O)(py)3] shows that the charging energy of the highspin state is 260 meV lower than that of the low spin state, yielding a 6% difference in capacitance. A magnetic field of ~40T can switch the spin state, thus changing the molecular capacitance. A smaller switching field may be achieved using nanostructures

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86 with a larger moment. Molecular magnetocapacitance could lead to revolutionary device designs, e.g., by exploiting the Coulomb blockade magnetoresistance whereby a small change in capacitance can lead to a huge change in resistance.

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87 CHAPTER 6 FIRST PRINCIPLES STUDIES O F C60 MONOLAYERS ON METAL SURFACES 6.1 Fedoped monolayer C60 on h BN/Ni(111) Surface2 Following intensive research activities in the 1990s ,125135 investigations of physical properties of C60 assemblies on surfaces continue as an active research area.136144 Scientists aim to understand fundamental physical process es in these complex systems that may have applications in nanomolecular electronic materials, for example, electrical or optical sensor s145 and photovoltaic devices .142 Like carbon nanotubes, the electronic structure of C60 and C60 assemblies can be tuned via contact with metals or organic molecules. The best known examples are perhaps the superconductivity of alka li metal doped C60 solids146, 147 and soft ferromagnetism in C60nonpolymeric organic molecule complexes .148 With high electron affinity, C60 is ofte n an electron receptor, which alters the electronic properties of the adherent molecule and molecular assemblies. Employing this characteristic, recent study of quantum transport136 shows that one can alter the I V characteristics of metal doped fullerene thin film deposited on gold substrate by controlling the thickness of potassium doped C60 thin films, eight years after the first report of a single C60 molecule transistor In the abovecited experimental investigations, various surfaces have been used as support. Our previous theoretical effort focused on surfacesupported monolayer C60 without doping.149151 We investigated and characterized thoroughly the interface electronic structure of these systems before and after adsorption of the C60 monolayer assemblies, and provided insights to experimental observations of C60 on Au(111), Ag(111), Cu(111) ,149, 150 and h BN/Ni(11 1) .151 More recently, Hebards group 152 2 This work has been published in Journal of Chemical Physics.124

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88 investigated Fedoped (by vapor deposition) C60 monolayers on silica. Inplane transport properties of the Fe thin film were investigated and a universal scale dependent weak localization correction to the anomalous Hall effect was observed. Compared to metal surfaces, insulator surfaces allow studies of dopant effects without interference from relatively strong metal surface fullerene interactions. However, Hebards experiments do not further understanding of the system in the ultrathin Fefilm limit since the transport measurements w ere conducted in the Boltzmann regime. First principles calculations such as we have performed can provide observations and understanding complementary to the experiments. In general, metal doping in C60 thin films can be prepared by evaporationdeposition procedure, and C60 can be the under layer (exohedral doping) ,129, 131, 153 or an over layer, or sandwiched between two layers of metal according to the order of deposition.125, 129, 133, 134 The electronic properties and the metal C60 interaction are found to depend strongly on the concentration of dopants and the type of metal. T his chapter report s our study of the structure, electronic and magnetic structure of irondoped the C60h BN/Ni(11 1) systems. We focus on the trend of charge transfer as a function of Fe concentration. We choose h BN/Ni(111) systems as a way to reduce complexity compared with amorphous silica. With current techniques the hBN/ Ni(111) system can be prepared experiment ally. In fact, the h BN monolayer is commensurate with the Ni(111) surface and the system has been studied by several experimental groups .154156 It was found that the h BN physisorbs on the metal surfaces and the electronic structure of the h BN monolayer is independent of the metal substrate. Later, this surface was also prepared in experiments to study C60 monolayer motioninduced

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89 charging and electron phonon coupling.143 The separation of electronic structure and the inert nature of the h BN provide an insulating layer that separates the C60 molecules from the metal surface. As a result, the fullerene layer has a relatively weaker interaction151 with the surface compared to noble metal surfaces .157 This feature is very useful for preparing quasi 2D electronic systems. Our previous calculations agree very well with experiments with and without C60 adsorption.157 Before moving to the next section, a brief summary of previous work on the metal C60 interaction is necessary. In processes of tailoring electronic structure, charge transfer at metal C60 interface has been a focus of many investigations. Numerous studies, both experiment and theory, have investigated the charging of the lowest unoccupied states of C60 molecules in gas phase,158160 solids ,146, 147, 161 and thin films 126, 129, 132, 134, 138, 162 (including monolayers). In some experiments, metal atoms are deposited over C60 monolayers or thin films ,129, 133, 135, 162 akin to systems we have used in this chap t er. It is wellestablished that C60 charge transfer from metallic surface to the molecules is about 1 electron or less in systems for a C60 monolayer deposited on the surface. The situation is different when metal atoms are doped into a C60 thin film or solid, in which case more than one electron can transfer to a fullerene molecule. Nevertheless, theoretical studies of C60transition metal and nonalkali metal system are mainly done for gas phase, that is, a single molecule complex. 6.1.1 Method and Computational Details As in previ ous chpaters, we used the VASP package and PAW methodoloty. Two types of exchangecorrelation approximation were used to calculate the exchange and correlation potentials: the spindependent PW91 GGA41 and the local spin density approximation (LSDA) as parameterized by Perdew and Zunger .29 In some cases,

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90 GGA+U and LSDA+U calculations were performed to check possible gap opening at the Fermi surface. For the DFT+ U calculations, we used the rotationally invariant approach by Liechtenstein et al. ,34 with the effective onsite Coulomb value ( U ) of 4.5eV and the effective onsite exchange ( J) value of 0.89eV for Fe atoms as proposed by Cao et al .163 We utilized convergence tests performed in our previous work on a C60 monolayer supported by h BN/Ni (111)151. In geometry optimization, surface Brillouinzone integrations were performed on a k mesh of 2 2 1 for h BN/Ni(111) (4 4) subst rate upon C60 adsorption. An 8 8 1 kmesh was chosen for calculating the density of states of Fendoped C60/ h BN/Ni (111). The energy cutoff wa s 5 00 eV. A Gaussian smearing of 0.1eV was used for the Fermi surface broadening. Finally, the force convergence wa s 0.01eV/ and energy convergence was 10 meV/unit cell. The equilibrium lattice constant151 is 2.50 and 3.52 for h BN and bulk Ni The system has a layered configuration, which contains a Fendoped C60 monolayer, a h BN monolayer and a sevenlayer Ni (111) slab. For the most stable adsorption sites of a C60 monolayer on h BN/Ni (111), the distance between the bottom hexagon of C60 and the h BN surface is 3.6 F urthermore the wall wall distance between two C60 molecules is 3.2 while their center center distance is 9.9 The thickness of vacuum between two slabs is ~15 In order to search for the stable structure of Fendoped C60 monolayer on h BN/Ni (111), all the at oms were allowed to relax except for the two bottom layers of Ni, which were fixed at the ideal bulk geometry In the equilibrium state, the FeC60 binding energy is a crucial indicator to determine the effect of Fe. It is computed as n nE E E EFe undoped total doped total b/, ( 6 1)

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91 where the first two terms refer to the total energies of C60 adsorbed on h BN/Ni (111) with and without Fe doping, respectively, n specifies the number of dopant Fe atoms, and EFe is the energy of an individual Fe atom with spinpolarization cor rections. The spindependent density of states (DOS) and spindependent projected density of states (PDOS) are crucial quantities for estimating charge transfer and magnetic moments. In VASP, the PDOS is obtained via projecting the DOS in spheres of Wigner radii centered at nuclei. The total DOS is the sum of the spinup and spindown DOS, and the density of spin states (DOSS) is the DOS difference between spinup and spindown DOS. The projected density of spin states (PDOSS) is the DOSS projected on subsy stems in the same way as for the PDOS. Electronic charge transfer at the FeC60 interface is also an important quantity in this chap t er. To estimate charge transfer, we have integrated the PDOS of each subsystem from the bottom of the valence band to the Fermi level, and thus obtained the total charge of the subsystem. For example, the charge transferred to C60 is calculated by subtracting the total charge of a C60 molecule f or a pure C60 monolayer from the total charge projected on the C60 monolayer in the FeC60/ h BN/Ni(111) complex. All integrations are scaled such that the sum of electrons is the total number of electrons of the system. We have also used the Bader analysis164 to compute the charge transfer The two methods give slightly different results but the trend a re the same. The Bader analysis is advantageous because it provides a uniqu e definition of charge on each atom and it is easy to compute. However, the PDOS analysis offers information of orbital specific information. We therefore present both results in this chapter

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92 The charge difference before and after doping is calculated as n i Fe Ni hBN C totalr r r ri1 ) 111 ( /) ( ) ( )( ) (60 ( 6 2) and spin density is the density difference between spinup and spindown electrons. Finally, the magnetic moment is computed from the population difference between spin up and spindown electrons. The magnetic moments projected on subsystems are the integrated PDOS s from the bottom of the valence band to the Fermi level. 6.1.2 Results and Discussions 6.1.2.1 Structure This section presents the ground state and some isomeric geometric structures of Fendoped C60 monolayer supported by h BN/Ni(111). The h BN/Ni(111) surface is a composite surface in which the Ni(111) surface is covered by monolayer hexagonal boron nitride ( h BN). The concentration of Fe doping ( n /C60) is 1 4 and 15 atoms. For n =1 4, we exhausted initial configurations in which the Fe atoms sit at different interstitial sites of C60. F or n =15, two initial configurations were tested. The procedure for geometry optimizations is described in the foregoing section. Here we report structures and energies of the most stable configurations found in our search. Figure 6 1 to 65 depict the lowest energy FenC60 structures supported by h BN/Ni(111). Binding energies of various configurations and systems are given in T able 61 Fig ure. 6 1 a shows the most stable structure of an Fe1doped C60 monolayer on the surface. The Fe atom is located in the center of the interstitial space of the fcc C60 molecular lattice, and forms two bonds to each adjac ent C60 molecule. The bond lengths between the Fe atom and its nearest C atoms are 2.22 and 2.29 respectively. The three C FeC angles shown in Figure 61a are about 38. The distance between the Fe

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93 Figure 61 Fe1doped C60 monolayer configurations on an h BN/Ni (111) surface. Panel (a) (d): Structure of the configurations according to binding energy (a) is the most stable and (d) the least. Left column is side view and right is top view. The red, grey, dark blue, pink and indigo balls represent iron, carbon, nitrogen, boron and nickel atoms, respectively. Note only parts of C60 (a) (d) (c) (b)

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94 molecules in 4 unit cells are not shown for a clear view. View of w hole molecules is presented in Figure 6 6 to 68 atom and the h BN surface is 6.86 The binding energy of 3.04eV/Fe atom is a result of the strong FeC bonding. The other possible sites are shown in Figure 61 ( b ) ( d ) listed from high to low binding energies. For n =2, the two Fe atoms energetically prefer to stay apart in the interstitial space of the C60 molecules. As shown in Figure 6 2 ( a ) Fe1 binds to three neighboring C60 molecules with the lengths ranging from 2.25 to 2.27 nearly equal. This Fe atom locates in the middle of the interstiti a l site. The three C FeC angles shown in Figure 62 (a), are all 38. Even though the Fe1 atom is a neighbor of three C60 molecules, Fe2 positions itself off the middle of the interstitial site. This Fe atom only binds to two C60 molecules. The two neares t bond lengths are 2.16 and 2.23 respectively. The two C FeC angles are 39 On this offmiddle site the Fe atom is too far away to form bonds with the third C60 molecule, where the shortest FeC distance is 2.40 The distance between the Fe (1 or 2) and h BN is 6.85 The relaxed Fe2C60 structure is similar to FeC60, except for one more Fe atom adding on another separate interstitial site. The distance between the two Fe atoms is 5.81 The binding energy in Fe2C60 is 3.11 eV/Fe, only 0.07 eV higher than that in FeC60, implying a small FeFe interaction mediated by a C60 molecule. Other doping sites are depicted in Figure 62 ( b ) ( d ) Table 6 1. Binding energy of Fe atoms at various sites in eV. FeC 60 Fe 2 C 60 Fe 3 C 60 Fe 4 C 60 Fe 15 C 60 a 3.04 3.11 3.12 3.23 3.41 b 2.95 2.85 2.87 3.18 3.21 c 2.02 2.30 2.71 3.05 d 1.99 2.18

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95 Figure 6 2 Same as Figure 6 1 for Fe2doped C60 m onolayer on an h BN/Ni (111) surface. Next, a third Fe atom was added into the optimized structure of Fe2C60 followed by further structural relaxation. Figure 6 3 ( a ) shows the most stable structure of Fe3C60. (a) (b) (c) (d)

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96 The additional Fe atom locates further away from the BN layer compared to the other two Fe. It binds to tw o nearest C60 molecules via 4 FeC bonds and to the adjacent Fe atom. The lengths of the FeC bonds are 1.97 2.19 The two C Fe3C angles are 41 and 42 respectively and the length of the Fe1Fe3 bond is 2.28 For Fe1, the Fe1C bond lengths range from 2.07 to 2.23 which are shorter than those in the Fe2C60 complex. The two C Fe1C angles are 38 and 39, respectively. On the other side, the Fe2 atom binds to three C60 molecules, two bonds each to two C60 and one bond to the remaining one. The bond lengths are 2.06 2.33 The two C Fe2C angles are both 39 (Figure 3a). The distance between Fe1 and the h BN surface is the same as in FeC60 and 0.01 more than in Fe2C60 (Figure 6 1 ( a ) and 6 2 ( a ) ). The binding energy in Fe3C60 is 3.12eV/Fe atom, contributed by the FeC and FeFe bonds, very close to the value of Fe2C60. The other two configurations are shown in Figure 6 3 ( b ) ( c ). The Fe4C60 system is the largest one for which an extensive energy minimum search was performed. Figur e 6 4a shows the most stable structure for Fe4C60 supported by h BN/Ni (111). It is notable that three iron atoms Fe1, Fe3 and Fe4, form a trimer with the FeFe bond lengths ranging from 2.40 to 2.89 The one atom (Fe2) is separated from the cluster by the C60, the same as in previous cases. All Fe atoms form Fe C bonds, with bond length ranging from 1.95 to 2.32 to adjacent C60 molecules. The distance between Fe1 at the lowest position and h BN is 7.03 a substantial increase from the 1 3 Fedoped systems. The binding energy in Fe4C60 is higher than that in FeC60 by 0.19eV/Fe atom and Fe2C60 by 0.12eV/Fe atom, suggesting

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97 that the Fe3 cluster has high stability in the interstitial spaces of C60 monolayer. Again, the other two structures are shown in Figure 64 b and c. Figure 6 3. Fe3doped C60 monolayer on an h BN/Ni (111) surface. Similar to Figure 6 1 with (a) being the most stable on and (c) the least. Finally, we investigated the behavior of a heavily doped FenC60h BN/Ni(111), with n =15. Starting with the optimized structure of Fe4doped system, 11 Fe atoms were added randomly over the C60 monolayer. We repeated the process twice with different (a) (b) (c)

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98 Figure 6 4. Same as Figure 6 3 for Fe4doped C60 monolayer on an h BN/Ni (111) surface. initial configurations of the 11 adatoms T he configuration with the lower energy is presented here. Note that this configuration may not be the most energetically stable state. However, our interest is in the morphology of the Fe15C60 complex and its impact on electronic and magnetic properties. Figure 6 5b shows the positions of the first 4 Fe atoms in interstitial sites, and the extra 11 Fe atoms distributed over the C60 monolayer. All of the Fe atoms form bonds with at least one other Fe atom, except for one (Fe2). Out of the 15 Fe atoms, seven form bonds to the neighboring C60 molecules. The (a) (b) (c)

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99 distance between Fe1 and h BN is 7.23 0 .2 higher than that in the Fe4C60 complex. Other low lying atoms, Fe1, Fe3 and Fe4, are shifted slightly away from the BN layer due to FeFe interaction. The binding energy in Fe15C60 is 3.41eV/Fe atom, higher than that in Fe4C60 by 0.18eV/Fe atom. The s econd structure with a little less binding energy is shown in Figure 6 5b. 6.1.2.2 Electronic and Magnetic Properties The bonding of Fe to C60, in FenC60/ h BN/Ni(111) complexes ( n =1 4, 15) may involve two possible types of interaction: ( i ) ionic bonding, with the characteristic feature being charge transfer from the metal towards C60 and (ii) covalent bonding, characterized by hybridization between metal s and d states and C60 orbitals. Following structural relaxation, we have calculated t he spindependent DOS and PDOS, and analyzed charge transfer, spin distribution, and spin populations projected on the individual orbitals. All the electronic calculations shown below were performed with both spin dependent GGA and LDA. For n =4 and 15, we used GGA+U and LDA+U to examine possible the gap at the Fermi level. Changes in charge transfer and magnetic moments as obtained from GGA+U (LDA+U) are compared with those from GGA (LDA). Upon obtaining the structure of various systems, we have performed B ader analysis to study the charge rearrangement. Table 6 2 provides information about the charge c hange in subsystems as a function of the number of Fe atoms and doping sites. It is clear that charge transfer to C60 increases monotonically as the number of Fe atoms increases, ranging from about 1eto 5e-. One also concludes that Fe atoms and the Ni(111) substrate always lose electrons, while C60 and the monolayer h BN always gains electrons. In addition, the magnetic moment of C60 is always in the direction

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100 opposite to the total moment of the systems. In the following sections, we discuss a detailed analysis based on an integration of PDOS, which is different from the Bader method, and focus on the most stable structures. Figure 6 5. Two isomers of a Fe15doped C60 monolayer on an h BN/Ni (111) surface. Panel (a) is more stable then panel (b). 6.1.2.2.1. FeC60 We performed the DOS calculations on the optimized FeC60 configuration. Figure 6 6a shows the siteprojected DOS of C60 and the Fe atom. The energy for PDOS ranges from 5eV to 2eV and the Fermi level is shifted to zero in each case. In the GGA calculations, four main features are observed in the spectrum for C60: peaks at 2.58eV (HOMO 1), 1. 43eV (HOMO), 0.29eV (LUMO) and 1.33eV (LUMO + 1). The LUMO (b) (a)

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101 peaks remain above the Fermi level. The LUMO level is partially filled. This partial filling of the LUMO level is a consequence of charge transfer from the metal to the molecule. Integrating the PD OS of doped C60 up to the Fermi energy, the estimated charge transferred toward C60 is 0.95 electrons/C60 molecule, slightly lower than the number given by the Bader method (see T able 6 2 ). Compared to a C60 monolayer on a metal surface without dopants, the charge transfer from the Fe atom to C60 per C Fe contact150, 157 is much more significant. This large value is a result of the difference in the el ectron affinity between the Fe and C60.127, 129 Thus, ionic character contributes to the FeC60 bonding. Close examination of the C60 HOMO state in the PDOS also shows that the electron density is shared by the C60 HOMO and the Fe s and d orbitals A small amount of hy bridization also occurs at the C60 HOMO1 band (see Figure 6 6 a). This level mixing signals the covalent nature of the FeC60 bonding. The spindensity of states of the C60Fe system is obtained by taking the difference of DOS and PDOS of the spinup and spin down electrons. As seen in Figure 6 6 b, the spin up electron, by convention, is the majority spin of Fe and also of the system, but more electrons with the minority spin (spin down) occupy the bands near the Fermi level. Consequently, the spindown electrons are preferentially transferred from Fe towards C60. The projected density of spin states (PDOSS) of C60 indeed shows that the spin down electrons dominate the bands of C60 near the Fermi level. Integrating the PDOSS distributions on C60 and Fe, r espectively, we list the spin populations for the individual orbitals in Table 6 3 These numbers indicate an intraatomic charge transfer from 4s to 4p and 3d within the Fe atom. Note that the spin projected onto the C60 is

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102 Figure 6 6. Projected densities of states (PDOS) panel and projected densities of spin states (PDOSS) on monolayer C60 and a Fe atom for FeC60/ h BN/Ni(111): (a) PDOS of C60 in GGA (black solid) and LDA (red dash), and PDOS of Fe in GGA (blue dot), Fermi level is at energy of 0; (b) PDOSS of C60 in GGA (black solid) and LDA (red dash), and PDOSS of Fe in GGA (blue dot). Panel (c) and (d) are isosurfaces of charge density difference and spin density, with yellow representing charge accumulation in (c) and positive spin in (d), and red charge depletion and negative spin. down, opposite to the majority spin that is dominated by the Fe 3d electrons. Also note that a pure C60 has the same populations of spinup and spindown and thus zero net spin. The negative magnetic moment on C60 ( 0.37 ( B)) that aligns anti parallel to that of the Fe atom is a direct consequence of Fermi level alignment and charge transfer. The contribut ion to charge transfer from the spin minority exceeds that from the spin majority by roughly 40%. The total spin projected on the Fe is 2.45 and 1.96 B using (b) (c) (d) (a)

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103 GGA and LDA, respectively. Further analysis of orbitals indicates that spin density transfer is f rom Fe 3d to C 2p orbitals (Table 6 3 ). In a study of Fedoped single C60 Table 62: Charge transfer and magnetic moments as computed by the Bader method Fe n C 60 / h BN/Ni Q Fe Q C60 Q h BN Q Ni(111) m Fe ( B ) m C60 ( B ) n=1 Config. a GGA 0.87 +1.04 +1.29 1.46 +2.47 0.38 LDA 0.84 +1.00 +1.22 1.38 +2.20 0.39 Config. b GGA 0.89 +1.08 +1.16 1.35 +2.23 0.51 Config. c GGA 0.95 +1.12 +1.27 1.44 +2.45 0.33 Config. d GGA 0.95 +1.12 +1.28 1.46 +2.47 0.32 n=2 Config. a GGA 1.70 +1.87 +1.30 1.47 +4.73 0.74 LDA 1.66 +1.84 +1.21 1.38 +4.54 0.70 Config. b GGA 1.38 +1.57 +1.27 1.45 +4.56 0.40 Config. c GGA 1.65 +1.87 +1.26 +1.48 +4.13 0.42 Config. d GGA 1.90 +2.07 +0.73 0.90 +5.04 0.07 n=3 Config. a GGA 2.46 +2.66 +1.27 1.47 +7.00 0.73 LDA 2.32 +2.50 +1.19 1.37 +6.56 0.62 Config. b GGA 2.50 +2.66 +1.28 1.44 +7.11 0.54 Configuration c GGA 2.46 +2.67 +1.25 1.46 +6.34 0.40 n=4 Config. a GGA 3.10 +3.29 +1.27 1.46 +9.49 0.57 LDA 2.92 +3.10 +1.19 1.37 +9.14 0.49 Config. b GGA 3.02 +3.23 +1.25 1.46 +8.62 0.62 Config. c GGA 3.28 +3.45 +1.23 1.40 +9.16 1.03 n=15 Config. a GGA 5.09 +5.24 +1.24 1.39 +29.93 0.69 LDA 4.76 +4.89 +1.19 1.32 +25.39 0.30 Config. b GGA 4.55 +4.69 +1.24 1.37 +40.34 0.16

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104 where the Fe atom is inside the C60, Tang et al 165 found that the magnetic moment of the Fe atom is 2.19 B and the spin projected on the C60 is 0.19 ( B), opposite to that projected on the Fe at om. In our investigations, the Fe atom is incorporated at the interstitial site of the C60 molecules (exohedral doping). Evidently, the negative sign of spin of the molecule is independent of the doping site of the Fe atom. The c harge density difference as computed via Equation 6 2 is depicted in Figure 6 6 c where the iso surface value is 0.04e /3. Interestingly, the charge transfer to C60 mainly stay s at the interface between the Fe atom and the molecule, similar to the monolayer C60 on metal surfaces150, 157. The s pin dens ity distribution also was computed and the isosurface at a value of 0.01e /3 is plotted in Figure 6 6 d. As can be seen, the down spin is mainly distributed at the contact point of Fe and C60. 6.1.2.2.2. Fe2C60 Compared to single Fe atom doping, two Fe atoms have more valence electrons available for charge transfer to C60. Integration of the PDOS from the bottom of the valence band to the Fermi level shows a charge transfer of 1.76efrom two Fe atoms to the molecule, 0.81emore than the single Fe doping. This number again is slightly lower compared to the Bader analysis. These electrons are accumulated at the tail of the LUMO band, as seen in Figure 6 7 a. Due to the orbital energy matching, the HOMO and HOMO 1 bands of C60 hybridize with the Fe orbital s. As the PDOS of Fe orbitals broadens, the HOMO and HOMO 1 bands of C60 spread out. This broadening enhances the hybridization (or vice versa) and the covalent character of the FeC bonds. The large charge transfer to C60 doubles the magnetic moment of C60, compared to the FeC60 system, in both GGA and LDA calculations (Table 63 ). Figure 6 7 b shows

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105 Fig ure 6 7 Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and two Fe atoms for Fe2C60/ h BN/Ni(111). P ane l s (a) (d) are for the FM state: (a) PDOS of C60 in GGA (black solid) and LDA (red dash), and PDOS of Fe in GGA (blue dot), Fermi level at energy of 0; (b) PDOSS of C60 in GGA (black solid) and LDA (red dash), and PDOSS of (a) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) ( h )

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106 Fe in GGA (blue dot). Panel (c) and (d) are isosurfaces of charge density difference and spin density, with yellow representing charge accum ulation in (c) and positive spin in (d), and red charge depletion and negative spin. Panel (e) (f) present the same information as panel (a) (d) for the AFM state. the PDOSS distributions on C60 and two Fe atoms. The PDOSS on C60 in Fe2C60 is similar to t hat in FeC60 (Figure 66 b), except for the higher population of electrons with the spinminority elements observed on the energy region from 0.9eV to the Fermi level, a consequence of enhanced charge transfer to C60. We again have computed the charge den sity difference and spin density. The isosurfaces of values 0.04e /3 and 0.01e /3 for charge and spin, respectively, are depicted in Figure 67 c and d. Compared to Figure 61, charge transfer to C60 is enhanced and still concentrated at contact point. The spin density spreads out more compared to one Fe doping. For Fe2 doping, there is an AFM state with energy almost the same as the FM state (~104 eV higher in ener gy). For this state, the charge transfer to C60 is 1.74eand 1.27efrom Bader and PDOS (Figure 6 7 e) analysis, respectively. The net spin of the system and net spin on C60 (Figure 6 7 f) are zero. Figure 6 7 g and h are the same plots as in Figure 6 7 c and d but for the AFM states. The two charge difference patterns are very similar ( c and g) but the two spin patterns are substantially different. 6.1.2.2.3. Fe3C60 Figure 6 8 ( a ) shows the PDOSs of C60 and 3 Fe atoms. The distortion in HOMO and LUMO bands suggests a large charge transfer from Fe to C60 and strong hybridization between the Fe and C60 orbitals. We obtain a charge transfer of 1.95 electrons/molecule to C60, increasing by 11% from Fe2C60. Compared to the 2.66 efrom the Bader analysis, the PDOS under estimated the charge transfer by more than half of an electron. However, this discrepancy does not change the observed trend.

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107 C60 doped with three Fe atoms has a rather high spindown population near the Fermi level, as seen in Figure 6 8 b. The negative magnetic moment of C60 originates from the majorit y of transferred electrons being spin down. These electronic and magnetic phenomena imply that heavy metal doping broadens states in C60 to the extent that features of the C60like orbitals start to vanish. The magnetic moment of C60 retains the same orientation (anti parallel to that of the Fe atoms). Its absolute value, calculated in GGA is doubled from Fe1doped C60, and slightly larger than that in Fe2doped C60 (see Table 6 3 ). In contrast, the absolute value calculated in LDA becomes smaller by 0.07 ( B), compared to that in Fe2doped C60. Figures 6 8c and d depict charge difference, and spin density, respectively. A close view of the spin density (Figure 6 8e) shows that while spin on each Fe atom is positive (or spinup) (yellow) there is a spin down distribution (red) between the two Fe atoms. This is a result of interaction with C60 molecule since there is no spindown population between two isolate d Fe atoms The molecule transfers some spindown electrons back to Fe to enhance Fe bonding. 6.1.2.2.4. Fe4C60 Incorporation of four Fe atoms enhances the interaction between Fe and the C60 monolayer. Figure 6 9 a shows the PDOS of C60 and 4 Fe atoms. Similar to the Fe3C60 system, the C60like feature vanishes because of the large charge transfer from the Fe orbitals to the like orbtials of C60 and the strong hybridization between the Fe 3d and C60 orbitals. The charge transfer toward C60 is 2.10 electron/molecule, an increase of 8% compared to that in the Fe3C60 system. Note again, this number is smaller than that from Baders method (3.29e-).

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108 For heavily doped C60, Li et al.162 studied the electronic states of an Yb doped C60 monolayer on Ag(111) using the synchrotron radiation photoemission technique. Doping Figure 68 Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and three Fe atoms for Fe3C60/ h BN/Ni(111): (a) PDOS of C60 in GGA (black solid) and LDA (red dash), and PDOS of Fe in GGA (blue dot), Fermi level at energy of 0; (b) PDOSS of C60 in GGA (black solid) and LDA (red dash), and PDO SS of Fe in GGA (blue dot). Panel (c) and (d) are isosurfaces of charge density differ ence and spin density, with (b) (c) (d) (e) (a)

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109 yellow representing charge accumulation in (c) and positive spin in (d), and red charge depletion and negative spin. Panel (e) is a close view of spin density. with more than 3 Yb atoms/C60 causes the electronic propert ies of t he sample to change from metallic to semi conducting because of the modulation in the LUMO+1 band.162 Lis observations suggest that calculations beyond GGA and LDA should be performed. T herefore, we have added an additional U term for the t reatment of strong onsite 3d electronelectron interactions on Fe (U = 4.5 eV ) as suggested by Cao et al 163. Our focus here is on the electron density of the band of C60 near the Fermi level. Our results show that neither GGA+U nor LDA+U produces a band gap near EF (see Figure 6 9 a ), which means t he system stays metallic. Similar to previous systems, the minority spin levels again dominate the Fermi level, as seen in Figure 6 9 b. The calculated magnetic moment on C60 is 0.52 ( B) in GGA+U (or 0.54 ( B) in LDA+U), which is slightly smaller than the GGA (or LDA) result, as seen in Table 6 1 Inclusion of Coulomb interactions on the Fe 3d orbital has a strong effect on the spin states of the system. The magnetic moment of Fe using GGA+U increases by 20% from the GGA result while the value calculated using LDA+U increases by 23% from the LDA. The charge difference and spin density are also similar to Fe3doped C60, except that the detailed patterns are more complicated due to the added Fe atoms. Therefore, they are not shown in the Figure 6 9. 6.1.2.2.5 Fe15C60 The next questions are: how many electrons at most can a C60 accept? And would the electronic signature (valence bands) of the C60 molecules be seen when the system is heavily doped? To understand the large n limit, we have investigated Fe15C60 on the

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110 same supporting surface. Figure 6 10 a shows the PDOS of the heavily doped C60 and 15 Fe atoms. Substantial charge transfer 4.61 electrons/molecule (compared to 5.24 efrom the Bader analysis), and strong hybri dization smear the C60like valence band completely. It is difficult to distinguish the HOMO and LUMO peaks in the spectrum of C60. The doped C60 retains metallic characteristics Saturation in charge transfer to C60 is not observed. Fig ure 6 9 Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and four Fe atoms for Fe4C60/ h BN/Ni(111): (a) PDOS of C60 in GGA (black solid), LDA (red dash), GGA+U (green dashdot) and LDA+U (orange dashdot dot), and PDOS of Fe in GGA (blue dot), Fermi level at energy of 0; (b) PDOSS of C60 in GGA (black solid), LDA (red dash) and GGA+U (green dashdot) and LDA+U (orange dashdot dot), and PDOSS of Fe in GGA (blue dot). Fig ure 6 10 b shows the spindifferent distributions of C60 and 15 Fe atoms. The magnetic moment of Fe obtained within either GGA or LDA becomes much larger than those in FenC60 ( n = 1 4) as expected. In c ontrast, the magnetic moment of C60 drops slightly drops down from Fe4C60 from GGA calculations and maintains near ly the same value using LDA (Table 63 ). For the same reason as for the 4 Fe system, we performed calculations with inclusion of the Hubbard U. Our results show again that the system is metallic. We note (a) (b)

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111 that the magnetic moment on Fe atoms increases substantially as a results of inclusion of the U term. Table 63. Spin popul ations on the individual orbitals of Fe and C60, spin up, spin down, m magnetic moment. Fe C 60 s p d m Fe ( B ) s p m C60 ( B ) FeC60/ h BN/Ni GGA 0.01 0.01 2.42 +2.45 0.01 0.35 0.37 LDA 0.04 0.01 1.92 +1.96 0.01 0.38 0.38 Fe 2 C 60 / h BN/Ni GGA 0.03 0.02 4.66 +4.71 0.03 0.68 0.71 LDA 0.07 0.01 3.38 +3.46 0.03 0.67 0.70 Fe 3 C 60 / h BN/Ni GGA 0.03 0.03 6.76 +6.82 0.04 0.71 0.76 LDA 0.03 0.03 6.40 +6.46 0.02 0.60 0.63 F e 4 C 60 / h BN/Ni GGA 0.04 0.03 9.14 +9.21 0.04 0.66 0.70 LDA 0.03 0.05 8.74 +8.82 0.03 0.55 0.58 GGA + U 0.02 0.01 11.00 +11.03 0.04 0.48 0.52 LDA+U 0.01 0.01 10.75 +10.77 0.03 0.51 0.54 Fe 15 C 60 / h BN/Ni GGA 0.02 0.13 29.47 +29.36 0.05 0.53 0.58 LDA 0.02 0.02 24.29 +24.29 0.03 0.49 0.51 GGA+U 0.08 0.01 38.80 +38.87 0.06 0.49 0.54 LDA+U 0.05 0.11 30.70 +30.54 0.05 0.62 0.66 6.1.3 Summary and Conclusions T he structural, electronic and magnetic properties of the FenC60 complexes ( n = 1 4 and 15) were investigated in this chapter via DFT. The amount of charge transfer increases monotonically as the number of Fe atoms increases. At low Fe doping concentration ( n = 1 or 2), individual Fe atoms diffuse in the interstiti al spaces of the C60 monolayer and bind to C60. The binding energy is mainly due to the FeC bonds. When n =3 or more, Fe atoms cluster around the C60 molecule as well as bind to C60

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112 molecules. The binding energy is therefore from both FeFe and FeC bonds. Large Fe concentration gives rise to higher binding energy in the FenC60 complex but far less than the cohesive energy, 4.28eV/Fe atom ,166 of crystalline Fe (fcc). Charge transfer, as a function of number of metal atom s, was investigated by both the Bader analysis and PDOS, with the former giving a larger transfer to C60 than the latter. Doping with Fe atoms results in modifications of the C60 LUMO derived bands, which are enhanced by Figure 6 10. Projected densities of states (PDOS) and projected densities of spin states (PDOSS) on monolayer C60 and fifteen Fe atoms for Fe15C60/ h BN/Ni(111): (a) PDOS of C60 in GGA (black solid), LDA (red dash), GGA+U (green dashdot) and LDA+U (orange dashdot dot) and PDOS of Fe in GGA (blue dot), Fermi level at energy of 0; (b) PDOSS of C60 in GGA (black solid), LDA (red dash) and GGA+U (green dashdot) and LDA+U (orange dashdot dot), and PDOSS of Fe in GGA (blue dot). increasing electron occupation. Analysis of PDOS shows that the charge transfer occurs mainly between the Fe3 d orbitals and the t1u orbitals of C60, indicating that ionic character dominates the FeC60 bonding. Hybridization also is observed among the C60 and Fe orbitals (mainly with Fe s and d or bitals). Therefore, a covalent effect also exists in the FeC60 bonding. Furthermore, doping with Fe induces a negative magnetic moment in C60, aligning it anti parallel with Fe. The value of this moment varies roughly within a fact or of two. In contrast to c harge transfer, magnetic moment values obtained (a) (b)

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113 by Baders method and PDOS analysis are very close in value. Heavy Fedoping does not destroy the structure of C60, but deforms the electronic structure of C60. Unlike alkali metal doped system, the Fedoped systems do not undergo a metal semiconductor transition. Our results present a comprehensive description of the interaction between the transitionmetal atom/cluster and the C60 thin film, which is important for understanding nanostructured, quasi two dimensional conducting systems. 6.2 C60 on defected Au(111) surface3 Surfaces with a regular array of energetic sites can be utilized as templates for growing nanosca le structures from site nucleation.168172 For example, the Au(111) surface with the socalled herringbone reconstruction173, 174 can form a surface dislocation network, and can guide the growth of ordered twodimensional arrays of metal islands. The reactive elbow sites,169, 175 where the nucleation of the metal islands takes place, are the key for such a guiding function. In terms of the C60 molecule adsorption on Au(111) at room temperature, the formation of closedpacked mol ecular monolayers long has been without any preferential attachment of C60 molecules to the elbow sites.176179 However, recent investigations have shown that the adsorption of C60 molecule on noble metal surface involves vacancies at room temperature.180 Sing le atom vacancy pit has been proposed for Ag(111) adsorption,180 while the s evenatom pit model has been proposed for Cu(111).181 Our collaborators have performed an experiment involving C60 molecule adsorption on the Au(111) surface.167 The adsorption of C60 molecule on Au(111) is 3 This work has been published in Physical Review B.167

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114 quite different at low temperature compared to room temperature. On one hand, at 46 K, the C60 molecules, either single molecule or small clusters, prefer to attach on the elbow sites. As the temperature goes up, the molecules sitti ng at the elbow sites start to move away and aggregate to island. Most elbow sites are depleted of individually adsorbed molecules by 284K. On the other hand, surprisingly, if the C60 molecules are deposited onto Au(111) at room temperature, single molecul e adsorption at the elbow sites is observed. This controversy can only be explained by different bonding configurations at low temperature and room temperature. Following the observation that the Au atoms can jump out of the elbow site, it is proposed that a pit vacancy model may apply to the system i.e. the C60 molecules are adsorbed onto the vacancies formed from moving of the Au atoms at room temperature. We thus performed DFT calculations to investigate the adsorption of C60 molecules onto pit vacanc y model on Au(111) surafce. 6.2.1 Method and Computational Details As before, we performed plane wave PAW53, 83 DFT28 calculations with VASP84 85 to investigate adsorption energies of C60 molecules on one, three and seven atom pits. We used local density approximations29 30 (LDA) A sevenlayer slab with the two bottom layers fixed was employed to simulate the Au(111) surface, on top of which a monolayer C60 was adsorbed. The thickness of the vacuum between the molecule and neighbor metal surface was larger than 15 and a ( 3 3 1) M onkhorst Pack kpoint mesh107 was used. 6.2.2 Results and Discussions On all three defect sites, the most stable configurations correspond to a hexagon of the C60 in contact with the surface (shown in Figure 6 11). The center of the hexagon

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115 aligns with the center of the vacancy along the zdirection. Geometry optimization shows very little change in C60 molecular structure in all three cases. We follow the same method by Li et al.180 to describe the surface reconstruction in the vertical ( z) direc tion (See Table 6 4 ). On average, atoms in the bottom hexagon of C60 molecules ar e 1.80 1.68 and 0.15 above the first Au layer in oneatom, threeatom, and sevenatom pit s, respectively. Vacancies shorten the distance between first layer and second layer Au atoms locally compared to a perfect Au(111) surface. We also calculated the intra i). The largest buckling occurs in the second layer in systems with oneatom and threeatom pits, and the first layer in the system with a sevenatom pit which causes larger buckling in layer 35 35) than the two smaller pits. To investigate surface reconstruction, we also examined the inplane displacement of atoms near the three vacancies. Table 6 4 lists the calculated deviations of lateral positions (compared to positions in a perfec t Au (111) surfac e) of those atoms right next to the centers of vacancies. Only atoms in the first and second layers show significant lateral displacement, denoted as xy1 and xy 2 respectively. Interestingly, atoms in the first layer appear to be repelled from the cen ter of vacancies, especially near the sevenatom pit where atoms move by 0.18 ; but in the second layer, atoms near a vacancy are attracted inward in the oneatom and three atom cases. Atoms in the second layer near a sevenatom pit move outward just as ones in the first layer because of the direct contact with a C60 molecule.

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116 Figure 611. C60 molecules on vacancy sites of Au(111) surface. Panel (a) is side view of C60 on a 7atom pit. Panel (b), (c) and (d) are top view of C60 on 1, 3 and 7atom pits, respectively, in which first, second and third Au layers are shown in dark/red, grey/blue, and light/yellow, respectively. Calculated adsorption energies are 2.07 eV, 2.33 eV and 2.56 eV for oneatom, threeatom and sevenatom vacancies, respectively. This trend agrees with our intuition. Compared to a perfect A u surface of 1.2 eV ,172 the energy difference signals the cause of defect trapping of C60 since the translational motion of the molecule on Au(111) is nearly barrierless. At 46 K, both individual molecules and small clusters appear 6.2 (c) (d) ( b ) ( a )

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117 above the substrate surface in STM images. The individual molecules observed at room temperature are about 4 above the surface which is in good agreement with the sevenatom pit model shown in Figure 6 11 a and d. Height measurement in STM is not always a reliable method for measuring the position of adsorbed species because of electronic effects. However, the 2.2 difference in measured heights is too large for a pure electronic effect This height difference is in very good agreement with that measured for C60 molecules sitting on the upper terrace and those on the lower terrace sharing a common step.182 Table 64 Structure and energies: dz is the average inter planar average intralayer buckling amplitudxy the average lateral displacement of atoms surrounding a vacancy (see Figure 6 11) 1 atom pit 3 atom pit 7 atom pit dz ( ) 1.80 1.68 0.15 dz 1 ( ) 2.32 2.27 2.25 dz 2 ( ) 2.34 2.35 2.36 dz 3 ( ) 2.35 2.35 2.35 dz 4 ( ) 2.35 2.35 2.35 1 ( ) 0.04 0.05 0.10 2 ( ) 0.09 0.11 0.03 3 ( ) 0.01 0.01 0.03 4 ( ) 0.01 0.01 0.04 5 ( ) 0.01 0.01 0.01 xy 1 ( ) 0.04 0.18 0.02 xy 2 ( ) 0.05 0.06 0.03 Adsorption Energy (eV) 2.07 2.33 2.56 In conclusion, our DFT calculations on the strong observed at room temperature show that the molecules are trapped by sevenatom pits with adsorption energy 2.56 eV This value is higher than the energy for a C60 sitting on a defect free Au(111), in a good agreement with the experiment.

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118 CHAPTER 7 SYMMETRY BREAKING INDUCED BY EPITAXIAL STRAIN IN BIMNO3 THIN FILM Multiferroics normally refers to materials exhibiting the coexistence of magnetic and electric orders .183 Due to both potential device applications and the physical ess ence of this unique property, extensive experimental and theoretical effort has been drawn in multiferroics. The ferromagnetic and ferroelectric BiMnO3 (BMO) is one of the most fundamental members in the multiferroic family. Bulk BMO has been determined to be FM and FE by early experiments .184 The saturated magnetization of bulk BMO is 3.6 B at 5K and the electric polarization is observed as 62 nC/cm2 at 87K. Density functional studies showed that the ferroelectricity is caused by the displacement of Bi atoms from centrosymmetric positions, due to the 6s2 electron lone pairs185. This theory contradict s recent experiments, in which C2/c symmetry is observed by both electron and neutron diffraction186. The inversion symmetry is further supported by DFT with Hubbardlike onsite interaction correction.187 BMO thin film has been grown on SrTiO3 (STO), introducing a 0.77% strain.188191 The compressive strain greatly raise s the electric polarization (16 C/cm2 by Son et al .191 and 23 C / cm2 by Jeen et al190), yet reduces the magnetic moment s ( 2. 2 B/Mn by Eerenstein et al.188 and 1 B/Mn by Jeen et al .190). The strain effect on BMO thin film was studied by Hatt et al using density functional theory They found that the C2/c symmetry remains up through as high as 4% strain, and the calculated electric polarization is at most 0.1 C/cm2.192 The origin of the ferroelectricity in BMO, both bulk and epitaxial s trained thin film therefore remains in controversy T o understand the ferroelectricity a hidden longrange AFM interaction has been suggested to induce C2/c symmetry breaking.193, 194 In fact slight frustration in orbital ordering195 (OO) exists even

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119 in BMO bulk.196 Such frustration could be enhanced in the strained thin film, possibly alter ing the FM ground stat e.197 T he fact that the saturated magnetic moment decreases in thin film implies the existence of AFM ordering. In addtion, the ordered oxygen vacancy ha s been proposed to understand the ferroelectric behavior .198, 199 In this chapter we study the strain effect on BMO thin film as well as the magnetic properties, using DFT Note that Hatt et al. only considered FM order .192 We f ind that inversion symmetry break ing can be caused by the emergence of AFM order in a strain ed thin film Due to the limited experimental data of the structure of the BMO thin film, only C2/c symmetry is considered in our calculation. 7 1 Method and Computational Details Our calculations utilized the DFT+U method28 within LDA31 for structural optimization and the HSE06 hybrid functional36 for more refined energy and polarization calculation s, which is implemented in planewave based VASP package84, 85. For Mn 3d electrons, we used U =6 eV and J=0.8 eV A 6 6 4 k point mesh107 and 500eV energy cut off were used. Thresholds for self consistent calculation and structure optimization were set as 105 eV and 0.02 eV/, respectively. The polarizability calculations were based on the Berry phase method57, 58 as described in Section 2.5. 7 2 Results and Discussions Fig ure 7 1 a shows the conventional monoclinic unit cell with C2/c symmetry of bulk BMO. Its distorted perovskitetype 40atom unit cell is determined experimentally to have a=9.54 b=5.61 c=9.86 186. Experimentally the (111) plane of the monoclinic unit cell (also refers to pseudocubic (001)) is oriented to the SrTiO3 (001) surface, causing a compressive 0.77% strain190. For simplicity, we use

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120 the 20atom primitive cell instead, as shown in Figure 7 1b. The axis of the primitive cell ( indicated by subscript 0) can be linked to the monoclinic unit cell by 0 0b a a 0 0a b b and 0c c (7 1) Thus the (011) plane of the primitive cell (denoted as (011)pr) is then the (001)ps plane. We first fully optimized the primitive cell using LDA+U. We introduced the epitax ial strain onto (011)pr, following the method described in Hatts paper .192 The b0 and c0 are altered according to the strain, yet the volume and b0/c0 ratio are conserved. The a0 axis is varied in the direction perpendicular to the (011)pr plane to keep the volume. Figure 7 1 (a) Monoclinic 40atom unit cell of BMO. (b ) 20atom primitive cell of BMO. Bi atoms are in grey. Mn atoms are large blue (dark) spheres. O atoms are small red (dark) spheres. One primitive cell has four Mn atoms numbered as in Figure 7 1 b Besides the ferromagnetic configuration (FM), magnetic orders can possibly be antiferromagnetic, including (up, up, down, down) (AFM_1), (up, down, up, down) (AFM_2), and (up, down, down, up) (AFM_3), as well as ferrimagnetic, including (up, down, down, down) (Ferri_1), (down, up, down, down) (Ferri_2), (down, down, up, down) (Ferri_3) and (down, down, down, up) (Ferri_4). Note that AFM_1 is the G type AFM order. We need (a) ( b )

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121 to stress that the other AFM order s, AFM_2 and AFM_3, are similar to the E type AFM order which is observed in TbMnO3.200, 201 E type AFM features zigzag chains of the parallel spins in one plane, while AFM_2 and AFM_3 have zigzag chains in both inplane and out of plane directions. This is due to its highly distorted and incommensurate structure. For the same reason, C type and A type AFM configurations cannot be considered in this case. After optimization of internal coordinates by LDA+U, energies of each spin configuration wer e then refined with the HSE06 hybrid functional. We f ound that AFM_2 and AFM_3 are energetically degenerate, which can be understood by symmetry, so we only pr esent results for AFM_2 in the rest of the chap t er. The AFM_2 and AFM_3 orderings are energetically more stable than AFM_1 (G type). The Ferri_2 is the most stable among four ferr i magnetic configurations for all strains we have investigated. LDA+U calculat ions show that FM is the ground state from 0 to 4 percent of strain. Under 5 % strain, Ferri_3 is more stable than FM, by 5 meV per formula unit (shown in Figure 7 2 a). In addition, different U values in the LDA+U calculation (from 3 eV to 8 eV) do not change the prediction of the ground state under strain. However, the HSE06 hybrid functional predicts a different situation (shown in Figure 7 2 b ). According to the HSE06 calculations, AFM_2 state is more stable than FM and Ferri_2, in the range of 1.9% t o 2.7% compressive strain, while FM is the ground state under 0% to 1.9% strain. For higher strain, from 2.7% to 5%, Ferri_3 is more stable than the other two. Since the HSE06 hybrid functional has been proven to be more accurate at describing energeti c information,202 it is reasonable to believe the emergence of the AFM_2 state would occur under about 2% compressive strain.

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122 Figure 72. The energies per primitive cell for FM, AFM_2, and Ferri_2 magnetic configurations. (a) is the LDA+U result and (b) is from the HSE06 hybrid functional. Only the most stable AFM and Ferri orders are shown. It is interesting to discuss the inversion symmetry for different spin configurations. FM always keeps inversion symmetry, so we cannot expect inversion symmetry break ing when FM is the ground state. AFM_1 is G type AFM, which is also centrosymmetric, while AFM_2 breaks inversion symmetry. The lowest ferrimagnetic configuration, Ferri_2, is still centrosymmetric. Therefore, the origin of t he ferroelectric behavior of strained BMO can be explained by the switching of the magnetic order under compressive strain, from FM to AFM_2. Under appropriate strain ( between 1.9% and 2.7% ), the noncentrosymmetric spin configuration of the ground state induces structural centrosymmetry break ing leading to the emergence of ferroelectricity The nonuniformed strain explanation thus is supported by our calculation. Fig ure 7 3a shows the pattern of atomic displacements within a primitive cell under the 2.2% compressive strain, when AFM_2 is the ground state. Apparently the displacements are noncentrosymmetric. Fig. 7 3 b shows a fractal of the (001)ps plane of BMO under 2.2% compressive strain. The MnO Mn angles are in the range of 146 to 155, indicat ing a strong octahedral tilting. Spin density of the states within 2 eV (a) ( b )

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123 below Fermi level is also plotted. Compared to the spin density of the FM ground state in bulk form (not shown here), orientations of Mn eg orbital s remain about the same, except the o ne at Mn3 atom The orientation is initially the same as Mn4 atom in bulk (in plane), but switches to out of plane under strain. It is also elongated in the Mn2 Mn3 direction. In addition, such a patter does not change if a 40atom unit cell is used. We then calculated the electric polarization for all the spin configurations, using the Berry phase method combined with the HSE06 functional. The FM state shows quite low polarization under all strains, agreeing with Hatt et al. With the Ferri_2 configuration, BMO also shows nearly zero polarization for all strains. In contrast, the AFM_2 configuration shows higher electric polarization. Combined with the previous energy calculation, we plot the electric polarization of the ground states versus strain in Figure 7 3 c. Clearly, the BMO system with the AFM ground state has the electric polarization, which is comparable to experiment results. One may wonder that the strain created by STO may not be enough to switch the ground state from FM to AFM_2. Though STO only creates compressive 0.77% strain, the strain may not be evenly distributed over the BMO thin film. Thus, the local strain is still possibly higher than 0.77%, which would allow the FM to AFM_2 transition to happen locally. Overall, the electric dipole could still be observed and a lower magnetic moment per Mn atom would appear. We need to stress that we do not exclude the possibility of other magnetic orderings, for example, longrange order and spin waves. Both longrange AFM orders and spin waves are much more complicated and expensive to treat, especially for the HSE06 calcula tion. However, we have proven that FM is not the ground state. Inversion

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124 symmetry would be broken under certain strain, and the electric polarization is created. It is also possible that strain can induce a structural phase transition, which could destroy the inversion symmetry, however no experiment has observed such a behavior. Figure 73. (a) The atomic displacements of a pr imitive cell under 2.2% strain. (b) Isosurface of spindensity of the states within 2 eV below Fermi surface of BMO thin film under 2.2% strain. The isovalue is 0.17 e/ 3. Blue refers to spin up and Red refers to spindown. ( c) The calculated polarization of the ground state corresponding to the strain. (a) ( b ) (c)

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125 7.3 Summary and Conclusions This chapter reports the study of the magnetic and ferroelectric properties of epitaxially strained BMO thin film using DFT with LDA+U and a hybrid functional. The phase transition of magnetic order from FM to AFM around a certain range of strain ( 1.9% to 2.7%) is found to be a reasonable explanantion of the enhancement of the electric polarization in strained BMO thin film. Inversion symmetr y, which is incompatible with the ferroelectricity, breaks during the phase transition. The uneven strain model can justify the possibility that the phase transition happens on STO substrate. Our theory also explains the reduction of magnetization of Mn atoms. We suggest that more experiments are needed for further understanding of the structure and magnetic propert ies in the BMO thin film.

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126 CHAPTER 8 FIRST PRINCIPLES STUDIES O F TA2O5 POLYMORPHS4 Tantalum pentoxide (Ta2O5)204 or tantala, is a potential alternative to SiO2 because of its high breakdown voltage, high dielectric constant ,205 and excellent step coverage characteristics .206 In addition to its applications as dielectric films207 this material also has been used experimentally for optical coatings208 and corrosion coatings .209 The inspiration for the work in this chap t er arises from the need for optical coatings in ultraprecision measurements. In ongoing experiments conducted by the laser interferometer gravitational observatory (LIGO), alternating layers of SiO2 and Ta2O5 are used as coatings for the test masses in gravitational wave detectors. It is predicted that the limiting noise in wave detection will be the thermal noise that is closely related to mechanical dissipation and properties of the coating materials .210, 211 For the SiO2/Ta2O5 coatings studied, the mechanical dissipation appears to be associated with the Ta2O5 component of the coatings .212 In addition, the film elastic properties of SiO2/Ta2O5 coatings can significantly influence the expected level of coating thermal noise.210 The structural and mechanical properties of Ta2O5 films are thus of considerable importance. Tantala has many crystalline forms, including those which are oxygenrich and o xygen deficient. A phase transformation in pure bulk Ta2O5 at ~ 1360 C was reported213 long ago, but the nature of the structure for each phase continued to stimulate researc h activity. At low temperature, various polymorphs have been proposed because of the difficulty in growing single crystal s of the lowtemperature form Ta2O5 (L Ta2O5) using conventional hightemperature techniques. A variety of LTa2O5 structures can be st abilized by adding certain amounts of other oxides .214220 The first 4 This work has been published in Physical Review B .203

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127 one was reported by Stephenson et al .219 in 1971. The group used the X ray powder diffraction method and proposed a crystal structure with an orthorhombic unit cell that consists of 22 Ta and 55 O atoms with a large number of oxygen vacancies. Those were then identified as the main cause for a large leakage current .221 In a later study, a vacancy free LTa2O5 orthorhombic structure ( Ta2O5) which contains only 4 tantalum and 10 oxygen atoms in the unit cell was observed via x ray diffraction,214 and verified by calculations using DFT with both LDA and GGA222 functionals. Besides Ta2O5, experimenters also had reported a hexagonal structure for the low temperature phase of Ta2O5 ( Ta2O5) .223 Fukumoto et al.224 studied the crystal structure of hexagonal Ta2O5, which has the space group of P6/mmm using firstprinciples ultrasoft pseudopotential calculations. Interestingly, a very recent first principles study225 Ta2O5 and Ta2O5 show some instability with lar ge super cells used to optimize the structure. For the hightemperature form of Ta2O5 (H Ta2O5), orthorhombic, tetragonal and monoclinic models have been proposed.220, 226, 227 Similar to L Ta2O5, single crystal H Ta2O5 can be grown with the help of other oxides. Two types of modulation have been proposed for TiO2stabilized H Ta2O5. One was determined by Liu et al.228 in 2006 using the conventional solidstate reaction method and advanced laser irradiation technique to hold the pure H Ta2O5 structure at room temperature. They identified the tetragonal structure of pure Ta2O5 with the space group I41/amd usi ng transmission electron microscopy (TEM). For the second high temperature tantala, Makovec et al.229 constructed a complicated structural model from analysis of electron diffraction data and high resolution electron microscopy (HREM) electron images of the solid solutions in the

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128 series (1x)Ta2O5xTiO2 with x=0.0 0.1. The proposed monoclinic crystalline structure is based on edge sharing of an oxygen octahedronh exagonal bi pyramid octahedron molecule building block. This unit block repeats 4 times in a unit cell of high temperature of Ta2O5. So far, experimental information is mostly limited to structure determination. Crystal structure specific mechanical and optical properties are yet to be determined. Compared to the experimental work on Ta2O5 mentioned above, theoretical effort, in particular for the hightemperature phases of Ta2O5 is even further far behind. We will see later that even the low temperature phases are not well understood. In this chapter we present our theoretical results on tantala starting with the two relatively simple, low temperature Ta2O5 polymorphs and followed by a high temperature structure and a complicated low temperature structure. Both have partial oxygen occupation issues that have been studied using the virtual crystal approximation (VCA) .230 In addition, we present our results for a model amorphous structure. The rest of the chapter is org anized as follows: In the second section, w e discuss theoretical methods. In the third section, we present our results on structures ( 8.3.1), cohesive energy and density of states ( 8.3.2 ) and elastic moduli ( 8.3.3) and finally, in the forth section we discuss and conclude our investigations. 8. 1 Me thod and Computational Details As before, all calculation were done with plane wave PAW53, 83 and VASP package. 83 84, 229 The exchange and correlation potentials were calculated using the GGA with PW 91 parameterization41. The energy cutoff was 520 eV for wave functions. Gaussian smearing of 0.1 eV was used for Fermi surface broadening. Surface Brillouin zone integrations were performed on a Monkhorst Pack107 k mesh of 8 8 8 in the

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129 Methfessel Paxton scheme231 for two low temperature crystal structures of Ta2O5, as well as 8 8 1 for the hightemperature structure of Ta2O5. With these k meshes, the total energies are converged to 0.01 eV. The equilibrium lattice constants for different Ta2O5 structures were obtained through total energy minimization. The force tolerance for geometry relaxation was 0.02 eV/. For electronic properties, the HSE06 hybrid functional232, 233 also was used to calculate the density of states and for comparison with results obt ained from PW91. To simulate systems with partially occupied sites, we constructed pseudopotentials via the virtual crystal approximation.230 In VCA, partially occupied sites are assumed to be fully occupied by pseudoatoms with the pseudopotential modified by occupancy. Without VCA, many possible configurations can be generated and hence a large supercell is needed to investigate the system. In the equilibrium state, the cohesive energy is calculated as O Ta O O Ta Ta total cn n E n E n E E / ( 8 1) where Etotal is the total energy per cell of a configuration. nA specifies the number of a particular element (A= Ta, O, Si) in that unit cell, and EA (A= Ta, O, Si) is the energy of an individual atom calculated with spindependent DFT. Elastic properties can be obtained by calculating the elastic tensor ijc that relates the stress with the strain for a given system234, that is, 6, 1 j j ij ic ( 8 2) where are stress components, are small strains. We used the ab initio calculation to relate stress response to strains E lastic constants can be obtained by solving a set

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130 of linear equations of ijc .235 For all crystalline systems we studied, 1.5% strain is added in six patterns To obtain sufficiently accurate elastic constants, we adopted stricter convergence and relaxation criteria than the ones used for total energy determination. The force tolerance in relaxation in this study is set to 0.01 eV/, and the stress values vary less than 0.02 kBar. The number of independent elastic constants varies depending on crystalline symmetry. In the case of orthorhombic Ta2O5, nine independent elastic constants are involved according to Ravindran et al .236 Both the polycrystalline bulk modulus (B) and shear modulus (G) can be determined based on cij. Two methods, the Voigt approximation237 (index V) and the Reuss approximation238 (index R), were proposed. Hill239 proved that these two methods correspond to upper and lower bounds on the true polycrystalline elastic constants. An approximation can be made by averaging the Voigt and Reuss method results, v RG G G 2 1 and v RB B B 2 1 ( 8 3) The average Youngs modulus Y and Poissons ratio can be determined by G B BG Y 3 9 and G B G B 3 2 2 3 ( 8 4) Reuss moduli (GR and BR) and Voigt moduli (GV and BV) are written as236, ) ( 2 ) ( 123 13 12 33 22 11s s s s s s BR ( 8 5) 23 13 12 33 22 119 2 9 1 c c c c c c BV ( 8 6) 66 55 44 23 13 12 33 22 113 4 4 15 s s s s s s s s s GR ( 8 7)

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131 66 55 44 23 13 12 33 22 115 1 15 1 c c c c c c c c c GV ( 8 8) where ijs are the elastic compliance constants, and s is the inverse of the elastic tensor c. The directional dependence of Youngs modulus in an orthorhombic crystal can be calculated by240 44 2 3 2 2 55 2 3 2 1 66 2 2 2 1 23 2 3 2 1 23 2 3 2 2 12 2 2 2 1 33 4 3 22 4 2 11 4 12 2 2 1 s l l s l l s l l s l l s l l s l l s l s l s l Y ( 8 9) where il are direction cosines to x, y, z, and ijs are the elastic compliance constants. Youngs modulus along axis i then becomes ii is Y 1 ( 8 10) where i can be 1,2,3. For some systems, we also computed directly the bulk modulus according to the second derivative of energy as, 2 2 0 dV E d V B ( 8 11) where V0 is the volume in the equilibrium state, and d2E/dV2 denotes the second derivative of energy with respect to volume. Youngs modulus and Poissons ratio are 2 2 0 0dL E d A L Y ( 8 12) dL dA A L0 02 1 ( 8 13)

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132 where L0 is the original length of the material. A0 is the original cross sectional area though which the force is applied. d2E/dL2 denotes the second derivative of the energy with respect to length. Comparison of the direct approach and approximated approach are made for some (001) directions to confirm our results. The primary reason for performing the direct calculations is check the validity of the VCA. The values of modulii and Poisson ratio from Equations 8 11 to 813 are different from the ones from Equations 8 2 to 8 10 because of some constraints applied during calculations. For obtaining mechanical properties, one should use Equations 8 2 to 810. 8. 2 Results and Discussions 8. 2 .1 Structures This section present s the four equilibrium structures of low and hightemperature tantala. In addition, an amorphous model also is discussed. 8. 2 .1.1 and L Ta2O5 The two low temperature structures are Ta2O5 and Ta2O5, which crystallize in a hexagonal structure and an orthorhombic structure respectively (see Figure 8 1 ab). Both of them contain 4 tantalum and 10 oxygen atoms per cell, thus the Ta:O ratio is 2:5. The experimentally reported space groups are p6/mmm for the phase and pccm for the phase, respectively However, the optimized structures show large distortions when the symmetry constraint is lifted. This phenomenon was discussed in a recent work225 where large super cells were used. Similarly, negative phonon frequencies were obtained if symmetry was kept, indicating an unstable system. After structure optimizations without the symmetry constraints, these two structures distort further and the phonon frequencies at the Gamma point become positive. The optimized lattice

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133 parameters are listed in Table 8 1. In the hexagonal structure ( Ta2O5), the optimize d lattice parameters are slightly larger than the experimental values, leading to a density 4% less than the experimental value. The results with the symmetry constraint (Table 8 1, numbers in parenthesis) are comparable to the values proposed by Sahu et al222 and the density is 2% smaller compared to experiments. Note that the calculated a and c parameters are larger than those from the experiment, but the c/a ratio from both calculations remains the same as the experimental ratio at 0.535. The fully optimized orthorhombic structure ( Ta2O5) has larger lattice parameters a and b but a smaller c, compared to the experiments223. Calculations with symmetry constraints show better agreement with experimental data. In particular, the calculated c/a ratio agrees well, again, with the experimental value 1.25. Densities from our calculation are 4% lower than the measured values, the same as in Ta2O5. Our results do show that Ta2O5 has higher density than Ta2O5, which is in agreement with experiment. Tables 8 1a and 1b provide detailed descriptions of TaO bond length and TaO Ta bond angle before and after distortion which are also depicted in Figure 8 2 a b. However, one of the four basis oxygen atom sites in the unit cell has a fractional occupancy of 75%, which makes the actual total number of oxygen atoms 30. As a result, The O/Ta ratio is still 2.5. The positions of the two missing oxygen atoms are random. To address the partial occupancy issue, we performed a number of calculations with different approaches including using an oxygenrich crystal, the virtual crystal approximation (VCA)230, and various combinations of two selected vacancy sites.

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134 Figure 81. Panel (a): Structure of phase Ta2O5. Panel (b): Structure of phase Ta2O5. The grey (light) atoms are Ta and red (dark) ones are oxygen. Yellow (light small) atoms are distorted Ta atoms and blue (dark small) ones are distorted oxygen atoms. Distorted atoms are labeled with primes. Panel (c): Structure of tetragonal phase Ta2O5. The blue (grey) atoms are partially occupied oxygen sites. Panel (d): Structure of low temperature phase 77atom Ta2O5. Blue (grey large) are oxygen atoms with 75% occupancy and yellow (grey small) atoms are oxygen atoms with 25% occupancy. (a) (b) (c) (d)

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135 Table 81. The optimized lattice constants, a, b, c Numbers in parenthesis are calculated with symmetry constraints. All lengths are in unit of a () b () c () Density (g/cm 3 ) L Ta 2 O 5 phase Expt. 7.25 3.88 8.37 Sahu et al. 7.12 3.80 8.86 Ours 7.39 (7.31) 3.90 (3.91) 8.01 (8.17) L Ta 2 O 5 phase Expt. 6.22 3.68 7.79 8.29 Sahu et al. 6.19 3.63 7.66 8.59 Ours 6.38 (6.30) 3.82 (3.72) 7.58 (7.89) 7.99 (7.99) H Ta 2 O 5 Liu et al. 3.86 36.18 8.23 O rich 3.91 35.82 8.20 with VCA 3.85 37.45 7.95 Defect_1 3.85 3.96 36.23 8.02 Defect_2 3.85 3.96 36.21 8.02 77 atom L Ta 2 O 5 Expt. 6.20 40.29 3.89 8.37 Ours 6.74 40.37 3.84 7.79 Amorphous 11.92 12.88 13.14 8.79 with VCA 11.61 11.96 12.29 6.82 Note that we fully expect explicit vacancy models to fail completely for electronic structure characterization. However, for mechanical properties, they can provide a reference point to check the VCA results. In it, a 0.75 factored modification of the oxygen pseudopotential was made and applied to 8 oxygen sites where vacancy can possibly occur, so that the number of oxygen atoms is effectively 30. In Table 8 1, the crystal without oxygen vacancies is denoted as O rich while two with two specific oxygen vacancies are denoted as defect_1 and defect_2, which are typical ones among 28 different combinations of vacant sites. We have fully relaxed the lattice and internal atomic positions for all four tetragonal models. Unl ike the Ltantala, this H tantala crystal does not show visible distortion from the experimentally proposed structure when oxygen sites are either 75% or 100% occupied.

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136 Table 8 2 Comparison of bond lengths and angles of symmetric and optimized phase structure. All lengths are in unit of and angles in degree. The optimized geometry parameters are shown in Table 1 and structure in Figure 8 1 c. Both tetragonal structures, one with 32 O and the other with 30O effectively, maintain the tetragonal unit cell, whereas the two with explicit vacancies do not. For Tetragonal_32O the optimized a and c are 3.91 and 35.82 respectively, which leads to a density of 8.20 g/cm3. The internal atomic coordinates do not show substantial deviation from experimental data. When VCA was used, the optimized lattice parameter a (3.85 ) is very close to the experiment result (3.86 ), but c is elongated, which lea ds to a density 2.9% smaller than the measured one. Systems with two randomly selected vacancies also show lower densities than nonvacancy O rich model according to our calculation. This trend indicates that VCA is not the reason of the discrepancy between calculations and experiments. Symmetric Distorted Ta1 O2 1.96 1.99 Ta2 O1 1.99 1.95 Ta2 O6 1.99 1.94 Bond Length Ta2 O4 1.96 1.94 Ta3 O7 1.99 2.23 Ta4 O5 1.99 2.10 Ta4 O9 1.99 1.90 Ta2 O1 Ta3 132.8 121.4 Ta2 O6 Ta4 132.8 157.9 Ta1 O2 Ta1 122.2 145.3 Angle Ta3 O5 Ta4 180.0 121.9 Ta3 O9 Ta4 132.8 160.8 Ta3 O7 Ta2 132.8 157.2 Ta2 O4 Ta2 180.0 164.9

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137 Table 8 3 Comparison of bond lengths and angles of symmetric and optimized phase structure. All lengths are in unit of and angles in degree. Symmetric Distorted Ta1 O5 1.84 1.93 Ta2 O1 2.14 2.19 Ta2 O2 1.98 2.19 Bond Length Ta3 O7 1.94 2.18 Ta3 O8 1.96 2.19 Ta4 O6 1.93 1.85 Ta4 O9 1.93 1.95 Ta1 O5 Ta1 180.0 138.7 Ta2 O1 Ta1 122.2 112.7 Ta2 O2 Ta1 122.2 177.5 Angle Ta2 O3 Ta3 180.0 159.2 Ta4 O6 Ta4 180.0 138.7 Ta3 O7 Ta4 122.5 107.8 Ta3 O8 Ta4 122.5 180.0 Ta1 O9 Ta4 180.0 158.7 8. 2 1 .3 77atom LTa2O5 With the capability of VCA and reasonable test results from H tantala, it is computationally feasible to study the W2O5stablized low temperature Ta2O5 phase proposed by Stephenson and Roth219. In the unit cell, there are 22 Ta sites and 62 O sites. However, due to oxygen vacancies, 4 O sites have 75% occupancy and 8 O sites have 25% occupancy, which effectively makes 55 O atoms in a unit cell and keeps the Ta:O ratio as 2:5. Experimentally, the unit cell is determined to be orthorhombic with lattice constants a=6.198 b=40.29 and c=3.888 For the two different occupancies of O vacancy sites, we have constructed two types of VCA potentials, 75% and 25% respectively. After full relaxation without symmetry constraint, the orthorhombic unit cell is maintained but the lattice constants change to a=6.74 b=40.37 and c=3.84 The optimized unit cell is shown in Figure 8 1 d. (different vacancy sites ar e in different colors.) The density decreases from 8.37 g/cm3 to 7.79

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138 g/cm3 after relaxation. Similar to the H Ta2O5, the symmetry of the structure is lower than for the other two LTa2O5. The internal coordinates have also been fully relaxed without imposing any symmetry constraints. 8. 2 1 .4 Model amorphous Ta2O5 The last system we present in this chapt er is a model structure for the amorphous phase of Ta2O5. Because there is no published for ce field, we have not been able to perform a molecular dynamics calculation to obtain amorphous structures. Instead, we used Monte Carlo method to assign positions to Ta and O atoms randomly, with constraints on the range of interatomic distance and coordi nation numbers. We constructed a cubic unit cell, with 48 Ta and 120 O atoms. Full structure relaxation was performed and an orthorhombic structure with lattice constants a=11.92 b=12.88 and c = 13.14 was obtained; see Figure 8 2 a. To examine this amorphous structure, we calculated the pair correlation, which reflects the radial distribution of TaO and TaTa distances for (Figure 8 2 b and 8 c). The pair correlation shows quite good amorphous characteristics, and is in good agreement with experimental data.241 However, the calculated density of this model (8.79 g/cm3) is much higher than for all other models. Therefore, we have to take oxygen vacancies into consideration. According to Cooks et al .211 and Penn et al .,212 the density of Ta2O5 amorphous thin film ranges fr om 74% to 93% of bulk density, depending on different experimental conditions. The decrease in density may come from partial oxygen occupancy (or vacancy) in thin films (similar to the situation in the tetragonal H Ta2O5). To include vacancy effects, we followed the same procedure for constructing amorphous structures, but used an 80% occupied VCA potential for oxygen atoms. In a unit cell of 32 tantalum and 100 oxygen sites, that gives effectively 80 O atoms, were positioned

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139 randomly. After full relaxation, an 11.61 11.96 12.29 unit cell was obtained, with density 6.82 g/cm3. It is 82.3% of the density of phase LTa2O5, within the foregoing range of 74% 93%. (a) (b) (c) Fig ure 8 2 Panel (a): Structure of amorphous Ta2O5. Ta atoms are in grey (light) and oxygen atoms are in red (dark). Panel (b): pair correlation function of TaTa in a 2 2 2 super cell. Panel (c): Pair correlation function of TaO in a 2 2 2 super cell. 8. 2 .2 Energetics and electronic structure We calculated the TaO cohesive energies for the models mentioned above, relative to the energies of the isolated Ta and O atoms. Results are listed in Table 8 4 Cohesive energies of the hexagonal Ta2O5 (7.46 eV) and the orthorhombic Ta2O5

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140 (7.41 eV) are quite similar with only a 50 meV/atom difference, indicating that the two phases are isomeric. Coincidently, the difference between the two phases seems to be the same regardless whether or not one imposes symmetry condition on the crystal during calculations. Our results from symmetry constrained calculations agree with those of Sahu et al For H Ta2O5, tetragonal structure with VCA structure has a cohesive energy 7.21 eV/atom, which is lower than the two L Ta2O5 phases as expected. Similar to the value within VCA, the two structures each with two O vacancies both have a cohesive energy of 7.26 eV/atom, indicating that VCA is a reasonable approximation. Interestingly, the orthorhombic 77atom LTa2O5 calculated using VCA has the second lowest energy, that is, 7.33 eV. Our amorphous model without vacancy shows a 7.23 eV/atom cohesive energy, which is even higher than for the tetragonal VCA model (not a surprise). However, the cohesive energy of the amorphous VCA model is 6.77 eV/atom, which is lower than any other model discussed here. One should not take the exact value of this particular number too seriously, because it merely provides a clue on how the cohesive energy decreases as the systems have part ially occupied oxygen sites, or equivalently, the energy decreases when there are vacancies. Densities of state (DOS) of were calculated, including the phase, phase, tetragonal H Ta2O5 and the 77atom low T model. To obtain a sensible band gap, which i s known to be underestimated by DFT with conventional LDA or GGA, we used the HSE06 hybrid functional to calculate the DOS of the first three systems. Figure 8 3

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141 Table 84 Numbers of oxygen and tantalum atoms, and cohesive energy. Values in parentheses are calculated with symmetry constraints. All lengths are in unit of n Ta n O O/Ta E c (eV/atom) L Ta 2 O 5 phase 4 10 2.5 7.46 (7.20) L Ta 2 O 5 phase 4 10 2.5 7.41 (7.15) O rich H Ta 2 O 5 12 32 2.67 7.18 with VCA 12 30 2.67 7.21 Defect_1 12 30 2.5 7.26 Defect_2 12 30 2.5 7.26 77 atom L Ta 2 O 5 22 55 2.5 7.33 168 atom Amorphous 48 120 2.5 7.23 with VCA 32 80 2.5 6.77 shows the DOS for Land H Ta2O5 structures. As shown in Figure 8 3 a and b, GGA gives a 1.1 eV gap and 0.1 eV gap for phase and phase, respectively, both in a good agreement with Sahu et al .222 However, the HSE06 functional widens the gaps to 2.0 eV and 0.9 eV for the two phases. For tetragonal H tantala, GGA predicts a metallic system whereas HSE06 opens a gap of 0.9 eV, a substantial improvement over the GGA results (Figure 8 3 c). GGA with VC A shows metallic property for tetragonal H tantala, which is the same as the O rich (Figure 8 3 d). Since we currently do not have a VCA potential suitable for use with the HSE06 hybrid functional and construction and test of such potential would constitut e a major separate effort, we do not have DOS information on the 77atom low T systems (Figure 8 3 e). 8. 2 .3 Elastic Moduli In this section, we discuss the mechanical properties of the Ta2O5 models. Our focus is on the elastic moduli, including bulk modul us, Youngs modulus and Poissons ratio. All of the calculations started with the equilibrium states for the Ta2O5 structures. Table 8 5 presents elastic tensors and derived mechanical functions using Equations 8 3 to 813. Because of symmetry, the number of independent elements varies from

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142 system to system. In this case, we remove symmetry constraints and allow all six distortions for every system. Based on the elastic tensors thus obtained, the symmetry is basically maintained in our calculation, though s ome small deviations exist. For example, c11 and c22 are different by 5 GPa for O rich HTa2O5. All elements of elastic tensor are taken into calculation of elastic constants, which effectively is an average of those elements corresponding to symmetry. Our calculations show that the lower and upper limit of the shear modulus, GR and GV from all systems range 4997 GPa, and those of bulk moduli range 105221 GPa, Youngs moduli 152233 GPa, and Poissons ratio 0.250.34. We take this as an indication that these crystals have moderately similar mechanical properties when averaged over all directions. It can be seen that the H Ta2O5 (both for the O rich and VCA model) has higher elastic moduli and Poissons ratio than the three LTa2O5 structures, indicating t hat H Ta2O5 is stiffer. The virtual crystal approximation makes tetragonal H Ta2O5 have a higher shear modulus and Youngs modulus than the O rich model, but lower bulk modulus. Among the three LTa2O5 crystalline structures, hexagonal Ta2O5 shows lower moduli and Possions ratio than the other two. The last three rows in Table 8 5 are Youngs moduli along the x, y, and z directions, respectively. We will compare Yz with the direct approach below. 8. 2 .4 Technical remarks: VCA and direct approach In the direct approach, the conventional way to compute the bulk modulus using firstprinciples calculations is via Equation 8 11. The volume change for each Ta2O5 structure discussed above was achieved by changing the lattice constants. For the two L Ta2O5 systems with small unit cells, all of the atoms were allowed to relax in the cells with the selected volumes, while for all large systems, the internal coordinates were

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143 Figure 8 3. Panel (a): DOS of Ta2O5. Panel (b): DOS of Ta2O5. Panel (c): DOS of tetragonal H Ta2O5. Panel (d): DOS of tetragonal H Ta2O5 with VCA. Panel (e): DOS of low temperature 77 atoms. GGA results are in blue and HSE06 results are in red. Green l ines indicate the Fermi level, which is adjusted to be zero. (a) (c) (b) (d) (e)

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144 Table 85 Elastic constants, Bulk moduli, Shear moduli, Youngs moduli and Poissons ratio of different phases of Ta2O5. Yx, Yy and Yz are Youngs modulus along a, b and c axis. Orthorhombic Ta2O5 (GPa) Hexagonal Ta2O5 (GPa) Tetr agonal O rich (GPa) Tetragonal_VCA (GPa) Orthorhombic 77 atoms L Ta 2 O 5 (GPa) c 11 290 175 379 445 224 c 12 126 104 173 67 106 c 13 52 169 131 130 44 c 22 263 167 374 443 256 c 23 115 3 130 132 42 c 33 428 403 366 326 430 c 44 78 47 65 64 51 c 55 39 46 65 64 54 c 66 52 57 62 62 36 G R 64 49 77 81 57 G V 80 72 84 97 76 B R 171 105 221 207 140 B V 174 110 221 208 144 Y 189 152 215 233 172 0.32 0.26 0.34 0.34 0.30 Y x 229 108 283 392 179 Y y 188 97 280 389 204 Y z 378 398 304 259 418 scaled proportionally to the changes in lattice constants. The O rich H Ta2O5 is found to have the largest bulk modulus, 246 GPa (Table 8 6 ). All other vacancy doped tetragonal H Ta2O5 models, including defect_1, defect_2 and VCA, are calculated to be 217227 GPa. The both moduli of two LTa2O5 structures are nearly the same values (228 GPa and 223 GPa), which are quite close to those reported by Sahu et al. .222 However, they are quite different from the values obtained from the elastic tensor, especially in the case of the phase (108 GPa). A possible explanation for the discrepancy is that relaxation of shape could lower the bulk modulus, whereas we do not let the unit cell shape relax in our calculation. The bulk modulus of the amorphous model is 234 GPa and the lowest bulk modulus is 163 GPa for the amorphous VCA model.

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145 Direct calculation of Youngs modulus involves much more effort compared to that for the bulk modulus. We focus only on the (001) direction for all systems for comparison with values listed in Table 8 5 (Yz). To determine the ratio of tensile stress to tensile strain, we helongated and compressed the lattice parameter c. For all systems, Young's moduli were calculated statically. Our calculation shows that high Youngs modulus (667 GPa for phase and 606 GPa for phase) is a characteristic of the two LTa2O5 crystals. These values are much higher than the Yz (398 GPa for phase and 378 GPa for phase), which is caused by relaxation of internal coordinates. The high Y values suggest that (001) is a very stiff direction, which is consistent with results in Table 8 5 The H Ta2O5 structures show relatively lower Young's moduli than L Ta2O5 and agree well with Table 8 5 A range of values has been obtained for tetragonal H Ta2O5 models. The O rich system has a value of 430 GPa, while the two specific vacancydoped models have 360 GPa and 380 GPa, respectively Young's modulus H Ta2O5 with VCA is obtained as 345 GPa. The calculated Youngs modulus of the amorphous phase is about 320 GPa, which is much higher than the experimentally measured 140 GPa (No amorphous phase was calcuated in Table 8 5 ). Our results su ggest that the amorphous VCA model may be useful for systematically approaching experimental value. Based on our calculation, oxygen vacancies and disorder in crystals cause the materials to soften. Finally, Poissons ratios, the ratio of transverse contraction strain to axial extension strain, of these systems, was computed. As seen in Table 8 2, the values of Poissons

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146 Table 86 Bulk moduli, Youngs moduli and Poissons ratio of different phases of Ta2O5. Y are Youngs modulus along c axis, calculated via direct method for validating VCA. B (GPa) Y (GPa) Exp. (amorphous thin film) 140 0.23 O rich H Ta 2 O 5 246 430 0.23 with VCA 224 345 0.22 Defect_1 217 360 0.23 Defect_2 227 380 0.23 168 atom Amorphous 234 320 0.27 with VCA 163 263 0.20 ratios in the (001) direction range from 0.09 to 0.23 in the low and hightemperature crystalline phases, which are lower compared to poly crystalline averaged values. Poissons ratio of amorphous system is determined as 0.27. Note that the Possions rati os calculated directly are in the z direction, so it can be possible that the value is lower than the values in Table 8 5 The amorphous system with oxygen defects described by VCA structure has a Poissons ration of 0.20. The experimental value of 0.23 li es between 0.20 and 0.27, suggesting that fine tuning the amorphous VCA model can possibly improve the agreement between theory and experiment. Since results from VCA and explicit defect models yield similar mechanical properties as listed in Table 8 6 the VCA seems to be a good approach for modeling vacancies. However, the values from the direct approach are quite different from the ones obtained by using Equation 8 2 to 810. The reason is the constraints applied during calculations For example, uni form lattice scaling in all directions is used for bulk modulus calculations, and fixed angles (among three lattice vectors) are assumed when loading the strain in one direction for calculating Youngs modulus. These constraints lead to inaccurate values f or systems as sensitive as Ta2O5.

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147 8. 3 Summary and Conc lusion In conclusion, the calculated mechanical properties of tantalum peroxide systems show that the Ta2O5 crystalline polymorphs and its derivatives have similar mechanical properties but are highly anisotropic. The (001) direction of the hightemperature phase is rather soft compared to all three low temperature phases, but the (100) and (010) directions are stiffer. All L Ta2O5 phases are energetically more stable than the H Ta2O5 and the amorphous phase. In systems such as H Ta2O5, 77atom orthorhombic LTa2O5 and the amorphous structure, oxygen vacancies (or partial occupancy), soften the materials and lower the cohesive energy. Mass density and Youngs modulus also decrease when the systems have ox ygen vacancies. By comparison with experimental data, we conclude that the amorphous structure with oxygen vacancy is a good structure model for investigation of a Ta2O5 thin film but a thorough investigation is needed in future studies. We also suggest that more experimental measurements should be carried out to understand various crystalline phases.

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148 CHAPTER 9 CONCLUSIONS We have employed DFT and the PBE and HSE06 functional s to study the adsorption of N2 and O2 molecules and Agn + cluster cations. The HSE06 hybrid functional is able to predict the right trends of 2D to 3D transition, while PBE fails. For adsorption of nitrogen molecules, polarization dominates the process resulting in exclusively physisorption states. We have found that the physisorption to sites at obtuse corners is stronger than to those at acute ones, which can be explained by the difference of their distance to the charge center. For oxygen molecules, a transition from physisorption to molecular chemisorption oc curs with n >3, which is accompanied by charge transfer from cations to the O2 and O2 bond elongation. We have also used Ag4 +O2 as an exampl e to understand cooperative nitrogen and oxygen coadsorption. First, the N2 adsorption energy of some sites far from O2 increases when there is an O2 adsorbed on a Agn + without inducing extra charge transfer to O2. S econd, when sites near O2 are also occupied, the charge transfer to O2 and the O2 bond length increases but the adsorption energy stay comparable to pure N2 adsorption. T hird, an N2induced extra charge transfer results in enhanced binding for the second adsorption shell. To understand the enhancement of Ag nanofractal fragmentation by chlorine pollution, we have performed a detailed density functional study of (Ag55)2 dumbbell structures with and without chloridization, as well as surface diffusion of Agn and AgnClm (n = 1 to 4) clusters on Ag(111) surfaces. For a single Ag55 cluster, surface adsorption of Cl atoms tends to loosen up the first two layers on t he surface. We have show n that the binding energy between two Ag55 icosa hedrons in the dumbbell structure is reduced by 17% with surface coverage of chlorine (19 Cl atoms on each Ag55). We also

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149 demonstrated that AgnClm clusters are generally more mobile than Agn clust ers on Ag(111) surfaces for n= 1 4. The formation energies of AgnClm imply that Ag3Cl3 and Ag4Cl4 are good candidates as basic units for Ag surface diffusion in addition to the monomer A g C The energy barrier calculations indicate that AgCl and Ag4Cl4 have barriers substantially lower than their corresponding pure silver clusters, and Ag3Cl3 has a barrier slightly lower than Ag3. Finally, our investigations have also uncovered diffusion paths of the clusters. W e have demonstrated the concept of molecular magnetocapacitance, in which the quantum part of the capacitance becomes spindependent, and is tunable by an external magnetic field. This molecular magnetocapacitance can be realized using single molecule nanomagnet s and/or other nanostructures that have antiferromagnetic ground states. As a proof of principle, first principles calculation of the nanomagnet [Mn3O(sao)3(O2CMe)(H2O)(py)3] shows that the charging energy of the highspin state is 260 meV lower than that of the low spin state, yielding a 6% difference in capacitance. A magnetic field of s~40T can switch the spin state, thus changing the molecular capacitance. A smaller switching field may be achieved using nanostructures with a larger moment. Molecular m agnetocapacitance could lead to revolutionary device designs, e.g., by exploiting the Coulomb blockade magnetoresistance whereby a small change in capacitance can lead to a huge change in resistance. We have also investigated the electronic and magnetic pr operties of the FenC60 complexes ( n = 1 4 and 15) The amount of charge transfer increases monotonically as the number of Fe atoms increases. At low concentration of Fe doping ( n = 1 or 2), individual Fe atoms diffuse in the interstitial spaces of C60 monolayer and bind to C60.

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150 The binding energy is mainly due to the FeC bonds. When n =3 or more, Fe atoms cluster around the C60 molecule as well as bind to C60 molecules. Large Fe concentration gives rise to higher binding energy in the FenC60 complex Dopin g with Fe atoms results in modifications of the C60 LUMO derived bands, which are enhanced by increasing electron occupation. Analysis of PDOS shows that the charge transfer occurs mainly between the Fe3 d orbitals and the t1u orbitals of C60, indicating t hat the ionic character dominates the FeC60 bonding. A covalent effect also exists in the FeC60 bonding. Furthermore, doping with Fe induces a negative magnetic moment in C60, aligning in anti parallel with Fe. The value of this moment varies roughly within a fact of 2. Heavy Fedoping does not destroy the structure of C60, but it deforms the electronic structure of C60. Unlike alkali metal doped system s, the Fedoped systems do not undergo a metal semiconductor transition. We have studied the strong bonding of C60 molecule on Au(111) surface at room temperature, as well as the weak bonding at low temperature. We calculated the strong at room temperature, i.e., molecules are trapped by a sevenatom pit, with adsorption energy 2.56 eV, which is higher than the energy for a C60 sitting on a oneatom pit, or weak bonding on Au(111) We have also investigated the magnetic and ferroelectric properties of epitaxially strained BMO thin film, using DFT with LDA+U and a hybrid functional. The phase transition of magnetic order from FM to AFM around a certain range of strain ( 1.9% to 2.7%) is found to be a reasonable explanation of the enhancement of the electric polarization in strained BMO thin film. Inversion symmetry, which is incompatible with the ferroelectricity, breaks during the phase transition. The uneven strain model can

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151 justify the possibility that the phase transition happens on STO substrate. Our theory also explains the reduction of magnetization of Mn atoms. We suggest that more experiments are needed for further understanding of the structure and magnetic properties in the BMO thin film. Finally, we have studied the structure, energetics, elastic tensors and mechanical properties of four crystalline forms of Ta2O5 with exact stoichiometry as well as a model amorphous structure. The calculated mechanical properties of tantalum peroxide systems show that the Ta2O5 crystalline polymorphs and its derivatives have similar mechanical properties but are highly anisotr opic. The (001) direction of the hightemperature phase is rather soft compared to all three low temperature phases, but the (100) and (010) directions are stiffer. All LTa2O5 phases are energetically more stable than the H Ta2O5 and the amorphous phase. In systems such as H Ta2O5, 77atom orthorhombic LTa2O5 and the amorphous structure, oxygen vacancies (or partial occupancy), soften the materials and lower the cohesive energy. The m ass density and Youngs modulus also decrease when the systems have oxygen vacancies. By comparison with experimental data, we conclude that the amorphous structure with oxygen vacancy is a good structur al model for investigation of a Ta2O5 thin film but a thorough investigation is needed in future st udies. We also suggest that more experimental measurements should be carried out to understand various crystalline phases.

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166 BIOGRAPHICAL SKETCH Yuning Wu was born in Yihuang, Province of Jiangxi, P R. China. He attended University of Science and Technology of China (USTC) in 2002. During his undergraduate student, he started to get interested in condensed matter physics. In 2006, he got his bachelors degree in physics. In the same year, he became a graduate student in the Department of Physics, University of Florida. In 2007, he joined Prof. Hai Ping Chengs group and started to work in computational condensed matter physics. He received his Ph. D. from the University of Florida in the summer of 2012. During his Ph. D., he has been working on structure, energetics, electronic structures of various systems, including nanoclusters, molecular magnets, noble metal surfaces, and crystals.