High Frequency Electron Paramagnetic Resonance Studies of the Anisotropy of Molecular Magnets and a Spin Dimer Compound

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High Frequency Electron Paramagnetic Resonance Studies of the Anisotropy of Molecular Magnets and a Spin Dimer Compound
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Koo,Changhyun
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Doctorate ( Ph.D.)
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University of Florida
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Physics
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Tanner, David B
Committee Co-Chair:
Hill, Stephen O
Committee Members:
Hirschfeld, Peter J
Lee, Yoonseok
Christou, George

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Subjects / Keywords:
anisotropy -- dimer -- electron -- epr -- magnet -- molecule -- paramagnet -- phase -- quantum -- resonance -- single -- smm -- spin -- transition
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Abstract:
This dissertation presents the experiments and simulation/calculation to investigate the magnetic anisotropy in molecular magnets and spin dimer compound. High frequency electron paramagnetic resonance (HFEPR) measurements were performed on the samples, and the obtained data were analyzed using spin Hamiltonians. In the results of these experiments, the magnetic properties of the studied materials will be further clarified. Results presented for the antiferromagnetic triangular MnIII3 complex demonstrate that the spin frustration in the complex is relieved. The well isolated spin ground state is confirmed by the EPR data and the simulation using the multi spin Hamiltonian. The small total spin number of the complex causes a big anisotropy value. Moreover, the anisotropy is negative easy-axis type, resulting in a significant anisotropy barrier to the magnetization reversal in the complex. High spin cluster Cu17Mn28 complex, which is geometrically spin frustrated, demonstrates the slow relaxation and hysteresis in the magnetization data. However, the structural symmetry (Td) of this complex doesn?t allow the second order anisotropy. High frequency EPR experiments and simulation for the Cu17Mn28 complex suggest that the complex has fourth order anisotropy following the structural symmetry of the complex. However, the simulation reveals that anisotropy is not strong enough to give rise to an effective barrier to the magnetization reversal. Thus, the slow relaxation and hysteresis of the magnetization observed in the complex is not caused by the magnetic anisotropy, but by the geometric frustration of the core structure. The magnetic anisotropy in the Ba3Mn2O8 compound contributes the magnetic phase of the compound in competition with the exchange interaction. High frequency EPR data of the Ba3Mn2O8 compound determines the sign and magnitude of the single ion anisotropy of MnV ions. The calculation of dipolar couplings between neighboring MnV ions demonstrates that the dipolar couplings contribute to the anisotropy value in the compound.
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by Changhyun Koo.
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Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Tanner, David B.
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Co-adviser: Hill, Stephen O.
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1 HIGH FREQUENCY ELECTRON P ARAMAGNETIC RESONANCE STUDIES OF THE ANISOTROPY OF MOLECULAR MAGNETS AND A SPIN DIMER COMPOUND By CHANGHYUN KOO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PART IAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011

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2 2011 Changhyun Koo

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3 To my family

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4 ACKNOWLEDGMENTS Foremost, I am deeply indebted to my advisor, Prof. Stephen Hill, for his patience, en couragement, and guidance. For the past 5 years, he has provided a world class environment for cutting edge experimental research in physics. He has always been a strong mentor, teaching me how to be a professional scientist, and giving me the opportunity to learn what an independent research should be like. I would like to thank my other committee members, Dr. David Tanner, Dr. Yoonseok Lee, Dr. Peter Hirschfeld, and Dr. George Christou, for overseeing the completion of my research work. I am also grateful to my collaborators, Dr. Enrique del Barco, John Henderson, and Asma Amjad for their valuable suggestions and help for the experiments during this course in PhD. I was always able to learn something new as I worked with them. I want to thank Patrick Feng, and Chris Beedle of the Hendrickson group in the chemistry department at the University of California at San Diego. They supplied great samples of anti ferromagnetic Mn 3 Zn 2 complexes and Cu 17 Mn 28 complexes used in the researche s presented in this dissertat ion. I also wish to thank Eric Samulon from the Fisher group in the applied physics department at the Stanford University. He supplied the single crystals of Ba 3 Mn 2 O 8 spin dimer compound studied in my research. I wish to express my deepest gratitude Marc L ink, Bill Malphurs, Ed Storch in the machine shop in the physics department at the University of Florida. When I built a probe for the experiments, they always provided helpful design suggestions and world class craftsmanship. Without their help and advice none of the research presented in this dissertation would have been possible. I also want to thank Greg Labbe and John Graham working for the cryogenic service in the physics department at the University of

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5 Florida. They kept providing the liquid helium which is critical to the magnetic resonance measurement. Also they gave lot of comments and help for the aspects of cryogenic equipment implemented in our lab. They were vastly knowledgeable and always willing to help with any matter. I thank to my collea gues, Jon Lawrence, Saiti Datta, and Junjie Liu in Hill group. They gave lots of valuable comments, help and discussion for my research. I also acknowledge my fellow students in the physics department at the University of Florida; Hyoung Jeen Jeen, Sung So o Kim, Ramsey Lundock for their help and friendship and especially Jesus Escobar for reading and commenting this manuscript. Finally, support, encouragement, and love. Specially, I would like to thank to my wife for giving me happiness, and loving me. A ll of my accomplishments would not be possible without my wife support and love.

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6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 ABSTRACT ................................ ................................ ................................ ................... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 14 1.1 Single Molecule Magnets ................................ ................................ .................. 15 1.2 Quantum Phase Transition in a Spin Dimer C ompound ................................ ... 19 1.3 Origin and Environment of Magnetic Anisotropy ................................ ............... 23 1.4 Spin Hamiltonian ................................ ................................ ............................... 28 1.4.1 Multi Spin Hamiltonian ................................ ................................ ............. 29 1.4.2 Giant Spin Hamiltonian ................................ ................................ ............ 32 1.5 Magnetization and Quantum Magnetic T unneling in SMMs .............................. 34 2 EXPERIMENTAL INSTRUMENTS ................................ ................................ ......... 47 2.1 Electron Paramagnetic Resonance ................................ ................................ ... 47 2.2 EPR Experiments Set up ................................ ................................ .................. 50 2.2.1 Probe for the EPR Measurements ................................ ........................... 51 2.2.2 Cavity for the EPR Measurem ents ................................ .......................... 54 2.2.3 Millimeter Vector Network Analyzer ................................ ......................... 58 2.2.4 Magnets for the EPR Measurements ................................ ....................... 61 2.2.5 Set up for the High Frequency EPR Measurements ................................ 61 2.3 Summary ................................ ................................ ................................ .......... 63 3 RELIEVING OF FRUSTRATION: THE CASE OF ANTIFERROMAGNETIC Mn 3 Zn 2 COMPLEX ................................ ................................ ................................ 73 3.1 Background ................................ ................................ ................................ ....... 73 3.2 Overview of Structure ................................ ................................ ....................... 76 3.3 Experiments and Simulation ................................ ................................ ............. 78 3.3.1 Prediction of Anisotropy using Projection Operator Calculation .............. 78 3.3.2 High Frequency Electron Paramagnetic Resonance Studies .................. 81 3.3.3 Magnetic Torque Measurement ................................ ............................... 85 3.3.4 Simulation of Energy Level Diagram ................................ ....................... 86 3.4 Summary ................................ ................................ ................................ .......... 89

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7 4 MAGNETIC ANISOTROPY BARRIER OF HETEROMETALLIC HIGH SPIN Cu 17 Mn 28 CLUSTER WITH T d SYM METRY ................................ ......................... 102 4.1 Background ................................ ................................ ................................ ..... 102 4.2 Overview of Structure ................................ ................................ ..................... 103 4.3 H igh Frequency Electron Paramagnetic Resonance Experiments ................. 105 4.4 Simulation of Angle Dependent EPR Data ................................ ...................... 109 4.5 Summary ................................ ................................ ................................ ........ 113 5 SINGLE ION ANISOTROPY IN AN ANTIFERROMAGNETIC SPIN DIMER SYSTEM: Ba 3 Mn 2 O 8 ................................ ................................ ............................. 122 5.1 Background ................................ ................................ ................................ ..... 122 5.2 Overview of Ba 3 Mn 2 O 8 ................................ ................................ .................... 122 5.3 High Frequency Electron Paramagnetic Resonance Studies in the Low Field Range ................................ ................................ ................................ ................ 126 5.3.1 HF EPR for a Big Crystal ................................ ................................ ........ 128 5.3.2 HF EPR for a Small Crystal ................................ ................................ .... 131 5.4 Analysis Using Single Ion Hamiltoni an ................................ ............................ 131 5.4.1 Simulation of Energy Level Diagram ................................ ..................... 131 5.4.2 Calculation of Dipolar Contribution to Anisotropy ................................ .. 132 5 5 Summary ................................ ................................ ................................ ........ 134 6 CONCLUSION ................................ ................................ ................................ ...... 145 LIST OF REFERENCES ................................ ................................ ............................. 148 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 156

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8 LIST OF TABLES Table page 1 1 The second and fourth order Stevens operators [ 49 ]. ................................ ........ 46 2 1 Available frequencies depending on the harmonic number and the ASA extension. ................................ ................................ ................................ ........... 72 3 1 Comparison of structural parameters fo r the AFM and FM Mn 3 Zn 2 complexes. ................................ ................................ ................................ ....... 101 3 2 ZFS parameters obtained via simulation of HFEPR data of the AFM and FM Mn 3 Zn 2 complexes. ................................ ................................ ........................... 101 5 1 Projection coefficients of the anisotropy for the spin multiplet S [51]. ............... 144 5 2 D 12 values for the four kinds of dipolar coupling. J 0 J 1 J 2 and J 3 indicate the dipolar coupling s associated with the pairs. ................................ ..................... 144

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9 LIST OF FIGURES Figure page 1 1 Schematic representation of anisotropy barrier ................................ .................. 37 1 2 Schematic representation of the spin frustration in a triangular and a tetrahedral structure of spins ................................ ................................ .............. 38 1 3 Phase diagram and m agnetization curve of a spin dimer system ....................... 39 1 4 Schematic representation of the two legs spin ladder structure for an antiferromagnetic dimer system ................................ ................................ .......... 40 1 5 Zeeman splitting of the triplet state and d ispersion of triplons in the momentum space ................................ ................................ ............................... 41 1 6 Schem atic representation of the field dependent ground spin configuration of an antiferromagnetic dimer system ................................ ................................ ..... 42 1 7 Schematic representation of the metal ion in octahedral environment ............... 43 1 8 Schematic representation of the d orbital lobes ................................ .................. 44 1 9 Schematic representation of the o rbital bonding type ................................ ......... 44 1 10 Quantum magnetic tunneling (QMT) in a energy barrier of S = 6 system ........... 45 2 1 Simulated energy level diagram and typical EPR spectrum with a negative anisotropy and external magnetic field applied to the z axis .............................. 64 2 2 Simulated energy level diagram and typical EPR spectrum with a negative anisotropy and external magnetic field applied to the xy plane .......................... 65 2 3 Simula ted energy level diagram and typical EPR spectrum with a positive anisotropy and external magnetic field applied to the z axis .............................. 66 2 4 Schematic representation of cylindrical cavities ................................ ................. 67 2 5 Schematic representation of the coupling of a cavity to the waveguide .............. 68 2 6 Schematic representation of MVNA ................................ ................................ .... 68 2 7 Picture of the harmonic generator (HG) and the harmonic mixer (HM) for 670 965 GHz frequency range ................................ ................................ ........... 69 2 8 Schem atic representation of a trans mission probe ................................ ............. 70 2 9 HF EPR spectra of BaCuSi 2 O 6 spin dimer compound ................................ ........ 71

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10 3 1 Core structure of the AFM Mn 3 Zn 2 complex ................................ ....................... 91 3 2 Magnetization hysteresis for the AFM Mn 3 Zn 2 complex ................................ ..... 9 2 3 3 Structure of the AFM Mn 3 Zn 2 complex ................................ ............................... 93 3 4 Two differently oriented molecules of the AFM Mn 3 Zn 2 in a unit cell .................. 94 3 5 Temperature dependent EPR spectra for the AFM Mn 3 Zn 2 complex at 344 GHz and 500 GHz ................................ ................................ .............................. 95 3 6 Temperature dependent EPR spectra for the AFM Mn 3 Zn 2 complex at 2 44 GHz and 419 GHz ................................ ................................ .............................. 96 3 7 Frequency dependence data for the AFM Mn 3 Zn 2 complex ............................... 97 3 8 Magnetic torque data for the AFM Mn 3 Zn 2 complex ................................ ........... 98 3 9 Zero field eigenvalue diagram for the AFM M n 3 Zn 2 complex ............................. 99 3 10 Simulated energy level diagram for the AFM Mn 3 Zn 2 complex ......................... 100 4 1 Molecular structure of the Cu 17 Mn 28 comp lex ................................ ................... 114 4 2 Structure of Mn 28 core cluster ................................ ................................ ........... 115 4 3 Spin frustration in the core structure of the Cu 17 Mn 28 complex ......................... 116 4 4 Temperature dependent EPR spectra for the Cu 17 Mn 28 complex ..................... 117 4 5 Angle dependen t EPR spectra for the Cu 17 Mn 28 complex ................................ 118 4 6 Angle dependence data for the Cu 17 Mn 28 complex ................................ .......... 119 4 7 Polar presentation of the transverse anisotropy terms ................................ ..... 120 4 8 Representation of the eigenstate solutions for the Cu 17 Mn 28 complex ............. 121 5 1 Schematic structure of the Ba 3 Mn 2 O 8 compound ................................ ............. 135 5 2 Energy level diagram of an antiferromagnetic dimer composed of S = 1 ions .. 136 5 3 Phase diagram of Ba 3 Mn 2 O 8 single crystal for the magnetic fiel d applied perpendicular to the c axis ................................ ................................ .............. 137 5 4 Pictures of the Ba 3 Mn 2 O 8 crystals used for the EPR experiments .................... 138 5 5 Frequency dependent EPR spectra for the big crystal of the Ba 3 Mn 2 O 8 compound ................................ ................................ ................................ ......... 138

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11 5 6 Schem atic representation of multiple reflections of wave radiation in a sample ................................ ................................ ................................ .............. 139 5 7 Angle dependent EPR spectra of a big crystal of the Ba 3 Mn 2 O 8 compound ..... 140 5 8 Temperature dependen t EPR spectra for the big crystal of the Ba 3 Mn 2 O 8 compound ................................ ................................ ................................ ......... 141 5 9 Angle dependence data for the small crystal of the Ba 3 Mn 2 O 8 compound ....... 142 5 10 Energy level diagram of a dimer model of tw o S = 1 ions ................................ 143

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12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy HIGH FREQUENCY ELE CTRON P ARAMAGNETIC RESONANCE STUDIES OF THE ANISOTROPY OF MOLECULAR MAGNETS AND A SPIN DIMER COMPOUND By Changhyun Koo August 2011 Chair: David Tanner Cochair: Stephen Hill Major: Physics This dissertation presents the experiments and simulation/calcul ation to investigate the magnetic anisotropy in molecular magnets and spin dimer compound. High frequency electron paramagnetic resonance ( HF EPR) measurements were performed on the samples, and the obtained data were analyzed using spin Hamiltonians. In th e results of these experiments, the magnetic properties of the studied materials will be further clarified. Results presented for the antiferromagnetic triangular Mn III 3 complex demonstrate that the spin frustration in the complex is relieved. The well iso lated spin ground state is confirmed by the EPR data and the simulation using the multi spin Hamiltonian. The small total spin number of the complex causes a big anisotropy value. Moreover, the anisotropy is negative easy axis type, resulting in a signific ant anisotropy barrier to the magnetization reversal in the complex. High spin cluster Cu 17 Mn 28 complex, which is geometrically spin frustrated, demonstrates the slow relaxation and hysteresis in the magnetization data. However,

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13 the structural symmetry ( T d High frequency EPR experiments and simulation for the Cu 17 Mn 28 complex sugg est that the complex has fourth order anisotropy following the structural symmetry of the complex. However, the simulati on reveals that anisotropy is not strong enough to give rise to an effective barrier to the magnetization reversal. Thus, the slow relaxation and hysteresis of the magnetization observed in the complex is not caused by the magnetic anisotropy, but by the g eometric frustration of the core structure. The magnetic anisotropy in the Ba 3 Mn 2 O 8 compound contributes the magnetic phase of the compound in competition with the exchange interaction. High frequency EPR data of the Ba 3 Mn 2 O 8 compound determines the sign a nd magnitude of the single ion anisotropy of Mn V ions. The calculation of dipolar couplings between neighboring Mn V ions demonstrates that the dipolar couplings contribute to the anisotropy value in the compound.

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14 CHAPTER 1 I NTRODUCTION Magnetism of c ond ensed matter is strongly relevant to quantum mechanics. The magnetic property of a solid state material is induced by the magnetic moments of constituent atom s which originates from the spin and orbital angular moment um of the unpaired electron around the nucleus [ 1 ]. Magnetic materials demonstrate various magnetic phase s such as paramagnetic, diamagnetic, and ordered phases, depending on the intrinsic characteristics of the material and external disturbance. The coordinate s of the atom, the interaction b etween the ions of the material, and the lattice (or molec ular) structure are examples of the intrinsic characteristics of the materials and the external magnetic field, pressure and temperature are examples of the external disturbance Thus, the magnetis m of materials has been extensively studied under the various controls of intrinsic characteristics and external disturbances. Among magnetic materials, single molecule magnets (SMMs) are a favorable system for s tudy ing the magnetism of a low dimensional m agnetic material. Compar ed with a bulk magnet, SMM complexes are relatively similar to a magnetic atom in size The bistability of magnetic moments is caused by the anisotropy barrier of SMM complexes [ 2 ]. Since a magnetic molecule in SMM complexes is isol ated from the neighbor molecules, the consideration of the interactions between m agnetic ions can be restricted to the cor e of a molecule. Therefore, SMM studies can lend insight into the magnetism of a low dimensional magnetic material. Magnetism in a la ttice compound system is more difficult to understand than that of the zero dimensional material due to the interactions between magnetic ions in the compound. Magnetic ions are interconnected via the ferromagnetic or antiferromagnetic

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15 exchange interaction s, resulting in various quantum magnetic phases of the compound. The transition between the quantum phases is tuned by the external parameters, such as pressure, magnetic field, and delocalization in the lattice at zero temperature The spin fluctuation ca used by the thermal energy dilutes the quantum phase, and leads to the paramagnetic phase in the compound. The magnetic phase of a magnetic compound can be displayed in the phase diagram drawn in the T vs external parameters plot [3]. In this dissertation, two types of magnetic materials are studied. One is a single molecule magnet (Chapter 3 and 4) and the other is a spin dimer compound which shows a quantum phase transition (Chapter 5) In order to understand the magnetic properties of these complexes an d compound, the exchange interactions are discussed, and the magnetic anisotropy and spin state are investigated using high frequency electron paramagne tic resonance ( HF EPR), which is a strong tool for unambiguously determining the spin Hamiltonian paramet ers of a system. 1.1 Single Molecule Magnets Single molecule magnets demonstrate intermediate magnetic properties between classic bulk magnets and individual spins, providing evidence of quantum size effects in magnets [2] Furthermore, the structure of a SMM complex is simple enough to investigate its magnetism in detail. The SMM bottom up synthesis strategy provides several benefits to magnetism studies. i) One can ideally add one magnetic center at a time to increase the size of a molecular magnet. Al though this is not practically possible, the synthesis strategy is successfully applied to gradually increa se the manganese cluster size [4 8 ]. ii) A cluster of large spins can be easily synthesized using a magnetic molecule as a building b lock [9, 10 ]. ii i) One can conveniently change the structural

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16 environment of magnetic molecules, exchanging the ligand molecules surrounding the magnetic center. The structural change induces the change of the magnetic property [11 13 ]. A SMM consists of mainly two parts : the magnetic core and the nonmagnetic ligand molecules The SMM complex has a magnetic core as a source of the magnetic prop erties. The magnetic core is predominantly built from 3 d transition metal ions, such as Mn, Fe, Ni, and Co ions. The spin angular momentum of the unpaired electrons in the metal ions couples with its orbital an gular momentum, resulting in magnetic anisotropy. This magnetic anisotropy is discussed in detail later. Recently 4 f rare earth ions are used to build a core structure by combi ning with 3 d transition metal ions [ 14, 15 ]. Since 4 f rare earth ions have larger single ion anisotropy than the 3 d transition metal ions due to the strong spin orbit coupling, one can expect a strong anisotropy barrier of the complex in the study of 4 f SM M complexes [ 16]. The ligand molecules surround the magnetic core structure and prevent the interaction between the neighboring cores, resulting in the isolation of the magnetic core. Even though t he y are nonmagnetic the ligand molecules contribute to the magnetic property of a SMM due to the ligand field Furthermore the ligand molecules play an important role in preserving stability of the molecular structure of the SMMs and in exhibiting the magnetic property without interaction between neighboring mag netic cores, unlike the bulk magnets. The main characteristic of SMMs is a significant barrier between up and down magnetization states which is attributed to a large spin ground state value S and significant negative uniaxial magnetic anisotropy D (Figur e 1 1 ) [ 17 19 ]. From the

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17 quantum mechanical prospective, the simplest spin Hamiltonian of SMMs can be written as (1 1 ) where D represents the dominant, second order axial anisotropy of molecules, and is the z component of spin operator. When the eigenfunction of the Hamiltonian in Equation 1 1 is expressed as the spin projection , the energy eigenvalue can be expressed as where m s is the spin projection along th e z axis. Since D is negative for SMMs, the ground state of spin projection is determined by spin number: Thus, the magnitude of the barrier between up and down magnetization states is written as and for the integer and half integer S values, respectively, with a negative value of D The barrier can be modeled with an inverse parabolic shape, resulting in bistability. Figure 1 1 shows the anisotropy barrier between spin up and spin dow n states of the S = 6 system with a negative D The spin ground state value S of the SM Ms is determined by the sum of the unpaired electrons number in the metal ions and the Heisenberg exchange interaction between the metal ions in the magnetic core. When the metal ions are coupled through a ferromag netic (FM) exchange interaction, the unpaired electron numbers of the metal ion is added in the summation for S On the other hand when the exchange interaction is antiferromagnetic (AFM), the unpaired electron s numbers of the metal ions are subtracted in the summation for S If two sublattices with unequal magnetic moments in a system are coupled though the AFM exchange interaction (ferrimagnetism), the system has an intermediate spin ground state value S For example,

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18 [Mn 12 O 12 (CH 3 COO) 16 (H 2 O) 4 ] (hereafter Mn 12 acetate ) complex has eight Mn III ( S = 2) ions and four Mn IV ( S = 3/2) ions in its core [20] If all of the metal ions are coupled through FM exchange interactions, the S value of this complex is 22. Wherea s, if AFM exchange interactions cause all of the metal ions to be coupled, this complex has S = 0. However, since both of FM and AFM exchange interactions contribute to coupling of the metal ions in this complex, Mn 12 acetate has an S = 10 ground spin stat e. Antiferromagnetic exchange interactions between identical ions in a specific geometry, such as triangular and tetrahedral structures, are noticeable, because the geometrical spin frustration occurs in this case. Triangular and tetrahedral spin structur es are common building blocks for the molecular complexes and lattice compounds. If the exchange interaction is significantly strong, then the ground spin state of a system can be well determined. For example, when the exchange int eractions in a triangular structure are ferromagnetic, the spin configuration of the ground state is clearly determined by the up or down state of all three spins, maximizing the spin number S When the exchange interactions are antiferromagnetic, the grou nd spin state is determined as the minimum spin number, S = 0. Although this ground state is a highly non classical superposition of product states, it corresponds rigorously to S = 0 in that limit. However, if the exchange interaction is weak and compara ble to anisotropy, the ground spin state of an antiferromagnetic system is not c learly determined. Figure 1 2A demonstrates the spin frustration in a collinear spin configuration for a triangular structure. The same spin frustratio n occurs in the tetrahedral spin structure constructed by four edge sharing triangles (Figure 1 2B ). The geometric spin frustration results in an ambiguous ground spin state of the molecular

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19 complexes and the multiple magnetic phases of the lattice compoun ds [21]. The spin frustration in a Mn III triangular SMM is discussed in Chapter 3, and the metastability caused by the geometric frustration in a high nuclearity complex is discussed in Chapter 4. 1.2 Quantum Phase Transition in a Spin Dimer Compound A sig nificant component of condensed matter physics is the study of the phases of matter. Particularly, the phase of quantum electron systems in solids provides a clue to understanding various magnetic states, such as ferromagnetic and antiferromagnetic states. Accordingly, the study of the phase transition between the two different phases induced by the external perturbation is necessary in order to understand the change in magnetic property. The thermal classical phase transition occurs due to thermal fluctua tions, and so the phase transition happens at finite temperature, but not at T = 0. In contrast to the thermal phase transition, the quantum phase transition (QPT) occurs at T = 0. Since there are no thermal fluctuations at T = 0, the quantum phase transit ion is not driven by principle [22]. The ground state of the system is changed at the quantum critical point (QCP) (Figure 1 3). Some of the possible external parameters that tune the quantum phase transition are the pressure applied to the system, the magnitude of the applied magnetic field, and the ratio of the dopants in a compound. An a ntiferromagnetic dimer compound is an example which shows the quantum phase transi tion. Here we think of a two leg spin ladder of dimers consisting of S = 1/2 metal ions along the rungs (Figure 1 4 ) [23, 24]. Metal ions interact with each other via the antiferromagnetic exchange interactions; intradimer interaction J 0 and interdimer

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20 in teraction along the ladder J 1 Between them, the intradimer interaction between two metal ions in a dimer is dominant. The Heisenberg Hamiltonian of this antiferromagnetic dimer system can be written as follows, (1 2 ) where i and in dicates a dimer site, and m = {1, 2} is the metal ion site index in a dimer The external magnetic field is applied along the z axis. Since the intradimer interaction is antiferromagnetic, J 0 > 0, the singlet S = 0 is the ground state and the triplet state S = 1 is the excited state at zero field (Figure 1 5A ) The energy gap between the singlet and the triplet is determined by the exchange interactions. Inelastic neutron scattering studies for the antiferromagnetic dimer compounds report energy momentum de pendence (dispersion) of the triplet (Figure 1 5B ) [ 25 27 ]. The dispersion of the triplet state induced by the weak interdimer interactions causes the energy band of the triplet state [ 28 ]. The energy gap is big enough for us to focus on the singlet and th e low energy state of the triplet at zero temperature. For this system, the external parameter leading the QPT is the external magnetic field applied along the z axis As the magnetic field increase s the low state of the triplet is lowered by the Zeeman effect and eventually crosses the singlet state resulting in the gap closing at the critical field H c1 (Figure 1 5A ). The magnetization curve of the antiferromagnetic dimer system at zero temperature exhibits the spin density popul ated in the triplet states of sites and the magnetic phase as the magnetic field increases (Figure1 3) Below H c1 the magnetization is zero and only the singlet states are populated Between H c1 and H c2 the magnetization increases linearly with the incre asing magnetic field as more triplet

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21 states are populated Therefore the spin state at an individual site can be written as a coherent superposition of the singlet and triplet state (Figure 1 6) [ 28, 29 ]. In this magnetic field rang e, the spins show the long range canted antiferromagnetic ordering. The transverse components of the interdimer interaction cause the ordering of the spin component perpendicular to the external magnetic field [ 30 ]. Above H c2 the triplet state of each sit e is occupied and the magnetization is saturated per site. The long range ordering between H c1 and H c2 can be interpreted as a Bose Einstein Condensation (BEC) of magnetic quasiparticles [ 31 33 ]. It is convenient to identify the triplet state with the tri plon, which is a bosonic quasi particle with S = 1, and the singlet state with the absence of the triplon. With this assumption, Equation 1 2 can be rewritten to the effective Hamiltonian, which has the effective spin 1/2 operators based on the new singlet triplet subspace; singlet and triplet [23, 24] Original spin operators in Equation 1 2 are expressed in terms of the effective spin operators. (1 3) Using these effective spin operators, the effective Hamiltonian is written as follows, (1 4)

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22 (1 5) where is the effective magnetic field, and i s an energy shift. The X and Y components of the effective spin operators in Equation 1 4 are written using the raising and lowering operators in Equation 1 5. These terms cause the exchange of triplet and singlet states between neighbor sites, describing the delocalization of triplons. Thus, the first two terms in the first summation in Equation 1 5 represent the kinetic energy of the triplons. The third term in the first summation in Equation 1 5 corresponds to the potential energy of the triplons, descri bing the repulsion of the triplons in neighboring dimer sites. The effective spin operators at each site commute to each other [ 31 ]. The second summation term is the effective Zeeman term and represents the chemical potential controlling the density of tr iplons in the system by the external magnetic field. Below H c1 all dimers are spin singlets, i.e. all sites are empty, because the chemical potential is large. The density of the triplons increases when the magnetic field exceeds H c1 Therefore, the tripl ons are diluted and the repulsive interaction between the triplons is suppressed near H c1 (= for the ladder discussed in here). As the temperature co ols down, the triplon can crystallize or condensate, depending on the balance betwe en the kinetic energy and the potential energy [34]. If the kinetic term is dominant and the system has U(1) symmetry along the applied magnetic field direction to conserve the finite quantity of the triplons, the system undergoes BEC corresponding to the coherent superposition of the triplet and the singlet state at each and every dimer site. Antisymmetric interaction or transverse

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23 anisotropy can destroy the BEC [ 35 ]. Although the antiferromagnetic ordering spontaneously breaks the rotational symmetry in t he dimer compounds, the rotation of spin is realized by changing the phase of the transverse component of the spin operators [ 29 ]. The BEC in a spin dimer system can be detected according to the relation between critical temperature and magnetic field. Clo se to the QCP, the critical temperature follows a power law with a universal critical exponent 2/d for the BEC where d represents dimension ality (Figure 1 3) [36 ]. Ba 3 Mn 2 O 8 antiferromagnetic spin dimer compound shows the field ind uced long range ordering. Since this compound has Mn V ions ( S = 1) in a dimer the magnetization curve has additional magnetization p lateau associated with the quintet state S = 2 for this compound Therefore, two distinct ordering phases appear in the pha se diagram of this compound with different magnetic field range s [ 37 ]. In Chapter 5, Ba 3 Mn 2 O 8 is investigated using the HFEPR measurement and the anisotropy of this compound is clarified 1.3 Origin and Environment of Magnetic Anisotropy Magnetic anisotro py causing magnetic properties in a material is induced from two sourc es, such as spin orbit coupling and spin spin dipole interaction S pin orbit coupling is usually the dominant sour ce of magnetic anisotropy. S pin orbit coupling results from the interact momentum. This coupling is explained with the interaction between a nucleus and an electron orbiting the nucleus. The nucleus has radial static electric field due to its charge. When an electron moves across in the electric field, special relativity demonstrates the magnetic field detected in the frame of the electron; The

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24 interaction energy between the magnetic moment of the spin and the magnetic field is writ ten as follows, (1 6) The interaction energy is modified by the Thomas factor, 1/2, in order to include an additional relativistic effect due to the acceleration of the electron [38]. The electric field can be described using the st atic Coulomb potential, and substituting this expression for the electric field into Equation 1 6 gives (1 7) where Rewriting for and using the relation between and , permits the interaction energy H to be rewritten in terms of the equivalent operators. (1 8) where is the spin orbit coupling constant. Thus, t he spin magnetic moment couples with the induced magnetic field, resulting in spin orbit coupling [ 39 41 ]. The dipole interaction between two spins is also explained in a similar way. When two spins are sufficientl y distant from each other, a spin experiences the dipole magnetic field induced by the other spin. Thus, two spins couple through the dipole magnetic field, resulting in the anisotropy contribution. Details of the dipole interaction are further discussed i n Chapter 1.4.

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25 The structural local environment, such as the crystal field and Jahn Teller distortion also contribute to the magnetic anisotropy. Isotropic m etal ions have fully degenerate spin states. However, when these ions have unpaired electrons, an d they are located in the geometric structure of negatively charged ions, such as the octahedron or tetrahedron of oxygen ions, the partially spin filled ions experie nce a breaking of the orbital degeneracy due to the crystal field effect. In contrast, th e spin fully filled ions or empty ions do not show the breaking of the degeneracy in the d orbitals, in the center o f the octahedron of oxygen ions. T he overlap of the 3 d orbitals of the metal ion and the 2 p orbitals of oxygen ions induces the splitting of 3 d orbital states to ( and ) level and ( , and ) level due to the electrostatic energy (Figure 1 7 B )[ 16, 42 ]. Figure 1 8 shows the five 3 d orbital shapes of the transition metal ion in the octahedron of oxygen ions. Two orbitals, and h ave lobes which point to the neighboring oxygen positions w hereas three orbitals, , and have the nodal planes in these directions. These orbitals are convention ally called to the and states, respectively, according to their symmetric behavior in the O h symmetry of the ideal octahedron [ 43 ]. The C oulomb repulsion between the 3 d orbitals of the metal ion and the 2 p or bitals of oxygen ions explains which orbital state has higher energy as the degeneracy is broken. In the octahedron of oxygen ions, the lobes pointing in the metal oxygen bonding direction, whereas the

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26 (Figure 1 9 p orbital of oxygen than the hus, the orbital state has higher energy than the in the octahedral oxygen structure (Figure 1 7 B ), and for the tetrahedral structure, vice versa. This phenomenon is known as the crystal field effect wherein the degeneracy breaking depends on the local structural environment [ 44 ]. The splitting of and orbital energy leve ls is caused by the environment of the atom. However, the magnetic properties o f the atom can also cause the symmetry of the environments to approach an energetically favorable state. A non linear molecule, which has orbital degeneracy in its electronic state, prefers to remove the degeneracy in order to reduce the electric energy. Simultaneously, in order to conserve the energy, the molecule distorts the molecular structure and lowers the structure symmetry. This phenomenon is known as the Jahn Teller effect [ 45 ]. The orbital states in a system are [16]. T he Jahn Teller effect is significant in the partially filled ions, such as Mn III ( S = 2), Mn V ( S = 1), and Cu II (S = 9/2) ions. However, t his phenomenon is not apparent in the fully filled (Cu I S = 0), half filled (Mn II S = 5/2), or empty ions, b ecause there is no net reduction of energy. Figure 1 7C and D ) shows the case of Mn III ion ( 3 d 4 ) in an octahedron of oxygen ions. The structure distortion changes the overlap of the 3 d orbitals of Mn III ion and the 2 p orbitals of O ions. The elongation of the octahedron along the z axis (Figure 1 7 D ) induces the orbital energy state to become higher than the energy state, and the energy state is lifted up above the and orbital energy states due to the coulomb

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27 repulsion. When the octahedron is compressed along the z axis (Figure 1 7 C ) the orbital energy states tend to move in opposite directions. Spin orbit coupling removes the degeneracy of the spin states and induces the magnetic anisotropy of a metal ion in crystal field. F or the 4 f rare earth ions, the spin orbit coupling is stronger than the crystal field interaction In contrast, t he crystal field interaction is much stronger than the s pin orbit coupling in the 3 d metal ions. This contr ast comes from the orbital size of an ion. The 3 d orbitals are relatively more extended away from the nucleus than the 4 f orbitals because the 3 d ions have smaller atomic number Z than the 4 f ions Furthe rmore, a nucleus of small Z induces a relatively weak magnetic field, resulting in weak spin orbit coupling. For a system where the spin orbit coupling is weak, the spin orbit coupling is considered as a perturbation which leads to the mixing of the states into the ground state with the non zero orbital angular momentum, resulting the coupling between the orbital ground state and the spin [ 46 ]. The spin orbit coupling causes the spin to be sensitive to its crystalline environment, because the orbital energy state configuration depends on the crystalline environment, and the spin couples to the orbital. Therefore, the energy level configuration of the orbitals caused from the crystal field and Jahn Teller distortion determines the sign of the magnetic anisotr opy D When the orbital energy state is lower than the orbital energy state due to t he elongation of the octahedron structure along the z axis the spin prefers to be aligned along the elongated axis, corresp onding to the easy axis type ( D < 0) magnetic anisotropy. On the other hand, when the energy state is lower than the orbital energy states due to t he compression along the z axis, the spin prefers to be align ed in the compressed plane, corresponding to the easy plane

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28 type ( D > 0) magnetic anisotropy. Thus, the type of the magnetic anisotropy is determined in the system. 1.4 Spin Hamiltonian A SMM conventionally consists of multiple magnetic ions. As discussed above, the ferromagnetic/antiferromagnetic exchange interaction between ions determines the ground spin state among the possible spin multiplet states of the complex. If the exchange interaction J is strong enough ( ), then the groun d spin state is much lower in energy than the fir st excited spin multiplet state resulting in the well defined ground spin state Thus, one can assume that only the ground spin state is populated in the complex I n other words, the complex can be consider ed to be a single ion having the ground spin multiplet as the spin number S This approximation is called as the giant spin approximation (GSA). Since the GSA considers only two terms, the Zeeman term and anisotropy term, the spin Hamiltonian is relatively simple. Thus, the role of the anisotropy terms in a complex is clearly elucidated in the spin Hamiltonian. Furthermore, since the giant spin Hamiltonian has the Hilbert space using the total spin number of a complex (for example, the size of the Hilbert s pace of Mn 12 S = 10 is ), one can completely solve the Hamiltonian using a recent conventional computer. A lthough the GSA has the benefit that the role of the anisotropy is clearly shown, the giant spin Hamiltonian cannot be applied to the lattice compound. Since the lattice compound has various and complicated exchange interactions between the metal ions arrayed in a lattice and the Hamiltonian matrix of the lattice compound is infinite, even though one can approximate them to a gian t spin, it can never be exact. Under the condition of the GSA cannot be applied for a SMM either. When the exchange

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29 interaction between the magnetic centers is not so strong the first excited spin multiplet state is pretty close to the ground spin state, and the ground spin state is not isolated from the excited spin multiplet states any more. The giant spin Hamiltonian is not a proper way to explain the system in this condition [ 47 ], because the giant spin Hamiltonian focuses on an approximated single ion with a spin multiplet state Accordingly a Hamiltonian is required which considers the anisotropy of the individual magnetic ions and the exchange interaction between them. In this dissertation, both giant spin Hamiltonian and multi spin Hamiltonian are used to analyze the experimental data. 1.4 .1 Multi Spin Hamiltonian The individual magnetic centers in a material have their own anisotropy due to the spin orbit coupling, and they interact with each other v ia the various interactions, such as the exchange interaction, and dipolar interaction. The multi spin Hamiltonian includes these terms as following: (1 9 ) (1 10 ) Equation 1 10 is the Zeeman term where is the Land g tensor is the Bohr magneton, is the external magnetic field, and is the spin number of single ion For the systems discussed in this dissertation, is considered isotropic. (1 11 )

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30 E quation 1 11 is t he anisotropy term of the single ion. The single ion uniaxial anisotropy should be distinct from the system anisotropy used for the GSA. is the second order transverse anisotropy value of single ion. (1 12 ) Equation 1 12 is the magnetic dipolar coupling term, where is a unit vector parallel to t he i j direction and is the distance between two ions. For a pair of spins, the magnetic dipolar coupling tends to orient the spins parallel to the line connecting two spins and parallel to each other, leading to the easy axis type anisotropy. The magnitude of the dipolar coupling is proportional to and the anisotropic component is negative. In Chapter 5, the contribution of dipolar coupling to anisotropy is further discussed. (1 13 ) E quation 1 13 is the exchange interaction term, where and represent the spin operators of the i th and j th ion. is the isotropic exchange interaction constant is the tensor for the anisotropic exchange interaction, and is the Dz y aloshinsk ii Mori y a (DM) vector

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31 The first term of Equation 1 13 is the isotropic exchange interaction term, which is usually the dominant t erm. When the total spin of the system is t his term breaks the degeneracy of the spin states by the spin state energy as following: (1 14 ) The isotropic exchange term keeps the spins either parallel or antip arallel to each other depending on the sign of for a given i and j If the spins are aligned parallel to each other for the ground state via the ferromagnetic exchange interaction. Whereas if the spins are aligned antiparallel to each other for the ground state via the antiferromagnetic exchange interaction. The second and third term s of Equation 1 13 are the aniso tropic exchange interaction and antisymmetric exchange interaction ter m (known as DM interaction ), respectively. The anisotropic exchange term causes the spin to be oriented along a given orientation in space, and the DM term ind uces a canting or non collinear spin alignment The spin orbit coupling leads to the admixture of the excited states into the ground state, and it is the physical origin of the anisotropic exchange interaction and the DM term s like the other anisotropy terms [ 48 ]. For the systems discussed in this dissertation, the anisotropic exchange interaction and DM interaction is much smaller than the uniaxial anisotropy, and therefore they are ignored. The multi spin Hamiltonian describes the fundamental characteristics of a given complex. The Hamiltonian includes single ion anisotropy of individual magnetic cen ters and the exchange interactions between the magnetic centers. The model is more likely to consider the complex itself. Therefore, when one observes a phenomenon during an

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32 experiment, this Hamiltonian provides a clue to the physical origin of the phenome non. But, since the Hamiltonian includes all magnetic centers in the complex, leading to the extension of the Hilbert space, i f the cluster has many magnetic centers it is difficult to completely solve the Hamiltonian. For example, the size of the Hilbert space of Mn 12 acetate for using the multi spin Hamiltonian is 1.4 .2 Giant Spin Hamiltonian The degeneracy of the spin states in a single ion is broken due to the anisotropy and an external magnetic field. When a single ion has a nisotropy and an external magnetic field is not applied, the S+1 (S+1/2) spin multiplet states are split for the integer spin (half integer spin) system showing the zero field splitting due to anisotropy The external ma gnetic field breaks the remaining de generacy of the spin states. The giant spin Hamiltonian including the anisotropy of the approximated single ion and the Zeeman effect can be written as following: (1 15 ) (1 16) (1 17 ) Equation 1 16 is t he Zeeman term for the GSA. In contrast to Equation 1 10, this term is for the approximated single ion. Equation 1 17 is a summation of anisotropy terms. The is the Stevens operator built by the z component of the spin operator, and the r a ising and lowering spin operators [ 49 ]. The is the coefficient of the Stevens operator corresponding to the anisotropy value obtained from the experiments. The k index indicates the order (rank) of the opera tors, and the q index depends on the point

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33 group symmetry. The q va riation has a limit according to k The second and fourth order Stevens operators are introduced in Table 1 1. The odd order terms of spin except the Zeeman split t erm are forbidden in the spin Hamiltonian Because the origin of the anisotropy, such as spin orbit coupling and dipolar interaction, is time reversal invariant, but the odd order terms violate the time reversal symmetry. Among those anisotropy terms, the second order terms are worthy of detailed inspection. As shown in Table 1 1, the second order uniaxial term is diagonal in the spin matrix based on the where S is a total spin quantum number and S z is a spin projection to the z ax is, and the transverse term mixes the spin states through the rising and lowering operators by Conventionally, the second order uniaxial anisotropy value, is represented using D /3 an d the rhombic transverse anisotropy term is represented using E The E variation has a limit according to: [50] The uniaxial anisotropy term depending on determines the magnitude of t he barrier to magnetization reversal in SMMs, whereas the transverse terms mix the spin energ y states. Generally, the second order term is dominant among the anisotropy terms in the spin Hamiltonian, and the contribution of the high er order terms depends on the system. Since the giant spin Hamiltonian and multi spin Hamiltonian are two models describing a single system, they should be the same in the strong exchange limit [ 51 ]. The system anisotropy can be wr itten as the combination of the single ion anisotropy of magnetic ions and the dipolar contribution. This is discussed in detail in

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34 Chapter s 3 and 5. Simulation studies using the multi spin Hamiltonian report that the higher order anisotropy terms relate t o the isotropic exchange interaction [ 42, 47, 52]. 1.5 Magnetization and Quantum Magnetic Tunneling in SMMs As mentioned above, a barrier of to magnetization reversal with a negative uniaxial anisotropy ( ) is an essential property of SMMs. The barrier can be modeled with an inverse parabola resulting in the bistable magnetic moment. Figure 1 1 shows the anisotropy barrier of the S = 6 system with a negative D The lowest energy states are the up and down magnet ic moment states. If there is no thermal energy contribution the magnitude of the Zeeman energy corresponding to the anisotropy barrier is required to change the magnetic moment from up state to down state. Thus, the magnetization data of SMMs display the hysteresis behavior due to the bistability caused from the anisotropy barrier. Quantum magnetic tunneling is another specific magnetic property of SMMs. For quantum magnetic tunneling to occur, it is necessary that the terms in the spin Hamiltonian mix th e spin states of both sides of the barrier and they with S z [ 2 ]. Since the spin states are mixed into linear superposition due to t he spin mixing terms, although there is the anisotropy barrier between two states, there is a probability for the spin vector to be measured in either state, i.e. up and down states. This is quantum magnetic tunneling (QMT) (Figure 1 10) There are several factors which mix the states and cause the tunneling: transverse component of the external magnetic field and transverse anisotropy in a complex [ 53 ]. Transverse anisotropies come from the symmetries of the complex. Depending on the symmetry, possible QMTs are determined. For example, when the complex has three fold symmetry, one can

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35 expect the QMT at k = 3 or 6; k = m s + m s he m s and m s the each side of the anisotropy barrier [54] The transverse component of the applied field to the complex affects the QMT of the complex. Because t he transverse field causes the spin state mixing. When these factors strongly contribute to QMT, the tunneling rate can be faster. In contrast to the classical magnet, a noticeable feature of the magnetic hysteresis of a SMM is the discrete steps on it. The steps indicate the quantum tunneling of the magn etic moment through the anisotropy barrier. When there is no spin state mixing, the tunneling between the degenerated spin states, i.e. m s = +6 and m s = 6 in Figure 1 1 is not allowed. However, the mixing of spin states makes quantum tunneling possible between the degenerate spin states (Figure 1 10 A ) Furthermore, the external magnetic field along the magnetic moment orientation causes the quantum tunneling between other degenerate spin states, such as m s = +6 m s = 5 and m s = +6 m s = 4 giving a bias to one side of the double potential well (Figure 1 10 B ) A relationship between possible quantum tunneling and the external magnetic field is described in following equation [ 53 ]. (1 18 ) Thus, the magnetic hysteresis of SMMs has discrete steps on it due to the quantum magnetic tunneling. The tunneling effect observed in SMMs provides advantages and also disadvantages in using these systems for the quantum devices [ 55 59 ]. When the spin states are mixed, the states are superpos ed, and the superposition of spin states is an essential phenomenon for the realization of quantum computing devices. Meanwhile,

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36 the qu bit of information can be distorted as a spontaneous reorientation of the magnetization vector occurring under the tunnel ing effect, therefore i t is a critical obstacle for quantum storage devices.

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37 Figure 1 1. Anisotropy barrier between spin up and down state of S = 6 system with negative D anisotropy parameter.

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38 Figure 1 2. Scheme of the spin configuration in a A) triangular and B) tetrahedral structure of spins. Every ions are connected via antiferromagnetic (AFM) exchange interactions ( J > 0). Total spin of these structure s is ambiguous due to the spin frustration.

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39 Figure 1 3. Phase diagram with paramagn etic (PM), ferromagnetic (FM), a nd canted antiferromagnetic ( AFM) phase, where BEC of triplon occurs. In vicinity of the critical field H c1 the phase boundary follows a power law with a universal exponent 2/ d for a BEC. Magnetization curve for each dimer site in a spin dimer system is overdrawn. Below H c1 and above H c2 the curve has a plateau at m z = 0, 1. See the main text for detail.

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40 Figure 1 4. Scheme of the two legs spin ladder structure for an antiferromagnetic dimer system consisting of half s pin ions. J 0 is intradimer interaction on the rungs and J 1 is interdimer interaction along the ladders

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41 Figure 1 5 A) Zeeman splitting of triplet state with energy gap and bandwidth 4 J B) Dispersion of triplons in momentum space at the critical magnetic field H c1

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42 Figure 1 6. Scheme of the field dependent ground spin configuration of an antiferromagnetic dimer system consisting of half spin ions. represents the spin state at a dimer site. is the field dependent proportion of the triplet (singlet) state [29].

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43 Figure 1 7 The d orbital splitting of Mn III ion induced by the oxygen octahedron crystal field and J ahn Teller distortion (see text for detail.). Magenta circle in a structure scheme represents a Mn III ion and blue circles represent oxygen ions. A) An isotropic Mn III 3 d 4 ion. B) Crystal field induced by the oxygen octahedron splits the orbital states of Mn III ions to the and the state. The Jahn Teller distortion breaks the degeneracy of and orbital state depending on the distortion type: C) com pression along the z axis, and D) elongation along the z axis. Arrows indicate the four spins of a Mn III ion.

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44 Figure 1 8 Schematic representation of the d orbital lobes in th e C artesian coordinate. Blue spheres represent the oxygen ions in the octahedron. Figure 1 9 Orbital bonding type between the d orbital and p orbital A type and B type bonding. The magenta lobes represent the d orbital, and the blue lobes represent the p orbital.

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45 Figure 1 10. Quantum magnetic tunneling (QMT) in a energy barrier of S = 6 system A) at zero field and B) when the magnetic field is applied along the magnetic moment orientation. The dotted arrow represents the QMT.

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46 Table 1 1. The second and fourth order Stevens operators [ 49 ] k q 2 0 1 2 4 0 1 2 3 4

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47 CHAPTER 2 EXPERIMENTAL INSTRUM ENTS 2. 1 Electron Paramagnetic Resonance Various experimental methods have been developed to i nvestigate the properties of magnetic materials. When the spin Hamiltonian parameters such as anisotropy value, exchange interaction, dipolar coupling, and the ground sp i n state are obtained through experiments, one can understand the given magnetic system by solving the spin Hamiltonian. For the single molecule magnets ( SMMs ) experimental measurements are used to obtain the magnetic anisotropy parameters and the spin gr ound state, because these two factors give rise to the anisotropy barrier. Structural information, such as Jahn Teller axis and excha nge interaction is also required due to their effect on the anisotropy and the spin ground state. However, it is difficult to obtain all parameters with one experimental method. Magnetic susceptibility data inform the ground spin state S and exchange interaction J but not anisotropy parameter D On the other hand, magnetization data provides estimation of S and D but not exc hange constant J According to this point of view, electron paramagnetic resonance (EPR) measurements have the advantage of elucidating the ground spin state and extracting the much more precise magnitude and the sign of the quadratic axial and rhombic ani sotropy parameters ( D and E ), and as well as, the EPR measurements unambiguously determine the higher order anisotropy parameters ( i.e. , etc.) [ 60 63 ] EPR is a branch of spectroscopy which uses microwave radiation (10 GHz 1 THz) as a magnetic dipole tr ansition source. Compared to other optical spectroscopy measurements, EPR experimentation has an additional requirement that the external magnetic field breaks the degeneracy of the sp in states and modifies the energy level of

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48 interest, causing the desired splitting of the spin energy levels. The magnetic field sweep EPR experiment using a narrow band resonant cavity is conventionally performed at fixed microwave frequenc ies The applic ation of a resonant cavity increases sensitivity of signal in measurements. Additionally, the magnetic field is convenient to change its magnitude and orientation without disturbing measurement conditions such as the cavity mode. Although it is also possib le to use a frequency sweep method in the EPR experiments, this method is not very common because of several problems with the method, such as reduced sensitivity due to small quality factor Q, poor signal to noise ratio caused from the variation of the de tected power, and non flat frequency characteristics of microwave systems In this dissertation, e ach of the EPR experiments is done using the field sweep measurement method. M agnetic reso nance can be explained using an energy level di agram. At zero field, an isotropic system with spin number S has 2 S + 1 degenerate spin states. When the external magnetic field is applied the degenerate spin states are split in to 2 S + 1 spin states due to the Zeeman effect. The Zeeman term is already shown in the spin Hami ltonian, Equation 1 10 of Chapter 1. When the DC magnetic field is applied along the z axis, t he energy difference between the spin states can be written as follow s : (2 1) where B z is the applied magnetic field and is the difference in the projection of the spins to the z axis If the energy difference between spin states matches the microwave radiation energy magnetic resonance occurs. In other words, a spin absorbs the radiation e nergy and transits from the lower spin state to the higher spin state. The selection rule for the magnetic dipole transition is S represents the

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49 total spin number of the system, and m s represents the projec tion of the spin, S onto the magnetic quantization axis. The resonance appears as a dip on the microwave transmission spectrum obtained through the field sweep measurement. In this dissertation, the dip on the transmission spectrum is call ed a resonance p eak or an absorption peak. The discrete resonance peaks observed in a n EPR spectrum are caused by the anisotropy in a sample. If there is no anisotropy, even though the degeneracy of the spin states is broken by the Zeeman effect, all resonances appear in the same magnetic field position. However, if there is anisotropy in the sample a nisotropy determines the z ero field splitting (ZFS ) between spin states resulting in the discrete 2 S resonance peaks in the EPR spectrum The separation of resonance peaks depends on the magnitude of anisotropy and the applied magnetic field orientation. In Chapter 4, the relation between the resonance peak separation and the magnitude of anisotropy value is used to investigate anisotropy in a given complex. On the other han d, t he order of the resonance peaks in a spectrum depends on the sign of the anisotropy value and the applied magnetic field orientation When a complex has easy axis type anisotropy, i.e., negative uniaxial anisotropy and the exte rnal magne tic field is applied along the axial direction ( z axis ) of the complex, the ground transition peak appears at the lowest field position (Figure 2 1) On the other hand, if the applied magnetic field is in the plane ( xy plane) perpendicular to the z axis, the ground transition peak appears at the highest field position (Figure 2 2). The gro und transition peak in the easy axis data shifts approximately twice as far from the g = 2.00 ( B = 7.14 T for 200 GHz) position, as c ompared to the peak in the h ard plane data. This confirms a uniaxial anisotropy with a

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50 negative D parameter [5 0 ] When a complex has easy plane type anisotropy, i.e., positive uniaxial anisotropy and the external magne tic field is applied along the z axis of t he complex, the ground transition peak appears at the highest field position (Figure 2 3) Therefore, the EPR measurement can elucidate the sign of anisotropy parameters [ 64 66 ]. In Chapter 5, the order of resonance peaks is investigated to clarify the sig n of the anisotropy value in a given system. 2.2 EPR Experiments Set up As mentioned above, EPR is a strong tool to determine the precise magnitude and unambiguous sign of magnetic anisotropy parameters and the spin ground state. However, there are a few technical obstacles for performing EPR experiments First, the good coupling of the microwave radiation with a sample is essential in order to observe the EPR resonance peaks. The usual dimension of SMM samples is less than the order of 1mm 3 Therefore, th e experimental set up should be designed in order to increase the coupling between the small sample and the transmitted microwave ( = 0.3 mm 3 cm) for the EPR experiment The long distance between the sample and the microwave radiation source and detect ion due to the magnets is also problematic. In order to put the sample in uniform magnetic field, one needs to find a way to ensure that the sample is loaded in the magnet center. Moreover, when the waveguides are used for carrying the microwave radiation, the long distance of the waveguides reduces the signal amplitude, because the attenuation of microwave radiation depends on the traveling distance in the waveguides (see below). The experimental set up for the EPR studies introduced in this dissertation consists of the probe, the resonance cavity, a Millimeter Vector Network Analyzer (MVNA), and a magnet [ 67 ]. This set up was designed to overcome the technical

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51 obstacles m entioned above and to obtain strong amplitude of microwave coupled with a sample. 2.2 .1 Probe for the EPR Measurements The probe consists mainly of a ca vity and waveguides, though the cavity will be disc ussed in Chapter 2.2.2. R ectangular waveguides are used to enable the EPR probe to carry the microwave radiation to the cavity that sits i n the center of a magnet. Because the distance between the top and the bottom of the probe is about 2 m, the signal amplitude loss by the waveguides is inevitable. Also the probe should work for the wide frequency range to reduce the experimental conditio n change. T his section will introduce the cutoff frequency and the attenuation constant of the rectangular waveguide, w hich was used for our EPR probe An electromagnetic wave inside a waveguide equations (2 2) and appropriate boundary condition. When the inside wall of a waveguide is considered to be the boundary between two media, the tangential component of electric field and the normal component of magnetic field must be c ontinuous across the boundary. Concerning electromagnetic wave p ropagation along the z direction there are three types of solutions: transverse electromagnetic (TEM) mode, transverse electric (TE) mode, and transverse magnetic (TM) mode [ 39 68 ]. The TEM mode has only transverse component s to the propagation direction, i.e. and The TEM mode can not propagate in a hollow waveguide such as ours because the waveguide surface is an equipotential, resulting in the absence of the electric field inside of the

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52 waveguide. Thus, the TE and TM modes are available within the rectangular waveguide. Compared with the TEM mode, the TE (TM) mode has a z component of a z c omponent of an electric field (magnetic field). Considering the boundary condition on the inside of the waveguide wall, therefore, t he boundary conditions o f the two modes are as follows [39 ] : TE mode: and TM mode: and (2 3) Thus, t he magnetic field lines in a waveguide form continuous closed loops surrounding a conduction current or displacement current and the magnetic field lines never start or end on the inside surface of the waveguide. On the other hand, the electric field lines in a waveguide may form continuous closed loops surrounding a changing magnetic field and it may end normally on a charge on the inside surface of the waveguide. The boundary condition of electromagnetic field in a hollow waveguide provides a cutoff frequency of the waveguide. Let us consider a rectangular waveguide with dimensions a and b where a > b for the width and height of the cut plane perpendicular to the propagating direction of the electromagnetic wave in the waveguide, respectively. By the boundary condition, the cutoff frequency of this waveguide is given as follows [39 ]: (2 5)

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53 where and are permeability and perm ittivity of the media inside the waveguide, and m and n are positive integers corresponding to the number of half waves in the width and length direction. The cutoff frequency indicates the low er limit of the frequencies wh ich will propagate in a given mode in a given waveguide condition. When the microwave radiation propagates in a waveguide, the microwave power is attenuated by a factor of due to the ohmic losses in the conducting waveguide inside walls. The attenuation constant is proportional to the ratio of the power dissipated in walls per unit length to the transmitted power. Thus the attenuation constant for the TE and TM mode in a rectangular waveguide is as follows r espectively [ 69 ]: TE mode: (2 8) TM mode: (2 #) where is the surface resistivity of the waveguide wall. A material which has low surface resistivity e.g., a co nducting material, reduce s the loss of the signal amplitude. Additionally, since the sample in a magnet is usually located at low temperature, the waveguides should reduce the heat flow from the top of the probe at room temperature to the bottom of the pro be at pumped 4 He temperature (~1.5 K). Fo r this reason, copper and stainless steel (SS) waveguides are used in turn for the probe used in the EPR experiments. Copper waveguides demonstrate a small loss of microwave power due to the small surface resistivit y, however they have high thermal conductivity.

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54 SS waveguides lose more microwave power than the copper waveguides, but they have low thermal conductivity. Moreover, the impurity EPR signal caused from the rust at the surface of the SS waveguide is problem atic. In order t o reduce power loss through the SS waveguides and remove the impurity signal the SS waveguide s are gold plated on the inside Thus, the combined waveguides with two materials reduce the power loss in the probe, and create thermal stability to maintain low temperature in the cavity and reduce the unnecessary consumption of liquid 4 He [ 70 ]. 2.2.2 Cavity for the EPR Measurements A resonant cavity stores the e nergy of microwave radiation, which has a narrow frequency bandwidth. The microwave a t resonance shows a standing wave configuration which has zero amplitude of electric and magnetic field at the inside walls of the cavity. The cavity is made of a good conducting material (e.g. copper for our cavity sets) in order to reduce the loss of the microwave power on the inside wall and to increase the quality factor Q and sensitivity. For the EPR measurements in this dissertation, two types of cylindrical cavities are employed: a vertical cavity and a rotating cavity which allows us to rotate a sam ple respective to the externa l magnetic field. Figures 2 4A and B exhibit the schematic representation of two cavities. In this section, the cutoff frequency of a cylindrical cavity and the quality factor of the cavity are discussed. The cylindrical cavity is a cylindrical waveguide with the end planes perpendicular to the axis of the cylinder Therefore, the boundary condit ion of the waveguides can be applie d to the cavity. The solution to is the m th order Bessel function where k c is the cutoff wavenumber [ 39 ] For TE

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55 mode, the radial component of the transverse magnetic field should be zero at r = R and for the TM mode, the radial component of the transverse electric field is zero a t r = R (2 9) When the radius and length of the cylindrical cavity are given as R and d the resonance frequencies in the cavity are written as follows [39 ] : (2 10) where m n and p are integer numbers, is the nth root of the equation and is the nth root of the equation Thus, the resonance modes can be described using the m n and p integer numbers, and the possible resonance modes in the cavity are determined by the dimension of the cavity. The resonant cavity is characterized by a quality factor Q. When the resonance occurs in the cavity, the energy will be st ored within the narrow bandwidth of the frequency centered at the resonance frequency in the cavity. The stored energy is smeared out due to the dissipation of the energy in the cavity walls. Q of the cavity is the ratio of the stored energy to the power l oss per cycle in the cavity. The frequency distribution of the energy in the cavity follows a Lorentzian shape [39 ] (2 11)

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56 where is the center frequency of the resonance is the full frequency width at half maximum corre sponding to the Q of the cavity; (2 12) The magnetic resonance of the sample is measured using a cavity perturbation method. A sample loaded in a resonant cavity acts as a small perturbation of the electromagnetic field in the cavity. The perturbation causes the change of the resonance characteristics, i.e. the change in the quality factor Q and the resonant frequency which are related to the change in the complex elect romagnetic response of the sample [ 71, 72 ]. Since the measurement sensitivity is enhanced by the confinement of the microwave radiation energy in a resonant cavity with a high Q value (typically on the order of 10 4 ) a high Q value of a cavity is criticall y required to carry out the EPR measurement using the cavity perturbation method. The leak of the cavity and the coupling hole to the waveguides cause the electromagnetic energy loss in the cavity, resulting in a reduction in Q value and measurement sensit ivity. Therefore, these mechanical factors are a relevant concern for the design of a good cavity. Our cavity sets were designed to have a fundamental mode (lowest frequency mode) TE 011 near 50 GHz, but it is also able to opera te at higher frequencies up t o 6 00 GHz (Chapter 3) The diameter and length of the cavities are determined to optimize the Q value of the cavity to the fundamental mode; the Q value for the TE 011 mode is nearly 21000 in the low temperature range, where we perform our experiments. In order to do EPR measurements using these cavities, the sample should be carefully loaded inside the cavities. In the fundamental mode of the cylindrical cavities, the magnetic field has a maximum at approximately the halfway position of the radius

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57 on the e nd plate of the cavity (Figure 2 4 C ) and along the rotational axis of the cylindrical cavity (Figure 2 4 D ) Since the EPR signal is proportional to the square of the average magnetic field of a microwave at the sample, the sample should be loaded at th at po sition. Additionally, since the magnetic field component of microwaves is strong along the cavity axis, the sample can be loaded on the axial line of the cavity, using a quartz pillar mounted on the end plate of the cavity (Figure 2 4 C ) The quartz pillar p rovides another benefit to preserving the cylindrical symmetry and the coupling in rotation of the endplate for the angle dependence experiments. After coupling with the loaded sample in the cavity, the microwave radiation is returned through the waveguid e. Since the microwave radiation comes in and out from the waveguide to the cavity, the cavity should be coupled with the waveguides via the coupling holes in order to reduce power loss in the microwaves radiation and preserve the resonant mode in the cavi ty [ 68, 70 ]. When the waveguides are terminated by the cavities, the tangential magnetic field inside the waveguides has a maximum at the coupling hole. In order to maximize coupling to the cavity, the coupling hole should be located where the magnetic fie ld inside the cavity is strong and oriented in the same direction as the magnetic field inside the waveguides (Figure 2 5). The coupling strength is determined by the diameter and thickness of the apertures. Large apertures result in strong coupling and la rge dynamic range, but decrease the Q factor of the cavity and sensitivity due to the power loss. On the other hand, small apertures provide a high Q factor for the cavity, transmit strong microwave radiation into the cavity. Our cavity s ets have t he coupling holes with diameter for a frequency of approximately 50GHz In order to reduce the radiation power loss as the microwave

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58 radiation passes through the coupling holes, the coupling plate should be sufficiently th in, [73 ]. 2.2.3 Millimeter Vector Network Analyzer The millimeter vector network analyzer (MVNA) is employed as a microwave source and detector [ 74 ]. Two continuously tunable yttrium iron garnet (YIG) oscillators provide the source frequencies in the range of 8 18 GHz. Two sources (S 1 and S 2 in Figure 2 6) are phase locked to each other through the phase locked loop (PLL) to stabilize the difference between two microwave sources The frequency difference between two sources is dete rmined by a built in programmable 0 14 MHz synthesizer. In order to stabilize the absolute frequencies a n external frequency counter is additionally phase locked to one of the sources [ 75 ]. The passive non linear devices called Schottky diodes are used t o multiply the source frequencies (F 1 and F 2 ) to the frequency range of 45 16 0 GHz. The Schottky diodes are optimized to produce the desired harmonic number N of the tunable microwave source. Each diode multiplies the frequency by a harmonic number: N = 3 and 4 for V band di ode (~45 70 GHz), N = 5 and 6 for W band diode (70 110 GHz), 1 generated from the S 1 is multiplied at the harmonic generator (HG) Schottky diode by harmonic integer number N 1 (Figure 2 6 ) The microwave signal F mm is transmitted to the sample in the cavity through the waveguides. After the coupling of the microwave signal to the sample in the cavity, the returned microwave signal is beat with a second signal at the other diode, the harmon ic mixer (HM). F 2 generated from the S 2 is multiplied at HM by harmonic integer number N 2 resulting in the second signal. Thus, t he beat frequency is given by

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59 If is the phase noise of S 1 ( S 2 ), then the beat phase noise is given by Since S 1 and S 2 are identical and have same phase noise due to the phase locking between two sources, appropriate choice of the harmonic number s cancels the phase noise associated with the beat signal, resulting in a low noise level [75 ]. With the above mentioned choice of harmonic numbers, t he beat signal is ( is intermediate frequency). The intermediate frequency carries the amplitude and phase information to the vector receiver in the MVNA. The signal detection is allowed at the vector receiver when the intermediate frequency corresponds to the operating frequenc y of the MVNA receiver ; There are three different operating frequencies ( 9.01048828125 MHz 34.01048828125 MHz and 59. 01048828125 MHz ), and one of them is chosen depending on the harmonic number N When one chooses N for a measuremen t, the difference in frequency between two sources should be considered, because the PLL is unstable at frequencies around or below 1 MHz. Therefore, 9.01048828125 MHz is chosen for 3 4.01048828125 MHz is chosen for and 59. 01048828125 MHz is chosen for is chosen between 3 and 7 according to the best signal to noise at N = 3, 4, 5, and 6. i s 54. After choosing the operating frequency, the intermediate frequency is down converted to the available signal which can be converted to the DC signal. For example, 34.3249023 MHz is beat with the 25 MHz frequency given from the 50 MHz reference

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60 freque ncy taken at the quartz reference oscillator of the PLL between S 1 and S 2 9.01048828125 MHz is then beat with 9 MHz from the 50 MHz reference frequency The entire down conversion process leaves a 10.488 KHz signal. This final signal can be converted to a DC signal by a lock in amplifier or a data acquisition card. Since the same 50 MHz reference oscillator provides all the frequencies used for the down conversion process, the amplitude and phase information of the beat signal is conserved and transferred to the DC signal. Recent development of the automatic source a ssociation (ASA ) extension extends the possible frequency range to 1 THz [ 74 ]. Moreover, one can electronically tune the frequency over the full frequency range. The ASA extension is built as a chain composed of multipliers (doubler, triplers, and tunable harmonic multiplier), the sextupler (N = 6), and the amplifier (Table 2 1). Figure 2 7A and B are pictures of the harmonic generator (HG) and harmonic mixer (HM) for 670 1000 GHz frequency rang e, respectively. When the sextupler is used without any multipliers, it covers 62 112 GHz. In order to increase the signal power for high frequencies, the amplifier is employed. Since the power generated by the sextupler (S/N 1004) is strong enough to dama ge the amplifier, the 1dB attenuators are alwa ys attached to the sextupler [74 ]. Depending on the multipliers attached to the sextupler, the available frequency range is determined, e.g., 220 336 GHz with a tripler ( N = 3) and 670 1000 GHz with two trip ler s (N = 9) (Figure 2 7 A ) Different detectors are used for the HM, depending on the frequency range. Table 2 1 exhibits available frequencies depending on the harmonic number N and the ASA extension for HG and HM. The HM for 670 1000 GHz has a mechanically tunable harmonic multiplier attached to the sextupler (S/N 127) (Figure 2 7 B ) For this

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61 HM, the intermediate frequency is amplified and directly carried from the HM to the receiver of the MVNA [74 ]. The active sextupler s and the amplifier s are powered by t he external voltage. 2.2.4 Magnets for the EPR Measurements Two s u perconducting magnets are used for the experiments in this dissertation. The split pair magnet of the physical property measurement system (PPMS) produced by Quantum Design [ 76 ] generates a horizontal field up to 7 T. The magnet produced by Oxford Instruments [ 77 ] generates a vertical field up t o 17 T. When the rotating cavity is used in the Oxford magnet, the sample can be investigated on a single rotational axis. When the r otating cavity is used in the PPMS magnet, the sample can be investigated o n two rotational axes using a stepper motor associated with the PPMS. The stepper motor allows us to rotate the sample in the xy plane, while the rotation in the plane perpendicular to the xy plan e is done using the rotation mechanism of the end plate of the rotating cavity [73 ]. The temperature is controlled by a variable flow cryostat in the range of 1.5 400 K with 4 He. 2.2.5 Set up for the High Frequency EPR Measurements The use of the EPR pro be set which consists of the resonant cavity and rectangular waveguides, becomes problematic for high frequency experiments. First, since the resonant cavity is overmoded at high frequencies, it is impossible to determine the microwave field configuration in the cavity for a given frequency. Moreover, the signal loss due to the finite conductivity of the waveguides in creases for high frequencies [39 ]. As frequency increases, the induced current at the inner surface of the waveguides increases, resulting in increasing ohmic loss.

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62 In order to compensate for this problem at a high frequency range, a transmission probe using cylindrical waveguides is employed (Figure 2 8). For this set up, the microwave radiation power from the source is transmitted to the samp le space and returned to the detector, using the cylindrical waveguides. Compared to a sample loaded in the resonant cavity, in this probe set, the sample is loaded on a thin mylar film and is directly coupled with the transmitted microwave, therefore the resonant mode is not important in this probe set. The absence of the cavity results in the simple probe set design and the reduction of effort of delicate mechanical machining. Additionally, compared to the rectangular waveguides, the cylindrical waveguide s reduce the signal loss at high frequencies because the attenuation of the radiation power in waveguides de pends on the waveguide shape [68, 78 ]. In order to reduce the signal loss at the joint, brass cylindrical waveguide is used for the probe without jo int from top to the sample position. The transmitted microwave is reflected by the optically polished copper mirrors in the U shape copper waveguide at the bottom of the probe and returns to the MVNA. The tops of the waveguides are designed to match the py ramidal transition horns associated with high frequencies. The transition horns are employed to carry the microwave radiation from the rectangular waveguides to cylindrical waveguides. Indeed, an EPR signal of BaCuSi 2 O 6 spin dimer compound was observed at high frequencies (864.7 GHz and 934.1 GHz) and at 1.3 K (Figure 2 9). The ASA extensions for the frequency ra nge of 670 1000 GHz (Figure 2 7 A and B ) and the transmission probe (Figure 2 8) are used to take measurements. A 31 T resistive magnet in the Nati onal High Magnetic Field Laboratory is used for the external magnetic field. Thus,

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63 one is able to do EPR measurements in a broad frequency range, from 40 GHz to 1000 GHz, in the EPR group at the National High Magnetic Field Laboratory in Tallahassee. 2.3 Summary In this chapter, we explained our experimental techniques for perform ing EPR experiments Two different probe sets for the experiments are discussed in detail. In addition to the probe set using the cavity perturbation method, the transmission prob e set up also proved to be useful in high frequency EPR measurements. These probe sets allow us to carry out the EPR measurements in a broad frequency range from 40 GHz to 965 GHz.

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64 Figure 2 1. A) Energy level diagram of S = 6 system with g = 2 .00 and negative anisotropy D = 0.5 K. External magnetic field is applied to the z axis. B) Simulated EPR spectrum with parameters used in the energy level simulation at f = 200 GHz at T = 11 K. The r esonance peak has a Gaussian peak shape. Arrows in the energy level diagram associate with the resonance peaks.

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65 Figure 2 2. A) Energy level diagram and B) simulated EPR spectrum of S = 6 system with negative anisotropy D = 0.5 K. External magnetic field is applied to the xy plane. All the parameters for the sim ulation is same with that of Figure 2 1.

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66 Figure 2 3 A) Energy level diagram and B) simulated EPR spectrum of S = 6 system with positive anisotropy D = 0.5 K. Except anisotropy, all the parameters for the simulation is same with that of Figure 2 1.

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67 Figure 2 4 Schematic representation of cylindrical cavities: A) a vertical cav ity and B) a rotating cavity [73 ]. Magnetic field of the TE 011 mode and loaded samples in a cylindrical cavity with C) the side view and D) the bottom view. Arrows represen t the magnetic field direction. A sample is loaded on the bottom (endplate) of the cavity in D) and on the quartz pillar mounted on the endplate in C). Reprinted with permission from Susumu Takahashi and Stephen Hill, Review of Scientific Instruments 76 023114, (2005). Fig. 2, pg. 023114 4. Copyright 2005, American Institute of Physics

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68 Figure 2 5 Coupling of the cavity to the waveguide through the coupling hole located A) at the side of the cavity and B) at the top of the cavity. Dashed arrows repr esent the magnetic field. The TE 011 mode is assumed for the resonance mode in the cavity. Figure 2 6 Schematic representation of MVNA

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69 Figure 2 7 Picture of A) the harmonic generator (HG) and B) the harmonic mixer (HM), respectively, for 670 965 GHz frequency range.

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70 Figure 2 8 Scheme of a transmission probe using cylindrical waveguides for high frequency EPR measurements. A sample is loaded on the thin mylar film. See main text.

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71 Figure 2 9 HF EPR spectra of BaCuSi 2 O 6 spin dimer co mpound with frequency of A) 864.7 GHz and B) 934.1 GHz at 1.3 K. Magnetic field is applied approximately close to the c axis of the crystal. The sharp peak is observed around g c = 2.307 position expected for this compound [ 79 ].

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72 Table 2 1. Available fre quencies depending on the harmonic number and the ASA extension The harmonic generator and mixer are built using the combination of the sextupler, amplifier, and harmonic multiplier N Frequency (GHz) Harmonic Generator Harmonic Mixer 6 62 112 Sextupler amplifier W band Schottky diode 12 124 224 Sextupler amplifier doubler Mechanical tunable detector I (only for N = 12) 18 220 336 Sextupler amplifier tripler I Mechanical tunable detector II (only for N = 18) 24 336 416 Sextupl er amplifier tunable harmonic multiplier 30 416 520 36 520 624 Sextupler tunable harmonic multiplier 42 624 680 54 670 965 Sextupler amplifier tripler I tripler II

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73 CHAPTER 3 RELIEVING OF FRUSTRA TION: THE CASE OF AN TIFERROMAGNETIC M n 3 Z n 2 COMPLEX The results presented in this chap ter can be found in the article Relieving Frustration: the Case of Antiferromagnetic Mn 3 Triangles J. Liu, C. Koo A. Amjad, E. S. Choi, P. L. Feng, E. del Barco, D. N. Hendrickson and S. Hill, Phys. Rev. B (s ubmitted May 2011). 3.1 Background The Mn III ion is frequently used as a basic component of single molecule magnets ( SMMs ) because the clusters composed of Mn III ions often show large spin ground state and an appreciable negative anisotropy [ 1 7, 18, 80 ] Accordingly complexes having a triangular [Mn 3 ( 3 oxo)] core ha ve been extensively studied due to the ir potential as a basic block to make a better SMM 11, 12, 52, 81 89 ] However, for a long time the Mn 3 to SMM characteristics, because of the spin frustration caused by the antiferromagnetic (AFM) exchange interaction in the triangular structure (Figure 1 2 A ) [ 81, 82 ]. Recent studies on the [Mn III 3 O(mpko) 3 (O 2 CR) 3 ](ClO 4 ) (R = Me, Et, Ph, and mpkoH = methyl 2 pyri dyl ketone oxime), and [Mn III 3 O(sao) 3 (O 2 2 O)(py) 3 and saoH 2 = salicylaldoxime ) complexes reported that the sign of the exchange interaction between Mn III ions is determined by a structure distortion [ 83, 90 ]. These complexes have a tria ngular [Mn 3 III ( 3 oxo)] 7+ unit as a magnetic core, and three Mn III ions are connected by the oximate (Mn N O Mn) bridging ligands (Figure 3 1 ). When the oximate bridge is distorted by a significant distortion angle and /or the 3 oxo center shifts out of center of the triangular plane the core structure is distorted. The orbital overlap between Mn III ions through the oxo center and the oximate ligands plays

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74 a significant role in determining the sign of the exchange interaction. Large distortion angle of the oximate bridge (>30) is observed in a Mn 3 III complex which has ferromagnetic (FM) exchange interaction between Mn III ions [ 13 ]. Similarly, the large displacement of the 3 oxo center is observed on the structure of the FM Mn 3 III complex [ 12 ]. Even tho ugh it is not elucidated yet which one is a critical factor to determine the sign of the exchange interaction, the distortion of triangular magnetic core structure clearly causes the FM exchange interaction in a complex. Thus, the FM exchange interaction c aused by the structure distortion leads high spin ( S = 6) ground state of Mn 3 III triangle complex, resulting in SMM characteristics [ 11, 12, 52, 54, 83, 84, 86 ]. Meanwhile, the AFM Mn 3 III complexes were examined as SMMs in order to compare with the FM Mn 3 III complexes. However, t he spin frustration in the triangle of the AFM Mn 3 III complexes prevents reasonable investigations using EPR and magnetization measurements [ 52, 84 ] In the weak exchange interaction limit, the spin frustra tion causes the low lying excited spin states close to the ground state, leading to the ambiguous ground spin state. Furthermore, the spin frustration induces the spin state mixing, resulting in reduction of anisotropy barrier. Therefore, unlike the FM Mn 3 III complex ( S = 6), the AFM Mn 3 III state and the significant bistability either. Accordingly, the EPR spectra has a broad overlapping resonance peaks, and the magnetization hysteresis. Thus, th e spin frustration in the triangle structure is an obstruction to synthesizing and investigating an AFM Mn 3 III SMM. The study of the AFM Mn 6 III complexes suggests the possibility of a well isolated ground state for AFM triangular complex. The Mn 6 complexes consists of two identical

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75 [Mn 3 III ( 3 oxo)] 7+ triangle units which are ferromagnetically coupled to each other, resulting in a maximum possible spin ground state, S = 12 [ 80 ]. The Mn 6 complex, [Mn III 6 O 2 (Me sao) 6 (O 2 CCPh 3 ) 2 (EtOH) 4 ] (Me saoH 2 = 2 hydroxyetha none oxime, and HO 2 CCPh 3 = triphenylacetic acid), composed of the coupled AFM Mn 3 triangles has S = 4 spin ground state, confirmed by the best fitting of the susceptibility data and the EPR data [ 91, 92 ]. T he intrinsic asymmetry of the coupled triangle str ucture of Mn 6 III complexes relieves the spin frustration in the AFM Mn 3 III triangle, resulting in the rise of a significant barrier for the magnetization reversal. Thus, the Mn 6 complex with S = 4 shows reasonably resolved resonance peaks with a big anisot ropy in the EPR measurement s [ 93 ]. A similar asymmetry of the structure was observed in the AFM Mn 3 Zn 2 complex, [NEt 4 ] 3 Mn 3 Zn 2 (salox) 3 O(N 3 ) 8 ] MeOH (here after complex 1 ) where [NE t 4 ] + = tetraethylammonium as a cation ion saloxH 2 = salicylaldoxime, and MeOH as solvate molecules [ 86 ]. The susceptibility data and fitting of the data suggest the S = 2 spin ground state of this complex. Moreover, the magnetic hysteresis measurement suggests that complex 1 has the SMM characteristics. Figure 3 2 shows the hysteres is of magnetization for a single crystal of complex 1 with 0.4 T/min sweep rate at different temperatures from 30 mK to 1.3 K, with the magnetic fiel d applied parallel to the average easy axis direction. The average easy axis of complex 1 will be precisely explained in Chapter 3.2. The hysteresis shows up at the lowest temperatures, with an onset at a blocking temperature, T B 0.8 K. This behavior implies bistability due to a magnetic anisotropy barrier between spin up and down states. Moreover, a clear step in the hysteresis

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76 appears at zero field (see derivatives in the inset of Figure 3 2) indicating the quantum tunneling o f the magnetization (QTM) The magnetization relaxation becomes temperature independent below ~0.2 K which suggests that the relaxation is not caused from the thermal activation, but from the quantum tunneling and/or spin lattice relaxation. It should be noted that these properties, which are normally associated with SMM behavior, have not been reported for the many other extensively studied AF M Mn 3 complexes; indeed, similar investigations of a related high symmetry AF M Mn 3 complex ( complex 4 in the study by Feng et al. [86 ]) could detect no hysteresis to the lowest temperatures (35 mK) investigated. The slow relaxation of the magnetization observed in complex 1 implies that the anisotropy barrier of this complex is strong enough to give the bistability to the magnetization reversal. Finally, a fitting of the high temperature magnetization suggests a ground spin state S = 1.7 ( 2) which is consistent with the estimate from the fi tting of susceptibility data [86 ]. In Chapter 3, complex 1 is investigated using EPR, and magnetic torque measurements to see the relieving of spin frustration and a strong anisotropy in the distorted triangle structure. 3.2 Overview of Structure The AFM Mn 3 Zn 2 complex, [NEt 4 ] 3 [Mn 3 Zn 2 (salox) 3 O(N 3 ) 8 ] MeOH (complex 1 ) and the FM Mn 3 Zn 2 complexes, [NEt 4 ] 3 [Mn 3 Zn 2 (salox) 3 O(N 3 ) 6 X 2 ] where X = Cl (complex 2 ), Br (complex 3 ) have an identical structural shape : a planar [Mn 3 III ( 3 oxo)] 7+ triangle unit as a magnetic core, and non magnetic Zn II ions as a capping ion above and belo w the triangle plane (Figure 3 3 ). Each Mn III (3 d 4 S = 2) ion has a nearly octahedral coordination geometry and an axial Jahn Teller (JT) distortion. Th e triangle of Mn III ions

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77 has a 3 oxo center, and the neighboring three manganese ions are bridged by the peripheral oximate (Mn N O Mn) ligands. The plane of Mn III ions is connected to the Zn II ions through the vertical azido groups (N 3 ) and the Mn azido bonds lie along the axial JT distortion axis of the Mn III ion The nonmagnetic Zn II capping ion doesn t directly contribute to the magnetic property of the complex, but it constrains the JT distortion axis, resulting in the reduced displacement of the 3 oxo center in contrast to the non capped Mn 3 triangle complexes [ 87 ]. The structure of complex 1 (Figure 3 3) is, however, different from the structure of complex 2 and 3 in respect of the Mn Mn distance, Mn N O Mn torsion angle, 3 oxo out of plane shift, and JT tilt angle (Table 3 1). The FM Mn 3 Zn 2 complex has a high symmetric trigonal structure ( C 3 symmetry, R3c space group), resulting in an identical local structural environment for three Mn III ions. On the other hand, complex 1 crystallizes in the mono clinic space group, P2 1 /n where each Mn III ion has a n inequivalent local structural environment. Three different Mn Mn bonding distances of complex 1 indicate that the Mn III ions form a non equilateral triangular core structure. The displacement of the 3 oxo center in complex 1 0.012 is less than that of the FM Mn 3 Zn 2 complexes, 0.025 0.029 Also, the Mn N O Mn (oximate bridge) torsion angles of complex 1 ( = 5.18 9.81 13.49 ) are relatively smaller than those of the FM Mn 3 Zn 2 complexes, indi cating a less distorted oximate bridge. Additionally the JT tilt angle ( = 3.47 3.68 1.35 ) is not so significant that the JT axes of three Mn III ions are almost parallel due to the less distortion of the oximate bridge. Additionally, the low symmetr y structure of complex 1 causes the two molecules within the unit cell to be differently oriented. Figure 3 4 displays the molecules in a unit

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78 cell of complex 1 Since the [Mn 3 III ( 3 oxo)] 7+ plane is almost flat and the JT axes of three Mn III ions are alm ost perpendicular to the Mn 3 III triangle plane, the easy axis of the molecule is perpendicular to the Mn 3 III triangle plane. Since the tilt between the easy axes of two molecules is relatively small, ~5, two easy axes can be considered to be on the same p lane. Accordingly, one can estimate the angle between the easy axes of two molecules from the angle information given in Figure 3 4. The easy axes of two differently oriented molecules are misaligned by ~ 32. There are a few specific structural character istics of complex 1 which contribute to its magnetic property. First, the different local structural environment of the three Mn III ions implies three different exchange interactions between Mn III ions [ 86 ]. Second, the reduced distortion of the structure and the out of plane shift of the 3 oxo center indicates that Mn III ions in complex 1 are connected via the AFM exchange interaction [ 94 ]. Finally, even though this complex has a low symmetric structure, the well aligned JT axes of Mn III ions contribute to the strong magnetic anisotropy o f the complex, as shown in the magnetic hysteresis data. 3.3 Experiments and Simulation 3.3.1 Prediction of Anisotropy using Projection Operator Calculation In the strong isotropic exchange interaction limit, the spin numbe r is a good quantum number, and the relation between the molecular anisotropy and the single ion anisotropy can be derived from the single ion spin Hamiltonian. The relation can be written by the projection operator technique as follows [ 51 ]: (3 1)

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79 represents the molecular uniaxial anisotropy tensor with the ground spin multiplet state, S is the projection coefficient for the single ion uniaxial anisotropy, and is the single ion uniaxial ani sotropy tensor of the i th ion in the N metal ions for the cluster. Although, generally, the dipolar coupling between metal ions also contribute to the (approximately 0.5 K for Mn 3 III ), it is neglected for the Mn 3 III triangle complex es due to an order of magnitude weaker than the single ion anisotropy of Mn III ions (approximately 4 K) The projection coefficients were calculated using multiple techniques in the literature In order to calculate the projection coefficient for clusters having multiple atoms, Bencini and Gatteschi suggest a method using the recurrence relationship which uses the products of coefficients relative to the coupling of two spins [ 42, 51 ]. The projection coefficients for a dinuclear cluster can be written as f ollows: (3 2) (3 3) where S is a total spin number, and S 1 and S 2 are spin number of each atom. In order to calculate the projection coefficients for a trinucle ar cluster from that of the dinuclear cluster, one need s to couple the third atom to the intermediate atom which take the total spin number of the dinuclear cluster as its own spin number S 12 Thus the total spin

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80 numbers for the ferromagnetic trinuclear cl uster ( S = 6) and the antiferromagnetic cluster ( S = 2) are S = ( S 1 + S 2 ) + S 3 = S 12 + S 3 and S = ( S 1 + S 2 ) S 3 = S 12 S 3 respectively [42 ]. According to this method, the projection coefficients of the trinuclear cluster are as follows: (3 4) where is the projection coefficient of i th ion in a trinuclear cluster, is that for a dinuclear cluster, and is that for t he coupled cluster. Thus, one is able to calculate the projection coefficients of the trinuclear cluster and then calculate the molecular anisotropy from the single ion anisotropy values [42 ]. Equation 3 1 and the projection coefficient calculation show t hat when complexes have the same magnetic ions as a magnetic core unit of a complex, the small total spin number of the complex leads to the large molecular anisotropy. The anisotropy value D calculated for a S = 2 Mn 3 III triangle complex and S = 6 Mn 3 III triangle are D S=2 = (69/49) D i and D S=6 = (3/11) D i respectively [ 42 ]. The molecular D value of A FM Mn 3 III complexes with S = 2 is 5 times bigger than that of the FM Mn 3 III complexes due to the small total spin number caused by the AFM exchange interaction [42, 52 ] The anisotropy value calculated using the projection operator method can be used to estimate the zero field splitting (ZFS) of the ground state transition in EPR spectra. Magnetic propert y measurements on complex 1 provide evidence that complex 1 possesses a reasonably isolated S 2, m S 2 ground state [86 ]. As stated above, therefore, D S=2 for the S = 2 Mn 3 III complex is 6.9 K with D i = 4.9 K (this is the best fit

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81 value given below in the simulation part. ) Accordingly, the ground state ZFS D S=2 ( 2 m s +1) 20.7 K, or 430 GHz for the S = 2 ground spin state. 3.3.2 High Frequency Electron Paramagnetic Resonance S tudies High frequency electron paramagnetic resonance (HFEPR) measurements were performed on a singe crystal of complex 1 using a cavity perturbation technique with the frequency range from 50 GHz to 600 GHz and in the temperature range from 1.8 K to 20 K. Two superconducting magnets were used t o apply the magnetic field: a 17 T vertical field magnet for the high field experiments and a 7 T horizontal field magnet for the high frequency experiments due to its more compact size, reducing the optical path to the magnetic field center In situ sample rotation is possible in both syst ems with the rotating cavity [73 ]. As mentioned in Chapter 3.2 c omplex 1 has two different orientations of Mn 3 III molecules in a unit cell leading to the two magnetic easy axis orientations [ 86 ]. Thus, angle dependent EPR experiments were first performed, and then the crystal could be aligned for tempera ture and frequency dependent experiments with the external magnetic field applied roughly to the intermediate orientation which is 16 away from the easy axis of each molecule. It will be shown that better sample alignment was achieved in the high field m agnet. Temperature dependence of EP R spectra taken in the 7 and 15 T magnets are displayed in Figure 3 5 and 3 6, respectively The frequencies which were used for magnetic resonance are indicated in the figures Even though the spectra with the low freque ncies and low fields have various transition peaks due to the low lying states close to the ground spin state, most of this intensity vanishes, leaving behind only a few isolated peaks as the temperature is reduced to 2 K. T he temperature dependence of

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82 eac h peak helps us pick out the transition peaks which have the ground state as an initial state, i.e., a ground state transition. Since the spin distribution of the energy states is changed due to the thermal energy, the ground state transition is significan tly strong at low temperature, and the peak intensity decreases as temperature increases. In the plots, three peaks are relatively strong, which are labeled as , and A weaker peak, labeled is seen only at the lowest two frequencies. Only a single peak ( ) is observed for all temperatures at frequencies of 500 GHz and above. On the basis of the temperature dependence, the resonances labeled , and a re associated with the ground state of complex 1 ; is discussed further below. The magnetic resonance selection rules, = 0 and s = 1 normally allow only a single ground state transition within a spin multiplet state. Accordingly, the observation of three strong resonances, as T approaches to 0, is unusual. However, the weak exchange interaction and the remained appreciable spin frustration cause the spin state mixing, resulting in more than one ground state transition, i.e., the inter spin multiplet ground state transitions, s = 1. The two peaks labeled are separated by about 0 .8 T at 344 GHz in Figure 3 5A meanwhile, only a single peak is observed at both frequencies in Figure 3 6. A similar behavior is found for the peak at the spectra with 593 GHz (Fi gu re 3 5B inset). The observation of double peaks is caused by the two molecular orientations of complex 1 and it indicates that the sample was not perfectly aligned for the high frequency experiments (Figure 3 5). Frequency dependence of the ground transi tion peaks is shown in Figure 3 7 The labeled transition peaks in the temperature dependence plots are tracked in the mul ti frequency EPR spectra from 67 GHz to 600 GHz, and at 2 K. The center position of the

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83 peak is indicated as a data point in Figure 3 7 The error bar is estimated from the linewidth of the transition peak s In the case of the double peaks, the data point represents the average positions with error bars reflecting the associated uncertainty, because we are interested only in the zero fie ld intercepts (ZFS) of the resonance branches. The branches of each resonance are assigned using same labeling as that of the temperature dependence plots. The , and resonance branches are simply linear fitted. The branch shows noticeable curvature, and therefore it was fit using a 2 nd order polynomial. These fitting to the resonance branches, which are displayed as a solid line in the plot, provides the specific y intersection, corresponding to the zero field splitting (ZFS ): = 4 59(2) GHz = 196(2) GHz = 66(2) GHz = 245(2). The average slope of the , and resonance branches shows that g = 2.00, as expected for Mn III in a fiel d misal ignment of ~16. The curvature of the branch suggests the state mixing, which is caused by the weak exchange in complex 1 Compared with the slope of other resonance branches , and we see that the resonance has a bigger slope, ~ 3. 60(7) The dif ference in the slope implies that the resonance is a double quantum transition, S = 2. The double quantum transition is n ot usually observed on the easy axis, but it is observed on the hard plane EPR data because of the state mixing. However, in this complex, the spin state mixing caused by the weak exchange and the field misalignment leads to the double quantum transition. The resonance, moreover, shows weak peak intensity at high fields (Figure 3 6 B ) and it disappears at a relatively lower tempera ture (2 K) than the other peaks. Consequently, t his indicates that peak has a d ifferent origin from the others; it cannot possibly be a ground state transition. Therefore, the resonance will be ignored in the following

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84 discussion. Finally, the resonances are related to the and resonances, respectively, through inversion of the applied magnetic field. These resonances correspond to the ground state transitions within the metastable potential well in a SMM. Therefore, ( ) have the same ZFS (Figure 3 7). In the frequency dependence plot, one can recognize that the ZFS value for the resonance branch is significantly large in comparison with the other resonances and it is close to the predicted ZFS from the anisotropy value cal culated using the projection operator calculation For these reasons, we analyzed and simulated the data by assuming that the resonance branch corresponds to the transition with the nominal S = 2 ground spin state (nominal in the sense that S is not exact due to the state mixing). There is other evidence to support this assignment in the temperature dependence EPR data. The intensity of the excited state and resonances ver y quickly (Figure 3 6 B ) This implies that the matrix element for the resonance, which is the same as that for the resonance, is stronger than those of the and resonances. Although spin state mixing breaks the magnetic dipole selection rules, the m atrix element of the intra spin multiplet transition is reasonably stronger than the inter spin multiplet transitions. Assuming the strong exchange interaction, the magnetic molecular anisotropy value, D can be calculated from the ZFS using the equation D ( 2 m s +1). T he calculated D value from the ZFS value (459(2) GHz) of the branch with the nominal S = 2 ground spin state is about 7.347(2) K. This value is remarkably larger in comparison with the other SMM complexe s: ~1.2 K for a Mn 3 Zn 2 complex with FM

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85 exchange interaction ( S = 6) [ 86 ]. The molecular anisotropy value of complex 1 obtained from the EPR data is six times bigger than that of the FM Mn 3 Zn 2 complex (Table 3 2). Indeed, this huge anisotropy value for an AFM Mn 3 III triangular complex w as estimated in the calculation using the projection operator technique and the exact diagonalization method in the strong exchange interaction limit [ 42, 52 ]. The small total spin number of complex 1 results in a bigger anisotropy value than the other man ganese complexes. The calculated anisotropy value for the AFM Mn 3 III complex is about 6.9 K The discrepancy between the experimental value and the calculated value is due to an assumption of the calculation, such as a strong exchange interaction or a sin gle exchange interaction for all coupling between the magnetic ions. Even though the spin number of complex 1 is relatively small ( S = 2), the anisotropy is big enough to give a rise of a significant barrier leading to the slow relaxation of the magnetic m oment. 3.3.3 Magnetic Torque M easurement The magnetic torque measurement was performed to investigate the chan ge in the ground state moment in the high magnetic field range of up to 35 T using a resistive magnet in National High Magnetic Field Laboratory ( NHMFL). Since the low lying state is close to the ground state due to the weak exchange interaction the ground state can be shifted from the nominal S = 2 state to the high spin multiplet state, S > 2, due to the Zeeman effect. A harmonic cantilever beam torquemeter with capacitative sensing was used for the measurements. The torque on the sample caused from the applied magnetic field ( ) induces the deflection of the cantilever beam, which is proportional to the measured capacitance of the torquemeter. A crystal of complex 1 was loade d on the edge of the cantilever, and then the direction of the applied magnetic

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86 field wa s aligned close to the magnetic easy axis of the crystal of complex 1 Temperature was controlled using a 3 He refrig erator in the range from 0.3 K to 10 K. Figure 3 8 shows the plot of the capacitance of the torque cantilever magnetometer versus the magnetic field for a series of different low temperatures from 300 mK to 10 K. The magnetic field was swept at a rate of 3 T/min. Since the capacitance is approximately proportional to the torque signal, the capacitance change is indicative of the magnetic moment change. change, i.e., the spin ground state is well isolated from excited spin sta tes, the low temperature torque signal displays a monotonic magnetic field dependence. In Figure 3 8, t display significant feature s up t o 28 T, indicating a well defined magnetic moment of the ground state. The oscillation of the signal i s observed from ~28 T to high magnetic field, and it is significant at lower temperature. The signal oscillation demonstrates the magnetic moment change, implying the multiple spin crossover transition for complex 1 3.3.4 Simulation of Energy Level Diagr am The observed EPR resonance modes and the spin crossover transition were simulated using a multi spin Hamiltonian as follows: (3 2) The first term is a single ion anisotropy term. represents uniaxial anisot ropy value The second term is a Zeeman term with g = 2.00, and the last term is the isotropic exchange interaction term. Since the magnetic field is applied to the average easy axes, the contribution of the transverse anisotropy to the simulation result i s not significant, so it wa s ignored in the simulation. Additionally, the tilting of the JT axis of single ions is not

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87 considered in the simulation for the simplification of the matrix calculation. Indeed, since the JT axes are almost parallel in complex 1 the ignorance of the JT axis tilt is acceptable in this simulation. T he variable parameters of the simulation are the uniaxial single ion anisotropy of Mn III ion and the exchange interactions between ions. Since the single ion anisotropy is relatively i nsensitive to the structure distortion as shown in the EPR studies of related FM Mn 3 Zn 2 complexes [ 86 ], the D i parameter was restricted in the range from 4 6 K during searches of the best simulation. Thus, the best combination of four parameters ( D i J 1 J 2 and J 3 ) in Equation 3 2 has been searched to account for the observed experimental data: the ZFS values of three resonance modes observed in EPR spectra, the spin crossover transition in the appropriate magnetic field (from 25 T to 36 T), and the differ ent resonance intensity between the , and resonances. The comparison of the resonance intensity was allowed by the computation of the magnetic dipole matrix elements for all transitions from the ground spin state. A cut off of was employed; for reference, the matrix element for the resonance is ~2.5. The best simulation was achieved with the following parameters: D i = 4.9 K, J 1 = 6.9 K, J 2 = 7.3 K, and J 3 = 11.8 K. As expected from the structure of complex 1 all of the exc hange intera ctions are the AFM interactions and their magnitude are different. The negative value of D i is indicative of the easy axis type anisotropy and the magnitude of D i is close to those found for the related FM Mn 3 Zn 2 complexes [86 ]. Figure 3 9 di splays the plot of energy levels as a function of the m s value using E quation 3 2 with the parameters mentioned above Because of the AFM exchange interactions, the S = 6 state appears at the high energy level, and the low spin states

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88 are positioned at the low energy level. Figure 3 9 B shows the zoom in plot of low energy state levels responsible for most of the low temperature EPR intensity; the colors and sizes of the data points have been coded according to the expectation value of with radii proportional to this value Since transverse anisotropy is ignored in the simulation, m s remained an exact quantum number, whereas S clearly did not. The energy difference of the each spin states from the ground spin state correspo nds to the ZFS value of the three resonance branches in frequency dependence dat a (Figure 3 7). Thus, the resonance is the ground state transition in the same spin multiplet, The and resonances are assigned to the inter spin multiplet g round transition, Since the degeneracy of the spin states (e.g. S = 1 state) is broken due to the weak exchange interaction, two ground transitions to the state are allowed. The intensity difference between the resonances observed in temperature de pendent EPR spectra (Figure 3 6 B ) is explained with the matrix element computation. The matrix element associated with the and resonances is significantly weaker than the resonance by a factor of 4 for and by two orders of magnitude for Although the spin frustration causes the complicated overlap of the spin states at the low energy level, the state is separated from the first excited state, by 3.3 K. The energy gap between the split degenerated S = 2 states is 3.9 K. Thus, the ground spin state is well isolated from the excited states. Figure 3 10 shows the plot of the energy level as a function of the DC magnetic fi eld along the easy axis. Three resonance modes are indicated by the color arrows.

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89 The resonance is the ground spin state transition in the nominal S = 2 spin multiplet state. On the other hand, the resonance and are the inter spin transitions from the nominal S = 2 state to the nominal S = 1 state. Also, the change of the spin ground state from to is well represented in the simulation as the crossover of th e energy levels occurs around 28 T. Thus, the simulation broadly agrees with the experimental results. The contrast of the anisotropy value betw een the FM Mn 3 Zn 2 complexes and complex 1 is noticeable. Although the single ion anisotropy for the Mn III ion is same due to the almost identical local structural environment, the molecular anisotropy for complex 1 is different from that of the FM Mn 3 Zn 2 c omplexes. Indeed, the D i = 4.9 K from the best simulation is almost the same as that of the FM Mn 3 Zn 2 complexes. However, the molecular anisotropy value, D S=2 = 7.34 7(2) K, of complex 1 is significantly bigger than that of the FM Mn 3 Zn 2 complex, D S=6 1.2 K. The projection operator calculation already shows that the big anisotropy of complex 1 comes from the small total spin number. In addition to that, this difference of molecular anisotropy values can be explained by the JT axis of the Mn III ions. Th e distortion of the triangle structure causes the tilting of the JT axes for the FM Mn 3 Zn 2 complexes, whereas, the JT axes for complex 1 are almost parallel. In addition to the small total spin number, the parallel JT axes of the Mn III ions also contribute to the strong molecular anisotropy [ 95 98 ]. 3.4 Summary The AFM Mn 3 Zn 2 triangular complex which shows the spin frustration was investigated by the HFEPR measurement and the magneto torque measurement. The

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90 intrinsic asymmetry of complex 1 relieves the spin frustration in the triangular spin structure, resulting in the well isolated spin ground state, S 2. The EPR data demonstrate that the anisotropy of complex 1 is the easy axis type, and the anisotropy value is bigger than that of the FM Mn 3 Zn 2 complexes due to the small total spin number, S 2, of the AFM Mn 3 Zn 2 The simulation of the EPR data and t he magneto torque data confirms the relieving of the spin frustration and the huge anisotropy value. Even though complex 1 has state mixing due to the spin frustration, the anisotropy barrier is strong enough to block the magnetization at low temperatures

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91 Figure 3 1. Structure of the Mn III triangular core with the oximate bridge (Mn N O Mn) and the 3 oxo center. Mn : magneta, O : red, N : blue. The out of center shift of the 3 oxo center is shown with the dashed arrow lines. A plane by Mn N O and the other plane by Mn O N are drawn as transparent green planes with dotted line. The angle between two transparent green planes is indicated with corresponding to the oximate bridge distortion angle.

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92 Figure 3 2. Magnetization hysteresis at different temperatures below the blocking temperature as a function of the magnetic field applied along the z axis Magnetic field sweep rate is 0.4 T/ min. The inset shows the field derivative of the magnetization curves.

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93 Figure 3 3. Structure of the AFM Mn 3 Zn 2 mol ecule viewed A) near the ab plane and B) along the c axis. Mn : magen ta, Zn : purple, O : red, N : blue, C : grey. Hydrogen atoms are omitted for clarity.

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94 Figure 3 4. Two differently oriented molecules of the AFM Mn 3 Zn 2 in a unit cell (gray cube). Same color label with Figure 3 3. Hydrogen atoms are omitted for clarity. The solid green lines display the easy axis direction of the molecule. The easy axes of two differently oriented molecules are misaligned by ~ 32

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95 Figure 3 5. Temperature dependent high frequenc y EPR spectra obtained in the 7 T horizontal field, split pair magnet. The fr equencies are indicated in the main panel, and fine temperatures are indicated in the legend. The ground state resonances observed at the lowest temperatures have been labeled accordingly using Roman letters. The inset to B) shows the high est frequency (59 3 GHz) data obtain ed in this study, revealing a resonance at 2. 0 K.

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96 Figure 3 6. Temperature dependent high frequency EPR spectra obtained in the 17 T vertical field magnet. The frequencies are indicated in the main panel, and five temperatures are sho wn. Several of the main resonances have been labeled (see main text).

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97 Figure 3 7. Frequency dependence of the resonance peaks taken at many frequencies in the range from 60 to 600 GHz and at 2 K. The data points have been coded according to the assoc iated resonance branches. The solid lines are linear fits for , and and 2 nd order polynomial fit for

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98 Figure 3 8. Capacitance of the cantilever at various temperatures as a function of the magnetic field. The field sweep rate is 3 T/min.

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99 Figure 3 9. A) Best simulation of the zero field eigenvalue spectrum generated from Equation 3 2. The highest energy states belong to well defined spin multiplets, a few of which have been labeled accordingly. B) Zoom in of low energy region of the spectr um which is highlighted in A). The colors and sizes of the data points in B) have been coded according to the expectation value of with radii proportional to this value (see legend also). The presumed and resonances have been marked on the figure.

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100 Figure 3 10 Simulated energy level diagram as a function of the applie d magnetic field along the easy axis. The simulation parameters are introduced in the main text. The three arrows correspond to the three resonances by the color legend. The vertica l black arrows above 25 T indicate the locations of spin crossover transitions from | to

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101 Table 3 1. Comparison of structural parameters for the AFM and FM Mn 3 Zn 2 complexes. Parameters for FM Mn 3 Zn 2 are prov ided by Feng et al. [86 ]. Complex 1 2 3 S 2 6 6 Space group P2 1 /n R3c R3c Mn Mn bonding distance ( ) 3.270, 3.276, 3.260 3.276 3.274 3 oxo center out of plane shift () 0.012 0.029 0.025 Mn O N Mn distortion angle ( ), (deg) 5.1 8, 9.81, 13.49 32.05 32.08 Jahn Teller tilt angle ( ), (deg) 3.47, 3.68, 1.35 8.50 8.09 Table 3 2. ZFS parameters obtained via simulation of HFEPR data of the AFM and FM Mn 3 Zn 2 complexes. Parameters for FM Mn 3 Zn 2 are provided by F eng et al. [86 ]. Complex S D S (K) D i (K) J (K) 1 2 7.35 4.752 5.520, 4.008, 3.480 2 6 1.157 4.507 2.077 3 6 1.177 4.598 2.267

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102 CHAPTER 4 MAGNETIC ANISOTROPY BARRIER OF HETEROMET ALLIC HIGH SPIN C u 17 M n 28 C LUSTER WITH T D SYMMETRY Reproduced with permission from Journal of the American Chemical Society submitted for publication. Unpublished work copyright (2011) American Chemical Society. 4.1 Background Extensive experimenta l and theoretical studies on homo and heterometallic complexes have been carried out in order to synthesize a new single molecule magnet ( SMM ) complex [ 60, 61, 99 107 ]. In order to increase the anisotropy barrier, between the up and down magnetic moment states in a SMM complex, the molecular topology, which maximize s the overall spin and negative anisotropy, is required. The large heterometallic complexes consisting of various 3 d transition metals were studied to achieve a large spin ground state [ 10, 60, 61, 107 ]. However, as mentioned in Chapter 3 an d pointed out in the several detailed studies [ 92, 108 ], the total spin number increase does not necessarily improve the SMM c haracteristics due to the inverse relation between the molecular anisotropy and the total spin number. Furthermore, the complicate d structure of large molecules prevents the detailed study of the electronic structure and magnetic properties Nonetheless, there are still several reasons to study some of these systems as they may exhibit interesting topologies, symmetr ies, or interesti ng magnetic phenomena. Wang et al. reported a large heterometallic complex, [Cu 17 Mn 28 O 40 (tea) 12 (HCO 2 ) 6 (H 2 O) 4 ] 36H 2 O hereafter referred to as complex 4 and where H 3 tea is short for triethanolamine [ 10 ]. This complex is intriguing for a number of

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103 reasons: ( 1) the transition metal ions in complex 4 have five different metal oxidation states, Cu I 4 Cu II 13 Mn II 4 Mn III 12 Mn IV 12 Most polynuclear high spin molecules consist of metal ions in one or two oxidation states [ 17, 80 ]. A few h igh spin molecules have been synt hesized with three different oxidation states of manganese [ 10, 109 112 ]. However, complex 4 is the first high spin complex composed of manganese and copper with five different oxidation states. (2) complex 4 has a relatively large intermediate ground spin state S = 51/2 with low lying excited states The energy gap between the intermediate ground spin state and the low lying excited state S = 63/2 is = 5 K [10 ]. The intermediate ground spin state demonstrates that there are competing ferromagnetic and antiferromagnetic exchange interaction between the metal ions. (3) Complex 4 has T d molecular site symmetry which forbids second order magnetic anisotr opy [ 43 113 ]. However, magneti zation versus field data clearly shows temperature dependent hysteresis behavior, which can be caused by a magnetic anisotropy barrier. Moreover, analysis of magnetization relaxation data for complex 4 provides an estimate of the effective thermodynamic anisotropy barrier ~ 17 K [ 113 ]. Thus, the slow relaxation of magnetization in complex 4 cannot be explained by large spin in conjunction with the second order anisotropy. T herefore, an interesting question arises regarding the origin of quantum magnetization relaxation behavior and magnetic metastability. In this chapter, single crystal high frequency electron paramagnetic resonance ( HF EPR ) measurements are employed to study the exhibited SMM behavior of complex 4 and to elucid ate the origin of magnetic metastability. 4.2 Overview of Structur e Complex 4 has T d molecular site symmetry and crystallizes in the cubic space group I 43m Beedle et al. provide the detailed x ray data collection parameters,

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104 structural coordinates and a description of solving techniques for complex 4 [ 113, 114 ]. The [Cu 17 Mn 28 O 40 ] 42+ core of complex 4 contains four Cu I ions, thirteen Cu II ions, four Mn II ions, twelve Mn III ions, and twelve Mn IV ions (Figure 4 1). The main core of complex 4 consists of si x [Mn IV 2 Mn III 2 O 4 ] 6+ distorted cubane units (Figure 4 2 C ) and four [Mn IV 3 Mn II O 4 ] 6+ distorted cubane units (Figure 4 2 D ) Each [Mn IV 3 Mn II O 4 ] 6+ unit shares its three Mn IV ion s with three neighboring [Mn IV 2 Mn III 2 O 4 ] 6+ units These cubane units form a symmetric tetrahedral cage through the in terconnections between cubane units (Figure 4 2 A and B ). One of the thirteen Cu II ions is at the center of the manganese cage, and the rest of them are connected to each [Mn IV 3 Mn II O 4 ] 6+ cubane unit and capped by the twelve a minoalcohol ligands (tea 3 ). E ach of the four Cu I ion s connect s three [Mn IV 2 Mn III 2 O 4 ] 6+ cubane units together through their oxygen ions. The neutral complex sphere is accomplished by six formate (HCO 2 ) ligands which bridge the six pairs of tetragonally e longated Mn III ions and four water molecules that are coordinated to Cu I ions The intermediate ground spin state S = 51/2 ( the fully ferromagnetic moment would be S = 117/2 ) of complex 4 demonstrates that there are competing ferromagnetic and antiferro magnetic exchange interactions between metal ions. The competition of exchange interactions induces spin frustration. Figure 4 3 displays the interconnection and exchange interactions between the metal ions in a part of the manganese core cage of complex 4 A [Mn IV 2 Mn III 2 O 4 ] 6+ cubane unit interconnects to a [Mn IV 3 Mn II O 4 ] 6+ cubane unit through Mn IV sharing and Cu II ion bridges Mn IV Mn IV Mn IV Mn III and Mn IV Mn II interactions tend to be an t iferromagnetic [115 ]. T he type of the Mn III O Mn III exchange interac tion depends on the bridging M O M bond angle e.g., when the bridge

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105 angle approaches 90 the interacti on tends to be ferromagnetic [113 ]. Since t he Mn III O Mn III bond angle in complex 4 is 95.0(3), the exchange interaction between Mn III ions is expected to be ferromagnetic In Figure 4 3, two Mn III ions in a [Mn IV 2 Mn III 2 O 4 ] 6+ cubane are ferromagnetically coupled, leading to the up spins (blue arrows), and one of the Mn IV ions in the cubane unit has the down spin (blue arrow) due to the antiferromagnetic exchange interaction. However, the other Mn IV ion cannot anti ferromagnetically couple with the both a neighboring Mn IV and Mn III ions simultaneously. This same frustration also occurs in the [Mn IV 3 Mn II O 4 ] 6+ cubane units where the antiferromagnetic exchang e interactions are frustrated between the Mn II and the three Mn IV ions. Thus, complex 4 is severely spin frustrated resulting in the intermediate spin ground state S = 51/2. 4.3 High Frequency E lectron Paramagnetic Resonance E xperiments High frequency (4 0 160 GHz) electron paramagnetic resonance (HFEPR) measurements were carried out on a single crystal of complex 4 using a millimeter wave vector network analyzer (MVNA) and a sensitive cavity perturbation technique [ 70 ]. A 7 T horizontal field supercondu cting magnet wa s used to apply a DC magnetic field to the crystal, which was loaded in a cylindrical cavity. The angle dependence of EPR data was investigated by chang ing the angle between the orientation of the crystal and the magnetic field using a room temperature step motor in the magnet set up Temperature was controlled using a liquid 4 He flow system. EPR measurement is a powerful method to determine the anisotropy value and the ground spin state of a given complex. As shown in Chapter 2, the anisotro py splits the 2 S resonance peaks in an EPR spectrum performed on a given crystal The separation of resonance peaks depends on the magnitude of anisotropy and the applied

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106 ma gnetic field orientation. In high magnetic fields the esti mate of the peak separation is about for a magnetic field parallel to the easy axis where D is a uniaxial second order anisotropy On the other hand, if a complex is isotropic, a single sharp peak is observed instead of the discret e 2 S peaks in the spectra. For example, an EPR study on the Mn 13 complex, which is isotropic, shows a sing le sharp resonance peak with 0.4 T linewidth in the spectra at 2.5 K [ 116 ]. When the anisotropy value is weak, a single broad peak is observed i n a n E PR spectrum and the 2 S resonance peaks are not clearly resolved However, the broad linewidth of the resonance peak suggests that a complex has anisotropy. Even though the anisotropy splits the 2 S resonance peaks, the peak separation is still not wide enou gh to resolve individual peaks, so that what emerge s is a br oad peak feature composed of overlapping 2 S resonance peaks. The EPR study on the powder sample of the Mn 25 complex, which has anisotropy D = 0.024 K, reported a broad peak feature due to a weak zero field splitting ( ZFS ) in the spectra [ 109 ]. Unlike an EPR data for a crystal, the EPR spectra for a powder sample have th e transition peaks for the easy axis and hard plane simultaneously. The separation between the ground trans ition peaks for the eas y axis and hard plane depends on the anisotropy in the system. For the Mn 25 complex, the anisotropy value is relatively small, leading to overlap of the groun d transition peaks for the easy axis and hard plane. Therefore, the EPR spectra for the Mn 25 compl ex have the broad peak feature with the 1 T linewidth at 1.4 K. A broad single peak feature is observed in HF EPR spectra for a single crystal of complex 4 Figure 4 4 shows the temperature dependent EPR spectra in the temperature range of 3 20 K with 91 .4 GHz. The external magnetic field is applied

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107 close to one of the extrema observed from the angle dependent measurements (see below). Even though complex 4 has a high spin number S = 51/2, only a broad peak feature is seen in the EPR spectra, and no other peak feature is observed in the wide magnetic field range up to 5 T. As mentioned earlier in a system where the anisotropy is weak, a single broad peak is typically observed instead of discrete peaks. Thus, the broad linewidth suggests that complex 4 has weak anisotropy The evidence for the anisotropy of complex 4 can be also seen in the temperature dependence of the EPR spectra. As the temperature decreases from 20 K to 3 K, a single symmetric peak at 20 K becomes asymmetric, causing the peak position to shift to a higher field The peak shape change is most likely due to the Boltzmann depopulation effect and anisotropy. Since the spin population in the excited states and the ground state is uniform a t high temperatures e.g., T ~ 20 K the EPR spectrum is most likely to be the paramagnetic resonance spectrum, which shows a sharp symmetric peak at the position corresponding to g = 2.00, i.e., 3.26 T for f = 91.4 GHz. However, as the temperature decreases, the peak becomes asymmetric due to the non unifor m spin population in the spin states. At low temperatures, the ground spin state and low lying excited states are more populated than the other excited states, and the intensity of the ground state and low lying state resonance is stronger than that of the other resonances in the broad peak feature Therefore, the portion of the spectrum that shift s away from the average g position of the complex produces both the asymmetric peak shape and the peak shift. The average g value ( g = 2.10, see below) of complex 4 is slightly greater than 2.00 due to the Cu II ion in the complex. Depending on the applied magnetic field orientation, the peak shift occurs either to the low or high field sides of

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108 the average g position of the complex. The peak shift is clearly shown at the low temperatures ( k b T < Zeeman energy), when only a few spin states, which are sensitive to the anisotropy, are populated. Thus, the asymmetric peak feature at low temperatures implies that complex 4 has magnetic anisotropy Angle dependent EPR spe ctra observed at 2 K with 60.0 GHz further confirms the anisotropy of complex 4 (Figure 4 5). Figure 4 6 demonstrates the angle dependence of the peak center (black squares) taken from the EPR spectra (Figure 4 5) The sample alignment within the cavity wa s not perfectly done due to the small size and irregular shape of the crystal T herefore, the rotation plane for the measurement was not known. Nevertheles s, a clear and systematic angle dependence is shown in the data. Four minimum peak positions app ear w ithin a 90 angle interval, and a mong these four peak positions, the two lowest field peak positions appear within a 180 angle interval (Figure 4 6 ). The extrema in the angle dependence data plot occur when the applied magnetic field is oriented near the principal axes of the ZFS tensor. For example, a complex, which has the second order transverse anisotropy associated with shows two fold periodicity in the angle dependence plot of the EPR resonance peak center a s the applied mag netic field orientation changes in a rotational xy plane. If a complex has the fourth order transverse anisotropy associated with it shows four fold periodicity in the angle dependent EPR data in a rotational xy plane Figure 4 7 s hows the geometric presentation of two transverse anisotropy terms and The polar presentation of two anisotropy terms clearly shows the two fold and four fold symmetry. Thus, the angle dependence of the peak center implies that complex 4 has anisotropy.

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109 4.4 Simulation of Angle Dependent EPR D ata Even though two modes of periodicity in the angle dependence data are observed, four fold anisotropy is allowed but two fold anisotropy is not allowed for complex 4 due to the T d point symmetry of the complex. When the T d site symmetry is strictly preserv ed in a complex, the leading ZFS term in the spin Hamiltonian would be the fourth order term given in E quation 4 1 : (4 1) with (4 2 ) and (4 3 ) where is the total spin operator with the x, y and z component of spin operator ; , and respectively. and are the fourth order axial and tetragonal Stevens operators, respectively, and is the coefficient of the cubic ZFS interaction. Using Equation 4 1, the giant spin Hamiltonian for complex 4 is as follows; (4 4 ) The angle dependence of the EPR resonance center is simul ated using the spin Hamiltonian with the fourth order spin operators allowed in the T d symmetry of complex 4 The energy of th e spin states are calculated by diago nalizing the spin Hamiltonian, E quation 4 4 When the energy difference between the ground state and the lowest excited state matches a given frequency one can calculate the ground transition peak

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110 position. As the appl ied magnetic field orientation changes in a rotation plane (Figure 4 6 inset), the ground transition peak position for each field orientation is calculated, and then the angle dependence of the peak position can be plotted as a function of the relative ang le of the applied magnetic field orientation to the initially aligned field direction The field rotational axis is determined using the Euler angle ( , ). The best simulation of the angle dependent EPR data is indicated by the red circles in Figure 4 6 As mentioned earlier, f our minimum peak positions appear in approximate intervals of 90 and a mong these four peak positions, the two lowest field peak positions appear in a 180 angle interval. The parameters for the simulation are as follows; S = 51/ 2, g = 2.1 0, = 1.92 10 7 K, = 45 = 33 = 0 f = 60 GHz. As noted above, the difference of the g value from 2.00 is likely due to the Cu II ions in the complex, which has g = 2.40 [117 ]. The parameters for the Euler angle s indicate that the assumed field rotational plane is m isaligned to the xy plane with respect to the cubic ZFS tensor (Equation 4 1). As seen in Figure 4 6 the two fold periodicity in the angle dependent EPR data of complex 4 does not require the second order uniaxial anisotropy. The fourth order anisotropy a nd the magnetic field tilt ed away form the principle axes are combined to cause the two fold periodicity in the angle dependent T d symmetry of complex 4 and it provides the estimate of anisotropy of complex 4 The magnitude of the anisotropy is, however, not strong enough to give rise to a significant effective barrier to cause the slow relaxation of magnetization observed in the magnetic hysteresis data. This is clearly shown in Figure 4 8, which demonstrat es a representation of the 2 S +1 ( S = 51/2) eigenstate solutions to Equ ation 4 4 at zero field

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111 using a basis. Figure 4 8 is the 3D plot, where the X axis of the plot is for the m s spin states, the Y axis is for the energy of the spin states, and the Z axis is for the probability of the m s spin states. The m s spin state is not an ideal basis as a representation of the spin states for a cubic system. This is because the probabilities of observing the spins along the x y and z axes in a cubic system are equal, i.e., the m s spin states are significantly mixed and degenerated. Nevertheless, this representation is used for comparing the cubic case to the axial cases (low symmetry Mn 12 SMM) [11 8]. The anisotropy barrier is exhibited in the XY plane of Figure 4 8. Since the fourth order ter m is a dominant ZFS source in this Hamiltonian, the barrier has an M shape, which is different from the parabolic bar rier shape caused by the second order term in the axial systems [ 11 8 ]. The energy differe nce between the lowest energy level and the highest energy level essentially corresponds to the magnitude of the anisotropy barrier. According to the magnitude of the barrier, 0.965 K, t he simulated anis otropy barrier using the fourth order spin operators is much smaller than the 17 K barrier calculated from the magnetic hysteresis data. Moreover, the severe spin state mixing caused by the transverse fourth order anisotropy terms reduces the strength of the effective barrier. As noted above, the symmetry ( T d ) of complex 4 results in the degeneracy of spin states; classically, six ground states corresponding to the spins along , and axes are expected, in contrast to the two ground states for an axial symmetric complex. However, the lowest six states are grouped into a doublet or quartet, separated by a small tunneling gap. Indeed, all of the states belong to either a degenerate doublet or quartet, which is rational ized in group theoretical consideration, e.g., a pair of quartet states is observed

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112 at the highest energy, where the classical counterparts would involve spins aligned along each trigonal axes. The probabilities of each degenerate spin state group are summ ed and represented as sticks in Figure 4 8. There are 8 doublets and 9 quartets demonstrated as red sticks and blue sticks, respectively. As exhibited in Figure 4 8, the spin state mixing in complex 4 is significant, in contrast to the axial case, even in axial cases with rather significant transverse anisotropy [11 8]. Finally, the tunnel splitting of ~30 KHz between the doublet and quartet ground states is calculated in the simulation of the EPR data. Thus, the anisotropy in complex 4 could not give rise t o magnetic bistability. Hence, instead of magnetic anisotropy, one must consider other possible explanations for the slow relaxation of magnetization observed in the magnetic hysteresis data for complex 4 Recent studies on zero dimensional systems [ 119, 12 0 ], the Kagome lattice [ 2 1 ], and larger quantum and classical spin lattices [ 121 ] have shown that the geometric frustration in a high symmetry system can give rise to metastability, leading to the slow relaxation and hysteretic behavior of the magnetizat ion without any magnetic anisotropy. When the spins on the vert ices of edge sharing triangles or other geometric shapes ( such as cubes and tetrahedrons ) in a highly symmetric system are antiferromagnetically coupled, the metamagnetic phase transition occur s as the external magnetic field forces another ground spin state configuration The geometrical frustration originating from the structure leads to competition between different spin configurations (different low energy spin states) in the system. If an a pplied magnetic field tends to favor one conf iguration over another, this give rise to the metastability and

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113 the slow magnetization relaxation behavior in the absence of any magnetic anisotropy [12 0]. Indeed, the cubane units in the core of complex 4 are c onnected via the edge shar ing and corner sharing, and the spins within the cubane units are geometrically frustrated due to the antiferromagnetic couplings between the metal ions in the cubane units (Figure 4 3 ). Thus, t he observed slow relaxation and hyst eretic behavior in the magnetization data of complex 4 could arise due to competition between the Heisenberg antiferromagnetic exchange coupling of geometrically frustrated spins and the susceptibility of those spins to align each other in an external magn etic field. 4.5 Summary HF EPR measurements w ere conducted on complex 4 to investigate the origin of its metastability, which is observed in the magnetization data Even though only a single broad EPR peak feature i s observed in the EPR spe ctra, temperature dependent and angle dependent EPR data suggest that there is magnetic anisotropy in complex 4 Simulations of EPR data using the spin Hamiltonian including the fourth order spin operators (Equation 4 4) indicates that the complex has fourth order anisotro py following the T d symmetry structure. However, the anisotropy barrier simulation demonstrates that the barrier height (~0.965K ) is significantly lower than the value obtained through magnetization relaxation experiments (~17K). Thus, the anisotropy in co mplex 4 is unable to account for the magnetic bistability exhibited in the magnetization data This study demonstrates that the magnetic hysteresis can be observed ev en in the absence of the second order axial anisotropy, and it implies that the magnetic h ysteresis is not a sufficient condit ion to categorize a complex as SMM.

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114 Figure 4 1. Oak Ridge Thermal Ellipsoid Plot (ORTEP) represents the molecular structure of complex 4 Atom color code: Mn, magenta; Cu, blue; O, red; N, sky blue; C, white. Hydro gen and water solvate molecules have been omitted for clarity.

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115 Figure 4 2. Stereo pair ORTEP illustration of the core Mn 28 cluster A ) and B) containing six [Mn IV 2 Mn III 2 O 4 ] cubanes C ) and four [Mn IV 3 Mn II O 4 ] cubanes D ). Atom color code: Mn IV (Mn1), vio let; Mn III (Mn2), wine; Mn II (Mn3), magenta; O, red.

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116 Figure 4 3. Exchange pathways and interconnectivity of [Mn IV 2 Mn III 2 O 4 ] 6+ cubane unit, [Mn IV 3 Mn II O 4 ] 6+ cubane unit and Cu II ions in the manganese core cage of complex 4 Blue arrows indicate spin orientation of ions, and blue question marks indicate spin frustration.

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117 Figure 4 4 EPR spectra of complex 4 with various temperatures with 91.4 GHz. The magnetic fie ld is perpendicular to the easy axis of the complex.

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118 Figure 4 5. Angle dependen ce of the EPR spectra for complex 4 taken at 2 K and 60.0 GHz. The data were collected over a 360 angle range in 10 steps.

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119 F igure 4 6 Plot of the EPR peak center position as a function of the angle of orientation of the applied magnetic field; the plane of field rotation was not known a priori Black squares indicate the experimental data obtained at 2 K and 60 GHz (Figure 4 5). The red circles represent the best simulation of the data. The line between data points is for guidance. The parameters fo r the simulation are introduced in the main text. The inset shows the magnetic field rotation plan e, corresponding to the Euler angle parameters used in the best simulation

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120 Figure 4 7 Polar presentation of two transverse anisotropy terms, A ) and B )

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121 Figure 4 8 R epresentation of the 52 eigenstate solutions to Equ ation 4 4 at zero field using a basis. The X axis is for the m s spin states, the Y axis is for the energy of the s pin states, and the Z axis is for the probability of the m s spin states. All of the states belong to either degenerate doublet (red stick ) or quartet (blue stick ) levels. There are 8 doublets and 9 quartets, giving a total of 52 states. The probabilities o f each degenerate spin state grouping are summed and represented as sticks. The black dashed line represents the classical anisotropy barrier.

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122 CHAPTER 5 SINGLE ION ANISOTROP Y IN AN ANTIFERROMAG NETIC SPIN DIMER SYS TEM: B a 3 M n 2 O 8 5.1 Background Antiferro magnetic dimer compounds have been extensively studied to understand the quantum phase transition [ 3, 33, 34, 122 ]. The magnetic phase of these compounds transits from the quantum paramagnetic phase to the long range ordering phase at the quantum critical point at the absolute zero temperature, tuned by the external parameters, such as pressure and magnetic field (Figure 1 3). Contrary to the classic al thermal phase transition, the quantum phase transition is caused by the spin fluctuation, so the thermal e nergy is not required for the quantum phase transition. Since the metal ions of a dimer compound are interconnected via the antiferromagnetic exchange interaction, which is dominant among the interactions between the nearest neighbor ions, the dimer compo und has the singlet ( S = 0) ground spin state in the quantum paramagnetic phase. As external parameters decrease below the quantum critical point, the compound shows the an tiferromagnetic Nel ordering [3 ]. At finite temperatures, the thermal energy distur bs the magnetic phase, and eventually the compound show s the paramagnetic phase above the temperature where the thermal fluctuation overcomes the correlation between spins. 5.2 Overview of Ba 3 Mn 2 O 8 Ba 3 Mn 2 O 8 compound, a spin dimer material has been studie d because of the possibility of a higher moment ordered phase by the quint upl et ( S = 2) spin state. In contrast to a typical spin dimer comprising S = 1/2 spins, a dimer of the Ba 3 Mn 2 O 8 compound consists of two S = 1 spins by favor of a rare oxidation stat e of the Mn ion; Mn V 3 d 2 The Ba 3 Mn 2 O 8 compound crystallizes in the rhombohedral structure. The

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123 vertical dimers of Mn V ions are arranged on a hexagonal layer in stacks of three [123 ] (Figure 5 1 ). Two manganese ions in a dimer are i nterconnected via the antiferromagnetic exchange interaction. Each manganese ion in a dimer interacts with the neighboring ions of other dimers in the same layer or in the next layer. Every exchange interactions are displayed using different line type s sho wn in Figure 5 1. The Hamiltonian including the exchange interactions of Ba 3 Mn 2 O 8 compound can be written as follows: (5 1) where represents the coordinate of a given dimer, and = 1, 2 indicate s each of the two spins in a given dimer. Quantities and represent the displacement of the coordinate of any manganese ion to its nearest neighbor dime r in the same layer, and in the next neighbor layer, respectively. The only represents the displacement of the coordinate of any manganese ion to its next nearest neighbor dimer in the next layer. The first term in the first summati on term is the anisotropy term of the manganese ion with the single ions anisotropy , and the second term in the first summation term is the Zeeman energy term. The remaining terms are the exchange interaction terms. Recent inelasti c neutron scattering (INS) studies revealed the exchange interact ion constants between Mn V ions in a dimer and between neighbor dimers [ 124 ]. Index for the exchange interactions is shown in Figure 5 1. The dominant intradimer interaction,

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124 J 0 is 1.642 meV. The exchange intera ction s between the Mn atoms of different dimers in the same hexagonal layer are described as J 2 J 3 = 0.1136 meV. J 2 and J 3 cannot be independently determined from the INS experiments because the result from the experimental data depen ds on their difference. The exchange Interaction with Mn atoms in the next neighbor layers is J 1 = 0.118 meV, and J 4 = 0.037 meV [ 124, 125 ]. The antiferromagnetic exchange interaction between two manganese ions in a dimer causes the singlet ( S = 0) state to be the ground spin and so the excited spin multiplet states are the S = 1 triplet and S = 2 quint upl et state at zero magnetic field and at zero temperature Figure 5 2 is the energy level diagram of an antiferromagnetic dimer composed of S = 1 ions wit h the intradimer exchange interaction J 0 = 1.642 meV and g = 1.96. As the external magnetic field, which is the external parameter for this compound, increases, the energy gap between the singlet and the triplet state becomes narrower, and it is finally cl osed at the critical field H c1 As the magnetic field increases above H c1 the ground spin state goes from the singlet to the triplet (Figure 5 2). The magnetization curve for Ba 3 Mn 2 O 8 single crystal clearly displays two plateaus, demonstrating the magneti c phase transition as the magnetic field increases (Figure 5 3) [37 ]. Since the phase transition of Ba 3 Mn 2 O 8 compound is a second order transition, the magnetization smoothly chang es. The magnetization is zero until H c1 ~ 9 T and then it linearly increase s up to half saturated magnetization value per site until we reach H c 2 ~ 27 T. So that i n the range of H c1 < H < H c2 the compound is in the long range antiferromagnetic ordering phase [3, 30 ]. After the second plateau according to the triplet ground stat e, the magnetization linearly increases again up to the saturated value demonstrating another quantum phase in the range of H c3 (~ 33 T) < H < H c4 (~ 48 T).

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125 Recent magnetic property studies on a single crystal of Ba 3 Mn 2 O 8 at finite temperatures have estab lished the phase diagram in a T vs H plot (Figure 5 3) [ 37, 126 ]. Two quantum phases are primarily observed, one at lower field and another at a higher field than the magnetic field where the tr iplet is saturated The lower field phase s (Phase I and Phase II) are associated with the ordered state superposed by a singlet ( ) and a triplet ( ) state, and the higher field phase (Phase III) is associated with the ordered phase superposed by a triplet ( ) and a quintuplet ( ) state. Depending on the applied magnetic field direction, there are two different types of low field phases. A single P hase I i s observed for the magnetic field parallel to the c axis, whereas two distinc t ordered phases (Phase I and Phase II) are observed within the external magnetic field perpendicular to the c axis in the magnetic field range between H c1 and H c2 The analysis of data using the spin Hamiltonian for Ba 3 Mn 2 O 8 (Equation 5 1) demonstrates th at the anisotropy is required to understand two distinct ordered phase s within the external magnetic field perpendicular to the c axis In deed, the study on the ordered P hase II reveals that it is determined by the competition between the single ion anisot ropy and interlayer exchange interaction [ 126 ]. Even though the importance of the anisotropy has been emphasized, the anisotropy for Mn V ions has not been clearly obtained. The electron paramagnetic resonance ( EPR ) study on the Mn V doped Ba 3 V 2 O 8 compound s uggests that the single ion anisotropy for Mn V ion is 0.279 K [ 127 ]. However, the compound structure is different from that of the Ba 3 Mn 2 O 8 and moreover the study on the Mn V doped Ba 3 V 2 O 8 provide the sign of the anisotropy.

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126 In this chap ter, the single ion anisotropy is investigated using the high frequency electron paramagnetic resonance ( HF EPR) measurements on a single crystal of Ba 3 Mn 2 O 8 at low field range H < 15 T. 5 .3 High Fre quency Electron Paramagnetic Resonance Studies in the Low Field R ange Two single crystal samples of Ba 3 Mn 2 O 8 with different size s are used fo r the EPR experiments (Figure 5 4 ). The EPR spectra for a big crystal (d ~ 0.2 mm) exhibit several extra peaks (Figure 5 5 ), instead of showing a single peak or two peaks e xpected for th is compound based on the brief energy level diagram (Figure 5 2 ). The extra peaks may be caused from the inter ference of microwave radiation i n the sample. When the transmitted radiation is multiply reflected between the flat faces of a samp le, one can expect to have a n interference patter n consist ing of a series of maxima nearly equally space in frequency [ 12 8 ]. In that case, the refractive index n can be determined from the sample thickness d and the separation between the maxima in frequen cy basis, In order to see the relation between the refractive index, sample thickness, and the ) to a sample with thickness d and the refractive index n (Figur e 5 6 ). Since we only consider the interference pattern in this discussion, the imaginary term in the complex refractive index is ignored, and only the real part of refractive index is consi dered. The traveling wave is described using a wave function, with amplitude A, wavenumber and frequency In Figure 5 6 T 0 indicates the transmitted wave without a reflection in the sample and T N indicates the transmitted wave after 2N time reflec tion s on the surfaces of the sample. The interference pattern is

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127 determined by the phase difference between the transmitted waves which is due to the differen ce of the optical path caused from the multiple reflections in the sample. In order to have maxim a, the phase difference satisfies the condition of constructive interference ; (5 2 ) where m is an integer number. Therefore, for the case of N = 1 and m = 1, the refractive index can be written as follows; (5 3 ) Considering the sample thickness d = 0.2 mm, the Equation 5 3 shows the relation between the refractive index and the maxima separation; = 75 GHz Although the refractive index of Ba 3 Mn 2 O 8 compound has not been obtained yet, t h e refractive index of a n insulating material can be approximately assumed to be 1.5 < n < 5.0 [ 129 ] Therefore, the maxima caused by the interference can be understood to imply a range for frequency separation, i.e., 15 GHz < < 50 GHz This separation approximately corresponds to 0.5 T < H < 1.8 T for Ba 3 Mn 2 O 8 compound with g c = 1.96 [ 127 ]. This approximation implies that the first maxim a caused by the interference in an insulating sample appears in the field range of 0.5 T < H < 1. 8 T Figure 5 5 shows the EPR spectra of the big crystal of Ba 3 Mn 2 O 8 compound taken at various frequencies and at 2 K with the magnetic field parallel to the c axis of the compound. At the frequencies a bove 150 GHz, many peaks appear around g c = 1.96 posi tion, where the appearance of main peak is expected. The deviation of peak position from g c is less than 1.8 T. This deviation of peak positions implies that the

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128 observed extra peaks could be caused from the interference of the microwave radiation in the b ig sample. Interference of radiation in a sample occurs when the wavelength of the radiation is smaller than the sample size. Therefore, if the wavelength of the radiation is longer than the sample size or if the sample is smaller than the wavelength of the radiation, the interference effect can b e reduced. Indeed, in Figure 5 5 the spectra at frequencies below 150 GHz ( = 2 mm) exhibit a single or two peaks, but not show many peaks. For 0.5 mm) of Ba 3 Mn 2 O 8 compound higher frequency than 150 GHz is expected to provide a single peak, but not extra peaks (see Chapter 5.3.2). Therefore, in order to do data analysis with the reduced interference effect, the EPR data with lower frequencies than 150 GHz are analyzed for stu dy with the big sample of Ba 3 Mn 2 O 8 compound 5.3 1 HF EP R for a Big C rystal Angle dependent EPR spectra for the big crystal of Ba 3 Mn 2 O 8 taken at 2.5 K and 57.77 GHz are shown in Figure 5 7 The spectra are plotted as a function of the relative angle between the applied magnetic field and the c axis of the crystal. The two peaks observed in a s pectrum are assigned as the resonances in the triplet state by the selection rule for the magnetic resonance, = 0 and s = 1. Although the ground spin state is the singlet state at this field range, the interference effect in the crystal and the spin state mixing cause d from the various exchange interactions of the compound allow the intra triplet transitions at low field range. As mentioned in Chapter 2, w hen the m agnetic field is parallel to the c axis (red spectra in Figure 5 7 A ) the peak intensity at h igh field is bigger than that of the low field range for the system with positive anisotropy D S=1 > 0 at low temperatures [ 64 ]. Temperature dependence data (Figure 5 8 ) supports

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129 the positive sign of the anisotropy as well. Thus, the red and black data points in Figure 5 7 B correspond to the resonances of m s = 1 m s = 0, and m s = 0 m s = +1 in the triplet state, respectively. As the relative field orientation to t he c axis is changed, the two resonance centers shift cross each other (Figure 5 7B ) The significant angle dependence of the two resonance centers in the triplet state provides the estimate of the axial zero field splitting (ZFS) value, D S=1 and g value. When the axial ZFS is relatively small compared with the Zeeman energy, the resonance fields for the transition ( m m+1 ) can be written as a function of a relative angle between the magnetic field and the c axis [ 50 ]. (5 4) wher e B 0 is the resonance field for a free electron, D is the uniaxial ZFS term, g is the Land g factor, B is the Bohr magneton, and m is the initial spin state of a resonance. The solid lines in Figure 5 7 B are the fitting curves on the data points of t he r esonance centers using Equation 5 4 A good fitting of the data is given with g = 1.96 and the axial ZFS value for the triplet state, D s=1 = +0.375 K. The positive sign of D S=1 is determined again by the fitting of the angle dependence data. Resonances in the triplet and quint uplet states are observed in the temperature dependent EPR spectra. Figure 5 8 shows the temperature dependence of the resonance peaks in the temperature range of 1.4 K to 7 K at 93.87 GHz with the magnetic field parallel to the c axis Two resonance peaks are clearly shown in the EPR spectra at 2 K and at 7 K. The peak separation in the spectra at 2 K is wider than that at 7 K, and that the ratio of the peak separation between the spectra at 2 K and at 7 K is ~0.5. Since the peak separ ation in the EPR spectra is proportional to the axial ZFS

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130 value, the different peak separations between the spectra at 2 K and at 7 K imply a different ZFS value for each spectra data. Thus, one can say that the resonance peaks in the spectra at 2 K and at 7 K originate from different spin multiplet states, i.e., the triplet and quint uplet state, respectively. Also, t he energy level difference between two spin multiplet states suggests that the resonances at 2 K are from the triplet state, and the resonance s at 7 K are from the quint upl et state. Although the zero field energy gap between two spin multiplet states is too big, 2 J 0 = 38 K, to observe a resonance of the quintuplet state at low temperatures, the resonances in the quint upl et state are shown at 7 K due to the spin state mixing from the various transverse exchange interaction of the compound and the interference effect i n the big crystal. Temperature dependent EPR spectra also elucidate the sign of the axial ZFS, D S ( S = 1 for the triplet and S = 2 f or the quint upl et). Since the ground spin state is mainly populated at low temperatures, leading to the strong intensity of the ground transition peak, the temperature dependent peak intensity allows us to determine the ground spin state transition. In Fig ure 5 8 the peaks indicated by the green arrows are the ground spin state transitions, and the peaks indicated by the black arrows are the first excited spin state transitions. The ground spin state transition peak in the 2 K spectra appears at a higher m agnetic field than the first excited transition peak, whereas the ground transition peak in the 7 K spectra appears at a lower magnetic field. This difference in the peak position of the ground spin state transition indicates that the ZFS values for the tr iplet and the quint upl et have different signs, i.e., D s=1 > 0 and D s=2 < 0. The sign of the ZFS values suggested by the temperature dependent EPR data is consistent with the result of the angle dependence data for the triplet.

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131 5.3 2 HF EP R for a Small C ryst al The EPR data for the small crystal of Ba 3 Mn 2 O 8 are simple r spectra than that for the big crystal due to the reduction of the interference effect on the sample. The inset of Figure 5 9 is the EPR spectra plot at 211.7 GHz at 1.8 K. Compared with the spec tra for the big crystal, only one sharp peak was observed for the small crystal, and no other peak feature was seen. Figure 5 9 shows the angle dependence of the resonance center with the black square symbols. The experimental data are well matched to the fitting curves (represented by a red line) using Equation 5 4 with the fitting parameters, D S=1 = +0.365 (0.002) K, and g = 1.96 The axial ZFS value obtained from the fitting is consistent with the value from the angle dependent EPR data for the big cryst al, suggesting that the peak observed for the small crystal is the ground state transition in the triplet spin state. 5.4 Analysis U sing Single Ion Hamiltonian 5.4 .1 Simulation of Energy Level Diagram The single ion anisotropy value of Mn V ions is estimat ed using the multi spin Hamiltonian for a dimer model of two S = 1 spins to simulate the EPR experimental result. The Hamiltonian used for the simulation is as follows; (5 5 ) The first term represents the Zeem an term, and the second term is the single ion anisotropy term. represents a diagonal tensor of the single ion anisotropy of i th ion in a dimer. The final term is the intradimer exchange interaction term, and J 0 is the intradimer ex change interaction constant. In the previous EPR and inelastic neutron scattering studies, the g value for the c axis of Mn V ion is given as 1.96 [ 127 ], and J 0 is

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132 given as +1.642 meV [ 124 ]. The simulation with D i = 0.375 K reasonably realizes the axial ZF S value for the S = 1 state, D S=1 = +0.375 K obtained from the EPR experiments. Figure 5 10 is the energy level diagram of a dimer model of two S = 1 ions using Equation 5 5 with the D i value obtained from the simulation. The magnetic field direction is as sumed along the c axis of the crystal. The black line indicates the energy level of the singlet state S = 0, and the red and blue lines indicate the levels of the triplet S = 1 and quint upl et S = 2 states, respectively. The insets show the zoom in of the e nergy levels of the triplet and quint upl et around the zero field, and it clearly shows that the anisotropy value for the triplet and quint upl et have opposite signs The axial ZFS values for the triplet and quint upl et spin states are D S=1 = +0.375 K and D S= 2 = 0.125 K, respectively. The D S=1 is about three times bigger than D S=2 Thus, the single ion anisotropy of Mn V ion, D i is 0.375 K. The obtained D i value is, however, bigger than the reported values, 0.279 K, in the previous EPR study of the Mn V dope d Ba 3 (VO 4 ) 2 single crystal [ 127 ]. Although our experiment elucidates the negative sign of the single anisotropy value of Mn V ion in the Ba 3 Mn 2 O 8 compound, the magnitude of D i is inconsistent with the previous study. This discrepancy c ould be explained by t he different coordinate of Mn V ion in two compounds. In the Mn V doped Ba 3 (VO 4 ) 2 compound a Mn V ion couple s with a non magnetic ion, a V V (3 d 0 ) ion However, the manganese ions in the Ba 3 Mn 2 O 8 compound compose a dimer via antiferromagnetic exchange interac tion. 5.4 .2 Calculation of Dipolar Contribution to Anisotropy The dipolar couplings between Mn V ions are also attributed to the discrepancy of the single ion anisotropy between the two studies. The anisotropy of a single ion in a dimer can be deduced from the anisotropy of the compound using a projection operator

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133 method [ 51 ]. A dinuclear model can be used for this compound. The relation between the compound anisotropy and the single ion anisotropy from Equation 3 1 is as follows; ( 5 6 ) D S is the compound anisotropy value for a spin multiplet state, S The quantities D 1 and D 2 are the uniaxial anisotropy values of the ions 1 and 2 in a dimer, respectively. The d 1 and d 2 are the projection coefficients for the anisotropy of the indivi dual single ion. D 12 represents the contribution of the dipolar coupling to the anisotropy. d 12 is the projection coefficient for the dipolar contribution. The EPR data provides the anisotropy value for the triplet state, D S=1 For a dimer of the identical ions like Ba 3 Mn 2 O 8 D 1 and D 2 should be identical. Projection coefficients for spin multiplet states, S of a dimer composed of S = 1 ions are introduced in Table 5 1. In Chapter 3, the contribution of the dipol ar coupling i s ignored for the Mn III ions be cause its magnitude is an order of magnitude weaker than the magnitude of the single ion anisotropy. However, for the Mn V ions of the Ba 3 Mn 2 O 8 compound, D 12 cannot be ignored because of a large magnitude. The contribution of dipolar coupling to the compoun d anisotropy, D 12 can be obtained by the tensor calcu lation of the dipolar couplings: (5 7 ) with (5 8 ) where T is the tensor term of the dipolar coupling, and g is the Land g factor. The sub indexed T an d g indicate their corresponding matrix elements. B is the Bohr magneton,

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134 and r is the distance between two ions in the dipolar coupling. Because of the hexagonal layered structure of Ba 3 Mn 2 O 8 D 12 is not only produced by the intradimer dipolar coupling (via J 0 pair), but it is also produced by the inter dimer dipolar couplings (via J 1 J 2 J 3 and J 4 pairs). Since the D 12 for the J 4 pair is too small due to the long distance between ions, the dipolar coupling for the J 4 pair is not considered here. The calculated D 12 for the four kinds of dipolar coupling s between neighbor ions are shown in T able 5 2. Although the D 12 for the J 1 J 2 and J 3 pairs are smaller than that for the J 0 pair, the number of those pairs in the compound is bigger, so they are comparable. As shown in the table, D 12 value is not a negl igible value compared with the single ion anisotropy and the compound anisotropy value. Although the combined contribution of various dipolar couplings to the anisotropy is not calculated exactly, the calculation clearly shows that the discrepancy of the s ingle ion anisotropy of Mn V ion between the two EPR studies is caused by the dipolar coupling in the compound, and the dipolar contribution does not only come from the intradimer dipolar coupling, but also comes from the interdimer dipolar couplings. 5 5 S ummary High frequency electron paramagnetic resonance ( HF EPR) experiments were carried out on single crystals of Ba 3 Mn 2 O 8 an antiferromagnetic dimer compound. Angle dependent and temperature dependent EPR data suggest that the single ion anisotropy of Mn V ion is a negative value confirmed by the energy level simulation using the single ion Hamiltonian. Also, the calculation of dipolar contribution to the anisotropy shows that the discrepancy in the magnitude of D i between two EPR studies is caused by the dipolar coupling of Mn V ions in a dimer and between dimers.

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135 Figure 5 1. Schematic structure of Ba 3 Mn 2 O 8 compound. Ba and O ions are neglect for simplicity. Each Mn V ion pairs associated with the exchange interactions are shown as different line types

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136 Figure 5 2. Energy level diagram of an antiferromagnetic dimer composed of S = 1 ions with the intradimer exchange interaction J 0 = 1.642 meV and g = 1.96. The black line represents the singlet level, and the red and blue represent the triplet and quintuplet, respectively.

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137 Figure 5 3. Phase diagram of Ba 3 Mn 2 O 8 single crystal for the magnetic field applied perpendicular to the c axis [37, 126 ]. Magnetization curve for T =0.5 K is overlaid (right axis). The light and dark yellow regions correspo nd to the phase I and the phase II, respectively. The blue region corresponds to the phase III. The phase boundary is determined using the heat capacity data and magnetocaloric effect (MCE) data. Reprinted figure with permission from E. C. Samulon, Y. Koha ma, R. D. McDonald, M. C. Shapiro, K. A. Al Hassanieh, C. D. Batista, M. Jaime, and I. R. Fisher Physical Review Letters 103 047202 (2009). Fig. 4, pg. 047202 4. Copyright (2009) by the American Physical Society.

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138 Figure 5 4 Ba 3 Mn 2 O 8 crystals use d in the HF EPR measurement. Pictures are not scaled, but the approximate diameters of the crystals are shown on the pictures. The c axis of the crystals is the out of the plane orientation. Figure 5 5. EPR spectra of a big crystal of Ba 3 Mn 2 O 8 compound at various frequencies at 2 K. The magnetic field is applied along the c axis of the crystal.

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139 Figure 5 6 Scheme of multiple reflections of wave radiation in a sample with the refractive index n and thickness d. indicates the incident angle of radi ation. T N represents the transmitted radiation after 2N times reflections in the sample.

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140 Figure 5 7. A) The experimental EPR spectra for the big crystal were taken at 2.5 K and with 57.77 GHz, as a function of the relative angle between the applied magnetic field and the c axis of the crystal. The angle increment is 8.4 The red spectra are obtained with the applied magnetic field parallel to the c axis. B) The angle dependence of two transition peaks obtained from the EPR spectra A). The solid lin es are the fitting lines using Equation 5 4 shown in the main text.

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141 Figure 5 8. Temperature dependence of EPR spectra for the big crystal with 93.87 GHz with the magnetic field parallel to the c axis of the crystal. The green and black arrows indicate the ground spin state transition and the first excited spin state transition, respectively.

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142 Figure 5 9 Angle dependence of the resonance peak taken from the EP R spectra for the small crystal at 1.8 K with 211.7 GHz. Only one sharp resonance peak is ob served in the spectra (inset). The black squares symbolize the resonance center, and the red line represents the fitting function using Equation 5 4 The fitting parameters are introduced in the main text.

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143 Figure 5 10 Energy level diagram of a dime r model of two S = 1 ions by Equation 5 5 with the D i value obtained from the simulation and the intradimer exchange interaction J 0 The magnetic field direction is aligned to the c axis of the crystal. The black line indicates the energy level of the sing let state S = 0, and the red and blue lines indicate the level of the triplet S = 1 and quint upl et S = 2 states, respectively. The insets show the zoom in of the energy levels of the triplet and quint upl et around the zero field.

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144 Table 5 1. Projection co efficients for the spin multiplet state S of a dimer composed of S = 1 ions [51 ]. S d 1 d 2 d 3 1 1/2 1/2 1 2 1/6 1/6 1/3 Table 5 2. D 12 values for the four kinds of dipolar coupling. J 0 J 1 J 2 and J 3 indicate the dipolar couplings associated with th e pairs. The distance between the ions is shown in the parenthesis. J 0 (3.985) J 1 (4.569) J 2 (5.711) J 3 (6.964) D 12 (K) 0.114 0.051 0.019 0.000189

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145 CHAPTER 6 CONCLUSION This Chapter provides the summary and conclusion of each Chapter in this di ssertation. This dissertation is focused on the anisotropy effect in the spin frustrated molecular magnets and spin dimer compound system showing the quantum phase transition. Chapter 1 is an introduction to the single molecule magnets (SMMs) and spin dimer compound system. We explained the origin and importance of anisotropy to magnetic behavior of both magnetic materials. Two spin Hamiltonians are proposed to model the given systems and to understand the magnetic property of the materials, su ch as quantum magnetization tunneling, and quantum phase transition. Chapter 2 outlines the experimental technique of electron paramagnetic resonance (EPR) measurements performed on the molecular magnets, and spin dimer compound. The fundamental principle of the EPR measurement is explained, and the simulated EPR spectra of a system with anisotropy are presented. We discuss the physical principles of the waveguides and the resonant cavities, which are the critical components of the EPR probe used in the EPR experiments for the research in this dissertation. The millimeter vector network analyzer used as a source and detector of microwave is also explained. Chapter 3 presents the relieving of spin frustration in the antiferromagnetic triangular Mn III 3 complex using the high frequency electron paramagnetic resonance (HFEPR) and magneto torque measurement. The relieving of spin frustration caused by the intrinsic asymmetry of the complex results in the well isolated spin ground state, S = 2, and negative easy ax is type anisotropy. The simulation of the EPR data and the

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146 magneto torque data confirm the relieving of the spin frustration, and determine the big anisotropy value of the complex due to the small total spin number. The anisotropy of the complex results in the significant anisotropy barrier to the magnetization reversal, leading the SMM behavior of the antiferromagnetic triangular Mn III 3 complex. In Chapter 4, the anisotropy barrier of a high spin magnetic cluster, Cu 17 Mn 28 complex, is investigated using th e HF EPR measurements. Although this complex has the geometric spin frustration anisotropy the slow relaxation and hysteresis of the magnetization are observed in the magnetization measurements. Th e temperature dependence and angle dependence of a broad peak feature observed in the EPR spectra demonstrate the anisotropy in the complex. The simulation using the spin H amiltonian including the fourth order spin operators proves that the complex has the fourth order anisotropy following the T d symmetry of the structure However, the simulated fourth order anisotropy is not strong enough to cause an effective barrier to the magnetization reversal. Thus, we conclude that the slow relaxation and hysteresis of the magnetization observed for the complex is induced by the geometric frustration of the core structure. In Chapter 5, the single ion anisotropy of Mn V ions in the Ba 3 Mn 2 O 8 compound is investigated using the HF EPR measurements. The magnetic anisotropy of Ba 3 Mn 2 O 8 compound contributes to the quantum phase transition in competition with the interdimer exchange interactions. The angle dependent and temperature dependent EPR data demonstrate that the single ion anisotropy of Mn V ion is a negative value. The magnitude and sign of the single ion anisotropy are confirmed by the energy level simulation using the single ion Hamiltonian. The calculation of dipolar couplings

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147 between neighboring Mn V ions demonstrates that the dipolar couplings contribute to the anis otropy value in the compound

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156 BIOGRAPHICAL SKETCH Changhyun Koo was born in Seoul, South Korea. He earned his B achelor of Science degree and M aster of Science degree in physics from Seoul National University in 2002 and 2004. He studied the metal insulator transition of the vanadium dioxide using the infra red spectroscopy during his master degree course. He came to the University of Florida in the fall of 2005 and joined summer. He studi ed single molecular magnets and spin dimer compounds using the high frequency electron paramagnetic resonance technique. He completed his Doctor of Philosophy program in physics in 2011.