This item is only available as the following downloads:
1 ELECTRON PARAMAGNETIC RESONANCE STUDIES OF SYMMETRY ENFORCED QUANTUM TUNNELING IN MOLECULE BASED MAGNETS By JUNJIE LIU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012
2 2012 J unjie Liu
3 To my parents
4 ACKNOWLEDGMENTS I would like to take this opportunity to acknowledge many people who have directly or indirectly helped me during my five years PhD life in the University of Florida and the National High Magnetic Field Laboratory at Tallahassee. I am deeply indebted to my PhD advisor, Prof Stephen Hill, for his patient guidance and continuous support during the past four y ears. I joined Prof Hill s group in 2008 knowing very little about experimental condensed matter physics. Prof Hill mentored me patiently through my graduate studies and gave me lots of helpful suggestions for my research Furthermore he also provided m e with financial support so I could focus on my research In addition I was afforded the opportunity to attend several conferences, both regiona l and international, where I was able to interact with some of the best known experts in our field. Prof Hill ha d a profound effect on my English skills, both written and spoken and for this I am truly thankful. I would like to thank Prof David Tanner for willing to serve as my PhD committee chair and helping me through the last three years. Without his support it is impossible for me finish my graduate study at UF. I would like to thank the other members of my PhD committee who monitored my PhD studies and provid ed me with valuable suggestions and comments on my dissertation : Prof Hai Ping Cheng Prof Christ opher Stanton and Prof George Christou. During my PhD studies, I was lucky to that I had chances to work with many research groups, both in condensed matter physics and in chemistry It has been a privilege to work with them. I would like to express my th anks to my collaborator Prof Enrique del Barco and his group at the University of Central Florida. We have collaborated in several projects and I was always able to learn a lot from our discussions. Prof Guillem Aromi, Prof Euan K. Brechin,
5 Prof David Hendrickson an d Prof Jeffery Long have provide d me with samples to study. I am really grateful to them for the collaborations. I want to thank all the form er and current group members, especially Dr. Saiti Datta and Dr. Changhyun Koo, who taught me the ba sic skills about being an experimental physicist I wish them have a great career. Dr. Christopher Beedle, who is the only expert in chemistry in our group, helped me a lot with chemistry and my written English. Dr. Alexey Kovalev is an excellent experimen tal physicist, who I have learnt a lot from our discussions. My labmates, Sanhita Ghosh and Muhandis Muhandis had provided me many helps for my experiments. I also want say thank you to the scientists in the EMR group at the NHMFL for their helps and all the meaningful discussions. I would like to acknowledge the US NSF (grant number DMR 0804408 and CHE 0924374 ) for the financial support s The NHMFL is supported by the NSF (DMR 0654118) and the S tate of Florida. Last, but not least, I woul d like to thank m y parents. Through all the years, they have always been supporti ve of me, encouraging me and loving me We are nothing without our family. I wish they have a happy life and I will always love them.
6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 8 LIST OF FIGURES ................................ ................................ ................................ ......................... 9 LIST OF ABBREVIATIONS ................................ ................................ ................................ ........ 12 ABSTRACT ................................ ................................ ................................ ................................ ... 13 CHAPTER 1 INTRODUCTION TO MOLECULE BASED MAGNETS ................................ .................. 16 1.1 Introduction ................................ ................................ ................................ ....................... 16 1.2 Spin Hamiltonian ................................ ................................ ................................ .............. 18 1.2.1 Giant Spin Approximation Hamiltonian ................................ ................................ 19 22.214.171.124 Second Order Anisotropies ................................ ................................ .......... 19 126.96.36.199 High Order Anisotropies ................................ ................................ .............. 23 1.2.2 Multi S pin Hamiltonian ................................ ................................ .......................... 25 1.3 Experimental Techniques in Studying SMMs ................................ ................................ .. 26 1.3.1 Electron Paramagnetic Resonance ................................ ................................ ......... 27 1.3.2 Quantum Tunneling of Magnetization ................................ ................................ ... 29 1.4 Single Chain Magnets ................................ ................................ ................................ ...... 35 2 INTERPLAY BETWEEN ANISOTROPY AND EXCHANGE IN DINUCEAR MOLECULAR MAGNETS ................................ ................................ ................................ ... 49 2.1 Introduction ................................ ................................ ................................ ....................... 49 2.2 Ferromagnetically C oupled D inuclear [Mn III ] 2 M olecule ................................ ................ 51 2.2.1 Discussions of Magnetometry Results ................................ ................................ ... 51 2.2.1 EPR S pectroscopy ................................ ................................ ................................ .. 54 2.3 Antiferromagnetically C oupled D inuclear [Mn III ] 2 M olecule ................................ .......... 58 2.3.1 EPR S pectroscopy ................................ ................................ ................................ .. 59 2.3.2 Magnet ic M easurements ................................ ................................ ......................... 67 2.4 Summary ................................ ................................ ................................ ........................... 69 3 QUANTUM TUNNELING OF MAGNETIZATION IN TRIGONAL SINGLE MOLECULE MAGNETS ................................ ................................ ................................ ...... 79 3.1 Introduction ................................ ................................ ................................ ....................... 79 3.2 Quantum Tunneling of Magnetization in the Mn 3 Single Molecule Magnet ................... 80 3.2.1 The Mn 3 Single Molecule Magnet ................................ ................................ ......... 80 3.2.2 Quantum Tunneling of Magnetization in Mn 3 ................................ ....................... 82
7 3.2.3 Berry Phase Inter ference in Mn 3 ................................ ................................ ............ 87 3.3. Quantum Tunnel of Magnetization in the Ni 4 Single Molecule Magnet ........................ 89 3.3.1 The Ni 4 Single Molecule Magnet ................................ ................................ ........... 89 3.3.2 Quantum Tunneling of Magnetization in Ni 4 ................................ ......................... 91 3.4 Summary ................................ ................................ ................................ ........................... 94 4 ELECTRON PARAMAGNETIC RESONANCE AND QUANTUM TUNNELING OF MAGNETIZATION STUDIES OF MN 4 SINGLE MOLECULE MAGNETS: REVEALING COMPETING ZERO FIELD INTERACTIONS ................................ ......... 108 4.1 Introduction ................................ ................................ ................................ ..................... 108 4.2 The Mn 4 Bet SMM ................................ ................................ ................................ ......... 111 4.2.1 The Structure of the Mn 4 Bet SMM ................................ ................................ ..... 111 4 .2.2 Discussion of Electron Paramagnetic Resonance Results ................................ .... 114 4.2.3 Discussion of Quantum Tunneling of Magnetization Results .............................. 119 4.3 Summary ................................ ................................ ................................ ......................... 124 5 SLOW MAGNETIC RELAXATION INDUCED BY A LARGE TRANSVERSE ZERO FIELD SPLITTING IN A MN II RE IV (CN) 2 SINGLE CHAIN MANGET ............... 135 5.1 Introduction ................................ ................................ ................................ ..................... 135 5.2 Introduction to the Structures of 1 4 ................................ ................................ ............... 138 5.3 Discussions of Magnetometry and EPR Results ................................ ............................ 139 5.3.1 Complex 1 ................................ ................................ ................................ ............ 139 5.3.2 Complex 2 ................................ ................................ ................................ ............ 142 5.3.3 Complex 3 ................................ ................................ ................................ ............ 143 5.4 Discussion of the Magnetic Relaxation Process ................................ ............................. 146 5.5 Summary ................................ ................................ ................................ ......................... 149 6 SUMMARY ................................ ................................ ................................ .......................... 161 APPENDIX STEVENS OPERATORS ................................ ................................ .................. 164 LIST OF REFERENCES ................................ ................................ ................................ ............. 166 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ....... 173
8 LIST OF TABLES Table page 3 1 Comparison of tunneling gaps obtained from the MS and GS models for resonances k = 0, 1, 2 and 3, for the two cases = 0 (top) and = 8.5 o (bottom). .............................. 84 5 1 Selected Interatomic Distances () and Angles () for compounds 1 4. ......................... 139
9 LIST OF FIGU RES Figure page 1 1 The molecular structures of several SMMs studied in this dissertation. ........................... 39 1 2 Schematic representa tion for the energy levels of an S = 6 molecule with easy axis type anisotropy ( D < 0). ................................ ................................ ................................ ..... 40 1 3 Potential energy surface corresponding to the 2 nd order anisotropy. ................................ 41 1 4 Potential energy surfaces corresponding to the fourth order Stevens operators ............... 42 1 5 Schematic sketches demonstrating the concept of the MS Hamiltoni an for a trinuclear molecule. ................................ ................................ ................................ ............ 43 1 6 Schematic plots illustrating the theory of EPR measurements. ................................ ......... 44 1 7 Quantum tunnel of magne tization in a S = 6 SMM. ................................ .......................... 45 1 8 The tunnel splitting associated with QTM. ................................ ................................ ........ 46 1 9 A schematic sketch of various tunneling paths o n a Bloch sphere. ................................ ... 47 1 10 The process of the relaxation of magnetization in SCMs. ................................ ................. 48 2 1 The molecular structure and magnetic co re of complex 1 ................................ ................ 70 2 2 Hard plane EPR spectra of complex 1 obtained at 171 GHz as a function of temperature from 20 K (top) to 2.5 K (bottom). ................................ ................................ 71 2 3 Representative easy axis EPR spectra of complex 1 obtained at 2 K (391.5 GHz) and 15 K (310.1GHz and 248.6 GHz). ................................ ................................ ..................... 72 2 4 Frequency dependence of EPR peak positions of 1 ................................ .......................... 73 2 5 The molecular structure of complex 2 ................................ ................................ .............. 74 2 6 EPR spectrum of complex 2 as a powder at 652.8GHz and 10 K, recorded in the first derivative mode. ................................ ................................ ................................ ......... 75 2 7 Single crystal EPR studies for complex 2 ................................ ................................ ........ 76 2 8 Powder EPR studies for complex 2 ................................ ................................ .................. 77 2 9 Plot of the isofield reduced magnetization measurements. ................................ ................ 78 3 1 The molecular structure and the magnetic core of the Mn 3 SMM. ................................ .... 95
10 3 2 Zeeman diagram for a spin S = 6 multiplet with easy axis anisotropy ( D < 0 in Equation 3 1) and H // z ................................ ................................ ................................ ...... 96 3 3 Zeeman diagram of Mn 3 generated by Equation 3 2 with different magnitudes of J ....... 97 3 4 The symmetries of the ZFS interactions and the tunnel splitting s ................................ .... 98 3 5 Tunnel splittings of the Mn 3 SMM as a function of the coupling constant J ................... 99 3 6 Color contour polar plot of the ground state gap as a function of H T cal culated using Equation 3 1 with ................................ ................................ ...................... 100 3 7 Shift of H L for as a function of the magnitudes of the applied H T ........................... 101 3 8 The BPI patterns for the ground QTM resonances of the Mn 3 SMM. ............................. 102 3 9 The structure and the magnetic core of the Ni 4 SMM. ................................ .................... 103 3 10 Zeeman diagram for the Ni 4 SMM simulated employing Equation 3 3. ......................... 104 3 11 The ground state QTM gaps for the Ni 4 SMM as a function of H T ................................ 105 3 12 Color contour polar plot of the ground gap as a function of H T calculated using Equation 3 3 with the parameters given in the main text. ................................ ............... 106 3 13 The effect of disorder on the ground state QTM gaps for the Ni 4 SMM. ........................ 107 4 1 The molecular structures and the magnetic cores of the Mn 4 SMMs. ............................. 125 4 2 Temperature dependent EPR spectra for Mn 4 Bet obtained at 139.5 GHz with the applied field close to the molecular easy axis. ................................ ................................ 126 4 3 Temperature dependent EPR spectra for Mn 4 Bet obtained at 67.3 GHz with the applied field in the molecular hard plane. ................................ ................................ ....... 127 4 4 Zeeman diagram for the ground multiplet ( S = 9) of Mn 4 B et with the field applied in the molecular xy plane. ................................ ................................ ................................ .... 128 4 5 Temperature dependence of the EPR spectra for Mn 4 anca. ................................ ........... 129 4 6 Plo ts of frequency versus field for Mn 4 Bet showing the observed ground state EPR peak positions. ................................ ................................ ................................ .................. 130 4 7 Hysteresis loops for Mn 4 Bet recorded as a function of H L at different temperatures. ... 131 4 8 Modulation of the QTM probabilities for resonances of Mn 4 Bet. ................................ 132
11 4 9 Contour plots of the QTM probabilities for resonances k = 0 and k = 1 of Mn 4 Bet as a function of H T and ................................ ................................ ................................ .... 133 4 10 Simulations of the magneto anisotropy and QTM probabilities of Mn 4 Bet. ................. 134 5 1 The structures of complexes 1 4 presented in Chapter 5. ................................ ................ 151 5 2 The EPR studies for complex 1 ................................ ................................ ...................... 152 5 3 Temperatu re dependence spectra for 1 collected at f = 126.9 GHz. ................................ 153 5 4 Frequency dependence of the high frequency EPR peak positions deduced from studies of a powder sample of 1 at 5 K. ................................ ................................ ........... 154 5 5 Zeeman diagram for complex 1 generated employing Equation 5 1 with D = +15.8 K and g z = 1.58. ................................ ................................ ................................ ................... 155 5 6 Frequency dependence of the EPR peak positions deduced from powder studies of complex 2 at 5 K. ................................ ................................ ................................ ............. 156 5 7 Frequency dependence of the EPR peak positions obtained from studies of a powder sample of 3 at 3.5 K. ................................ ................................ ................................ ........ 157 5 8 Structure and spin arrangement of chain compound 4 ................................ .................... 158 5 9 The ground tunnel splitting as a function of the size ( N ) of the SCM. ............................ 159 5 10 Classical magneto anisotropy energy surface corresponding to the zero field operator equivalent terms given in Equation 5 1 ................................ ................................ .......... 160
12 LIST OF ABBREVIATION S EPR Electron Paramagnetic Resonance GSA Giant Spin Approximation JT Jahn Teller MS Multi Spin QTM Quantum Tunneling of Magnetization SCM Single Chain Magnet SMM Single Molecule Magnet ZFS Zero Field Splitting
13 Abstract of Dissertation Presented to the Gradua te School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ELECTRON PARAMAGNETIC RESONAN CE STUDIES OF SYMMETRY ENFORCED QUANTUM TUNNELING IN MOLECULE BASED MAGNETS By Junjie Liu M ay 2012 Chair: David Tanner Cochair: Stephen Hill Major: Physics This dissertation presents studies of several molecule based magnets using both experimental and theoretical methods to understand the quantum tunneling and thermally assisted magnetic relaxation at the single ion level. The molecule based magnets presented in this dissertation can be sorted into two categories : a ) single molecule magnets (SMMs) and b ) single chain magnets (SCMs). In these systems, the quantum tunneling and thermally as sisted magnetic relaxation play critical roles both in terms of fundamental scientific reasons and due to the potential applications of SMMs and SCMs in information technologies. Electron paramagnetic resonance (EPR) and quantum tunneling of magnetization (QTM) were employed as two spectroscopic techniques to probe the rich physics of these magnet ic materials Detailed EPR measurements were performed on two dinuclear [Mn III ] 2 molecular magnets to study the interplay between magnetic anisotrop y and exchange interactions. These results show the importance of considering both interactions in interpreting magnetometry measurements performed at both high and low temperatures. The presented analyses shows that EPR is a particularly powerful technique for determini ng magneto anisotropies, as well as
14 magnetic exchange interactions, in weakly coupled systems; furthermore, these studies paved the way for using EPR to study magneto structural correlations under pressures in these compounds. Theoretical studies regardin g quantum tunneling in SMMs were conducted on a trinuclear [Mn III ] 3 SMM with idealized C 3 symmetry. We considered the origin of the three fold transverse anisotropy by mapping the spectrum of the ground spin multiplet generated by a multi spin Hamiltonian onto that of a giant spin approximation Hamiltonian. The rotation of the easy axes of the individual ions leads to the emergence of a three fold transverse anisotropy term, which unfreezes k odd resonance ( k = m + m where m a nd m are the spin projections of the states involved in the QTM resonance) and shifts the ground k = 0 QTM resonance away from zero longitudinal field. For comparison, theoretical studies of quantum tunneling in a tetranuclear [Ni II ] 4 SMM were performed. In these studies we considered the effects of disorder, which significantly change the QTM selection rules. EPR studies were performed on two tetranuclear [ Mn 2 II Mn 2 III ] SMMs, Mn 4 Bet and Mn 4 anca, to probe the magneto anisotropies of these molecules. The B erry phase interference (BPI) patterns were measured in QTM experiments as a function of the transverse field for the Mn 4 Bet complex, leading to the first observation of an intriguing motion of the BPI minima in the magnitude direction phase space of the applied transverse field in SMMs. This motion was attributed to the compet i tion between the zero field anisotropies of inequivalent magnetic ions, Mn III and Mn II within the molecule. Finally, we considered the thermally assisted magnetic relaxation dynami cs in a Re IV based SCM. EPR studies of molecules mimicking fragments of this chain suggest that the Re IV ions in these compounds possess an easy plane type anisotropy and a significant transverse anisotropy. Theoretical analyses reveal that quantum tunneli ng is suppressed in the one dimensional chain,
15 which leads to the observation of slow magnetic relaxation in an easy plane type system. Within this picture, it is the transverse anisotropy which gives rise to the anisotropy barrier for the relaxation of sp ins. These results demonstrate the first example of slow magnetic relaxation through transverse anisotropy; thus, it provides a new strategy for the design of single chain magnets.
16 CHAPTER1 INTRODUCTION TO MOLE CULE BASED MAGNETS 1.1 Introduction Rapid de velopments of molecule based magnets have attracted much inter est in the past two decades due to both fundamental scientific reasons 1 and their potential applications in information technologies 2 3 A particular ly appealing area in molecul e based magnets is the study of single molecule magnets (SMMs) which exhib it slow relaxa tion of magnetization 4 and magnetization hysteresis loop 5 at the molecular level at low temperatures. The SMMs capability of preserving spin polarizations without application of external fields places them as candidates for high density spin based information storage materials. Their discovery has motivated chemists in designing and synthesizing new SMMs as well as physi cists who work on interpreting their novel pr operties The magnetic properties of a SMM mostly originated from its magnetic core, which is constituted by magnetic ions with unpaired electrons. The core of a SMM is surrounded by large non magnetic ligands which minimize inter molecular interactions. T he most important feature of SMMs is that the slow relaxation of magnetization and hysteresis have a molecular origin, as confirmed by the observation of these phenomena in dilute frozen solutions. 6 7 T h e first SMM, [Mn 12 O 12 (C H 3 COO) 16 (H 2 O) 4 ].2CH 3 COOH.4H 2 O (hereafter Mn 12 Ac), was first synthesized in 1980. 8 However, it was not identified as a SMM until m ore than a decade later, when slow relaxation of magnetization was observed in ac susceptibility measurements. 4 Since then, many SMMs with various topolo gies and magnetic cores have been synthesized and studied 9 17 in order to increase the blocking temperature, T B below which the relaxation of magnetization in a SMM becomes slow. However, the record of T B set by Mn 12 Ac 18 19 remained unbroken until a Mn 6 SMM was s ynthesized by Brechin and his collaborators 15 Recently, studies on lanthanide based
17 SMMs have yielded several groundbreaking results which have led to substantial improvements of T B 20 27 The SMMs presented in this dissertation possess magnetic cores which are constituted by transition metal ions coupled through chemical bonds. Figure 1 1 shows the structures of several SMMs which will be studied in this dissertation; detailed descriptions of these molecules are presented in the following chapters. The sizes of atoms in F i gure 1 1 have been adjusted to emphasize the transition metal ions, i.e., the magnetic cores of these molecules. Instead of pursuing high T B which is critical for applications of SMMs in information technology, the studies presented in this dissertation focus on understanding fundamental quantum mechanical phenomena in SMMs. To achieve this, we studied SMMs with relatively simple magnetic cores (2~4 magnetic ions or 1D chains). The simple structures of these SMMs allow one to clearly understand several nanoscale quantum phenomena which are often obscured by structural complexit ies in large SMM clusters. A SMM is usually characterized by the two features: (a) a high spin ground state ( S ) (b) significant negative uniaxial magnetic anisotropy ( D ) As shown in Figure 1 2, the combination of S and D gives rise to a parabolic shape energy ba rrier ( ) between the spin up and spin down states which leads to a magnetic bistability at low temperatures. Supposing all the molecules are magnetized in one orientation (up), at zero field, in order for the spins to relax to the opposite orientation (down) they must obtain enough energy to jump over the relaxation barrier. S u ch a process is thermally activated and, thus, suppressed with decreasing temperature. The characteristic relaxation time for such a process follows the Arrhenius Law that: (1 1)
18 where is independent of temperature. Roughly speaking, equals to | D | S 2 for integ er spins and | D |( S 2 1/4) for half integer spins. As shown by Equation 1 1, the relaxation time grows exponentially as the temperature decreases; thus, at very low temperatures, a SMM can be treated as a permanent magnet. Another source of magnetization relaxation in SMMs is the quantum tunneling of magnetization (QTM). 28 Unlike the thermally assisted relaxation process, QTM is a p urely quantum mechanical effect which is temperature independent if only ground state tunneling is involved. The theorem of QTM will be discussed in more detail later in this chapter. 1.2 Spin Hamiltonian In order to explain the magnetic properties of SMMs one needs to introduce a proper Hamiltonian to describe a SMM. A widely used approach involves a spin Hamiltonian, which ignores the orbital angular momentum of the constituent atoms and treats the system as its total angular momentum is contributed only by spins. Such an approximation is valid when the orbital angular momentum is quenched, which is a good approximation for all the SMMs presented in this dissertation. The spin Hamiltonian can be divided into two parts: The firs t part is the zero field Hamiltonian which corresponds to the zero field splitting s (ZFS) originat ing from spin orbit interactions and the super exchange interactions. The second part characterizes the Zeeman interaction with an applied external magnetic f ield. In this dissertation we introduce two types of spin Hamiltonian: the giant spin approximation (GSA) Hamiltonian and the multi spin (MS) Hamiltonian. Both Hamiltonians are used in data interpretations and the correlations between these two types of Ha miltonian are discussed in this dissertation.
19 1.2.1 Giant S p in Approximation Hamiltonian For a polymetallic SMM, the total magnetic moment varies depending on the relative alignment of spins within the molecule. If the ground spin state is well defined and separated from other excited spin states, only the ground spin multiplet is thermally populated at low temperatures. In this case, it is reasonable to treat the spin of the molecule as a rigid number and neglect all other excited spin multiplets This app roximation is called the giant spin approximation (GSA). A convenient way for exploiting SMMs with a GSA Hamiltonian is by using the so called Stevens operators. 29 The zero field spin Hamiltonian of a molecule is given by: (1 2) are anisotropy parameters and are Stevens operators. The explicit forms of the Stevens operators (up to the 6 th order) are included in Appendix A; more generalized forms of Stevens operators can be found elsewhere 30 The subscript, p is a positive even integ er which stands for the order of the Steve ns operator. Due to the fact spin orbit interaction are invariant under time reversal, the order of the Steven operator must be even. For a molecule with spin S the highest order of Stevens operator is limited by S where 2 p 2 S The superscript, q denotes the rotational symmetry of the operator about the z axis. 188.8.131.52 Second O rder A nisotropies Even though high order Stevens operator s ( p 4) are allowed for a molecule with S it is not always necessary to include all of them up to p = 2 S For SMMs constituted by transition metal ions, the dominant energy scales are usually determined by the second order anisotropies. Therefore, t he simplest z ero field GSA Hamiltonian used in characterizing SMMs can be written as:
20 (1 3) Equation 1 3 includes only the second order anisotropies, where D is the second order uniaxial anisotropy ( D = ) and E is the second order rhombic a nisotropy ( E = ). In an axial system, D is the dominant anisotropy and the z axis is chosen as the quantization axis. The magnitude of E is restricted by the ratio between E and D where | E / D | < 1/3 ; otherwise the quantization axis is no longer the z axis and one can rotate the coordinate frame so that | E / D | < 1/3 is satisfied. As can be seen, when E = 0, Equation 1 3 is diagonal in the basis where m is the spin projection onto the molecular z axis, and m = S S +1 ... + S The energy of each state is then given b y ( m ) = Dm 2 where the ground states are the m = S states if D < 0; therefore, the z axis is the easy axis (preferred magnetization axis) and the xy plane is called the hard plane. In the situation where E is non zero (> 0), the x axis is called the hard axis and the y axis is the intermediate axis. One of the major objects of this dissertation is to understand the influences of symmetries on quantum tunneling in SMMs. Hence, it is important to exam ine the symmetry of Equation 1 3 since, strictly speaking, the symmetry of the Hamiltonian should be compatible to the symmetry of the molecule In general, a spin Hamiltonian must be invariant under time reversal, which is assured by the nature of spin or bit coupling. Therefore, any spin Hamiltonian should naturally possess C i symmetry, which means a spin Hamiltonian should remain invariant under the reversal of applied fields. Figure 1 3 shows the classical analogy of Equation 1 3, with using | E | = | D |/5 and D < 0. This analogy is performed by substitut ing the spin operators in Equation 1 3 by their classical counterpart as follows: (1 4)
21 wher e and are the inclination and azimuth al angles in spherical coordinates, respectively In this classical analogy, a spin is treated as a m a croscopic magnetic moment where all the three components ( S x S y and S z ) can be determined simultaneously. The surf ace shown in Figure 1 3 represents the energy of a spin as a function of its orientation, where t he radial distance to the surface corresponds to the spin s energy. As shown in Figure 1 3, Equation 1 3 contains the following symmetry elements: a) three ort hogonal C 2 axes, which are the x y and z axes and b) three orthogonal mirror planes, which are the xy yz and zx planes. These symmetry elements, together with the C i symmetry, give rise to a D 2 h symmetry for Equation 1 3, which is essentially much highe r than C i symmetry. Consequently, even though Equation 1 3 can successfully account for most of the low temperature magnetic measurements, such as the ac susceptibility and magnetization versus temperature data, it may fail to explain symmetry sensitive me asurements, e.g., Berry phase interference in quantum tunneling of magnetization and /or electron paramagnetic resonance experiments. As one may notice from the preceding discussions, the ( ) term was negl ected. The operator possesses only C 1 rotational symmetry about the z axis; therefore, the symmetry of a second order spin Hamiltonian may be expected to be lower than D 2 h when including However, as we will show, the term can be annihilated by choosing a proper coordinate frame; hence, a Hamiltonian with 2 nd order ZFS anisotropy always possesses D 2 h symmetry. To prove this, we write the 2 nd order ZFS terms in a more general fo rm which includes D E and : (1 5)
22 w here is a matrix corresponding to the full 2 nd order an isotropy tensor. In Equation 1 5, D and E are related to the diagonal elements of while appears as off diagonal elements, where The only restriction on is that it must be Hermitian in order to guarantee a Hamiltonian that is Hermitian. T herefore, as a property of a Hermitian matrix, can al ways be diagonal i ze d by choosing a proper Cartesian coordinate frame, i.e. all the mat rix elements except D xx D yy and D zz vanish. By doing so, can be expressed in a new Cartesian coordinate frame where F inally, because the absolute values of energy levels can be arbitrarily chosen, on e can subtract from (I is th e 3 3 id entity matrix) while the physical properties of the molecule should not be changed. The resultant zero field Hamiltonian can be written as Equation 1 3 with (1 6) Hence, Equation 1 5 is essentially equivalent to Equation 1 3, where only the axial and rhombic anisotropy terms are needed to describe the 2 nd order anisotropies; thus, the 2 nd order ZFS Hamiltonian must possess at least D 2 h symmetry. The preceding discussions show something considerably important. Even though the dominant energy scales for a SMM are determined by the second order anisotropy, it possesses an artificially high symmetry ( D 2 h ) which may not be compatible with the structural symmetry of a molecule. The consequences of this symmetry on quantum tunneling will be discussed in more detail in Chapters 3 and 4. Furthermore, in Chapter 4, we will show an example of a SMM which possesses only C i symmetry.
23 184.108.40.206 High O rder A nisotropies For a SMM with S being equal to or greater than 2, high order Stevens operators are allowed in the zero field spin Hamiltonian. The high order anisotropy parameters are usually of orders smaller than the 2 nd order anisotropies; however, they can still be clearly observed in spectroscopic type measurements. 31 33 Furthermore, high order anisotropies may introduce several features which cannot be generated by 2 nd order anisotropies. The high order axial anisotropies ( q = 0) lead to a non parabolic magnetic relaxation barrier which causes non even spacings between EPR and/or QTM resonance fields (see examples in Chapters 2 a nd 4), whereas the high order transverse anisotro pies ( q 0) can cause the zero field spin Hamiltonian to possess a symmetry other than D 2 h In this dissertation, we focus on the changes of symmetry introduced by the inclusion of high order transverse anisotropies. In order to demonstrate the symmetry of high o rder operators, we show the classical analogies for the potentials of the fourth order Stevens operators in Figure 1 4. The operator is not shown since it commutes with and posses ses C (c ylindrical ) ro tational symmetry. As shown in the figure, all the operators exhibit a rotational symmetry which is compatible with the superscript q However, one could note a systematic difference between the q odd and q even operators. It is clearly shown in the figure that the q even operators have the xy plane as an additional symmetry plane. In particular, the nodes of the and operators lie within the xy plane while the nodes of the an d operators point out of the xy plane. The symmetries of these operators can be understood in terms of the combinations of rotational symmetries and the intrinsic C i symmetry of spin Hamiltonians. For the q even operators, the di rect products of the rotational groups and inversion group lead to symmetry for and for ; thus, the xy plane is introduced as a new symmetry element for q even operators. By contrast, for the q odd operators,
24 for and for ; the resultant symmetry groups corresponding to these operators include an i mproper rotation ( C i can be treat as the improper rotation S 2 ). The absence of the xy mirror plane in the presence of q odd operators implies that the molecular hard plane may not coincide with the xy plane, which leads to several intriguing phenomena desr ibed in Chapters 3 and 4. The inclusion of high order Stevens operators has a significant influence on interpreting experimental results for SMMs, e specially the data collected in QTM measurements. If a system is limited to 2 nd order anisotropies, the zero field Hamiltonian can only lead to states mixing between energy levels wi th even | m | ( | m | is the difference between the spin projections of two states), which means quantum tunneling can only occur between states with even | m |. However, as we will show in Chapter 3, high order operators introduce new spin mixing rules, which can lead to quantum tunneling between odd | m | states. Moreover, in systems where the 2 nd order rhombic anisotropy is forbidden by structura l symmetry, m is strictly a good quantum number if the Hamiltonian is limited to 2 nd order anisotropy; hence, no quantum tun neling should occur. In this circumstance, it is necessary to include high order anisotropies in order to explain experimental results. The advantage of the GSA Hamiltonian lies in that one only needs to deal with a handful of parameters. The number of zer o field anisotropy parameters involved in data analysis can be restricted by considering the overall symmetry of a molecule. In the GSA Hamiltonian, the Hilbert space only includes the 2 S +1 states which belong to the ground spin multiplet; therefore, it ma kes data analyses for complicated large clusters computationally possible. However, t he GS A model ignores the internal degrees of freedom within a molecule thus completely fail ing to capture the underlying physics in cases where the total spin can fluctua te 34 36 On the other hand,
25 when a molecule only possesses C i or C 1 structural symmetry, the zero field anisotropy parameters cannot be restricted on the basis of symmetry, and all possible Stevens operators should be taken into account in data analysis. In these scenarios it is more reasonable to introduce another kind of spin Hamiltonian for studying SMMs, the multi spin Hamiltonian. 1.2.2 Multi S pin Hamiltoni an In the multi spin (MS) model a molecule is treated as a cluster of ions which are couple d with the other s through exchange or super exchange interactions The co rresponding zero field Hamiltonian is: (1 7) w here stand s for the spin operator of the i th ion is the 2 nd order anisotropy tensor of the i th ion. For the reason of simplicity, is written in the diagonal f orm: (1 8) where the local coordinate frame of is chosen to be the local principal anisotropy axes of the i th ion. is t he Euler rotation matrix specified by the Euler angles and which transform the local coordinate frame of the i th ion into the molecular coordinate frame. is the interaction between the i th and j th ions. Figure 1 5(a) shows a representative sketch for the MS Hamiltonian of a trinuclear SMM (Mn 3 in Chapter 3), where the molecule is treated as three coupled anisotropic spins. Figure 1 5(b) illustrate the transformation between the local coordinate frame and the molecular frame via Euler rotations. The black arrows in Figure 1 5(b) represent the molecular coordinate frame while the red arrows represent the local anisotropy axes
26 of the spin. It should be emphasized that all of the parameters in the MS should be constrained by the structure of a molecule, i.e., the overall symmetry of the Hamiltonian must be compatible with the molecular symmetry. The MS model considers the internal degrees of freedom within a molecule. Consequently, the MS Hamiltonian can naturally characterize t he ground spin multiplet as well as excited spin states, as shown in F i gure 1 5(c), which implies that a MS Hamiltonian is capable of describing phenomena where the total spin of a molecule fluctuates. In addition, all of the parameters in the MS Hamiltoni an have clear physical relevance, which allows one to understand some in depth physics in SMMs. By comparing the energy diagrams generated by GSA and MS models, one can track the origin of high order anisotropies in the GSA Hamiltonian and relate them to t he molecular structure. For instance, several studies on SMMs have shown that the high order anisotropies arise from the interplay between single ion anisotropy and magnetic interactions between the ions, which leads to mixing of excited spin states into t he ground spin multiplet. 35 37 In Chapters 3 and 4, we will discuss the emergence of high order anisotropies based on molecular symmetries to demonstrate the correlations between structural symmetry and anisotropy. However, the size of the Hilbert space involved in a MS Hamiltonian grows exponentially as the size of a magnetic core increase which makes it imprac tical to apply the MS model to very large systems. Hence, with the exception of Chapter 5, this dissertation is primarily restricted to simple molecules. 1.3 Experimental T echnique s in S tudying SMM s The novel properties of SMM have attracted many researche rs in the fields of both physics and chemistry to employ various experimental techniques to study them. One of the first landmark experiments was the observation of an unusual out of phase signal in the ac susceptibility measurements of Mn 12 Ac. 4 Since then, numerous types of thermodynamic and
27 spectroscopic techniques have been applied in the studies of SMMs. In this section we introduce two kinds of spectr oscopy, Electron Paramagnetic Resonance (EPR) and Quantum Tunneling of Magnetization (QTM), which are used in the studies presented in this dissertation. 1.3.1 Electron Paramagnetic Resonance Electron Paramagnetic Resonance (EPR) is a microwave spectrosco py technique which is used to study chemical species with unpaired electrons. To illustrate the theorem of EPR measurements on SMMs, we consider a molecule described with the giant spin Hamiltonian in the presence of an external field. The spin Hamiltonian is: (1 9) The second term is the Zeeman interaction where is the L and g tensor. Figure 1 6(a) shows a representative Zeeman diagram simulated with S = 6, D = 1.2 K and g = 2. With the application of an external magnetic field parallel to the z axis, the energy levels split into 2 S +1 non degenerated states. These states can be labeled according to their spin projections onto the molecular z axis, m Th e magnetic dipole radiation selection rule for EPR transitions is and which suggests that it probes transitions between two states within the same spin multiplet. In a typical continuous wave EPR exper iment, the frequency of the incident microwaves is fixed while a magnetic field is swept. The transmission microwave signal through the sample is recorded as a function of the applied magnetic field. When the selection rule is satisfied and the energy diff erence between two sta tes, e quals to the incident microwave frequency f ( = hf ), a molecule can absorb the incident microwave; this absorption can be observed as a dip in the magnitude of transmission microwave signal. Figure 1 6(b) shows the simulated single crystal EPR spect rum with f = 300 GHz at the temperature of 15 K upon sweeping the magnetic field up to 10 T. As shown in the figure, a series of equally spaced EPR
28 transitions is observed. These transitions are labeled A 6 A 5 ... A 2 where A i denotes the transition from m i i 1) where m is the spin projection onto the quantization axis ( z axis). In the low power regime, where there is no saturation effect caused by the incident microwaves, the intensity of an EPR transition, I i is proportional to the difference between the thermo population of the initial and final states of molecules In addition, I i is proportional to the transition matrix element connecting the initial and final states. Therefore, N i and N f stand for the number of mo lecules at the initial state and final state in the thermo equilibrium condition, respectively and represent the initial and final states of the molecule, respectively. descr ibes the Zeeman interaction associated with the magnetic fields of the incident microwaves. By comparing EPR spectra collected at various temperatures with the same frequency, the intensity shift between different transitions distinguishes the ground state EPR transition from excited transitions. In EPR performed at the NHMFL, field sweeps are carried out at multiple frequencies to generate a 2D frequency versus field plot where the slopes and intercepts of resonance branches are related to the g tensor and zero field anisotropy parameters, respectively The magnetic parameters of a molecule can thus be deduced by simulating the frequency dependence and/or the real spectra by using a proper Hamiltonian. EPR serves as a powerful technique for accurate ly probi ng the ground spin states ( S ) the zero field splitting parameters ( D E ...) and the Land g tensor for a given SMM. In the studies of SMMs, it is often combined with other magnetic measurements, e.g. susceptibility measureme nts, to determine the magnetic properties However, as we mentioned above, the general selection rule for EPR transitions is S = 0, which means the transition between different spin multiplets is not usually observed, unless strong S mixing presents. Ther efore, in the
29 situations where the interactions between the magnetic ions are strong ( ), an EPR measurement is insensitive to the coupling constant J where it can be measured by other spectroscopic techniques, e.g., inelastic neu tron scattering. 38 However, as we will show in Chapter 2, when the exchange coup ling is relatively weak, we have found that the influence of J on EPR transitions is non trivial; therefore, the interactions between spins can be deduced from EPR experiments. 1.3.2 Q uantum T unneling of M agnetization For the magnetic moment to reverse in a SMM, the spin can either acquire enough energy to jump over the barrier shown by Figure 1 2, or it can circumvent the barrier by quantum tunneling. Such tunneling is called quantum tunneling of magnetization (QTM) and was first observed in Mn 12 Ac. 5 QTM in SMMs has elicited great interest since it is related to superpositions of quantum states, which is one of the key ingredients for quantum computations. 2 Figure 1 7 shows the scheme of QTM for an S = 6 SMM. In general, QTM in SMMs can be understood with the following GS Hamiltonian: (1 10) where is the portion of the zero field Hamiltonian which does not commute with The exact expression should be determ ined by the symmetry of a molecule. For the reason of simplicity, we assume that can be treated as a perturbation ( ) and the external field is applied parallel to the molecular easy axis ( z axis). At thi s stage, we assume that the external field is applied along the z axis (the transverse field equals to zero). The effects of the transverse fields will be discussed later. QTM in SMMs occurs only between the degenerate states on the two sides of the potent ial barrier. At zero field, the energies of the 0 th order eigenvetors with opposite spin projections are
30 degenerate, where (m) = ( m) for all states. This corresponds to the energy levels crossing at zero field in Figure 1 7(a). Tunneling between and states is allowed if there is mixing between these two states caused by In a QTM experiment, the tunneling between these two states is observed as a step in the magnetization versus field plot at zero field ( k = 0 step), as shown in the hysteresis loop in Figure 1 7(b). It is noteworthy that usually an ensemble of molecules, e.g., a single crystal, is measured in experiments, where only the magnetic moments of a portion of the molecules will tunnel when the field is swept through a QTM resonance. The height of the QTM step is proportional to the probability of quantum tunneling, which will be discussed in the next section. When an external longitudinal field is applied, the states become non degenerate ; thus, QTM is switched off. However, at certain fields, the energies of and states can become degenerate again and QTM between these two states is allowed. This corresponds to the level crossings at non zero magnetic field as shown in Figure 1 7(a). If the axial anisotropy o f a SMM is restricted to the 2 nd order axial term, D these fields are given by the following equation: (1 11) Because the spin projection, m can only take discrete intege r or half integer values, the longitudinal fields correspond ing to QTM steps must be multiples of ; therefore, discrete QTM steps are observed with evenly spaced steps in SMM experiments, as shown in Figure 1 7(b). In the presence of high order anisotropies, such as the magnetic relaxation barrier deviates from parabolicity, which causes non even spacings between QTM steps.
31 Crucially speaking, the selection rules of spin tunneling in SMMs should be deter mined by and, hence, the symmetry of the molecule. Depending on the symmetry of the molecule, not all of the level crossings correspond to non zero tunneling steps. In order to have a better understanding about spin tunneling, on e needs to look more closely at the energy level crossings in the Zeeman diagram. F igure 1 8(a) shows the two possible scenarios for the energies of the two crossing states close to a QTM resonance field. For the sake of simplicity, we treat the molecule a s a two level system with the basis of the Hilbert space constituted by and states. The dash lines correspond to the energy levels of the unperturbed states ( ) where the red da sh line is the state and the black dash line is the state. When the two states are not connected by the two crossing states are not mixed and the eigenvectors of the molecule are exactly and i.e., the spin is strictly located on one side of the relaxation barrier. Thus, QTM between these states is forbidden For instance, supposing that the longitudinal field is initially at H and all molecules are polarized in the state, when the longitudinal field is swept through the QTM resonance field, H 0 to H + all molecules will follow the red arrows in Figure 1 8(a) and remain in the state. In such a scenario, no tunneling of magnetization will be observed at H 0 When the two states are connected by i.e. ( and/or higher order perturbation results are non zero), quantum tunne ling between these two states is allowed. The energy levels of these two states are shown as the thick solid curves in Figure 1 8(a). As shown in the figure, the two states do not cross at the resonance field H 0 Instead, the energy levels exhibits an avoi ded crossing, with one level has an energy minimum and the other has a maximum. The minimum energy difference between these two levels is called the tunnel
32 splitting, as shown in the figure. The state with lower (higher) energy is labeled as ( ), as shown in Figure 1 8(a). In this situation, the eigenvectors of these states are linear superpositions of and Figure 1 8(b) shows the projections of onto the basis eigenvectors where When the longitudinal field is at H ( H + ), ( ). At these fields, the spin is well localized on one side of the relaxation barrier, either in the or state, where no QTM occurs. However, at the resonance field, H 0 the spin is strongly delocalized with equal probabilities on both sides of the relaxation barrier, which leads to tunneling of the magnetization. In this situation, supposing that initially all the molecules are in the state and the field is swept from H to H + a portion of the molecules will tunnel through the barrier at H 0 and relax to the state, while the rest of the molecules remain in the state. The QTM rate betw een two states, and which corresponds to the height of the QTM st ep, is directly related to the tunnel splitting 39 40 Generally speaking, a non zero QTM step must correspond to delocalization of spins; hence, a non zero tunnel splitting. Experimentally, QTM has been used as a technique to measure smal l tunnel splitting s. 41 However, the probability for spin tunnelings at a QTM resonance is complicated by many factors. The simplest theorem for estim ating tunneling probabilities is the Landau Zener Stuckelberg formula 42 44 which treats molecules as non interacting spins. According to this theorem, the tunne ling probability at very fast sweeping rates is given by 45 : (1 12)
33 where dH L / dt is the longitudinal field sweeping rate. The dependen ce of the field sweeping rate can be understood intu itively. If dH L / dt the spins would not be able to tunnel at H 0 since the tunnel frequency which is proportional to is finite. In this case, P = 0, as predicted by Equation 1 12. On the other hand, if dH L / dt 0, which corresponds to the adiabatic limit, the magnetic moments of all the molecules will tunnel. This corresponds to P = 1, which is also consistent with Equation 1 12. However, between this two limits, the Landau Zene r Stuckelberg formula is valid only in the very fast sweeping rate regime. In most experiments, application of Equation 1 12 is somewhat naive due to limitations of field sweeping rates. For slower sweeping rates, the effects of both nuclear spins 46 48 and dipolar interactions between molecules 40 46 48 49 become significant; thus, the Landau Zener Stuckelberg formula, which is valid for an isolated spin, fails to explain experimental results. In addition, defects within a crystal can lead to a distribution of tunnel splitting s, with the experimental results giving an average of these splitting s. 50 51 So far, there is no generally accepted formula for tunneling probabilities in QTM experiments. F i nally, we discuss the influence of transverse fields on QTM. Besides the zero field transverse anisotropy the presence of an external transverse field H T (perpendicular to the z axis) also causes state mixing and contributes to tunnel splitting s. For instance, application of a transverse field parallel to x axis introduces a new term, which does not commute with the axial part of the Hamiltonian. The presence of a transverse field introduces the first order raising and lowering operators which can connect any two states with arbitra ry m through first or hig h order perturbations. Therefore, in principle, the application of a transverse field should unfreeze all tunnel splitting s regardless of the symmetry of a molecule.
34 A purely quantum mechanical phenomen on observed in QTM measurements is the Berry Phase Int erference (BPI) patterns observed upon applications of transverse fields. As we discussed previously, the presence of a transverse field should increase tunneling. However, it has been predicted that the tunnel splittings may not vary monotonically with in creasing transverse fields, especially when applied along the hard axis of the molecule. In this case, the tunnel splitting will vanish at certain magnitudes of the transverse field. 52 This phenomenon can be understood as an analogy of the Aharonov Bohm effect in spin space, as shown in F i gure 1 9. In the case of 2 nd order anisotropy, i.e., Equation 1 3, there are two different spin tunneling paths with the same least action, which correspond to two classical paths between two states on the Bloch sphere. These two paths should wind around the hard axis in opposite directions, as illustrated by the thick red lines in Figure 1 9. These two paths can interferen ce, where the destructive interference between the two paths leads to a vanishing tunnel splitting. 53 54 The difference in the phase between these two tunneling paths is related to the solid angle enclosed by the two paths, which is labeled in the figure. Furthermore for the ground state tunneling be tween and it has been shown that when a magnetic field is applied along the hard axis, the topologic phase between these two paths varies from 2 S to 0, where the tunnel splitting vanish es when the pha se difference is an o dd multiple of 52 55 The points where tunnel splitting s vanish are called diabolical points. 56 This oscillat ion of the tunnel splittin g upon application of a transverse field is called BPI, as shown by the inset of Figure 1 9. In a QTM experiment, the BPI appears as the height of a QTM step oscillates upon application of a transverse field, which ha s been observ ed in QTM experiments on SMMs. 39 41 The s ymmetry of a BPI pattern is determined by the transverse anisotropy of a molecule, consequently the molecular symmetry. Therefore, BPI provides an extremely sensitive technique
35 for probing the symmetry of a SMM. In this dissertation, we will discuss BPI pa tterns in two SMMs with different symmetries in Chapters 3 and 4. Both theoretical and experimental studies have been performed in order to understand the influences of molecular symmetry on QTM. 1.4 Single Chain Magnets Recently, a new type of one dimensi onal molecule based magnet has been synthesized which exhibits slow relaxation of magnetization at low temperatures. 57 63 This type of compound is called a sin gle chain magnet (SCM), which is named by analogy to SMM. SCMs are one dimensional isolated chains which are composed of anisotropic spin units repeatedly connected in series. Even though SCMs do not undergo long range order at finite temperatures, the com bination of the large anisotropy of the spin units and strong interaction between these units within a single chain causes an energy barrier for the relaxation of a chain s magnetic moment; therefore, the relaxation time of SCMs at low temperatures are so long that one can treat them as permanent magnets. Compared with SMMs, the obstacles to increasing the relaxation barriers in SCMs seem to be less severe. 58 64 Since the first discovery of SCMs in 2001, they have attracted much interest due to their potential application as high density information storage materials. Although the terminology, SCM, is analogous to SMM, the relaxation mechanisms in these two types of magnets are somewhat different. A SCM can be describe with the following Hamiltonian (1 13) where is t he spin operator of the i th unit. D is the anisotropy of a single spin unit and J is the Heisenberg interaction between spin units. The spin relaxation mechanism in SCMs was first predicted by Glauber based on an Ising type chain in contact with a thermal bath. 65 While the relaxation barrier in a SMM corresponds to
36 the energy required to flip the entire magnetic moment of the molecule simultaneously the relaxation barrier in SCMs cor responds to the energy to flip one spin unit within the chain. Glauber s theory predicts an exponential decay of the magnetization with a relaxation time of (1 14) w here is the correlation energy for creating a domain wall in the chain. i is the characteristic relaxation time for flipping an isolated spin unit and is given by (1 15) where is temperature independent and A is the anisotropy barrier of each spin unit. Therefore, at low temperature s where the relaxation time can be written as (1 16) As can be seen, the relaxation dynamics follows the Arrhenius law and the relaxation barrier is (1 17) The activation energy for spin reversal requ ires creation of two domain walls, which separates a domain structure s with different spin orientations. In the situation where the spin units in a chain possess easy axis type anisotropy as described by Equation 1 3, A = | D | S 2 The magnitude of the correlation energy for chains is = 2| J | S 2 in the Ising limit (| D / J | > 2 /3), while in the Heisenberg limit (| D |<<| J |) 66 However, between these two limits, the correlation energy is a complex function of S, D and J and the theorem for is still under investigation. Equation 1 17 corresponds to the energy required for the nucleation o f reversed spins i.e., the creation of a domain structure, in an infinite chain. In SCMs, once the domain structure is
37 created, the domain walls can propagate in the chain without causing additional energy, as shown in Figure 1 10. We illustrate this proc ess with a SCM in the Ising limit, where the domain structures are separated by narrow domain walls, i.e., the width of the domain wall is a single spin unit. The local field introduced by the interactions for the spins in inside a domain structure (as ind icated by the blue box in Figure 1 10(a)) is equal to 2 JS 2 which is due to the fact that the two nearest neighbors of this spin are parallel to each other (Figure 1 10(a)). However, this local field vanishes for the spins of the domain wall (the spins ind icated by the red boxes in Figure 1 10) because the two nearest neighbors of these spins are aligned anti parallel to each other; hence, the spins of the domain wall can flip with the characteristic time defined by Equation 1 15. At low temperatures, the relaxation of the magnetic moments of SCMs is mainly due to the random walks of the domain walls, where each step of the propagation of the domain structures process has the characteristic time The prev ious discussion is based on the assumption that the system is an ideal 1D chain, which means the length of a chain can be considered to be infinite. However, in a real system, due to the presence of small crystallographic defects, a SCM can only have a fin ite length, L The effect of finite lengths can have a significant influence on the magnetic dynamics of a chain at low temperatures. The correlation length which is defined as the characteristic length of the spatial decay of the two spin correlation, grows very fast as the temperature decreases. Below a certain temperature, T *, exceeds the average length of the chains, L When T > T *, the SCM behaves the same as an infinite chain and Equation 1 17 is valid. When T < T *, the probabilities for flipping spins at the ends of a chain are much larger since these spins are only connected to one nearest neighbor. 67 The activation energy for reversing spins at the ends of a chain requires t he creation of only one domain wall, and it is given by:
38 (1 18) The activation energy required by Equation 1 18 is smaller than that of Equation 1 17. Therefore, for a fin ite chain at very low temperatures, the nucleation of spin reversal always starts at the ends of a chain, where only one domain wall is needed for the domain structure. After that, the domain wall can propagate along the chain, which leads to the relaxatio n of magnetization of the chain. Therefore, small crystallograph ic defects lead to decreasing of the relaxation barrier in the SCMs at very low temperatures. 63 64 68 It is generally believed that there are th ree criteria for building a SCM : (a) Ferro or ferri magnetic interactions within a chain (b) The spin uni ts in a chain should possess large easy axis type anisotropy (c) Inter chain interactions should be small. While the first and the third criteria for building a SCM are absolutely required by definition the second criterion seems to be arguable. Kajiwara an d his collaborators have shown that a SCM can be built by a twisted alignment of easy plane type ions. 61 In Chapter 5, we will discuss a Re IV based SCM which is formed with positive D anisotropy. EPR measurements on the building blocks of this SCM indicate that the Re IV ions possess a strong biaxial anisotropy with D > 0, as well as a significant t ransverse anisotropy. We found that quantum fluctuations in this SCM are suppressed as the length of the chain increases, while the transverse anisotropy of the Re IV ions gives rise to an anisotropy barrier and leads to magnetic bistability. The results de monstrate for the first time that slow relaxation of magnetization can be achieve through anisotropy barriers created by transverse anisotropy, which suggests a new strategy in designing SCMs.
39 Figure 1 1. The molecular structure s of several SMMs studied in this dissertation The structures will be discussed in more detail in Chapter s 2 ( a ) 3 ((b) and (c)) and 4 (d). Colo r code: Mn III = purple, Mn I I = magenta, Ni = olive, Zn = green, Cl = dark gold, O = red, N = blue and C = black. H atoms have been omit ted for clarity.
40 Figure 1 2. Schematic representation for the energy levels of an S = 6 molecule with easy axis type anisotropy ( D < 0). The spin up and spin down states are separated by the energy b hich lead s to magnetic bi stability at l ow temperatures
41 Figure 1 3. Potential energy surface corresponding to the 2 nd order anisotropy. The surface is generated employing Equation 1 3 with | E / D | = 1/5 and D < 0. T he radial distance to the surface represents the energy of a spin as a functio n of its orientation.
42 Figure 1 4. Potential energy surfaces corresponding to the fourth order Stevens operators (a), (b) (c) and (d). As can be seen, the q even operators include the xy plane as a n extra symmetry operation while the q odd operators include a n improper rotation.
43 Figure 1 5. Schematic sketches demonstrating the concept of the MS Hamiltonian for a trinuclear molecule. (a) The mole cule is treated as a cluster of anisotropic spins coupled via intra molecular interactions. (b) The transformation from the local coordinate frame of a single spin ( x y and z ) to the molecular coordinate frame ( x y and z ). (c) Zeeman diagram for the M S Hamiltonian. The ground spin multiplet ( S = 6) is colored in red and the excited spin multiplets are colored in black.
44 Figure 1 6. Schematic plots illustrating the theory of EPR measurements. (a) Zeeman diagram for an S = 6 and D = 1.2 K SMM with the field applied parallel to the molecular z axis. The spin projections ( m ) for each state are labeled in the figure. The red arrows indicates the EPR transitions with f = 300 GHz. (b) Simulated EPR spectr um at f = 300 GHz and T = 15 K. The peaks are labeled according to m
45 Figure 1 7. Quantum tunnel of magnetization in a S = 6 SMM. (a) Zeeman diagram for an S = 6 SMM. (b) Magnetization hysteresis loop for a single crystal of [NEt 4 ] 3 [Mn 3 Zn 2 (salox) 3 O(N 3 ) 6 Cl 2 ] Detailed descriptions of this SMM are include d in Chapter 3.
46 Figure 1 8. The tunnel splitting associated with QTM. (a) Energy levels of two crossing states close to a QTM resonance field. The solid black lines show the energy levels with avoid crossing. The tunnel splitting is non zero and QTM bet ween these two states in allowed. The dashed red lines correspond to the energy level when tunneling is forbidden. (b) Spin projections of the ground state onto the basis functions. has equal projections onto and state at the resonance field, which is indicative of strong delocalizations of a spin as H 0
47 Figure 1 9. A schematic sketch of various tunneling paths on a Bloch sphere. The red thick arrows represent two possible t unneling paths from to The solid angle enclosed by these two paths is which is related to the phase difference between these two paths. (inset) The ground k = 0 tunnel splitting of a S = 10 SMM as a function of the applied transverse field, H x
48 Figure 1 10. The process of the relaxation of magnetization in SCMs. (a): The nucleation of spins reversal in an infinite SCM. (b) and (c): The propagation of the domain walls in an infinite SCM. The spins on the domain walls are labeled by red boxes. Note that the local fields introduced by the nearest neighbor interactions vanish for these spins.
49 CHAPTER 2 INTERPLAY BETWEEN AN ISOTROPY AND EXCHANG E IN DINUCEAR MOLECU LAR MAGNETS Portions of the work presen ted in this chapter can be found in the following articles: Inglis, R.; Houton, E.; Liu, J.; Prescimone, A.; Cano, J.; Piligkos, S.; Hill, S.; Jones, L. F.; Brechin, E. K., Accidentally on purpose: construction of a ferromagnetic, oxime based [MnIII2] dime r. Dalton Trans. 2011, 40 (39), 9999 10006. (reused with permission from The Royal Society of Chemistry ) 2.1 Introduction The magneto anisotropy barrier ( ) plays a significant role for single molecule magnets (SMMs) since this barrier determines the blocking temperature ( T B ). In the picture of the giant spin approximation (GSA) model, the anisotropy barrier is mostly defined by the molecular axial anisotro py, D and the ground spin state, S wher e | D | S 2 It is commonly believed that the magneto anisotropy of a molecule is mostly contributed by the anisotropies of the single ions within a molecule. Therefore, the axial anisotropy of a molecule is strongl y related to the anisotropy parameters of single ions. On the other hand, the ground spin state of a molecule is mainly determined by the interactions between magnetic ions. Moreover, when the interactions within a SMM are weak, the internal degrees of fre edom can have non trivial influences on its magnetic relaxation barrier. Consequently, it is important to understand the interplay between anisotropy and exchange interactions, as well as any correlations with the molecular structures, in order to build SM Ms with high T B Unfortunately, deducing single ion anisotropies and exchange interaction parameters for a large SMM cluster is not trivial. The interactions between spins within a molecule are usually determined by fitting the dc susceptibility measuremen ts carried out over a wide temperature range. 1 However, given the complexity of a large molecule, it is almost impossible to deduce a
50 unique parameter set owing to the presence of multiple interac tion parameters. This can be understood intuitively, where the procedure of data analysis involves deducing multiple parameters from one dc susceptibility versus temperature curve; hence, the Hamiltonian used in the analysis is likely to be over parameteri zed. On the other hand, in magnet ic measurements, the anisotropy of a molecule is usually deduced by fitting the magnetization measurements at various fields and temperatures conducted at the low temperature regime. This fit is strongly affected by the ini tial guess values used in the fitting program where, in most cases, both easy axis ( D < 0) and easy plane ( D > 0) type anisotropies can be obtained by fitting the same group of data, depending on the initial guess values. This dependence will be discussed in more detail in the data analysis for complex 1 in this chapter. Even though low temperature spectroscopic measurements such as electron paramagnetic resonance (EPR) can give a clear determination of the anisotropy of a molecule, it is still too ambitiou s to deduce a unique set of parameters for the exchange interactions of a large complicated molecule. Therefore, in order to understand the SMM behavior at the microscopic level, it is advantageous to study simple, low nuclearity molecules. Studying such s ystems provides information about magneto structural correlations, while the molecules may also be used as building blocks for larger clusters. The simplest (and still extremely useful) platform for one to study the interplay between anisotropy and exchan ge interactions is dinuclear molecules composed of two identical magnetic ions. Especially when the molecule possesses inversion symmetry the interaction parameters reduce to one exchange coupling tensor, which can be simplifie d to a Heisenberg interaction, J in most compounds. In these molecules, the single ion anisotropies and exchange interactions can be deduced unambiguously by combining magnetic measurements and low temperatures
51 EPR. The information learned from these dinu clear compounds is of great help in revealing the correspondence between bond structures and magnetic interactions. Among transition metal ions, Mn III has been used extensively in synthesizing SMMs due to its large spin ground state ( s = 2) and strong anis otropy caused by Jahn Teller (JT) distortions. In this chapter, we present EPR studies on two dinuclear Mn III molecular magnets, [Mn III 2 Zn II 2 (Ph sao) 2 (Ph saoH) 4 (hmp) 2 ] ( 1 ) 69 and [Mn III 2 ( L 2) 2 (py) 4 ] ( 2 ) 70 ( where L 2 is the trianion of 3 (3 oxo 3 phenylpropionyl) 5 methylsalicylic acid, and py is pyridine ). Compound 1 has a magnetic core constituted by two ferromagnetically coupled Mn III ions while the Mn III ions in the core of compound 2 are antiferromagnetically coupled. These compounds have attracted our interest s because a) they provide ideal platforms to study magneto structural correlations and b) they are proposed as candidates for pressure studies. The molecular structure, e.g., the bond lengths and bond angles, can be varied upon applying pressures, which will lead to variations in the magnetic pro perties of the molecule. EPR experiments were conducted on both compounds to explore the magnetic interactions and anisotropies associated with them. 2.2 Ferromagnetic ally C oupled D inuclear [ Mn III ] 2 M olecule 2.2.1 Discussions of Magnetometry Results Comple x 1 crystallizes in the triclinic space group P 1 and lies about an inversion cent re. The structure of compound 1 is shown in Figure 2 1(a). The magnetic core of this molecule is shown in Figure 2 1(b). The molecule contains a butterfly like core of four m etal ions with two Mn III ions at the body position and two Zn II ions at the wing tip positions. The Mn III are six coordinate in a Jahn Teller elongated octahedral geometry. The Zn II ions are diamagnetic which block any interaction through them. As a result the two Mn III are only coupled via the super exchange interactions through the two identical N O bridges.
52 Complex 1 was synthesized to study the magneto structural relationship in oxime bridged Mn III cluster compounds It is also a candidate for studyi ng the influences of large pressure on the magnetic properties of molecules. The bonds between the two Mn III ions are likely to be distorted in the presence of large pressure; thus, the interactions within the molecule should vary upon applying pressure. M agnetic measurements for complex 1 were carried out in Dr. Euan Brechin s group in the University of Edinburgh, Scotland. High frequency EPR measurements were carried out in the NHMFL and form the subject of this section. Complex 1 possesses inversion symm etry, which guarantees the anisotropy tensors of the two Mn III ions to be identical and parallel to each other. In the data analysis, we simplified the system by only considering the Heisenberg interaction between the two Mn III ions. The molecule is modele d by the following multi spin (MS) Hamiltonian: (2 1) where is the spin operator for the i th ion; d and e are the 2 nd order anisotropy parameters of the single Mn III ions; and J is the Heisenberg interaction between them. At this point, we would like to briefly review some discussions about the magnetic data made by Dr. Brechin s group. These discussions demonstrate several important issues involved in interpret ing magnetic measurement results; furthermore, these discussions emphasize the significance of EPR. In the magnetometry data analysis for most SMMs, the exchange interactions and the anisotropy parameters of a molecule are obtained by fitting different mea surement results to different models. The exchange interactions are usually obtained via fitting the dc susceptibility versus temperature da ta ( m T vs. T ) to an isotropic MS Hamiltonian, which treats the molecule as a cluster of isotropic ions coupled thro ugh an isotropic exchange interaction, i.e., it ignores the zero field splitting s (ZFS) of individual ions. This measurement is
53 usually carried out at a wide temperature range (from the helium temperature to room temperature), where the main features of th e obtained curve are dominated by the energy separations between different spin multiplets. On the other hand, the anisotropy parameters are usually deduced by fitting the low temperature magnetization measurements at various fields and temperatures, namel y reduced magnetization (RM), to a GSA Hamiltonian, which assume that only the ground spin multiplet is populated. In the data analysis for complex 1 Dr. Brechin s group tried to fit the magnetic measurements to Equation 2 1, which allows both the anisotr opy parameters d and e as well as the exchange interaction, J to vary simultaneously. The details of these discussions can be foun d in Ref. [ 31 ]. In th eir analysis, they showed that the m T vs. T data provides a tight constraint on the exchange interaction, wher e the best fit results alw ays converge at approximately the same J value with J = 6.4 K, regardless of the initial guess values used in the fit. By contrast the axial anisotropy values are not well constrained by the m T vs. T data, depending strongly on the initial guess values. Meanwhile, varying d has little influence on the obtained J value. This is a clear indication that m T vs. T data cannot be used as a reliable criterion in the determination of magnetic anisotropy. Therefore, in order to determine d the Brechin group fit the low temperatures RM data to Equation 2 1, fixing J = 6.4 K. The RM result provides a much tighter constraint on d with the best fit value of d = 4 .65 K. However, in their analysis, they clearly showed that the best fit of d still relies on the initial guess used in the fit. In fact, it is possible to fit the RM data with an easy plane type d (= +5.21 K) if the initial guess value for d is chosen to be positive, though the quality of fit clearly favors an easy axis type d (< 0). The preceding discussion indicates that the anisotropy parameters deduced from magnetic measurements are not always reliable, which also applies to the compound 2 discussed in the
54 second part of this chapter. T herefore, s ingle crystal EPR measurements were performed on complex 1 primarily with a view to determin ing the magnetic anisotropy associated with its ground state However, as shall be seen, it is also possible to estim ate the exchange interaction strength within the molecule from EPR measurements. The capability of deducing the exchange interaction parameter from EPR is of great importance, as it paves the way for future studies regarding the influence of high pressures on the magnetic interactions in this molecular magnet. 2.2.1 EPR S pectroscopy EPR Experiments were conducted in a spectrometer that enabled in situ rotation of the sample about a fixed axis; d etails can be found elsewhere 71 72 Variable frequency (60 410 GHz) and variable temperature measurements were thus performed with the field aligned both in the hard plan e and at an orientation close to the easy axis. Figure 2 2 shows representative spectra obtained at 171 GHz with the external field applied in the hard plane. The sharp transition seen at ~ 6 T at all temperatures is due to a paramagnetic impurity (with g = 2.00) in the sample and is ignored in the ensuing data analysis. A cluster of transition s seen in the 12 13 T range increase s in intensity with decreasing the temperature, indicating that it involves excitations from the ground state. Likewise, a broad, featureless dip centered at ~7.5 T (the red arrow in Figure 2 2) is also seen to strengthen with decreasing temperature. The observation of multiple excitations from the ground state can be attributed to weak disorder in the sample. Indeed, essentially ide ntical behavior is seen in the hard plane spectra of the Mn 12 acetate 31 73 74 and Ni 4 75 76 SMMs. The cluster of fine structure peaks at the highest fields is likely caused by a discrete disorder associated with either the ligand or solvent molecules that gives rise to distinct species with slightly different transverse anisotropy parameters, hence the observation of several resolved peaks in the hard plane spec tra (as denoted by the vertical arrows in Figure 2 2 ). 31 74 The broad feature can be attributed to a
55 very s light tilting of the zero field splitting ( ZFS ) tensors associated with the different species such that the field is not perfectly aligned in the hard plane for all of the molecules ( vide infra ) 31 Figure 2 3 shows several representative spectra collected with the field being close to the molecular easy axis at 15 K. W e note that the diminished sensitivity of the spectrometer at higher frequencies required data above 360 GHz to be recorded at 2 K, where only the A 4 transition (ground state transition) is observed The spectra also reveal clear evidence for th e same disor der, with the effect most pronounced f or the ground state transition as expected on the basis of D strain. 77 78 Figure 2 4 shows the frequency dependence of EPR peak positions observed with the field applied (a) close to the molecular easy axis and (b) exactly in the hard plane ; we include only peaks that can clearly be identified and assigned to known transition s As can clearly be seen in (a), four branches of resonances are observed which are labeled A 1 A 2 A 3 and A 4 where A i denotes the transition from m = i to ( i 1) states and m is the spin projection on to the easy axis (A 4 is the m = +4 to +3 transiti on and B 4 is a shoulder on A 4 attributed to disorder) The resonance s in Figure 2 4 (b) have been labeled according to the same scheme except that m now corresponds to the spin projection on to the high field quanti z ation axis, which necessarily corresponds to the applied field direction Th e observation of four resonance branches with positive zero field intercepts in Figure 2 4 (a) confirms the S = 4 ground state spin value, while the fact that the A 4 transition persists to the lowest temperatures suggests e asy axis anisotropy. I t is noticeable in Figure 2 4 (a) that the horizontal spacing between the four main branches is non uniform It is well documented that this is indicative of weak coupling, i.e., J ~ d 35 37 In the strong coupling limit ( ), only 2 nd order ZFS ( ) is expected within the S = 4 ground state, wh ich would result in equal spacing between the easy axis peaks. In the GSS picture, this
56 uneven spacing is captured via and higher order axial anisotropies, which is caused by spin mixing between different S multiplets in the weak coupling regime. Thus, the uneven spacing provides an additional handle on the exchange coupling parameter J The solid lines in Fig ure 2 4 represent the best simulations of the peak positions using Eq uation 2 1 with the following parameters: d = 5.52 K J = 6 71 K and g = 2.00. Figure 2 4 (a) required an 18 o field misalignment due to the fact that one cannot guarantee exact coincidence with the easy axis when rotating about a single axis Furthermore, i n order to obtain a good simulation for the hard pla ne data in Figure 2 4 (b) a rhombic anisotropy term was included for each Mn III ion with e = 0. 26 K ; the data points represent the strongest central resonance within each cluster of peaks, with the error bars spanning the adjacent peaks/shoulders (see vert ical arrows in Figure 2 2 ). It should be noted that the obtained e value represents a lower bound because the precise field orientation within the hard plane was not known. The broad feature around 7.5 T in Figure 2 2 is illustrated by the considering a sm all field misalignment as shown in Figure 2 4(b). When the applied field is in the molecular hard plane, the A 3 resonance goes through a minimum (~190 GHz) and then tends to a sizable non zero value as the field tends to zero; hence, this resonance should not be observed at 171 GHz, at which the spectra in F i gure 2 2 were recorded. The thin curves in Figure 2 4(b) show that when the field is slightly out of the molecular hard plane, the minimum of the A 3 resonance decreases below 171 GHz; thus, it is possib le to explain this transition at 171 GHz by including a small field misalignment, which leads to the broad feature close to 7.5 T in Figure 2 2. The slope of the A 3 resonance branch at 7.5T/171 GHz is extremely shallow, which is why the linewidth of the fe ature is so much broader than that of other EPR transitions. It should be clarified that this out of hard plane feature is not simply an experimental error, as careful angle dependence studies were performed
57 to ensure that the applied field was within 1 o o f the molecular hard plane. The observation of A 3 at 171 GHz requires a field misalignment for at least 3 o We attribute this to orientational disorder resulting in a fraction of molecules whose easy axes are slight tilted away from the others. As can be s een, t he simulation s provide a good overall agreement for all plotted EPR transitions observed in both orientations. We note the e xcellent agreement between these parameters and those obtained via analysis of magnetic data, thereby providing further confid ence in the conclusions of this work. We conclude the analysis for complex 1 by discussing the constraint of d and J based on EPR spectra. The ZFS associated with the easy axis ground state transition (A 4 ), 0 is dominated by the single ion anisotropy, d while it is extremely insensitive to J even in the weak coupling regime. With the aforementioned single ion anisotropies ( d = 5.52 K and e = 0. 26 K), 0 only changes by 0.7% (0.1 K) upon varying J fro m 1 K (~ d /5) to Furthermore, if e = 0, as long as the interaction is ferromagnetic, 0 is completely independent of J This is essentially a classical result which indicate s the fact that the in phase precession of 2 identical spins about a fixed axis is insensitive to the interactions between them A similar conclusion was obtained in other SMMs which can be generalized to any ferromagnetically coupled molecule with N identical spins. 37 Another semi quantitative approach for understanding this result is based on considering the symmetry of the molecule. We first consider the scenario where e = 0; in this case, m is a good quantum number. The molecule possesses inversion symmetry, which guarantees that parity is good quantum number for the system. The ground state, and the first excited state, are two symmetric states. Because is the only m = 4 state, it cannot mix with any other states if e = 0. There are two m = 3 states, namely and However, state is an anti symmetric state;
58 thus, there cannot be any mixing between the and s tates. Therefore, the and cannot mix with any other states and the transition between these two state (A 4 ) is independent of J By contrast, the can mix with which is a symmetric state. The mixing between these m = 2 states is affected by J ; hence, the transition between the m = 3 and m = 2 states (A 3 ) is J depende nt. When e 0, m is no longer a good quantum number and the state can mix with m = 2 states via the 2 nd order rhombic anisotropy. In this scenario, the ground state transition will be affected by J since the can mix with through However, this mixing is a high order perturbation because it involves both m mixing introduced by e and S mixing introduced by weak J Therefore, the ground state transition shows very weak dependence on J even in the extremely weak interaction regime. Ac cording to the preceding discussions, d can be accurately determined by only considering the ground state transition. The spacing between the ground transition (A 4 ) and the first excited transition (A 3 ) provides a constraint on the exchange interaction J In principle, the molecule can be characterized by these two transitions, which are the strongest transitions at low temperatures. This indicates that it is possible to study the influence of pressures on the magnetic properties of complex 1 by EPR, where the sensitivity of experiments is strongly decreased due to the limitations on the volume of the crystals that can be used in pressure studies. 79 2.3 Anti ferromagnetically C oupled D inuclear [ Mn III ] 2 M olecule Complex 2 crystallizes in a triclinic space group P 1 and lies about an inversion cent er. The structure of compound 2 is shown in Figure 2 5. 2 was originally proposed as a candidate for investigating the magnetic properties of molecular magnets under pressures. The Mn III ions in the
59 molecule are si x coordinated in a JT elongated octahedral geometry with the JT axes defined by the Mn N bonds, which is similar to complex 1 However, the exchange paths in this compound are more complicated, since the shortest contacts connecting two Mn III ions are two O N N N O bonds. Nevertheless the dc magnetic susceptibility measurement suggests a non zero interaction between the Mn III ions within the molecule 70 which is confirmed by EPR studies ( vide infra ). The structure and magnetic studies of complex 2 was originally reported by Dr. Aromi s group in 2002. 70 However, its magnetic properties were not fully understoodat the time since the anisotropies of the Mn III ions we re unknown, in spite of the fact that a powder EPR study had been performed. A MS Hamiltonian including anisotropic exchange was proposed in Ref. [ 70 ] in order to interpret the magnetic measurements However, the data analysis resulted in an unphysical, hi ghly anisotropic interaction ( J z / J x y 0.8) which is unlikely to occur in Mn based magnets. 8 0 To fully understand the magnetic properties of 2 further single crystal EPR measurements were performed to probe its magnetic parameters. Detailed analyses if the EPR spectra were performed, giving a comprehensive picture regarding the low temperature physics in weakly antiferromagnetically coupled Mn III based dinuclear molecules. The EPR experiments give an unambiguous determination of both the anisotropy and exchange parameters of 2 which also explain the previous magnetic measurements. 2.3.1 EPR S p ectroscopy The previous EPR measurements were performed on powder samples of 2 by Dr. Krzystek at the NHMFL However, the results were never understood. Therefore, we carried out EPR experiments on single crystals of 2 In the mean time, we also redid some powder measurements and analyzed the old powder spectra collected by Dr. Krzystek. Single crystal measurements
60 were performed in a 7 T superconducting magnet with temperature control achieved by a 4 He flow cryostat. A Millimeter Wave Network Analyzer serv ed as a microwave source and detector, and a cavity perturbation technique was employed to measure a needle shaped single crystal with 3 71 72 Unfortunately, larger sample was unavailable due to difficulties in growing large single crystals. The volume of this crystal is of an order smaller comparing with most single crystal s amples used in our experiments. Powder EPR measurements were carried out in a transmission type spectrometer based on a 1 5 T superconducting magnet, which differed from that described in 81 only by the use of a Virginia Diodes solid state source operating at 13 1 GHz, followed by a cascade of multipliers and amplifiers. The powder experiments were performed with both loose powders and finely ground powders restrained by n eicosane. We start by describing the experiments on a powdered sample of 2 ; an example of EPR spectra is shown in Figure 2 6 The sample produced a very strong and well defined EPR response over all the available frequency range. However, this rich spectrum was essentially uninterpretable. Qualitatively, it shows that the interactions between the Mn III ions create new spin states that are unlike those of the individ ual Mn III ions, as the spectrum does not resemble that of a spin S = 2 system At the same time, we did not succeed in interpreting the data in terms of a well defined S state. We thus performed HFEPR measurements on a single crystal. An example of a singl e crystal spectrum is shown in the inset of Figure 2 6 Single axis rotation was carried out first to align the applied external field close to the molecular z axis. Following that, extensive frequency and temperature dependence studies ( 380 510GHz, 2 20K ) were performed at this fixed orientation. Figure 2 7( a ) shows the peak positions of the observed EPR transitions in a crystal with the field applied close to the molecular z axis at 2 K. Temperature dependent
61 measurements (not shown) confirmed that the tw o transition branches observed at low field which are denoted as and in Figure 2 6 and 2 7( a ) originate from the ground spin state. As shown in Figure 2 6, due to the limitation that a large single cr ystal is not available for 2 the signal to noise ratio in the single crystal measurements is much lower than that of the powder experiments. However, as we will show, the single crystal experiments still provide sufficient information to fully characteriz e complex 2 For an antiferromagnetically coupled dinuclear molecule, the ground state is expected to be a non magnetic state if The fact that we observe well defined EPR transitions at low temperatures indicates that the J is s mall and the system cannot be describe d by a GS A model. Therefore, we employed Equation 2 1 in analyzing our EPR results. In the data analysis, we reduced the number of unknown parameters by making the following two assumptions: a) g = 2.00 and b) e = 0. T he a ssum ption of an isotropic Land g factor, g = 2.00, is a reasonable approximation for most reported Mn III ions in octahedral coordination. 82 The rhombic anisotropy for individual ions is ignored in our analysis due to the fact that, with the fields applied in the molecular xy plane, the low temperature EPR tr ansitions appear out of experimental field range (above 15 T). Fortunately, the experimental results indicate that such simplification is acceptable (see more details in powder data discussions). Th is is simplest Hamiltonian that one can adopt to describe complex 2 However the Hamiltonian can account for the two key experiment al results: ( a ) the EPR spectra (Figure 2 6 to 2 8) and ( b ) the reduced magnetization data (F i gure 2 9) Before presenting the detail ed analysis of the spectra, we should discuss the EPR selection rules that can be applied to 2 Due to the fact that 2 does not have a well defined spin state, one can not apply the normal EPR selection rules, and as we mentioned in C hapter 1 However since only the axial anisotropy is included in the Hamiltonian, m remains a good
62 quantum number wh en the field is applied along the molecular z axis. Thus, the selection rule still holds. Furthermore, due to the inversion symme try of the molecule, all of the eigenvectors must have definite parity upon exchanging two spins ( symmetric or anti symmetric ). The probability of EPR transitions, which are magnetic dipole transitions, is non zero only if the transition occurs between sta tes with the same parity. Therefore, we conclude that the EPR selection rules of 1 are: ( a ) ; and ( b ) the parity of the initial and final states is conserved. Figure 2 7( b ) shows the low energy part of the simulated Zeeman diagra m of 2 with the applied field along the molecular z axis. This simulation was performed employing Equation 2 1 with the following parameters: d = 5.76 K e = 0 and J = + 2.45 K. These parameters give good agreements for both single crystal and powder EPR e xperimental results ( vide infra ). In order to understand the origins of the observed EPR transitions, one needs to consider the symmetry of the eigenvetors corresponding to the energy levels in Figure 2 7(b). In order to do this, we treat the diagonal part of the zero field Hamiltonian, as the unperturbed Hamiltonian and the off diagonal part of the Heisenberg interaction, as the perturbation where a re the raising and lowering operators and As we will show, this perturbation treatment gives a quantitative agreement with the experimental results. If J = 0, the molecule is composed of two non inte racting spins; hence, the eigenvectors of the molecule can be written as direct products of the uncoupled single Mn III eigenvectors and (abbreviated as ) where and represent th e spin projection of the two Mn III ions and m = At zero field, the ground state of the molecule is quadruply degenerate, where these four states are: and These correspond to two uncoupled spins where the eigenvectors are not constrained by parity. When we include a small
63 antiferromagnetic interaction, the eigenvector s should be written as the symmetric /antisymmetric combinations of these states in order that they have definite parities. Thus, the eigenvectors for the ground states are: ( m = +4), ( m = 4), ( m =0) and ( m = 0). For the sake of simplicity, we abbreviate as and as With these eigenvectors, we co nsider the 1 st order correction to the zero field energies. The inclusion of a small interaction lifts the degeneracy of these states, where (2 2) Therefore, inclusion of J splits the m = states form the m = 4 states by 8 J Because the interaction is antiferromagnetic ( J > 0), when considering the 1 st order perturbation, the zero field ground states of the molecule are and as shown in Figure 2 7(b ). The small energy difference between these states originates from high order corrections, which will be discussed later. Now we consider the origin of the and transitions observed in single crystal EP R measurements. According to the preceding discussions, the ground states are two m = 0 states. Therefore, the final states associated with and transitions must be two m = 1 states. For reasons of simp licity, we only consider the m = 1 states in our analysis. W h en J = 0, the eigenvectors for the m = 1 states are and where and are the states which are lower in energy. Upon including a small interaction, the eigenvectors of
64 these low energy m = 1 states are and The energies of these states given by the 1 st order perturbation are It i s important to point out that, when only considering the 1 st order correction, and are degenerate, and the two excited states, and are also degenerate. Hence, the EPR transition associated with the symmetric states, to should be exactly the same ener gy as the transition between the anti symmetric states, to which suggests only one ground EPR transition should be observed The ZFS of these transitions predicted by the 1 st order correction equal s 3 d +2 J which is 22.2 K (462GHz) according to our parameters. The ZFS s correspond to the and branches are 451GHz and 438GHz, respectively which are in close agreements with the prediction. I n the higher order perturbation calculations, causes mixing between the states with the same parity and m value, resulting high order corrections to the eigenvectors. For instance, the eigenvector of the symmetric m = 0 state is composed of li near combination s of and while the eigenvector of the anti symmetric m = 0 state is composed of linear combination s of and These corrections lift the degeneracy of the states Consequently, two distinct transitions, and are observed correspond ing to the with the symmetric states and anti symmetric states, respectively. Ho wever, simulations shows that the energies of and only differ by 0.03 K, which means that both states have equal population in the experimental temperature high order corrections are dominated by which depend only on the coupling constant J In our simulations, we noticed that difference between the ZFS of
65 and is approximately proportional to | J | 3 which indicates that it is the result of a 4 th order correction in energy. The simulations of the EPR transitions are shown in Figure 2 7(a) The simulation was performed employing Equation 2 1 with the following p arameters: d = 5.76 K e = 0 and J = + 2.45 K. These parameters are constrained by trying to simulate the peak positions of the and branches. We note that the single ion axial anisotropy of the Mn III io ns is similar to the reported anisotropy values of Mn III ions in other elongated octahedrally coordinated complexes 37 69 82 and the coupling constant J is consistent with the previous dc susceptibility study 70 ( J z = +3.14 K, J x = J y = 4 K) The preceding discussions show that it is possible to deduce all of the zero field magnetic parameters from the two ground state transitions observed in single crystal EPR measurements. The ZFSs of the and transitions approximately equal to 3 d +2 J while the splitting between them provides a tight constraint on th e interaction J After obtaining the ZFS parameters from single crystal studies, we returned to the powder spectra to see whether these parameters can explain the many other EPR transitions Figure 2 8(a) shows the peak positions of the observed EPR transitions collected from a powder sample at 10 K. The transitions are sorted into four different resonance branches: and as labeled in Figure 2 8( a ) on the basis of the peak positions of the high frequency data. As shown in the figure, all of the resonance branches have roughly the same slope correspond ing to the g = 2.00, EPR transitions w hich affirms the simplification of including the axial anisotropy term only. The solid lines in Figure 2 8( a ) represent the simulations of EPR transitions while the Figure 2 7(b) is the simulation of the energy diagram. The simulations we re performed with the field applied parallel to the molecular z axis. We also performed simulations upon applying the
66 magnetic field in the molecular xy plane and the results indicate that the low temperature EPR transitions in the xy orientation will occur at fields above 1 5 T for the experimental frequencies ; thus they are not observed in the se experiment s The low field branches and have been observed and discussed in the single crystal studies. We focus on the tran sitions occurring at high fields. The parities of the low energy states are labeled by color in Figure 2 8(b) and the arrows indicate the observed EPR transitions at approximately 650GHz. The final state s of these transitions are determined by the EPR sele ction rules which we discussed previous ly The transition is the m = 4 3 transition between the symmetric states while the and transitions are the two m = 3 2 transiti ons associated to the symmetric and anti symmetric states, respectively As shown in Figure 2 8(b) all of the transitions are reasonably explained with the aforementioned selection rules We note that there are two symmetric m = 2 states, as labeled in F igure 2 8(b). However, simulations reveal that the matrix element associated with transition involving the higher energy m = 2 states (the thick red arrow) is more than 10 times stronger than that of the other transition (the thin red arrow) Therefore, o nly the stronger transition seems to be observed in the experiments. The powder spectra are consistent with the single crystal studies and confirm the parameters we deduced from single crystal experiments We notice that although the simulation successful ly explain s the high frequency results (above 550GHz), we still cannot fully reproduce all of the transitions observed in the lower frequency spectra (below 550GHz), such as the and resonance branches i n single crystal studies as well as several other high field transitions observed in powder spectra between 400GHz and 450GHz. We suspect that this may due to the fact that we oversimplified the Hamiltonian and neglect ed transverse anisotropy term s I nclu ding the se transverse term s will lead to mixing s between different m states lead ing to new EPR transitions. In our simulations, we
67 also noticed that the single crystal EPR spectra are extremely sensitive to the field orientations. The and transitions may emerge in single crystal spectra when a small field misalignment is includes. However, at high fields and high frequencies, the peak positions are relatively insensitive to small variance s of the experim ental parameters, where only well defined resonance branches are observed, as shown by the data collected above 600 GHz in powder experiments (Figure 2 8(a)). 2.3.2 Magnetic M easurements In order to validate the results obtained via EPR measurements, we at tempt ed to simulate the reduced magnetization (RM) measurement results with the obtained parameters. The original RM measurements were performed by Dr. Aromi in 2002. However, the result was never published because it could not be interpreted via a simple GSA or MS model, even considering anisotropic interactions 70 In magnetic measurements, if the high temperatures regime is mostly affected by the energy difference between different S multiplets set by J while the low temperature measurement is strongly influenced by the ZFS of a molecule. Therefore, as we discussed in th e data analysis for complex 1 the general strategy is to fit high temperature dc susceptibility ( m T vs. T ) measurements with an isotropic MS Hamiltonian to obtain J while the low temperature RM data are fit to a GSA model to deduce d However, a s shown in our EPR analysis d is two times larger than J in complex 2 If we consider an isotropic MS mod el, the energy scale set by J is about |8 J |, which corresponds to the energy difference between two spins aligned parallel and anti parallel to each other. On the other hand, the zero field splitting between the and states for a single Mn III ion equals |4 d |, which approximately equals to |8 J | for 2 i.e., the energy scales determined by d and J are virtually identical in 2 ; therefore, both d and J have equal influence on the magnetic properties of 2 in a ll temperature
68 regimes, which indicates the necessity of including the ZFS anisotropy in analyzing both m T and RM data However, as we discussed in the data analysis for 1 deducing these parameters from a simple magnetometry measurement can be unreliable For instance, as we discussed earlier in this chapter, the sign of d strongly dependes on the initial guess value used in the fit. Furthermore, i n the case of 2 the influence of d so strong that it is not possible to determine J without accurate informa tion about d T h e experimental and simulated RM results are shown in Figure 2 9 The squares are the experimental data while the solid lines are the simulations employing Eq uation 2 1 with the parameters obtain ed via EPR studies. The experimental results s how that, at a constant field, the magnetic moment of the sample goes through a maximum and then decreases upon reducing the temperatures. The reduction of the magnetic moment at low temperatures can be understood with the Zeeman diagram shown in Figure 2 8(b), where the molecular ground states are non magnetic ( m = 0) at fields below 3.7 T. Therefore, the magnetic moment decreases when lower temperatures. Even at 4 T, the m =4 state is very close in energy with the m = 0 states, which leads to a reduction of the magnetic moment at low temperatures, as shown in both experiments and simulations. The simulations reproduce the main featu res of each isofield data curve s and give quantitative agreements for the 0.5 T, 1 T and 2 T data. More importantly, the simul ations exactly reproduce the maxima positions of the reduced magnetization on the abscissa scale which indicate s that the simulations predict the temperatures at which the magnetization saturates with different applied field s The simulations confirm that the estimations of the anisotropies and the coupling between the Mn III ions are reliable, while the small deviation s of the absolute value may come from the Land g factor not equal to 2 and/or experimental error s such as the samples being partial ly aligne d by the magnetic field. From simulation s we also
69 noticed that, below 3.7 T, the ground state of the molecule is non magnetic ( m = 0) while above 3.7 T the ground state is m = 4 At low temperatures, when the field is above 3.7 T, a powder sample of compl ex 2 will face a much stronger torque compar ed with the torque at lower field. The strong torque can easily cause the sample to be partia l l y aligned We suspect that this might be the reason why the simulation and experimental data show a larger deviation at 4 T compared to lower field results. In the EPR experiments, when loose constrained powders were used, strong hysteresis loops were also observed due to the mechanical movements of micro crystals. Such hysteresis loops vanish when the powder samples are well constrained with n eicosane. 2.4 Summary In this chapter we presented EPR studies on two Mn III dinuclear molecules and compared the results with magnetic measurements. Complex 1 and 2 show two types of representative EPR spectra which one would expec t from ferromagnetic and antiferromagnetic dinuclear molecules, respectively Good agreements between the EPR measurements and magnetic measurements are obtained in both compounds. The effects of weak coupling are extensively discussed in both 1 and 2 ill ustrating the importance of considering both exchange and anisotropy in the analysis of magnetic measurements. In addition, the studies presented in this chapter have paved the road for future pressure studies of these compounds by a recently developed hig h frequency EPR technique.
70 Figure 2 1. The molecular structure and magnetic core of complex 1 (a) The molecular structure of complex 1 Colo r code: Mn = purple, Zn = green, O = red, N = blue and C = black. H atoms have been omitted for clarity. (b) The magnetic core of complex 1 Only the metallic atom, their coordination atoms and the chemical bonds between them are shown in the figure. The Jahn Teller axes (local easy axes) of Mn III ions are defined by the elongated Mn N bonds, as shown in figure. The two Mn III ions are connected by two identical N O bonds. (c) Schematic representation for the magnetic core of 1
71 Figure 2 2. Hard plane EPR spectra of complex 1 obtained at 171 GHz as a function of temperature from 20 K (top) to 2.5 K (bottom)
72 Figure 2 3. Representative easy axis EPR spectra of complex 1 obtained at 2 K (391.5 GHz) and 15 K (310.1GHz and 248.6 GHz). The observed EPR transitions are labeled according to the scheme described in the main text.
73 Figure 2 4. Frequency dependence of EPR peak positions of 1 The data were collected with the field applied (a) close to the molecular easy axis and (b) exactly in the hard plane The data were mainly collected at 15K ; however, data above 360 GHz were collected at 2 K The thick solid lines are the simulations of the peak positions employing the Hamiltonian and parameters provided in the main text. The thin solid curves in (b) illustrate the influence of applying the field slightly out of the molecular hard plane on the A 3 transition. Each c urve represents an increment of 1 o of tilting of the field out of the hard plane.
74 Figure 2 5. The molecular structure of complex 2 Colo r code: Mn = purple, O = red, N = blue and C = black. H atoms have been omitted for clarity.
75 Figure 2 6. EPR spectr um of complex 2 as a powder at 652.8GHz and 10 K, recorded in the first derivative mode. The EPR transitions have been labeled according to their behavior in the frequency dependence studies (see detailed discussion in the main te xt). The inset is the spectrum of complex 1 as a single crystal at 418 GHz and 2K recorded in absorption mode with the orientation of the field being close to the molecular z xis Note that this single crystal spectrum was collected with a frequency be low the ZFS which results in the order of and be ing reversed comparing with the high frequency spectrum.
76 Figure 2 7. Single crystal EPR studies for complex 2 (a) Frequency vers us field plot showing the EPR peak positions for a single crystal with the field applied close to the molecular z axis at 2 K. The squares are the experimental data while the solid lines are the simulat ions of the EPR peak positions The dash lines are gui des to the eyes. Simulations are performed with the Hamiltonian and parameters discussed in the main text. (b) Simulated energy level diagram. The parities of the states are indicated by color where red is even and blue is odd. The 0 th order eigenvector s o f the states associated with transitions are labeled in the figure.
77 Figure 2 8. Powder EPR studies for complex 2 (a) Frequency versus field plot showing the EPR peak positions in the powder sample stu dies at 10 K. The squares are the experimental data while the solid lines are the simulated EPR peak positions with the field parallel to the molecular z axis. (b) S imulation of the energy level diagram. The thick lines in (b) are the energy levels between which EPR transitions are observed The states associated with transitions are ignored for clarity. The states are color coded to denote their parities where red stands for even and blue stands for odd The thick arrows in (b) r epresent the observed EPR transitions at approximately 650 GHz.
78 Figure 2 9. Plot of the isofield reduced magnetization measurements The squares are the experiment al results while the solid lines are simulat ions employing the spin Hamiltonian and parame ters provided in the main text.
79 CHAPTER 3 QUANTUM TUNNELING OF MAGNETIZATION IN TRI GONAL SINGLE MOLECULE MAGNETS Portions of the work presented in this chapter can be found from the following article: Liu, J.; del Barco, E.; Hill, S., Quantum tunneling o f magnetization in trigonal single molecule magnets. Phys. Rev. B 2012, 85 (1), 012406. (reused with permission from American Physical Society ) 3.1 Introduction Quantum Tunneling of magnetization (QTM) in single molecule magnets (SMM) is important for both scientific reasons as well as applications in quantum information technologies. While QTM could be exploited for creating quantum superpos ition states necessary for quantum computation 2 it also leads to demagnetization in SMMs which is a disadvantage for classical information storage. Thus it is important to both understand and control the origin of QTM in SMMs. Extensive theoreti cal and experimental works have been performed to understand QTM in biaxial 39 83 84 and tetragonal 85 SMMs. In addition, QTM in spin systems with both rhombic and four fold transverse anisotropies has been discussed from various points of view. 55 86 88 An outstanding example is the observation of Berry phase interference (BPI) in SMMs 39 which signifies the importance of molecular symmetry in QTM. However, so far only scarce works have been conducted to understand QTM in SMMs with trigonal topology. 89 Crucially speaking, at zero transverse fields the symmetry of a SMM should determine the spin selection rules in QTM. Without applying a transverse field, the selection rules of QTM should be entirely decide d by the zero f ield transverse part of a Hamiltonian, As we discussed in C hapter 1, is constrained by the rotational symmetry of a molecule. Generally speaking, for a molecule with q fold rotational symmetry QTM can only occur between the
80 states with equal to multiples of q S urprisingly in most QTM experiments performed on SMMs, non zero tunneling steps are found for all level crossings regardless of the associated values Recently, a new SMM, [NE 4 ] 3 [Mn 3 Zn 2 (salox) 3 O(N 3 ) 6 Cl 2 ] (hereafter Mn 3 ) 16 37 90 91 with exact C 3 symmetry has been synthesized and studied. Clear evidences of quantum mechanical spin selection rules have been observed. 91 Motivated by the recent experimental studies of this Mn 3 SMM, we carried out theoretical investigat ions on QTM in SMMs with trigonal symmetry. We focus on studying the origin of the three fold transverse anisotropy ( ) as well as several fascinating phenomena generated by this interaction. For comparison, we also consider QTM in a [Ni(hmp)(dmb)Cl] 4 SMM (hereafter Ni 4 ) with S 4 symmetry 13 33 35 92 93 to show the uniqueness of C 3 symmetry. 3.2 Quantum T unneling of M agnetization in the Mn 3 S ingle M olecule M agnet 3.2.1 The Mn 3 S ingle M olecule M agnet The Mn 3 SMMs crystallize in a trigonal space group R 3 c with a ra cemic mixture of C 3 symmetric chiral molecules. The structure of the Mn 3 molecule is shown in Figure 3 1(a). The magnetic core is constituted by three ferromagnetically coupled Mn III ( s =2) ions, which form the Mn 3 plane, with two Zn II ions located above a nd below the the Mn 3 plane, forming a trigonal bipyramidal structure. The C 3 axis of the molecule is perpendicular to Mn 3 plane. Figure 3 1(b) shows the magnetic core of the molecule with the local easy axes of the individual spins defined by the Jahn Tell er (JT) elongation axes of the Mn III ions. The resultant spin S = 6 ground state experiences a relatively high barrier to magnetization relaxation ( U eff ~ 50 K). I mportantly, clear evidence s of quantum mechanical selection rules have been observed in QTM m easurements. 91
81 The Mn 3 SMM pr ovide s an ideal opportunity to ex plore the consequences of a trigonal spin topology in terms of the resultant QTM We do so via numerical comparisons between the giant spin approximation (GSA) and multi spin (MS) formalism The GSA treats the total spin S associated with the ground state of a molecule to be exact. For Mn 3 this results in 2 S + 1 (= 13) multiplet states that can be described by the following effective spin Hamiltonian : (3 1) The Stevens operators included in this Hamiltonian have been introduced in C hapter 1. Here, we consider only 2 nd and 4 th order axial ( p = 2, 4; q = 0) terms, and the leading trigonal ( ) and hexagonal ( ) operators. The first term in Eq uation 3 1 is the dominant 2 nd order axial anisotropy (where ) that gives rise to the energy barrier between spin up and spin down The advantage of the GS A lies in the fact that one need only deal with a few parameters and a small Hamiltonian matrix. However, t he GSA ignores the internal degrees of freedom within the molecule thus completely failing to capture the underlying physics in c ases where the total spin can fluctuate. 11 34 36 37 Apart from that, one of the main objectives of our work is to investigate the origin of high order transverse anisotropies. Therefore, a more physical model which takes into account the zero field spl itting ( ZFS ) tensors of individual ions and the coupling between them is given by the MS Hamiltonian: (3 2) Here, are spin operators associated with the uncoupled s = 2 Mn III ions The diagonal matrices, parameterize the 2 nd order ZFS in the local coordinate frame of each Mn III ion, with and where d i and e i are the respective axial and rhombi c ZFS
82 parameters The local coordinate frames are then transformed into the molecular frame by means of rotation matrices, specified by Euler angles i i and i where i is the angle between the local z axis (JT axis) and the molecular C 3 axis. The second term re presents the isotropic exchange between the i th and j th spins with J ij parameterizing the strength of this coupling on each bond, and the final term is the Zeeman interaction. Mn 3 is particularly attractive in the con text of the present investigation. The dimension of the MS Hamiltonian matrix for three s = 2 spins is just [(2 s + 1) 3 ] 2 = 125 125. The high ( C 3 ) symmetry then reduces the number of interaction parameters to just a single exchange constant, J and identi cal d and e values for each ion Two of the Euler angles are known from x ray studies 16 and the remaining parameters have been determined from EPR and QTM measurements 16 37 90 91 Lastly the structure contains no solvent molecules. This is rare among SMMs 35 and removes a major source of disorder 94 Consequently exceptional spectroscopic data (QTM and EPR) are a vailable against which one can test theoretical models 3.2.2 Quantum T unneling of M agnetization in Mn 3 We focus on the transverse ZFS operators in the GSA ( q > 0), particularly which we show to be responsible for several fasc inating results. T he effects of q > 0 ZFS terms typically manifest themselves at energy scales that are orders of magnitude smaller than those of the axial ( q = 0) terms. We thus focus on the tunneling gaps at avoided level crossings, as these are dominate d by the transverse terms in Eq uation 3 1 Due to symmetry restrictions ( q = 3 n for C 3 symmetry, where n is an integer), non zero tunneling gaps are limited to level crossings with where m is the projection of the total spin ont o the C 3 ( z ) axis. All such gaps, have been labeled in Fig ure 3 2 for QTM resonances k 3, where k (= m + m ) denotes an avoided crossing between pairs of levels with spin projections m and m (an overbar denotes
83 negative m ). Published ZFS parameters were employed for simulations involving Equation 3 2 i.e., d = 4.2 K and e = 0.9 K. 91 Mean while the exchange constant J (= 10 K) was set to a larger absolute value to isolate the ground state from excited multiplets thus simplifying analysis of higher lying gaps as shown in Figure 3 3. W i th J = 4.88 K, which is the value determined by experiments 91 the excited spin multiplets ( S < 6) overlap with the ground S = 6 mu ltiplet at low fields, which complicates analysis of excited QTM resonances, such as the resonance. The influences of J will be discussed later. The Euler angles we re set to 1 = 0, 2 = 120 o and 3 = 240 o ( all i = 0 ) to preserv e C 3 symmetry while i (= ) wa s allowed to vary in order to examine its influence on QTM selection rules We first consider the situation in which the Jahn Teller (JT) axes of the three Mn III ions are parallel to the C 3 axis i e ., = 0 In the top sect ion of Table 3 1, we give the magnitudes of even n tunneling gaps involving pairs of levels with deduced via diagonalization of Equation 3 2 in the absence of a transverse field, H T ( z ). The odd n H T = 0 gaps are identically ze ro, as can be seen from their dependence on H T (Fig ure 3 2 inset): t he power law behavior indicates no contribution from ZFS interactions (at H T = 0). Consequently one expects only even n ZFS terms of the form in the GSA: those satisfying this requirement have six fold rotational symmetry about the C 3 axis, i.e., a higher symmetry than the real molecule (further explanation is given below). To compare models for the = 0 case we calculated the non zero tunneling gaps, setting D = 1.096 K and in Equation 3 1 In the absence of a transverse field, the n = 2 gaps and are proportional to while the n = 4 gap, is proportional to This can be traced to the order of perturbation at which the gaps appear, e.g., by t reating
84 the m s = 3 states as a two level system, we find that based on a first order perturbation calculation. The best overall agreement between the two models is obtained by setting (Table 3 1) S mall differences may be due to our neglect of higher order six fold terms such as etc Table 3 1 Comparison of t unnel ing gaps obtained from the MS and GS model s for resonances k = 0, 1, 2 and 3, for the two case s = 0 ( top) and = 8.5 o (bottom). k n GS gap (K) MS gap (K) Ratio (GS/MS ) Jahn Teller axes parallel to the molecular z axis 0 2 ( 3,3) 2.6010 2 2.6610 2 0.98 0 4 ( 6 6 ) 1.05 2 2 ( 2 4 ) 1.01 Jahn Teller axes til ted = 8.5 o away from the molecular z axis 0 2 ( 3,3) 0.95 0 4 ( 6 6 ) 1.01 1 3 ( 4,5 ) 1.12 1 1 ( 1 2 ) 1.00 2 2 ( 2 4 ) 0.94 3 3 ( 3 6 ) 1.15 3 1 ( 0 3 ) 1.00 Next we consider the situation in which the JT axes are tilted = 8.5 o away from the C 3 axis as is the case for Mn 3 16 Both even and odd n H T = 0 tunneling gaps are generated in this situation, i.e., odd QTM resonances become allowed. Th is may be understood within the framework of the GSA as being due to the emergence of ZFS interactions possessing three fold rotational symmetry about the molecular C 3 axis, i.e., with n = 1 and p > 3; the leading term is We begin by considering ( k = 1) and ( k = 3), which depend only on to first order A perturbati on analysis gives and By c omparing with MS simulations ( Eq uation 3 2 0 we obtain ed The remaining gaps are then evaluated via diagonalization of Equation 3 1 using the optimum and parameters.
85 Excellent agreement is once again achieved (see Table 3 1) M inor deviations may, in principle, be corrected by introducing high er order transverse terms such as The emergence of the in teraction clearly signifies a lowering of the symmetry of the ZFS Hamiltonian upon tilting the JT axes To understand th is one needs to consider both the symmetry of the molecule and the intrinsic symmetry of the ZFS tensors of the individual ions. Conside ring only 2 nd order ZFS the Hamiltonian of a single Mn III ion possesses D 2h symmetry with three mutually orthogonal C 2 axes. When the JT axes are parallel ( = 0), the local z ax is of each Mn III coincides with the molecular C 3 axis. T he resultant ZFS Ham iltonian then possesses symmetry (see Fig ure 3 4( a ) ), requiring ; the additional C i symmetry arises from the time reversal invariance that guarantees an identical spectrum upon inversion of the total fie ld (or, in the classical limit, inversion of the total spin) In contrast, w hen the JT axes are tilted the C 2 and C 3 axes do not coincide The rotational symmetry then reduces to three fold and, hence, is allowed ; the symmetry i n this case is (see Fig ure 3 4( b ) ) The preceding arguments may be reinforced vi a group theo re tic considerations without involving an exact expression of the Hamiltonian. W hen the external magnetic field is applied along the mole cular z axis the C 6 h symmetry reduces to C 6 and the 13 basis functions of the S = 6 Hilbert space fall into six distinct one dimensional irreducible representations 95 By investigating how these basis functions behave under a C 6 rotation, we can sort them as follows: ; ; ; ; ; where are the six irreducible representations following the Bethe notation 95 Because the Hamiltonian operator belongs to the totally symmetric representation, is non zero only when and belong to the same representation 96 As can be
86 seen, such states have = 3 n with n even, which is the criterion for state mixing in C 6 symmetry. When the symmetry of the ZFS Hamiltonian is reduced to S 6 ( C 3 upon application of B // z ) the basis functions fall into three different irr educible representations : ; and Here, the selection rule for mixing is again in agree ment with the above calculations An important consequence of the preceding analysis is the demonstration of the existence of odd k QTM resonances, i.e., a quite realistic parameterization of Eq uation 3 2 generates ZFS terms in the GSA containing odd powers of and This dispels the notion that odd QTM resonances cannot be generated via ZFS interactions. 97 These ideas ought to apply quite generally, e.g., the disorder potential associated with the distortion of a symmetric molecule likely contains ZFS terms (e.g. ) that unfreeze odd QTM resonances. It rema ins to be seen whether this can account for the absence of selection rules in SMMs such as Mn 12 97 We note also that these arguments do not apply to zero field ( k = 0) QTM in half integer spin systems, 89 The J dependence of higher order ( p 4) coefficients in the GSA has been discussed in the context of other high symmetry SMMs 35 37 98 100 In these situations, the 2 nd order contributions to the transverse anisotropy ( q > 0) cancel exactly, emerging at higher order s as a consequence of the mixing of spin states. This is illustrated for Mn 3 in Figure 3 5 which plots the power law dependence of several tunneling gaps as a function of the ratio of J / d ; the parameter values given above were employed in these calculatio ns It is found that the gaps are proportional to i.e., and 37 Note that this implies a complete suppression of QTM in the strong coupling limit ( )
87 3.2.3 Berry Phase Interference in Mn 3 We conclude the studies on Mn 3 by focusing on the transverse field ( H T ) dependence of the tunneling gaps generated by and The influence of the fo rmer is rather straightforward: the C 6 h symmetry (see Fig ure 3 4( a ) ) guarantees six fold modulations of the tunneling gaps in all allowed resonances, regardless of whether a longitudinal field, H L (// z ), is present; also generat es hexagonal Berry phase interference (BPI) patterns (due to quenching of the tunneling 39 85 91 ) upon rotation of H T within the xy plane as shown in Figure 3 6 Figure 3 6 shows the k = 0 BPI pattern generated when only including ( ). This BPI pattern for the k = 0 resonance exhibits hexagonal symmetry, with the resonance field located at H L = 0, exactly. By contrast, the influence of is quite fascinating. In order to simplify the discussion, Fig ure 3 4( c ) was generated with We first examine the dependence of ( k = 0) and ( k = 3) for a fixed value of H T (see Fig ure 3 4( c ) ). As anticipated, exhibits a three fold modulatio n which rotates 60 o upon inversion of H L (dashed curves), as required on the basis of the time reversal invariance of Eq uation 3 1 i.e., is invariant to inversion of the total field. The figure does not convey the fact that it w as also necessary to vary H L in order to exactly locate the gap minima, i.e., H T influences the exact H L locations of the resonances, a behavior that is well documented for k > 0 resonances observed for other SMMs. The corresponding modulation of H L also e xhibits a three fold pattern for either polarity. The behavior of is yet more intriguing. One might expect a six fold behavior given the requirement that the spectrum be invariant under inversion of H T However, this assumes that H L = 0. In fact, application of a transverse field causes a shift of the k = 0 resonance away from H L = 0, as illustrated in Fig ure 3 4( d ) Only a very weak modulation of is observed upon
88 rotation of a 0.2 T transverse field; th e modulation pattern is indeed six fold (solid curve in Fig ure 3 4( c ) ). However, the corresponding modulation of H L exhibits a three fold pattern (dotted and dash dotted curves in Fig ure 3 4(c) ). One way to interpret this result is to view the operator as generating an effective internal longitudinal field, under the action of an applied transverse field; is then responsible for the shift of the k = 0 resonance from H L = 0. In deed, one can see this from inspection of the form of the operator, which, unlike even q interactions, contains an odd power of akin to the Zeeman interaction with H // z An alternative view may be deri ved from the S 6 surface depicted in Fig ure 3 4(b) where one sees that the hard/medium directions do not lie within the xy plane, contrary to the case for the C 6 h surface in Fig ure 3 4( a ) (or quite generally for any even q operator 84 ). In other word s, the classical hard plane is not flat, but corrugated with a 120 o periodicity. Consequently, application of a longitudinal field is required in order to insure that the total field is within the hard plane when rotating H T Figure 3 7 illustrates the sh ift of H L for upon applying H T at several representative orientations in the molecular xy plane. Upon the application of H T H L is shifted to + z at 0 o and z at 60 o which is consistent with the classical energy potential shown i n Figure 3 4(b). However, we note that this classical potential surface cannot explain the shifts of H L quantitatively. It is notable that H L is not a linear function of H T which indicates that the exact location of the molecular hard plane depends on the magnitude of the applied transverse field. The quantum molecular hard plane is not flat and also exhibits a 120 o periodicity which is similar to the classical energy potential analog. However, the hard plane is also field dependent: its exact
89 shape varie s as a function of the magnitude of H T which is different to the classical zero field energy potential surface. Finally, Fig ure 3 8 shows the patterns of BPI minima for k = 0 (a) and k = 3 (b), generated purely from the interact ion. The k = 0 pattern in (a) is hexagonal. However, the coloring indicates the polarity of the compensating longitudinal field, H L Thus, on the basis of the sign of H L one sees that the BPI minima exhibit a three fold rotational symmetry. In contrast, t he k = 3 BPI minima exhibit obvious trigonal patterns, regardless of the behavior of the compensating H L field. Observation of these BPI patterns in Mn 3 is complicated by several factors, including strong avalanches 91 and the existence of two molecular orientations with opposite chiralities, where the two species are rotated ~ 27 o respect to each other (with parallel C 3 axes); 16 90 we note that it may be possible to select and study one species via hole burning. 87 This technique involves sweeping H L in the presence of H T during initially polarizing the sample. With application of H T at a certain orientation, e.g., along the intermediate axis of one molecule sp ecies, these mol ecules will have a larger tunnel splitting; therefore, the magnetic moments of these molecules are more likely to relax when H L is swept through a QTM resonance. These molecules can them be studied on the negative side of the hysteresis loop ( H L ); therefo re, by manipulating H T it is possible to selectively polarize a portion of the sample for BPI studies. The primary motivation for the present theoretical study is to stimulate future measurements on Mn 3 or one of several other SMMs known to possess C 3 sym metry 89 3.3. Quantum T unnel of M agnetization in the Ni 4 S ingle M olecule M agnet 3.3.1 The Ni 4 S ingle M ol ecule M agnet We conclude this chapter by comparin g the preceding results with QTM in the Ni 4 SMM which possesses even fold rotational symmetry. The structure of the Ni 4 SMM is shown in
90 Figure 3 9(a). The molecule possesses S 4 symmetry with the S 4 axis show n in Figure 3 9(a). This complex crystallizes in an I 4 1 / a space group without any lattice solvent molecule. 33 35 92 The magnetic core of the molecule is a slightly distorted cube with four Ni II ions ( s = 1) located on opposite corners, as sketch ed in Figure 3 9(b). The four Ni II ions are ferromagnetically cou pled which leads to a spin S = 4 molecular ground state. The complex exhibits extremely fast quantum tunneling which significantly decreases the effective relaxation ba rrier which was unable to be directly measured by low temperature ac susceptibility meas urements ( | DS 2 | 3 K ) 93 The molecule can be described with the f ollowing spin Hamiltonian: (3 3) This Hamiltonian differs from Equation 3 2 only in that the rotational matrices, are replaced by which take i mproper rotations into account. The reason for this substitution is due to the different natures of rotations and improper rotations. An improper rotation can be achieved in two steps: i) a rotation about a given axis and a reflection about ii) a plane whi ch is perpendicular to the rotation axis. It should be noted that a reflection operation cannot be described by a rotation matrix. The simplest way to view this is that the determinant of a rotation matrix is 1, while the determinant of a reflection matrix is 1. Therefore, if the local coordination frame of the i th ion is related to the molecular frame by a rotation (identity or C 2 ), ; if the local coordination frame of the i th ion is transformed into the molecular coordination fr ame via an improper rotation ( S 4 or ), ( is reflection). The Ni 4 SMM is a particularly ideal platform for this comparison with Mn 3 The molecule possesses a well separated S = 4 ground state with the S = 3 excited spin multiplets located roughly 30 K above in e nergy. The 33 Hamiltonian matrix associated with a single Ni II ion
91 contains only two 2 nd order ZFS parameters, d and e ; i.e., any high order anisotropi es ( p ) are strictly forbidden. The ZFS of the individual Ni II ions, as well as their orientations, are directly measured through EPR s tudies on an isostructural Zn 3 Ni compound. 13 Due to the restriction of S 4 symmetry, only two independent Heisenberg interaction parameters, J 1 and J 2 are allowed; these interactions can be determined by d c susceptibility measurements 93 Therefore, all of t he parameters in Equation 3 3 can be determined separately On the other hand, the molecule possesses S 4 symmetry which prohibits the rhombic anisotropy term in the GSA Hamiltonian. The high symmetry of the molecule is confirmed by singe crystal EPR measur ements where exceptional ly sharp EPR transitions are observed with a four fold modulation pattern upon rotating H T 33 This clearly illustrates the presence of high order transverse anisotropy which is responsible for the fast quantum tunneling in the Ni 4 SMM. 3.3.2 Quantum T unneling of M agnetiz ation in Ni 4 We again discuss the transverse anisotropy in Ni 4 by calculating the QTM gaps. In the studies of Ni 4 we focus on the k = 0, 1 ... 4 ground state QTM tunnel splitting as shown in Figure 3 10 where the values associa ted with these gaps equal to 8, 7 ... 4, respectively. The simulations were performed with published ZFS parameters d = 7.6 K, e = 1.73 K and J 1 = J 2 = 10 K. 13 33 35 Previous EPR studies on Zn 3 Ni reveal that the easy axis of the local Ni II ion is tilted away from the molecule z axis with = 15 o In our studies, w e let vary ( = 15 o and 0) to illustrate the influence of on the symmetry of the molecular Hamiltonian. Figure 3 11 shows the QTM gaps as function of transverse field ( H T ) via exact diagonalization of Equation 3 3. As shown in the figure, and retain non zero values in the absence of a transverse field, while all other tunnel splitting s vanish at H T = 0. Such a result is of no surprises based on the S 4 molecular symmetry wher e only QTM between states with
92 ( n is an integer ) are allowed. However, unlike the Mn 3 SMM, the QTM selection rules corresponding to the = 15 o and 0 situations are exactly the same. In both scenarios only the ( ) and ( ) are non zero, while the other k even QTM gap, e.g., ( ), always vanishes when H T = 0. These simulation results imply that does not affect the symmetry of the Hamiltonian, which is different from the results we obtained for the Mn 3 SMM. This can be explained by the dif ference between the symmetry operations associated with even and odd fold symmetries. In an even fold system, the molecular z axis must be a C 2 axis of the molecule; thus, the parallelism between the local C 2 axes of the individual ions and the molecular z axis does not introduce an extra C 2 operation to the Hamiltonian. By contrast, the molecular z axis is not a C 2 axis in an odd fold molecule; therefore, the symmetry of the Hamiltonian can be changed if the local z axes are coincident with the molecular z axis. I n the preceding discussions of Mn 3 the spin selection rules can be simply understood in terms of the rotational symmetry of a molecule ( C 6 or C 3 ). F o r the Ni 4 SMM, due to the fast tunneling at k = 0, only the k = 0 QTM step was observed in experi ments. Nevertheless, it is useful to examine the selection rules for Ni 4 since they should be generally applicable to SMMs with the same symmetry. However, the QTM spin selection rules in the Ni 4 SMM cannot be fully accounted by the S 4 molecular symmetry; one must additionally consider the intrinsic C i symmetry of the spin Hamiltonian must be considered. Upon application of a magnetic field parallel to the molecular z axis, the S 4 symmetry group reduces to C 2 in which the tunnel splitting is allowed. This clearly contradicts to our simulation which shows that the spin Hamiltonian possesses a higher symmetry. W hen considering the C i symmetry, the consequential zero field spin Hamiltonian possesses symmetr y, which corresponds to the symmetry of the interaction, as shown by Figure 1 4(d) in Chapter 1. In the presence of a
93 longitudinal field, the C 4 h group reduces to the C 4 group in which the QTM selection rule is that equals to multiples of 4. We would like to remind that the C i symmetry is guaranteed by the nature of the spin orbit interaction which is not limited to spin Hamiltonians, i.e., for any Hamiltonian determined by crystal field and/or spin orbi tal coupling, the C i symmetry should be applied regardless of whether the orbital angular momentum is quenched or not. Thus, it is always necessary to consider C i symmetry in addition to the structural symmetry, especially when improper rotations are invol ved Observation of k > 0 QTM steps in Ni 4 is not practical due to the extremely fast tunneling at k = 0. We note that the k = 0 tunneling could be greatly suppressed if the ground spin state of the molecule was increased, e.g., if the molecule is constitu ted by four s = 2 ions. Then it would be possible to study the k > 0 QTM steps. The BPI pattern of the tunnel splitting is shown in Figure 3 12 which is simulated employing Equation 3 3. The simulation was performed with the incl usion of a 15 o angle between the local easy axes of Ni II ions and the molecular z axis. As shown in the figure, the tunnel splitting exhibits four fold modulation pattern, with H L = 0 (exactly). Such a result can be reinforced ba sed on considering the symmetry of the Hamiltonian: with the inclusion of C i symmetry, the resultant zero field Hamiltonian has C 4 h symmetry with the xy plane being a mirror plane; thus, the molecular hard plane must lie exactly in the xy plane and the k = 0 QTM gap should locate at H L = 0. F i nally, we discuss the influences of disorder on the QTM steps in Ni 4 In the presence of random crystallography defects, the symmetry of the molecules is expected to be lowered, leading to the absence of spin QTM selec tion rules. Figure 3 13 was generated by considering the orientation of the ZFS tensor of one of the Ni II ions in the molecule to be slightly different, i.e., the ZFS tensors of three Ni ions are tilted 15 o from the molecular z axis, while one is tilted
94 10 o from the molecular z axis. As shown in the figure, all resonances posses a non zero tunneling gap at zero H T The inset of the figure plots the k = 1, 2 and 3 gaps in the log log scale, which clearly indicates that these tunneling gaps, which are forbidd en in the S 4 symmetry, also saturate at a non zero value when H T tends to zero. These results show that a small disorder can effectively unfreezes all QTM steps without the assistance of a transverse field. This argument can be reinforced by group theoreti c considerations. With random disorder, the symmetry of a molecule is lowered to C 1 where the spin Hamiltonian only possesses C i symmetry. Upon applying a longitudinal field, the C i group reduces to the C 1 group, where all the states necessarily belong to the same one dimensional irreducible representation. 95 Therefore, mixing between any states is allowed. We note that this kind of disorder can be introduced by small crystallographic defects, which always exist in real samples. The preceding discussion shows that disorder may b e responsible for the observations of k odd QTM steps in SMMs with even fold symmetries. 3.4 Summary In this chapter, we discussed QTM in a Mn 3 SMM with exact C 3 symmetry. We discovered the origin of the operator in the GSA Hamil tonian by mapping the spectrum obtained via the MS Hamiltonian onto the GSA model. By investigating the tunnel splitting s generated by the operator, we found that it unfreezes the k odd QTM resonances and shifts the k = 0 resonan ce away from H L = 0. In addition, we discussed QTM in a Ni 4 SMM with S 4 symmetry. We found that the BPI patterns of the k = 0 QTM splitting s in the Mn 3 and Ni 4 SMMs behave remarkably different. This comparison shows the uniqueness of C 3 symmetry.
95 Figure 3 1. The molecular structure and the magnetic core of the Mn 3 SMM. (a) The molecular structure of the Mn 3 SMM Colo r code: Mn = purple, Zn = green, O = red, N = blue C = black and Cl = dark gold H atoms have been omitted for clarity. (b) Schematic repre sentation of the magnetic core of the Mn 3 SMM.
96 Figure 3 2. Zeeman diagram for a spin S = 6 multiplet with easy axis anisotropy ( D < 0 in Eq uation 3 1 ) and H // z All possible non zero tunnel ing gaps for C 3 symmetry are labeled according to the scheme dis cussed in the main text. The inset shows the H T dependence of the odd n tunneling gaps
97 Fig ure 3 3 Zeeman diagram of Mn 3 generated by Equation 3 2 with different magnitudes of J The diagrams are generated with (a) J = 10K and (b) J = 4.88 K. As show n in (b), when J = 4.88 K, the excited spin multiplets overlap with the ground state at low fields.
98 Fig ure 3 4 The symmetries of the ZFS interactions and the tunnel splitting s (a) and (b) illustrate the p otential energy surfaces corresponding to the and GSA operator equivalents with the presence of a negative axial anisotropy ( ) respectively (c) k = 0 (solid curve) and k = 3 (dashed curves) ground state tunneling gaps as a function of the orientation of H T (= 0.2 T) within the xy plane, calculated using Eqn (1) with The data have been normalized and offset to aid viewing: oscillates from 3.65 to 3.90 10 6 K (~ 6 %) a nd from 4.065 to 4.074 10 9 K (~0.2%). The inner curves correspond to the H L field (dotted H L > 0, dash dotted H L < 0) needed to compensate for the shift of the k = 0 resonance upon application of H T as illustrated in (d): for H T = 0.2 T, H L oscill ates about zero with an amplitude of and a three fold ( S 6 ) periodicity
99 Fig ure 3 5 Tunnel splittings of the Mn 3 SMM as a function of the coupling constant J Simulations are performed with the JT axes tilted away from the molecule z axis. T he splittings associated with same value are rendered in the same color. Note the results are plotted on a logarithmic scale.
100 Figure 3 6. Color contour polar plot of the ground state gap as a function of H T calculated using Equation 3 1 with The BPI exhibits a six f old modu l ation pattern with the gap locat ed exactly in the molecular xy plane ( H L = 0).
101 Fig ure 3 7 Shift of H L for as a function of the magnitudes of the applied H T Note that the slopes of each curve is not a constant (except 30 o where H L = 0), which indicates that the orientation of the molecular hard plane, which corresponds to H L / H T de pends on the magnitude of H T
102 Figure 3 8. The BPI patterns for the ground QTM resonances of the Mn 3 SMM. The Color contour plots show (a) and (b) as a function of H T calculated using Eq uation 3 1 wi th A compensating H L field was required in (a) that alternates between positive (red) and negative (blue) values. Both figures display BPI minima (dark spots) that exhibit three fold symmetry when the variation of H L is also tak en into account
103 Figure 3 9. The structure and the magnetic core of the Ni 4 SMM. (a) The molecular structure of the Ni 4 SMM Colo r code: Ni = olive O = red, N = blue C = black and Cl = dark gold H atoms have been omitted for clarity. (b) Schematic re presentation of the magnetic core of the Ni 4 SMM.
104 Figure 3 10. Zeeman diagram for the Ni 4 SMM simulated employing Equation 3 3. Only the ground s tate S = 4 multiplet is shown in the figure. The k = 0 to 4 ground state QTM splitting s are labeled in the f igure.
105 Figure 3 11. The ground state QTM gaps for the Ni 4 SMM as a function of H T The simulations were performed employing Equation 3 3 with the parameters given in the main text. The solid lines were generate d with = 0 and the dash lines were genera ted with = 15 o
106 Figure 3 12. Color contour polar plot of the ground gap as a function of H T calculated using Eq uation 3 3 with the parameters given in the main text. The BPI exhibits four fold moduation pattern with the gap located exactly in the molecular xy plane ( H L = 0).
107 Figure 3 13. The effect of disorder on the ground state QTM gaps for the Ni 4 SMM. The simulations were performed employing Equation 3 3 by considerin g the orientation of t he ZFS tensor of one of the Ni II ions in the molecule to be slight different from the others (see details in the main text).
108 CHAPTER 4 ELECTRON PARAMAGNETI C RESONANCE AND QUAN TUM TUNNELING OF MAGNETIZATION STUDIE S OF MN 4 SINGLE MOLECULE MAGNETS: RE VEALIN G COMPETING ZERO FIELD INTERACTIONS The work presented in this chapter can be found in the following articles: Liu J. ; Beedle C. C. ; Quddusi H. M. ; d el Barco E. ; Hendrickson D. N. ; Hill S., EPR and magnetic quantum tunneling studies of the mixed valent [M n4(anca)4(Hedea)2(edea)2]2CHCl3, EtOH single molecule magnet Polyhedron 2011, 30(18), 2965. (reused with permission from Elsevier); Hendrickson, D. N., Cationic Mn4 Single Molecule Magnet with a Sterically Isolated Core. Inorg. Chem. 2011, 50 (16), 7367 (reused with permission from American Chemical Society ); Quddusi, H. M.; Liu, J.; Singh, S.; Heroux, K. J.; del Barco, E.; Hill, S.; Hendrickson, D. N., Asymmetri c Berry Phase Interference Patterns in a Single Molecule Magnet. Phys. Rev. Lett. 2011, 106 (22), 227201. (reused with permission from American Physical Society ) 4.1 Introduction In the previous chapter we compared the giant spin approximation (GSA) with t he multi spin (MS) model in a high symmetry ( C 3 ) Mn 3 single molecule magnet (SMM) where excellent agreement is achieved In those studies, we showed that the GSA reproduces all of the important low temperature features of the molecule, including the quantu m tunneling of magnetization (QTM) results, which are extremely sensitive to the symmetry of a molecule. However, as we have mentioned in Chapter 1, the GSA model fails when the total spin of the molecule fluctuates. 34 36 In addition, there are several other scenario s in which the GSA model fails. One of these situations is the case of antiferromagnetically (AF ) coupled low symmetry SMMs. Several works on AF coupled low symmetry Mn 3 molecules 101 as well as st udies for the AF coupled dinuclear Mn molecule presented in Chapter 2, suggest that, in those systems, it is
109 meaningless to analyze experimental results with the GSA model since the ground spin state cannot be rigorously attributed to any S state. On the o ther hand, the GSA model seems to be a reasonable approximation in most ferromagnetically (FM) coupled SMMs, even when a molecule possesses very low structural symmetry and/or when the interactions within its magnetic core are weak. In FM coupled compounds the ground spin state can be estimated by optimizing the total spin of the molecule. Low temperature magnetic measurements, such as reduced magnetization and ac susceptibility, can be understood by considering the second order anisotropies within the GSA Even more precise spectroscopic measurement results, for instance, single crystal EPR, can also be interpreted by using the GSA model including consideration of a few high order anisotropy operators. Still, there are some important properties of FM coupl ed SMMs which are hard to explain with a GSA. One of them is the low symmetry zero field magneto anisotropy energy potential of a molecule. For instance, when a SMM possesses only C 1 (or C i ) structural symmetry, all possible Steve ns operators (up to p S ) should be included if the GSA Hamiltonian is applied. In such situations, it is still possible to interpr et experimental results with the GSA model since no spin fluctuations are involved; however, the obtained parameters provide little insight into the physics of the molecule. The magnetic symmetry of a molecule can be probed with QTM measurement where the Berry phase interference ( BPI ) pattern is directly related to the molecular symmetry. Observations of low symmetry BPI patterns are more likely to oc cur in weakly coupled systems for the following reasons: In the strong interaction limit, where high order ( p otropy terms vanish and the resultant GSA Hamiltonian, which includes only the second order anisotropies, pos sesses D 2 h symmetry 35 37 ; by contrast, when the interactions are weak, high order anisotropy terms start to em erge so that different BPI patterns can be observed.
110 In the past several years, our group has studied a family of mixed valent [ Mn 2 II Mn 2 III ] SMMs which have similar magnetic cores with various types of ligand. 102 104 Th o se studies provide important insights into the physics of SMMs having relatively weakly coupled magnetic cores. In this chapter we will discuss electron paramagnetic resonance (EPR) and QTM studie s on two similar Mn 4 SMMs, a) [Mn 4 (Bet) 4 (mdea) 2 (mdeaH) 2 ](BPh 4 ) 4 SMM 103 h enceforth Mn 4 Bet, where Bet is glycine betaine and mdeaH is N methyldiethanol amine and b) [Mn 4 (anca) 4 (Hedea) 2 (edea) 2 ].2CHCl 3 ,2EtOH 105 henceforth Mn 4 anca, where anca is the anion of 9 anthracenecarboxyl ic acid and Hedea and edea are the mono and di anions of N ethyldiethanolamine, respectively. Extensive EPR and QTM measurements were carried out to determine the magneto anisotropies of these molecules. In this chapter, we will focus on the data collect ed on Mn 4 Bet where an intriguing motion of the k = 1 BPI minima in the magnitude direction phase space of the transverse field, as well as an asymmetric BPI pattern with respect to the polarity of the applied transverse field, were observed in QTM studies We found that these features result from the competition between the zero field splitting (ZFS) tensors of the inequivalent Mn ions within the magnetic core of the molecule. Both the EPR and QTM experimental results are simulated with a MS Hamiltonian wh ich includes non collinear ZFS tensors for the different Mn ions. We will also show EPR spectra collected on Mn 4 anca, where very similar magneto anisotropies were observed. The work presented in this chapter was performed in collaborations with Dr. del Ba rco s group at the University of Central Florida (UCF) and Dr. Hendrickson s group at the University of California, San Diego (UCSD). The samples were synthesized at UCSD and the high temperature magnetometry measurements were also performed at UCSD. Low t emperature QTM measurements were performed at UCF. The EPR measurements, as well as the simulations
111 of both the EPR and QTM results were performed at the NHMFL and form the subject of this chapter. 4.2 The Mn 4 Bet SMM 4.2.1 The S tructure of the Mn 4 Bet SMM The Mn 4 Bet SMM crystallizes in a triclinic P 1 space group with half of the molecule in the asymmetric unit ; the other half of the molecule is generated via inversion. The structure of the molecule is shown in Figure 4 1(a). The [Mn 2 II Mn 2 III O 6 ] 4 + core re sembles two face sharing cubanes missing opposite vertices with two hepta coordinate d Mn II and two hexa coordinate d Mn III Due to the inversion symmetry of the molecule, all four Mn ions lie in the same plane, with the JT axes of the Mn III ions oriented along the Mn III N bonds, which lie about 55 degrees out of the plane. Figure 4 1(c) shows the schematic representation of the molecule illustrating the magnetic interactions within the core. The magne tic ions are labeled in the figure with the 1 st and 4 th ions being Mn II and the 2 nd and 3 rd ions being Mn III As shown in Figure 4 1(c), three independent Heisenberg interaction parameters are included based on the symmetry of the molecule. Fits to dc susc eptibility give a ground spin S = 9 state for the molecule. Ac susceptibility measurements were carried out at frequencies from 0.1 10 kHz where both temperature and frequency dependent out of phase signals are observed below 5.2 K with a relaxation barri er of 20.5 K, confirming the complex to be a SMM. 103 The structure of Mn 4 anca is shown in Figure 4 1(b). The Mn 4 anca and Mn 4 Bet molecules possess ve ry similar [Mn 2 II Mn 2 III O 6 ] 4 + magnetic cores with different structur al ligands. The magnetic cores of these two molecules possess the exactly same topology; therefore, they can be characterized with the same spin Hamiltonian with different parameters. Howev er, we note that the Mn 4 Bet SMM crystallizes without any lattice solvent molecule, while the Mn 4 anca SMM co crystallizes with EtOH as a solvent molecule. As will be shown by the EPR
112 spectra, the presence of solvent molecules introduces significant disord er which leads to a considerable difference in the linewidths of the EPR transitions observed in these two compounds. The Mn 4 Bet molecule possesses C i symmetry with two types of magnetic ions: two Mn III ( s = 2) and two Mn II ( s = 5/2). The Mn III and Mn II i ons are not related by any symmetry operation since they belong to the same asymmetric unit of the molecule; this implies that the anisotropy tensors of the Mn III and Mn II ions can be non collinear. In addition, dc susceptibility measurements suggest relat ively weak interactions within the magn e tic core which lead to considerable high order anisotropy terms in the GSA. 103 Such high order anisotropies are confirmed by EPR measurements where non even peak spacing is observed (see more details in data discussion). Therefore, in order to gain suitable insights into the underlying physics, we introduced the following MS Hamiltonian to explain the experimental results: (4 1) T he first term represents the local anisotropy of the i th ion where is the diagonal 2 nd order ZFS tensor s associated with the i th ion, with components and ; and the represent Euler rotation matrices defined by the Euler angles i i and i Due to the inversion symmetry of the molecule, the parameters are related as follo ws: and For the sake of simplicity, we choose the molecular coordinate frame to coincide with the local coordinate frames of the Mn III ions, i.e. 2,3 = 2,3 = 2,3 =0. The second term in Equation 4 1 is the Zeeman coupling to the applied field, where we assume an isotropic Land factor, g = 2 00. The third term is the Heisenberg interaction term with J 1,2 = J 3,4 = J a J 1,3
113 = J 2,4 = J b J 2,3 = J c and J 1,4 = 0, as shown in Figure 4 1(c). The last term is the dipolar interaction term, where the dipolar interaction matrices, have been chosen to exactly reproduce the dipolar interaction within the magnetic core based on structural in formation; the r i, j represent the distances between the i th and j th ions obtained via x ray crystallography studies where r 1,2 = r 3,4 = 3.259 r 1,3 = r 2,4 = 3.341 r 1,4 = 5.793 and r 2,3 = 3.161 for the Mn 4 Bet SMM. We ignored dipolar interactions i n the preceding chapters. However, as we shall see, we cannot them in Mn 4 Bet. The strength of the dipolar interaction is of the order of 0.1 K in the Mn 4 Bet molecule, which is small compared with the isotropic exchange (Heisenberg) interactions within th e molecule. However, the dipolar interactions are anisotropic, which means that they can directly affect the magneto anisotropy of the molecule. In our discussions of the BPI pattern of Mn 4 Bet, we find that the motion of the BPI minima is due to the compe tition between the ZFS tensors of the Mn III and Mn II ions ( vide infra ). The Mn II ion possesses a half filled shell (3 d 5 s = 5/2), where the ground state electron configuration has spherical symmetry; therefore, the ZFS of the Mn II ion only arises from hig her order mixing of excited spin states, which is generally very weak. 106 In fact, the largest d (Mn II ) value for Mn II we have found in literatures is 1.4 K 82 107 with values of order 0.1 K more typical, the effect of the dipolar interactions is likely non negligible compared to that of the ZFS of the Mn II ions. In Chapter 2 and 3, we focused on molecule s composed of Mn III ions or Ni II ions, wh ere each ion possesses a much stronger anisotropy itself; thus, dipolar interactions were justifiably ignored. The Mn 4 Bet SMM is an ideal candidate for single crystal EPR and QTM studies: the crystal contains no lattice solvent molecules, which removes a major source of disorder. 94 Furthermore, there is no apparent disorder associated with the ligands and only one molecule per unit cell, i.e., only one molecular orientation which greatly simplifies single crystal data analysis.
114 4.2.2 Discussion of Electron Paramagnetic Reso nance Results Single crystal EPR measurements were conducted in a transmission type spectrometer that enabled in situ rotation of the sample about a fixed axis Variable frequency ( 50 36 0 GHz) and variable temperature measurements were thus performed wit h the field aligned both in the hard plane and at an orientation close to the easy axis. Figure 4 2 shows the temperature dependen ce of EPR spectra obtained at 139.5 GHz with the applied field close to the molecular easy axis. Extremely sharp EPR transiti ons are observed which confirm the high quality of the crystal. Peak assignments can be made on the basis of the temperature dependence of the resonances and their relati v e spacings. Those resonances belonging to the S = 9 ground state have been labeled in the figure (A resonances, with the subscript denoting the absolute value of the spin projection associated with the level from which the transition was excited). Careful examination of the 20 K spectrum reveals nine successive peaks belonging to the A ser ies, distributed from 1.15 T to ~4.75 T, which is just below the g = 2.00 position for 139.5 GHz. This confirms that the molecule posseses an S = 9 spin ground state. However, it is notable that the spacings be tween the A series peaks are non uniform. In p articular, the peaks become very closely spaced at low fields, i.e., the spacing between A 9 and A 8 is 0. 21 T which is about 1/ 3 of the spacing between A 1 and A 2 (0. 64 T) Additional resonances are observed at elevated temperatures, confirming the existence of many low lying excited spin ( S < 9) multiplets. Simulations reveal the presence of many overlapping excited spin states not far above the lowest lying levels associated with the S = 9 ground state ( m = 9 and 8), making assignment of the excited state r esonances essentially impossible The existence of the low lying excited states is consistent with the relatively weak exchange coupling within the magnetic core of the molecule. We note that almost identical spectra have been obtained for a related Mn 4 co mplex, albeit with slightly enhanced coupling, resulting in a reduced density of
115 low lying excited spin states 104 Thus, comparisons between the spectra o btained for the two complexes provide add itional support to the assignments made in Fig ure 4 2 Figure 4 3 shows the temperature dependence of EPR spectra obtained at 67.3 GHz, with the field applied in the hard plane. These spectra further support the ass ertions made above on the basis of the easy axis measurements. Again, both the transitions from the S = 9 ground state and excited spin state are observed, with the A 9 transition persisting at the lowest temperature. The resonance s from the S = 9 state hav e been labeled according to the same scheme except that now the subscript corresponds to the spin projection on to the high field quanti z ation axis, which corresponds to the applied field direction Non uniform spacings between peaks are also observed in th e hard plane spectra, which is likely due to the presence of high order transverse anisotropy terms and/or the fact that the spectra were not recorded in the true high fields limit. Note that the spectra in Figure 4 3 were collected at a relative low frequ ency (67.3 GHz); thus, only the A 9 A 7 ... transitions are observed while the A 8 A 6 ... transitions are missing at this frequency. At frequencies above 100 GHz, a complete series of A transitions, A 9 A 8 A 7 ... are observed (see Figure 4 4(b)). The spectr a can be understood with the aid of the hard plane Zeeman diagram for the ground S = 9 multiplet, as shown in Figure 4 4. The observed high field EPR transitions are those between adjacent levels, i.e., | m | = 1, where m is the spin projection onto the direction of the applied field. However, if one follows these levels to low fields, one finds that the transitions from the m odd to m even states, which correspond to A 9 A 7 ... (indicated by the red arrows) decrease to zero as the field tends to zero. By contrast, the transitions from the m even to m odd states, which correspond to A 8 A 6 ... (indicated by the blue arrows), go through a minimum and then tend to a non zero value as the field tends to zero. Th us,
116 the A 8 A 6 ... transitions cannot be observed below certain frequencies, which is the reason for their absence in the spectra shown in Figure 4 3. High frequency EPR measurements were also performed on the Mn 4 anca molecule to determine its spin state a nd magneto anisotr o py. Fig ure 4 5(a) shows temperature dependence spectra obtained at 165 GHz with the field aligned close to the easy axis. The peaks have been sorted into two series (labeled A and B), based upon their temperature dependence and the relat ive spacings between peaks within each series. The stronger peaks, which persist to the lowest temperatures, are labeled A 9 A 8 etc., while the weaker peaks that emerge between the A peaks at higher temperatures are labeled B 8 B 7 etc. At the lowest temp erature only one strong transition is observed (A 9 ), indicat ing a reasonably well separated ground state. Looking more closely to the high temperature spectrum at 15 K one can clearly see nine successive peaks belonging to the A series, distribute d from ~1.8 T all the way to just below 5.9 T which is the g = 2 .00 position for 165 GHz. The se nine peaks are labeled according to the exact same scheme which is used to label the transitions observed in Figure 4 2. These spectra confirm that the approximate gr ound state spin value is S = 9. I t is also notable that the spacing s between the A series peaks are non uniform. In particular, the peaks become very closely bunched at low fields, i.e., the spacing between A 9 and A 8 i s 0. 16 T which is less than 1/4 of the spacing between A 1 and A 2 (0.75T). Fig ure 4 5(b) shows the temperature dependence of EPR spectra obtained at 50.5 GHz, with the field applied in the hard plane. The se spectra further support the assertions made above on the basis of the easy axis measurem ents Again, both the A and B series of peaks are observed, with the A series persisting to lower temperatures. In addition, all but one of the peaks (A 9 with the subscript referring to the m projection onto the applied field axis) vanish at
117 the lowest te mperature The B series of resonances behave exactly like another set of hard plane peaks with S = 8 and different ZFS parameters. We can compare the spectra collected on the Mn 4 Bet molecule with the spectra collected on the Mn 4 anca molecule. These spect ra support that both compounds possess an S = 9 ground states. Actually, the ZFS associated with the g r ound state easy axis transitions (A 9 ) for the both compounds is approximately equal to 110 GHz, which indicates that the Mn 4 Bet and Mn 4 anca SMMs have s imilar axial anisotropy. However, one can find a significant difference between the linewidths of the observed EPR transitions of these two compounds. The linewidth of the easy axis ground transition (~0.02 T) observed in the Mn 4 Bet SMM less than 1/5 of t hat of Mn 4 anca (~0.1 T). The difference is most likely due to the fact that that Mn 4 Bet does not have any lattice solvent molecules; therefore, this removes a major source of disorder. We focus the rest of this chapter on analyzing the EPR and QTM data c ollected on the Mn 4 Bet SMM, where extensive QTM measurements were performed to probe its BPI properties. Fig ure 4 6 shows the frequency dependence of the A resonance peak positions obtained with the field (a) approximately parallel (within a few deg r ees o f misalignment ) to the molecular easy axis at 3 K, and (b) in the hard plane at 5 K The solid lines in the f ig ure represent simulations of the EPR transitions within the S = 9 ground state; they have been color coded to denote the associated transitions. For some frequencies in the easy axis data many resonances are observed because the measurements were performed at an elevated temperature (20 K) The simulations were performed employing Equation 4 1 with the following parameters : d 2 = d 3 = 4.99 K and e 2 = e 3 = 0.82 K; d 1 = d 4 = 0.67K and e 1 = e 4 = 0, with the axes of the Mn II ions rotated with respect to the central spin by identical Euler angles 1 = 4 = 45 o 1 = 4 = 0 and 1 = 4 = 0, as required by the inversion symmetry of the molecule; finally, t he Heisenberg interaction
118 parameters J a = 3.84 K, J b = 1.20 K and J c = 3.36 K were used. It should be emphasized that these parameters were constrained by simulating both the EPR and QTM results simultaneously which is the reason why non zero d 1 d 4 1 and 4 were obtained ( vide infra ). The parameters are constrained by the best simulations of the A series of EPR peaks in both easy axis and hard plane orientations simultaneously. As shown in Fig ure 4 6 the simulations are in excellent agreement with a ll of the ground state EPR transitions. T he obtained anisotropy values for the Mn III ions are very similar to related Mn III complexes 16 37 91 101 104 while the d 1,3 value lies within the bounds reported for other Mn II systems 82 107 There are several important fe atures about these parameters that should be pointed at this stage. First, the inclusion of a small non collinear ZFS parameter for the Mn II ions ( d 1 = d 4 = 0.67K and 1 = 4 = 45 o ) is necessary for the subsequence discussions about the motion of the BPI minima observed in QTM studies. This motion is caused by the competition between the ZFS anisotropies of the Mn III ions and Mn II ions. Due to the inversion symmetry of the molecule, the ZFS tensors of the two Mn III ions must be collinear. If the Mn II ions are isotropic, d 1 = d 4 = 0, in this case, there cannot be any competition between the magneto anisotropies of the ions within the molecule; hence, no motion of the BPI minima should be observed. For the same reason, the ZFS tensors of the Mn II ions must be non collinear with those of the Mn III ions ( 1 = 4 = 45 o ). Otherwise, the anisotropies of the individual ions would simply add together and the resultant Hamiltonian would possess a high symmetry such as D 2 h Second, we note that the anisotropy of the M n II ions is weak ( 0.67 K), which indicates that the effect of dipolar interactions is not negligible. The dipolar interactions cause a non trivial effect and contribute 10% to the motion of the BPI minima, as will be shown later in this chapter. Finally, we need to point out that the obtained parameters are not necessarily unique; it may be possible to model the molecule with
119 another set of parameters. Nevertheless, these parameters have caught all of the important physics in this Mn 4 Bet SMM and give exce llent quantitative agreement between the experimental results and simulations. 4.2.3 Discussion of Quantum Tunneling of Magnetization Results QTM experiments were carried out by Dr. del Barco s group at UCF. A high sensitivity micro Hall effect magnetomete r, a 3 He / 4 He dilution fridge and a 3D vector superconducting magnet were employed to record magnetization hysteresis curves as a function of a longitudinal magnetic field ( H L ) applied parallel to the easy axis of the molecules, in the temperature range 35 1200 mK. The results are shown in Fig ure 4 7 where extremely sharp QTM resonances ( k = 0, 1, and 2), spaced by confirm the high quality of the crystal. Within the GSA, this spacing corresponds to an axial ZFS parameter, D = 0 28 K ( g = 2 .00 ). The observed blocking temperature below which open hysteresis loops are observed and crossover temperature below which the QTM becomes temperature independent, are ~ 1 2 and ~ 0 2 K, respectively. A transverse field ( H T ) was subsequently employed in order to study the symmetry of the QTM in resonances k = 0 and k = 1. Figure 4 8( a) shows the modulation of the QTM probability for resonance k = 0 as a function of H T applied along the magnetic hard axis ( = 0). The angle is defined as the azimuthal angle between the molecular hard axis and the applied transverse field. The QTM probability is defined as where M i and M f are the magnetizations before and after the resonance, resp ectively, and M sat is the saturation value. The orientation of the molecular hard axis which lies ~ 30 o away from one of the crystal faces, was deduced from the t wo fold modulation of P 0 as a function of the orientation of a 0.2 T transverse field within t he hard plane 87 as shown in the inset of Figure 4 8(a). We note that there was a small (~3 o ) misalignment of the field rotation plane during the measurements. The
120 influence of this slight misalignment is evaluated further below. The results show that the misalignment is not the primary cause of the observed phenomena ( vide infra ) The P 0 osc illations in Fig ure 4 8 (a) correspond to BPI for the k = 0 resonance with minima at regularly spaced field values ( ). A maximum in P 0 is found at H T = 0, as expected for an integer spin value. Within the framework of the GS A whe re 52 one would expect the 2 nd order rhombic anisotropy parameter to be | E | = 0.06 K Note that the regularly spaced k = 0 BPI minima are invariant under inversion of H T ; i.e., they are symmetric with respect to H T = 0. Interestingly, this is not the case in resonance k = 1, for which the behavior of the QTM probability is very different. This can be seen in Fig ure 4 8 (b), which illustrates the dependence of P 1 on H T with = 13.5 o (the angle for which the first BPI minimum at H T = 0 30 T is the sharpest). In fact, for resonance k = 1, different BPI minima appear at different field orientations, of the transverse field within t he xy (hard) plane of the molecule (see Fig ure 4 9 ); i.e., the first minimum ( H T = 0 3 T) appears at = 13 5 o while the second ( H T = 0 6 T) occurs at = 6 o contrary to what is found for the k = 0 resonance where all P 0 minima are seen most clearly at = 0 Such behavior has been predicted theoretically 87 108 though never observed experimentally. Before considering this aspect in detail, we first discuss the asymmetric nature of the BPI oscillation pattern in resonance k = 1. As seen clearly in Fig ure 4 8 (b), reversal of the longitudinal field, H L results in a reflection of the P 1 BPI pattern about H T = 0. In other words, the BPI minima are in fact invariant under a full magnetic field inversion, as required on the basis of the time reversal invariance of the spin orbit Hamiltonian responsible for this physics. As noted above, the symmetries of BPI patter ns must respect the symmetry of the zero field spin Hamiltonian. If one considers only 2 nd order ZFS within the GS A as we discussed in C hapter 1,
121 then the resulting Hamiltonian necessarily possesses the following symmetry elements: (i) three mutually orth ogonal C 2 rotation axes ( x y and z ); (ii) three mutually orthogonal mirror planes ( xy xz and yz ); and (iii) an inversion center. The presence the C 2 axis ( z axis) guarante es invariance with respect to reversal of H T ; i.e., it enforces symmetric BPI patte rns, irrespective of whether a longitudinal field is applied ( k > 0) or not ( k = 0). On the other hand, the horizontal mirror plane guarantees invariance with respect to reversal of H L As we show below, one must break the xy mirror symmetry in order to ob tain asymmetric BPI patterns with respect to inversion of H L In this case, reversal of H L results in different patterns; the time reversal symmetry then guarantees that these two patterns are mirror images. Nevertheless, no matter how many spatial symmetr ies are broken, the time reversal invariance of the spin orbit interaction guarantees that the BPI minima should be invariant under a full reversal of the applied field, i.e., simultaneous reversal of H L and H T as we have confirmed experimentally for the first time It is possible to reproduce the essential features of the experiments by introducing 4 th order terms into the GS A ; the xy mirror symmetry and C 2 rotational symmetry can then be broken by rotating the coordinate frames of the 2 nd and 4 th order t ensors. Interestingly, this approach also reproduces the complex motion of the P 1 minima within the H T phase space shown in Fig ure 4 9 However, a complete GS A analysis for Mn 4 Bet requires many parameters The difficulty of using the GSA model lies in the fact that all of the 4 th order tensors originate from the same order of perturbation in the MS model 35 37 109 ; therefore, in a system with no symmetry restri ctions, all ( q = 0 to 4) terms need to be equally considered. These parameters, obtained via the GSA approach, provide little insight, while the same physics can be naturally understood within a MS Hamiltonian such as Equation 4 1 Note that the emergence of significant higher order anisotropy terms within the GS A is a manifestation of mixing of the
122 ground spin state with excited states, which can only be captured within the MS model In this context, the xy mirror symmetry may be trivially broken by rotating (tilting) the ZFS tensors at the two inequivalent magnetic sites in the molecule so that their local z axes no longer coincide. This is similar to the case of the Mn 3 SMM discussed in C hapter 3 where the molecular hard plane i s a three fold corrugated plane. 109 However, in Mn 4 Bet, due to the low magneto symmetry of the molecule, the exact hard plane o f the molecule is an irregular corrugated plane with inversion symmetry only This is corroborated by the BPI results, where only C i symmetry is observed in QTM measurements. We simulated the BPI patterns for the k = 0 and k = 1 QTM resonances employing Equation 4 1 with the parameters obtained via EPR studies. As discussed in the previous paragraph the molecula r hard plane is corrugated, and literally impossible to map out computationally. In addition, similar to what we discussed in Chapter 3, this hard plane is field dependent; therefore, we first found the molecular easy axis and defined the perpendicular pla ne as the nominal hard plane. This exactly reproduces the experimental procedures for the QTM measurements. The orientation of the molecular easy axis is determined by searching for the direction corresponding to the minimum H L associated with the k = 1 st ep. Figure 4 10 (b) plots the locations of BPI minima obtained via diagonalization of Equation 4 1 (solid red symbols). The small misalignment of the experimental field rotation plane has been taken into account in our simulations. The quantitative agreemen t with experiment is also excellent. The motion of the P 1 minima can be understood as a result of the competition between different anisotropic interactions within the molecule, without a need to invoke unphysical 4 th and higher order GSA anisotropies. Imp ortantly, the angular positions ( ) of the k = 1 minima move with H T while the k = 0 minima remain stationary, as found experimentally (Fig ure 4 9 ).
123 Finally, the MS model perfectly reproduces the H T asymmetry of the k = 1 BPI pattern. As seen in Fig ure 4 10 (c), the asymmetry is reversed upon inversion of H L as required by the time reversal invariance of the anisotropic interactions in E quation 4 1 and observed experimentally ( Fig ure 4 8 (b) ) As we have shown in the preceding data analysis, the complex be havior of the BPI patterns are interpreted by employing Equation 4 1 with a set of standard parameters. The crucial ingredient is the tilting of the ZFS tensors of the external spins, s 1 and s 3 relative to the central spins s 2 and s 4 so that the xy mirro r symmetry is broken. This is illustrated in Fig ure 4 8 ( a ), where one observes that the classical energy landscape is invariant under full field inversion (blue v ersus black arrows), while this is not the case when only H T is reversed (red v er s us black arr ows). The Euler angle 1 3 = 45 o results in a significant projection of the relatively weak anisotropy associated with the Mn II ions into the hard ( xy ) plane. This, together with the finite e 2 4 parameters and the dipolar interactions, results in competing transverse interactio ns and to the complexity of the BPI patterns observed in Fig ure 4 9 We also performed simulations without including any field misalignment. The results show that the misalignment has little influence on the k = 0 BPI; the first and second k = 1 BPI minima are found at = 7.5 o and 3 o respectively without field misalignment; hence, the misalignment cannot be the major source for the motion of the BPI minima, especially the first k = 1 minimum. We note that the dipolar interaction has a very significant effect on the ene rgy levels of the molecule: the ZFS within the S = 9 multiplet varies by as much as 10% when dipolar interactions are omitted, and the location of the k = 1 QTM step is shifted by ~ 0 02 T. We conclude by noting that asymmetric BPI patterns have been seen i n other centro symmetric SMMs for which a clear explanation has been lacking. 36 80 110 111 In these previous studies, the asymmetric BPI pattern has been attributed to the anti symmetric Dzyaloshinski
124 Moriya (DM) interaction. However, in the Mn b ased molecules, DM interactions are expected to be weak; therefore, the effects of DM interactions are always expected to be much smaller than those of the single ion anisotropies. 80 In our work, we showed that the asymmetric pattern can originate simply from competition between ZFS tensors. The present resu lts may help shed light on the effect that symmetric anisotropic interactions can have in magnetic systems with inversion symmetry, where a net DM interaction is stric t ly forbidden. The present work clearly demonstrates how studies of simple low nuclearity systems can address fundamental symmetry considerations related to QTM in molecular nanomagnetism. 4.3 Summary In this chapter we presented EPR and QTM studies on a low symmetry Mn 4 Bet SMM showing asymmetric BPI in QTM measurements. A complex motion of B PI minima within the magnitude orientation phase space of the transverse field has been observed, which stands as the first experimental evidence for such motion in a SMM. These behaviors are attributed to the competition between the ZFS tensors of inequiv alent magnetic ions within the molecule. We explained all experimental results based on the parameters obtained via EPR and QTM measurements. An excellent quantitative agreement between simulations and experiments is achieved.
125 Figure 4 1. The molecular structures and the magnetic cores of the Mn 4 SMMs. (a) The molecular structure s of the Mn 4 Bet SMM. (b) The molecular structure s of the Mn 4 anca SMM Colo r code: Mn III = purple, Mn I I = magenta, O = red, N = blue and C = black. H atoms have been omitted for clarity. (c) Schematic representation of the magnetic core of these Mn 4 SMMs. The z axis of the molecule is defined by the Mn III N bond, as shown in the figure.
126 Fig ure 4 2 Temperature dependen t EPR spectra for Mn 4 Bet obtained at 139.5 GHz with the ap plied field close to the molecular easy axis. The peaks A 1 A 2 9 represent transitions within the S = 9 ground state multiplet.
127 Fig ure 4 3 Temperature dependen t EPR spectra for Mn 4 Bet obtained at 67.3 GHz with the applied field in the molecular hard plane The peaks A 9 A 7 5 represent transitions within the S = 9 ground state multiplet. Note that the A 8 A 6 ... transitions are not observed at this frequency.
128 Figure 4 4. Zeeman diagram for the ground multiplet ( S = 9) of Mn 4 Bet with the field appli ed in the molecular xy plane. For the reason of clarity, only a few of the lowest energy states are shown. The high field spin projections onto the quantization axis ( m ), which corresponds to the direction of the applied field, are labeled on the right han d side of the figure. The arrows correspond to EPR transitions from the m = i to i + 1 (A i ) where the red and blue arrows indicate that i is even and odd, re s pectively.
129 Figure 4 5. Temperature dependence of the EPR spectra for Mn 4 anca. The spectra were obtained with the field (a) close to the easy axis at a frequency of 165 GHz and (b) in the hard plane at a frequency of 50.5 GHz The peaks observed in the 15 K (a) and 12 K (b) spectr a have been labeled according to the scheme described in the main text
130 Fig ure 4 6 Plot s of f requency versus field for Mn 4 Bet showing the observed ground state EPR peak positions The data were collected with the applied field (a) close to the molecular easy axis at 3 K and (b) in the molecular hard plane at 5 K A dditi onal temperature dependence studies were performed at 139.5 GHz, 211.4 GHz and 289.6 GHz for the easy axis orientation The solid lines are the simulations of A series peak positions (red squares) using the Hamiltonian and parameters provided in the main t ext.
131 F igure 4 7 Hysteresis loops for Mn 4 Bet recorded as a function of H L at different temperatures. The QTM resonances are spaced by a step of
132 F igure 4 8 Modulation of the QTM probabilities for resonances of Mn 4 Bet. (a) k = 0 and (b) k = 1 as a function of H T applied at different angles, within the hard plane of the Mn 4 Bet SMM The inset to (a) illustrates the two fold angular modulation of P 0 for H T = 0 2 T, providing clear evidence for a significant 2 nd order rhombic anisotropy. The asymmetry of the BPI pattern of oscillations in res onance k = 1 is inverted upon reversal of H L
133 F igure 4 9 Contour plots of the QTM probabilities for resonances k = 0 and k = 1 of Mn 4 Bet as a function of H T and All of the k = 0 minima lie approximately along the = 0 axis, whereas the k = 1 minim a appear at different orientations for different H T values.
134 F igure 4 10 Simulations of the magneto anisotropy and QTM probabilities of Mn 4 Bet. (a) T he classical anisotropy barrier generated by the noncollinear ZFS tensors (see main text for explanatio n) and the different perspectives resulting from permutations of H T and H L (b) Experimental (open black symbols) and calculated (solid red symbols) H T dependence of the BPI minima for resonances k = 0 (circles) and 1 (squares) obtained from Eq uation 4 1 (c) Calculated tunnel splittings for resonance k = 1, for = 9 5 o as a function of H T for H L > 0 ( solid black line) and H L < 0 ( solid blue line).
135 C HAPTER 5 S LOW MAGNETIC RELAXAT ION INDUCED BY A LAR GE TRANSVERSE ZERO FIELD SPLITTING IN A M N II R E IV (CN) 2 SINGLE CHAIN MANGET The work presented in the chapter can be found from in the following article: Feng, X.; Liu, J.; Harris, T.D.; Hill, S.; Long, J.R. Slow Magnetic Relaxation Induced by a Large Transverse Zero Field Splitting in a Mn II Re IV (CN) 2 Single Chain Magnet. (In revision for submission to J. Am. Chem. Soc.) 5.1 Introduction In this chapter, a different type of molecule based magnet will be disc ussed, namely a single chain magnet (SCM), which is a one dimensional chain showing slow relaxation of magnetization at low temperatures. As discussed in Chapter 1, a SCM is a one dimensional isolated chain constituted of anisotropic spin units connected i n series. Compared to single molecule magnets (SMMs), the advantage of pursuing high magnetic relaxation barriers in SCMs stems from the exchange coupling ( J ) between the magnetic ions, which adds an additional energy term known as the correlation en ergy, to the expression for the relaxation barrier. The total magnetic relaxation barrier with respect to spin reversal can then be expressed as A + 2 for an infinite chain, or A + 2 for a finite chain. At low temperatures, a SCM typically fal ls into the second category due to the fact that the correlation length grows exponentially, thereby exceeding the finite chain length caused by small crystalline defects. In designing a SCM, it is commonly believed that the individual spin units should po ssess easy axis type anisotropy ( D < 0). A negative D value assures that states with maximum spin projection ( m = S ) onto the z axis lie lowest in energy with a ~| DS 2 | barrier between the m = S states, which leads to magnetic bistability at low temperatu res. In an easy axis type system, the presence of a rhombic transverse anisotropy ( E ) is considered to be antagonistic to magnetic bistability since it causes quantum tunneling 39 87 and reduces the effective relaxation barrier 31 32
136 74 94 Meanwhile, a positi ve D value results in ground states with minimal m ( 1/2 for half integer S and 0 for integer S ). In the pure axial case ( E = 0) the spin is completely delocalized with in the xy plane, i.e., Even in the presence of a sizable E term, which leads to an anisotropy in the xy plane, strong quantum fluctuation s prevent localization of the spin along any preferred direction in the xy plane. So far, apart from a few exceptions 112 114 (where slow relaxation is attributed to a phonon bottleneck effect or dipolar interactions), slow relaxation of magnetization has not been observed in D > 0 type molecule. Therefore, it is commonly believed tha t negative D values are essential in the design of single chain magnets. 115 116 The situation is quite dif ferent in the classical limit, i.e., a macroscopic spin, where quantum fluctuations are off I n the classical limit, with D > 0, the presence of a transverse interactions, causes an anisotropy in the xy plane, where t he energy of system equals to + ES 2 and ES 2 when the spin is parallel to the x and y axes, respectively. In this scenario, the easy axis would lie along either x or y depending on the sign of E and a n anisotropic energy barrier would prevent rotation of the magnetic moment from + x to x (or + y to y ); the maximum barrier height, corresponding to rotation of the moment through z would be dictated by D while the activation energy for the spin reversal, which is determined by the minimum of the barrier, c orresponding to the rotation within the xy plane would be dictated instead by E However, s uch a scenario has, so far, not been considered as a possible source of slow magnetic relaxation behavior in either a SMM or SCM The difficulties for achieving thi s in SMMs l ie in the strong quantum fluctuations associated with the relative small spins of SMMs. In SCMs, where the magnetic moment of the chain can be much larger, it is possible to achieve slow magnetic relaxation through a barrier generated by a trans verse anisotropy.
137 Recently a series of cyano bridged SCM s of the type (DMF) 4 MReCl 4 (CN) 2 (M = Mn, Fe, Co, Ni) have been synthesized with = 44.6 80.6 24.5 and 28.8 K respectively. 63 Hexa coordinated Re IV [ ReCl 4 (CN) 2 ] 2 was used as the building block owing to its sizable spin ground state ( S = 3/2) and strong magneto anisotropy originating from strong spin orbit coupling associated with the third row transitions metal ions 117 119 Magnetic analys i s reveal s that the chain compounds in this family fall neither in the Ising nor Heisenberg limit 63 where (as a function of S D and J ) is still under investigation. Given that the magnetic behavior of the chain compounds results from a combination of the magnetic anisotropy of the individual spin centers and the interactions between adjacent spin s the study of molecules mimicking small fragments of the chain (containing one, two, or three metal centers) may afford substantial insight into the mechanisms that lead to the observed magnetic data. Hence, we performed electron paramagnetic resonan ce ( EPR ) studies of three compounds, (Bu 4 N) 2 [ trans ReCl 4 (CN) 2 1 ), (DMF) 4 ZnReCl 4 (CN) 2 ( 2 ), and [(PY5Me 2 Mn) 2 ReCl 4 (CN) 2 ](PF 6 ) 2 ( 3 ) in order to elucidate the influence of the metal coordination environment and magnetic interactions on the magneto anisotropy of the Re IV ion in the [ReCl 4 (CN) 2 ] 2 magnetic core. Su rprisingly, 1 3 exhibit easy plane anisotropies ( D > 0) with significant E values, suggesting the presence of the same type of anisotropy ( D > 0) in the chain compound (DMF) 4 MnReCl 4 (CN) 2 ( 4 ). We describe a mechanism for slow magnetic relaxation in a single chain magnet that arises due to a barrier created by the transverse anisotropy E rather than the axial anisotropy D Based on this model and our EPR results, we calculated a relaxation barrier in 4 which is in excellent agreement with the results of magne tic studies. To the best of our knowledge, this is the first observation of slow relaxation arising from E in any coordination compound.
138 The studies presented in this chapter were conducted in collaboration with Dr. Jeffrey Long s group at the University o f California, Berkeley. The samples were synthesized by Dr. Long s group. X ray crystallography and magnetometry measurements were performed by Dr. Long s group. EPR measurements and the theoretical analysis was performed at the NHMFL and form the subject of this chapter. 5.2 Introduction to the Structures of 1 4 The structures for complexes 1 4 are shown in Figure 5 1. Compound 2 is a structural analogu e of 4 where the Mn II ions are replaced by diamagnetic Zn II ions. The crystal structures of 1 4 reveal a n octahedral coordination geometry around the Re IV center, with four chloride ligands in the equatorial positions and two axial cyanide ligands A comparison of the bond lengths and bond angles is included in Table 5 1, where one can find that the coordina tion of the Re IV is essentially a constant for all compounds. This reinforces the viability of 1 3 as models for the chain compound 4 Additionally, within each structure, no significant hydrogen bonding contacts between chains or molecules are evident in any of the structures, ruling out the presence of any significant pathways for long range magnetic interactions other than those along the chains in 4 There are several features of these structures which are noteworthy. We discuss these by using 1 as an e xample; the following discussion can be applied generally to complexes 1 4 Complex 1 crystallizes in a triclinic space group P 1 and lies about an inversion cent er The molecule only possesses an inversion symmetry, which indicates that a transverse aniso tropy, E is allowed by symmetry. In particular, we note that the four chloride ions are coplanar, which is guaranteed by the inversion symmetry of the molecule. These four chloride ions determine the molecular xy plane with the Re ion located approximatel y at the center. However, there are two different Re Cl bond lengths as indicated in Table 5 1. In addition, we note that the Cl Re Cl
139 bond angle is not 90 o ( 90.1 o ), while the Re C bonds are not perpendicular to the plane formed by the chloride ions (the C Re Cl bon d ang 9 0 o ). These features, especially the presence of two different Re Cl bond lengths, strongly suggest a transverse anisotropy associated with the Re IV ion, which is confirmed by the EPR experiments ( vide infra ). Table 5 1. Selected Interatomic Distances () and Angles () for compounds 1 4. 1 2 3 4 Re Cl 2.351(1), 2.341(1) 2.312(4), 2.316(3) 2.344(2), 2.330(1) 2.340(1), 2.343(2) Re C 2.148(4) 2.123(6) 2.134(5) 2.125(1) M N 2.121(2) 2.192(1) 2.228(1) M N py 2.224(4) M O 2.100(2) 2.181(1) Re C N 177.2(1) 175.0(1) 175.2(1) 175.8(1) M N C 158.8(1) 154.1(1) 158.8(1) Cl Re Cl 90.1 (4) 90.5(1) 90.1(4) 90.6(1) C Re Cl 88.5(1) 88.1(2) 89.0(4) 89.0(1) 5.3 Discussions of Magnetometr y and EPR R esults 5.3.1 Complex 1 Previous analysis of the reduced magnetization measurements on 1 suggested an easy axis type anisotropy with D = 27.4 K. 63 However, as we discussed in Chapter 2, extracting the sign of D from magnetic data can often be unreliable. In fact, recent reanalysis of the magn etic data shows that they can also be fitted with the following parameters: D = +10.4 K, E = 2.88 K and g = 1.88. Furthermore, slow relaxation of magnetization was not observed on 1 even upon application of a small dc field, which contrasts with expectati ons for a strong easy axis type anisotropy (| D |( S 2 1/4) = 54.8 K if D = 27.4 K, which is a sizeable relaxation barrier). Therefore in order to obtain a definitive determination of the magnetic anisotropy of the [Re(CN) 2 Cl 4 ] 2 building block ( 1 ) EPR meas urements were carried out in compound 1 at high fields. Both single crystal and powder EPR measurements were performed; experimental details can be found elsewhere 72 81 The EPR data can be describe d by the following spin Hamiltonian:
140 (5 1) with S = 3/2 and 0 < | E | < | D | /3. Figure 5 2 plots t he peak positions of EPR transitions observed via high field studies at 1.3 K ; the crystal was oriented in situ so that the field was aligned close to one of the principal axes of the magneto anisotropy tensor, determined to be the magnetic hard ( z ) axis ( vide infra ) The most notable feature of the se low temperature spectra is the fact that three resonances are observed in the frequency range from 60 to 130 GHz (see Fig ure 5 2 inset): t he sharp peak at low field (marked by a blue arrow) correspond s to a t ransition within the lowest Kramers doublet while the broad peaks at high field (marked by red arrows) correspond to inter Kramers transitions. The difference s in linewidth can be explained on the basis of D and E strain which primarily influence inter K ramers transitions. 77 78 Figure 5 3 shows the temperature dependence studies at 126.9 GHz which indicate that all three resonances correspond to excitations from the lowest lying energy level within the S = 3/2 manifold. The thick solid curves in Fig ure 5 2 correspond to the best simulation employing Eq uation 5 1 assuming a positive D value. The obtained paramete rs are: D = +15.8 K, | E | = 4.6 K and g z = 1.58, with the field being close (3 o misalignment) to the molecular z axis. The importance of the single crystal EPR results lies in the fact that the inter Kramers transitions were observed, which give a direct de termination for the zero field splitting (ZFS) between the m = 3/2 and 1/2 Kramers doublets; hence, the magnitude of D However, t here are actually two scenarios i) D > 0 and ii) D < 0, that can account for the high field single crystal data shown in Fi g ure 5 2 ; both sets of parameters involve a significant biaxiality, i.e., an E term approaching | D | /3. Therefore, In order to discriminate between these two cases, low field powder EPR measurements were performed (see Fig ure 5 4 ) The observed EPR transiti ons correspond to the three principal components of the effective g tensor associated with the lowest
141 Kramers doublet The powder results are also included in the bottom left part of Figure 5 2. First, note that the highest field (lowest effective g ) data points for the powder overlay exactly on the single crystal data, thus confirming that the sample was well aligned for the high field measurements. The thick solid curves in Fig ure 5 2 and 5 4 correspond to the best simulations of the combined single cryst al and powder data sets employing Eq uation 5 1 using the aforementioned parameter s with the addition of g x = g y = 1.89. A reasonable simulation can be obtained for the negative D case as well. However, it yields g values greater than 2.00, whereas the prin cipal Land values must be less than 2.00 for an orbitally non degenerate atom with a less than half filled d shell, as is the case for octahedrally coordinated Re IV (5d 3 with S = 3/2 and L = 0). In other words, the negative D (easy axis) simulation is un physical. We thus conclude that the D value is positive for 1 and we shall see below that the situation is even more definitive in the case of compound 2 We return to the discussion of the high field single crystal measurements. It is notable that the lo w field intra Kramers transitions in Fig ure 5 2 do not lie on a straight line that intersects the origin (thin solid line). This may be understood in terms of an avoided level crossing cause d by the rhombic E term in Eq. (1), as illustrated in Fig ure 5 5, which also explains the broadening of the intra Kramers resonance with increasing frequency In the absence of an E term, there would be no interaction between the m s = 3/2 and +1/2 states and the spectrum would be linear in B z ( orange curves in Fig ure 5 5 ), i.e., the intra Kramers data would lie on a straight line passing through the origin. Consequently, the departure of the low field data from the thin line in Fig ure 5 2 provides one of the main constraints on E the other being the splitting between th e x and y components of the powder spectrum. Meanwhile, the inter Kramers transitions above 15 T primarily constrain D and g z Finally, a small field misalignment of = 3 was considered due to
142 the fact that a single axis crystal rotation does not guarantee exact coincidence with the molecular z axis This misalignment accounts for the 2 nd avoided crossing at ~ 28 T, but it does not affect the low field data significan tly Thus even though the high field simulation contains four variables, the obtained parameters are well constrained. Moreover, the y are further constrained by the powder measurements. The obtained E parameter is quite significant, corresponding to | D | /3 .4, i.e., very close to the maximum allowable value. For the extreme biaxial case in which E = | D | /3, the D > 0 and D < 0 parameterizations are in fact equivalent. It is precisely for this reason that good simulations can be obtained for both cases, i.e., the two parameterizations are similar, though it should be stressed that the powder measurements clearly favor the positive D scenario. In any case the underlying magnetic properties resulting from either parameterization would be expected to be quite sim ilar. Moreover, such a strong E value is of no surprise, since the molecule only possesses C i symmetry. 5.3.2 Complex 2 Due to the difficulty associated with interpreting EPR data from a dynamic chain system, where collective spin wave resonance is observe d instead of discrete paramagnetic resonance, 16 zero field splitting parameters for Re IV were not directly obtained from the EPR data for the chain com pound 4 As a means of isolating the magnetic anisotropy of the Re IV ions within the chain the isostruct ural compound 2 was prepared. Here, the paramagnetic Mn II ions have been replaced by diamagnetic Zn II ions in order to prevent significant exchange interactions involving the Re IV centers, while preserving any effects that stem from connecting [ReCl 4 (CN) 2 ] 2 units to other metal centers via the cyanide ligands. To confirm the magnetic isolation of the Re IV centers, variable temperature d c magnetic susceptibility data were collected for a microcrystalline sample of 2 T he measurements shows that the M T vs T plots of 1 and 2 are essentially superimposable,
143 indicating an absence of inter and /or intra chain magnetic interactions in 2 Importantly, ac magnetic susceptibility measurement s as a function of both frequency and temperature on a polycrysta l l ine sampl e of 2 r evealed no slow relaxation behavior at or above 1.8 K. Due to the reason that sufficiently sized single crystals were not available, EPR measurements were carried out on a powder sample of 2 as shown in Figure 5 6. Just like complex 1 three spect ral features are resolved, corresponding to the principal components of the effective g tensor associated with the lowest Kramers doublet. In the case of easy plane anisotropy ( D > 0), the x and y components should occur at fields below the spin only g = 2.00 position (dashed line), while the z component should occur above; the opposite would hold for the easy axis ( D < 0) case. As seen in Fig ure 5 6 the powder measurements clearly indicate easy plane type anisotropy ( D > 0), although they do not permit a direct determination of the magnitude of D Meanwhile, t he sizeable splitting between the x and y components of the spectrum again signifies appreciable biaxiality, i.e., a significant E term Attempts to detect inter Kramers transitions were unsuccessful (sufficiently sized single crystals were not available). Therefore, for the purposes of simulation, we were forced to adopt the D value obtained for 1 i.e., +15.8 K The solid curves in Figure 5 6 represent the best simulation of the peak positions emplo ying Equation 5 1 with | E | = 3 K, g x = g y = 1.78 and g z = 1.94 As noted already, it is impossible to obtain any reasonable simulation with D < 0. However, t he obtained E value is rather well constrained by the splitting between the x and y components of t he powder spectrum. Therefore, these simulations unambiguously demonstrate that 2 possesses easy plane type anisotropy and appreciable transverse anisotropy, in analogy with complex 1 5.3.3 Complex 3 To exclude the possibility that magnetic exchange coup ling may act to invert the sign of D in the chain compound 4 relative to its constituent [ReCl 4 (CN) 2 ] unit, comp ound 3 was
144 prepared and investigated In particular, m agnetic and EPR measurements were carried out on polycrystalline samples of 3 to study the ZFS parameters of Re IV in the presence of magnetic exchange coupling. First, the sign and magnitud e of J were probed through variable temperature dc magnetic susceptibility measurement s. A fit to the dc susceptibility results with the following Hamiltonian, gives antiferromagnetic interactions between the Re IV and Mn II ions with J = 4.3 K. This value is comparable to the magnetic coupling observed in 4 where J = 7.2 K. The slight difference in the magnitude of J may be due to the different ligand field around Mn II and the small differences in the Mn N C angles. Thus, 3 shoul d provide a reasonable model of the motif in 4 We note that slow relaxation behavior was not observed in the variable frequency ac susceptibility measurements at temperatures above 1.8 K even in the presence of an applied field suggesting the p ossibility of easy plane type anisotropy in 3 EPR studies on 3 confirm t hat the Re IV ion possesses a positive D parameter, lending further weight to arguments that 4 also experiences easy plane type anisotropy Fig ure 5 7 plots the frequency dependence o f high frequency EPR peak positions obtained from studies of a powder sample of 3 at 3.5 K ; the inset displays a representative spectrum collected in first derivative mode at 208 GHz. T hree relatively strong features are observed, which we label y z and At high fields and frequencies, t he slopes of the y and z branches agree with expectation s for standard EPR transition s with m = 1 an d g 2. Meanwhile, the slope associated with the branch suggests that it is a double quantum transitions with m = 2 and g 4 We associate the y and z branches with the parallel ( z ) and perpendicular ( y ) extremes of the high field spectrum. We note that the low field spectrum would be rather more complex, thereby illustrating the importance of high field measurements.
145 Unlike the powder results for 1 and 2 the y and z resonance branches correspond to excitations within a coupled molecular spin state experiencing zero field splitting. It is for this reason that the high field portions of the spectrum extrapolate to finite zero field offsets, unlike the transitions within the lowest Kramers doublet in the case of 1 and 2 Most importantly, th e zero field offsets deduced from Fig ure 5 7 provide direct information on the anisotropy associated with the ReMn 2 molecular units of 3 The results may be interpreted by treating the molecule as a spin S = 7/2 object T his assignment is based on the magn etic measurements that indicate antiferromagnetic coupling between the Re IV and Mn II moments, giving a total ground state spin S = 25/2 3/2 = 7/ 2 The solid lines are the simulations employing E quation 5 1 and the following molecular zero field splitting parameters: D = +0.43 K, | E | = 0.043 K and g = 2 .00 As noted above, t he z branch corresponds to the ground state transition with the field being parallel to the molecular z axis while y corresponds to the same transition but with the field parallel to y The x component is buried within the very large signal close to g = 2.00. T he simulations also successfully account for the double quantum transition ( ) assuming that it belongs to the y component of the spectrum It is noteworthy that double quantum t ransitions usually appear when the field is applied perpendicular to the dominant quantization axis ( z axis) which is consistent with our observations and provides further confirmation for the peak assignments. Although fine structure peaks are not resol ved, observation of the x and y extrema provide an extremely robust constraint on the sign of the D parameter: the fact that the shift of the z component to the high field side of the isotropic g = 2.00 position is substantially greater than that of the y component to the low field side can be taken as a sure sign that the molecular anisotropy is of the easy plane type
146 We can estimate the anisotropy parameters associated with the Re IV ion using the projection method. 120 The z ero f ield s plitting associated with octahedrally coordinated Mn II may be assumed to be ne gligible; thus the anisotropy of 3 will be dominated by Re IV The projection method gives where and are the anisotropy tenors of the ReMn 2 molecule and the Re IV ion, respect ively Therefore, we can deduce that +30 K i.e., it is of the easy plane type and 3 K. However, due to the significant uncertainty associated with the resonance positions in Fig ure 5 7 and the many simplifying assumptions made in the analysis these values sho uld be considered as highly approximate Nevertheless, the data once again confirm easy plane anisotropy and considerable rhombicity. 5.4 Discussion of the Magnetic R elaxation P rocess High spin Mn II ions typically possess near negligible magnetic anisotrop y. 82 Consequently the magnetization relaxation barrier in 4 must arise primarily from the anisotropy of the Re IV ions However, the EPR studies reported here clearly indicate the presence of a positive D for Re IV Moreover, the theoretical relaxation barrier for 4 obtained using the magnitude of D deduced both by EPR and magnetic measurements is in stark disagreement with the experimental barrier of 17.28 K. The estimated anisotropy barriers are 54.8 K and 31 K using the magnitude of D (assuming D negative) deduced from the magnetic measurements and EPR, respec tively. These factors suggest that new physics may be at play in governing the magnetic relaxation of 4 If one considers the doubly degenerate M S = 1/2 ground levels of an isolated S = 3/2 molecule with positive D then extreme quantum tunneling effects prevent localization of the molecular magnetic moment within the xy plane, i.e. x = y = 0. However, the coupling of such spins, one by one, to form a ferrimagnetic chain, results in a gradual suppression of these tunneling effects. In such a descript ion, the chain possesses a giant spin, S which scales with the
147 chain correlation length, L In the case of 4 the coordination environments of the Re IV ions are collinear throughout the chains, with the axial z direction approximately parallel to the dire ction of chain propagation. As such, the preferred orientation of the giant spin, S lies in the plane perpendicular to the chain (see Figure 5 8 ). A transverse anisotropy then creates a preferred axis within this plane, although quantum tunneling may stil l prevent localization of S for small chain lengths. However, as the chain correlation length grows, these fluctuations diminish as the quantum tunneling is increasingly suppressed. T h e suppression of quantum tunneling is one of the key ingredients for ob serving slow magnetic relaxation in 4 To evaluate the correspondence between quantum tunneling and the size of the chain, we treated a finite length chain as a microscopic molecule whose spin is proportional to the number of units, N (= S ); each unit has s = 1 due to the antiferromagnetically coupled Re IV and Mn II s = 5/2 3/2 = 1. Figure 5 9 shows the dependence of the ground k = 0 tunneling gap as a function of the size of the chain. Because the Hilbert space associated with the chain grows exponential ly with increasing N we performed simulations with two different approaches (a) a multi spin (MS) Hamiltonian, which considers the chain as N coupled s = 1 units, and (b) a giant spin approximation (GSA) Hamiltonian, which treats the chain as a giant mole cule with S = N The simulations were performed to study the zero field ground state tunneling gap ( k = 0 gap) as a function of N The data points in the figure are simulation employing the MS Hamiltonian, treating the chain as s = 1 units connect ed in ser ies with a ferromagnetic interaction | J | = 7.2 K. The solid line is the simulation performed with the GSA in which the anisotropy parameters of the molecule is calculated via the projection m ethod. 120 Using this method, the chain possesses only second order anisotrop ies, D mol and E mol ; the high order anisotropies are zero because J is assumed to be infinite as discussed in Chapters 2 and 3.
148 I t is interesting to note that in both methods the shape of the magneto anisotropy potential energy surface of the system, i.e., the ratio between the molecular axial and tr ansverse anisotropies, D mol / E mol remains a constant when the size of the system grows. This is due to the fact that the ZFS tensors of all units in the chain are collinear; therefore, they directly add together forming the anisotropy of the chain. As show n in the figure, in both models, the tunneling gap decreases exponentially as the size of the system grows, which indicates that pure quantum tunneling quickly vanishes as the chain become sizable. Hence, in a chain with reasonable length, quantum tunnelin g can be neglected and the dynamics of the chain is completely governed by Glauber s theorem. Another method of visualizing such an effect is to consider the probability that all spins within the chain tunnel simultaneously, as would be required if the tot al spin were to tunnel coherently. Clearly, this probability decreases exponentially with increasing number of spins in the chain. Consequently, in the limit of large correlation length, S can be treated classically. Figure 5 10 depicts the classical poten tial energy surface in spin space corresponding to the zero field operator equivalent terms given in Eq uation 5 1 with D > 0 and | E / D | equal to the ratio found from the present EPR experiments. T he radial distance to the surface represents the energy of a spin as a function of its orientation. The minimum energy occurs when the spin points along y with an energy barrier separating these two orientations. The barrier maximum occurs in the yz plane, and is determined by D Meanwhile, it is E that sets the energy scale of the barrier minimum for rotation in the xy plane. In the large length limit, the anisotropy barrier against coherent rotation of an entire chain will be quite considerable, because the anisotropies of the individual Re IV centers sum togethe r. Nevertheless, we propose that magnetization dynamics can still proceed via the Glauber mechanism, except that the relevant anisotropy energy scale is
149 determined by E rather than D As we discussed in Chapter 1, the anisotropy barrier in the Glauber theo rem is characterized by the characteristic relaxation time for flipping an isolated spin unit. According to our previous discussions, the spin flipping process corresponds to the rotation of the spin in its xy plane; therefore, it is of no surprise that th is characteristic time is determined by E As such, t he anisotropy energy associated with the reversal of a single Re IV spin within the chain A = 2| E | S 2 Considering the value of | E| = 3 K deduced from the EPR measurements the anisotropy energy of 4 can be estimated at A = 13.7 K. Using the previously obtained value of the correlation energy of = 27.4 K, along with the express ion for the overall relaxation barrier for a chain in the finite size limit = A + a value of = 41.1 K is obtained for 4 Remarkably, this value is in excellent agreement with the experimentally determined value of = 44.6 K which has heretofore b een unexplainable in terms of uniaxial magnetic anisotropy ( D ) and correlation. Moreover, in the previous work on the isostructural chain compounds (DMF) 4 MReCl 4 (CN) 2 (M = Mn, Fe, Co, Ni) the experimental correlation energy of (DMF) 4 NiReCl 4 (CN) 2 was found to be = 12.7 K Taking the small contribution of the zero field splitting of octahedral Ni II ions and the anisotropy barrier ( A = 13.7 K ) estimated from EPR data into account, the overall relaxation barrier = 26.1 K agrees very well with the experimental v alue of = 28.8 K providing further support for the relaxation process via transverse anisotropy. Finally, note that the anisotropy energy of the CoRe and FeRe chain compounds are different, as the important contribution of zero field splitting from high spin Co II and Fe II enables a more complicated relaxation mechanism. 5.5 Summary The stidies presented in this chapter demonstrate that slow relaxation of magnetization can arise from transverse anisotropy. In particular, high field, high frequency EPR mea surements on
150 a series of model compounds ( 1 3 ) show that the magnetic relaxation in the single chain magnet (DMF) 4 MnReCl 4 (CN) 2 is governed by a nonzero E term, despite the presence of a positive D Based on our model, the theoretical barriers are in great agreement with the experimental results. The observation of slow relaxation arising from E is, to our knowledge, an unprecedented phenomenon in molecular magnetism, and it represents a fundamentally new design strategy toward constructing low dimensional m agnetic materials.
151 Figure 5 1. The structures of complexes 1 4 presented in C h apter 5. The figure illustrates the t eaction of trans [ReCl 4 (CN) 2 ] 2 (upper) with [Zn(DMF) 6 ] 2 + to form the one dimensional solids (DMF) 4 ZnReCl 4 (CN) 2 (left), with [PY5Me 2 Mn( CH 3 CN)] 2+ to form cluster [(PY5Me 2 ) 2 Mn 2 ReCl 4 (CN) 2 ] 2+ Orange, yellow, cyan, green, red, blue, and gray spheres represent Re,Mn, Zn, Cl, O, N, and C atoms, respectively; H atoms have been omitted for clarity
152 Figure 5 2. The EPR studies for complex 1 ( main panel) EPR peak positions observed for a single crystal of 1 at 1.3 K with the field aligned 3 o away from the molecular z axis; intra and inter Kramers transitions are marked with blue and red data points, respectively. Powder EPR data are included in the low field region; three components are observed at each frequency corresponding to the three components of the effective Lande g tensor associated with the lowest Kramers doublet (see legend). The solid lines represent the best simulation of the com bined data sets (both single crystal and powder measurements), employing Eqn. (1) and the single set of Hamiltonian and parameters given in the main text. The thin solid line would be the expectation for the intra Kramers transition in the absence of rhomb ic anisotropy ( E = 0); the data clearly depart from this expectation, providing a constraint on the E parameter. (inset) Representative single crystal EPR spectra for 1 collected at 1.3 K demonstrating the observation of three ground state resonances; intr a and inter Kramers transitions are marked with blue and red arrows, respectively
153 Figure 5 3. Temperature dependence spectra for 1 collected at f = 126.9 GHz All three resonances strengthen and persist to the lowest temperature where k B T << hf Thi s indicates that of all the transitions correspond to excitations from the ground state of the molecule with the blue and red arrows denoting intra and inter Kramers transitions, respectively
154 F i gure 5 4. Frequency dependence of the high frequency EP R peak positions deduced from studies of a powder sample of 1 at 5 K. A representative spectrum collected in the first derivative mode at 50.4 GHz is displayed in the inset Three branches of resonances are observed, corresponding to the three princip al co mponents of the effective L and g tensor associated with lowest Kramers doublet (field parallel to x y and z ). The fine structure seen in the y component is due to nuclear hyperfine splitting ; the y resonance position is chosen at the center of the fine s tructure spectrum The solid lines in the main panel are simulations of the three resonance branches employing Eq uation 5 1 and the parameters given in the main text.
155 Figure 5 5. Zeeman diagram for complex 1 generated employing Equation 5 1 with D = +15 .8 K and g z = 1.58. The figure is generated considering different E values and field misalignment angles (see legend) The approximate spin projection ( m s ) of each state is labeled in the low field region of the figure. The figure demonstrates that the a voided crossing at ~15 T is determined entirely by E while the one at ~28 T is determined entirely by Consequently, different regions of the data in Fig ure 5 2 constrain these two parameters. The blue and red arrows correspond to the intra Kramers and inter Kramers transitions ~120 GHz in single crystal experiments, respectively.
156 Figure 5 6. Frequency dependence of the EPR peak positions deduced from powder studies of complex 2 at 5 K. Three resonance branches are observed, corresponding to the thre e principal components of the effective Lande g tensor associated with the lowest Kramers doublet (field parallel to x y and z ). The solid lines are best simulations employing Eq uation 5 1 and the parameters discussed in the main text. The dash ed line rep resents the g = 2.00 position The observation of one resonance above g = 2.00, and two below, is indicative of easy plane type anisotropy (see main text for further explanation)
157 Figure 5 7. Frequency dependence of the EPR peak positions obtained from studies of a powder sample of 3 at 3.5 K. T he inset displays a representative spectrum collected in first derivative mode at 208 GHz. The strong truncated feature at g = 2.00 is likely due to paramagnetic impurities and/or uncoupled Mn II centers, while t h e sharp signals marked with asterisks are impurity signal s from molecular oxygen absorbed in the sample holder. The broader anisotropic signals labeled correspond to conventional m = 1 transitions, while the resonance corresponds to a double quantum t ransition (see main text for detailed explanation).
158 Figure 5 8. S tructure and spin arrangement of chain compound 4 The atoms are colored as follows: Re IV = orange, Mn II = yellow, Cl = green, N = blue and C = gray. The black dashed circles denote the local xy plane of each Re IV center, which is determined by the four coplanar chlorine atoms ; the z axis is parallel to the chain direction In the classical ground state the Mn II spins ( yellow arrows) are locked into an antiparallel arrangement relative t o those of the Re IV spins (orange arrows) ; the easy plane anisotropy associated with the Re IV centers then constrains the spins within the xy plane on both sub lattices
159 Figure 5 9. The ground tunnel splitting as a function of the size ( N ) of the SCM. The red squares are the numerical simulations performed with | J | = 7.2 K, and the blue solid line is calculated using the projection method. The gaps calculated by the two methods are essentially identical. Quantum fluctuations are exponentially suppressed as the length of a chain increases.
160 Figure 5 10. Classical m agneto anisotropy energy surface corresponding to the zero field operator equivalent terms given in Eq uation 5 1 The surface was generated with D > 0 and | E |/| D | equal to the ratio found fro m the present EPR experiments on complex 2 T he radial distance to the surface represents the energy of a spin as a function of its orientation ; z ero energy has been chosen to correspond to the case when the spin is parallel to y and only the z > 0 surfac e is shown in order to aid viewing of the cross section in the xy plane As can be seen, the spin experiences an anisotropic kinetic barrier against reversal from + y to y with the barrier minimum occurring along x
161 CHAPTER 6 SUMMARY This chapter gives a summary of the work presented in the preceding chapters. This dissertation is focused on studying both the quantum tunneling and thermally assisted relaxation of magnetization in two types of molecule based magnets, namely single molecule magnets (SMMs) a nd single chain magnets (SCMs). High frequency and/or high field electron paramagnetic resonance (EPR) was used as the primary experimental technique in these studies. We also performed extensive numerical and theoretical studies in order to understand the insights into the interesting physics in SMMs and SCMs. Chapter 1 gives a general introduction to SMMs and SCMs. We introduced two types of spin Hamiltonians, the giant spin approximation (GSA) Hamiltonian and multi spin (MS) Hamiltonian In particular, w e discussed the symmetries of the Stevens operators used in the GSA Hamiltonian, which decide the spin selection rules and Berry phase interference (BPI) patterns associated with the quantum tunneling of magnetization (QTM). We also discussed Glauber s the ory in the SCMs to illustrate the thermally assisted magnetic relaxation process in a one dimensional chain. In Chapter 2, we present ed EPR spectra for two [Mn III ] 2 dinuclear molecular magnets, one ferromagnetic and the other antiferromagnetic. The obtaine d spectra were analyzed both by exact diagonalization and symmetry based perturbation calculations. In both cases, we have show that EPR is a particularly powerful technique in determining the magneto anisotropy, d and exchange interactions, J in weakly coupled molecular magnets, where J is rarely obtained in EPR studies on strongly coupled systems. Furthermore, these studies paved the way for using EPR to study the magneto structural correlations in these molecules under pressure.
162 In Chapter 3, we presen ted several theoretical studies of QTM in a trinuclear [Mn III ] 3 SMM with idealized C 3 symmetry. By mapping the spectra obtained via a MS Hamiltonian onto a GSA Hamiltonian, we discovered that the three fold transverse anisotropy, arises when the Jahn Teller axes of the Mn III ions are tilted away from the molecular z axis. The emergence of the operator unfreezes k odd QTM resonances and shifts the k = 0 ground QTM resonance away from zero longitudinal fi eld. These studies demonstrated the correlations between molecular structure and the symmetry of the Hamiltonian. In addition, we considered QTM in a tetranuclear [Ni II ] 4 SMM with idealized S 4 symmetry. The studies for Mn 3 and Ni 4 illustrated that the ZFS anisotropies of a molecule can unfreezes k odd QTM resonances without the presence of a transverse field, which emphasizes the importance of disorder in QTM. In Chapter 4, we presented EPR and QTM studies on two mixed valent [Mn 2 III Mn 2 II ] SMMs. A motion of BPI minima was observed in the QTM experiments for Mn 4 Bet, which has been predicted, but never observed in SMMs. We found that this motion is due to the competition between the ZFS tensors of inequivalent Mn ions. We also showed that the asymmetric BPI p attern can originate from the competition between non collinear ZFS tensors within a molecule, where a net antisymmetric interaction is forbidden by the structural symmetry of the molecule. These results signify the importance of symmetry in QTM of molecul ar magnets. Finally, in Chapter 5, we presented our work on a Re IV based SCM. EPR measurements were performed on three related molecules mimicking small fragments of the chain to elucidate the influences of the local coordinations and magnetic interactions on the anisotropy of the [Re IV (CN) 2 Cl 4 ] 2 building block. The EPR spectra collected for all the three compounds indicate an easy plane type anisotropy for Re IV with a significant E term, which seems to be contradictory with the general strategy for buildi ng a SCM. Inspired by our findings, we developed a
163 theoretical model in which the relaxation barrier is determined by the transverse anisotropy, E rather than axial anisotropy, D Due to the macroscopic spin associated with the chain, QTM is suppressed, w hich leads to the observation of magnetic bistability induced by the transverse anisotropy. The theoretical barrier predicted by this model is in excellent agreement with the experimental results. This presents the first example of the observation of slow magnetic relaxation arising from E in either a SMM or SCM.
164 APPENDIX STEVENS OPERATORS The ( p operators are defined as:
165 where [ A B ] + = ( AB + BA )/2 and s = S ( S +1).
166 LIST OF REFERENCES  D. Gatteschi, R. Sessoli, and J. Villain, Molecular Nanomagnets (Oxford University Press, O xford, 2006).  M. N. Leuenberger and D. Loss, Nature 410 789 (2001).  M. Affronte, F. Troiani, A. Ghirri, A. Candini, M. Evangelisti, V. Corradini, S. Carretta, P. Santini, G. Amoretti, F. Tuna, G. Timco, and R. E. P. Winpenny, J. Phys. D: Appl. P hys. 40 2999 (2007).  A. Caneschi, D. Gatteschi, R. Sessoli, A. L. Barra, L. C. Brunel, and M. Guillot, J. Am. Chem. Soc. 113 5873 (1991).  J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo, Phys. Rev. Lett. 76 3830 (1996).  S. M. J. A ubin, M. W. Wemple, D. M. Adams, H. L. Tsai, G. Christou, and D. N. Hendrickson, J. Am. Chem. Soc. 118 7746 (1996).  M. R. Cheesman, V. S. Oganesyan, R. Sessoli, D. Gatteschi, and A. J. Thomson, Chem. Commun. (1997).  T. Lis, Acta Crystallographi ca Section B 36 2042 (1980).  A. L. Barra and et al., EPL (Europhysics Letters) 35 133 (1996).  A. L. Barra, A. Caneschi, A. Cornia, F. Fabrizi de Biani, D. Gatteschi, C. Sangregorio, R. Sessoli, and L. Sorace, J. Am. Chem. Soc. 121 5302 (1999)  W. Wernsdorfer, N. Aliaga Alcalde, D. N. Hendrickson, and G. Christou, Nature 416 406 (2002).  S. Hill, R. S. Edwards, N. Aliaga Alcalde, and G. Christou, Science 302 1015 (2003).  E. C. Yang, C. Kirman, J. Lawrence, L. N. Zakharov, A. L. Rheingold, S. Hill, and D. N. Hendrickson, Inorg. Chem. 44 3827 (2005).  A. J. Tasiopoulos, A. Vinslava, W. Wernsdorfer, K. A. Abboud, and G. Christou, Angew. Chem. 116 2169 (2004).  C. J. Milios, A. Vinslava, W. Wernsdorfer, S. Moggach, S. Parsons, S. P. Perlepes, G. Christou, and E. K. Brechin, J. Am. Chem. Soc. 129 2754 (2007).  P. L. Feng, C. Koo, J. J. Henderson, P. Manning, M. Nakano, E. del Barco, S. Hill, and D. N. Hendrickson, Inorg. Chem. 48 3480 (2009).
167  D. E. Freedman, W. H. Harman, T. D. Harris, G. J. Long, C. J. Chang, and J. R. Long, J. Am. Chem. Soc. 132 1224 (2010).  R. Sessoli, H. L. Tsai, A. R. Schake, S. Wang, J. B. Vincent, K. Folting, D. Gatteschi, G. Christou, and D. N. Hendrickson, J. Am. Chem. Soc. 11 5 1804 (1993).  D. Gatteschi, A. Caneschi, L. Pardi, and R. Sessoli, Science 265 1054 (1994).  F. Branzoli, P. Carretta, M. Filibian, G. Zoppellaro, M. J. Graf, J. R. Galan Mascaros, O. Fuhr, S. Brink, and M. Ruben, J. Am. Chem. Soc. 131 4387 (2009).  J. D. Rinehart, M. Fang, W. J. Evans, and J. R. Long, J. Am. Chem. Soc. 133 14236 (2011).  J. D. Rinehart, M. Fang, W. J. Evans, and J. R. Long, Nat Chem 3 538 (2011).  I. J. Hewitt, J. Tang, N. T. Madhu, C. E. Anson, Y. Lan, J. L uzon, M. Etienne, R. Sessoli, and A. K. Powell, Angew. Chem. Int. Ed. 49 6352 (2010).  R. Jeon, W. Wernsdorfer, R. Clrac, and S. Dehnen, Chemistry A European Journal 17 9605 (2011).  P. H. Lin, T. J. Burchell, R. Clrac, and M. Murugesu, Angew. Chem. 120 8980 (2008).  P. H. Lin, T. J. Burchell, L. Ungur, L. F. Chibotaru, W. Wernsdorfer, and M. Murugesu, Angew. Chem. Int. Ed. 48 9489 (2009).  C. Papatriantafyllopoulou, W. Wernsdorfer, K. A. Abboud, and G. Christou, Inorg. Chem. 50 421 (2010).  E. M. Chudnovsky and J. Tejada, Macroscopic Quantum Tunneling of the Magnetic Moment (Cambridge University Press, 2005).  A. Abragam and B. Bleaney, Electron paramagnetic resonance of transition ions (Do ver, New York, 1986).  C. Rudowicz and C. Y. Chung, J. Phys.: Condens. Matter 16 5825 (2004).  S. Takahashi, R. S. Edwards, J. M. North, S. Hill, and N. S. Dalal, Phys. Rev. B 70 094429 (2004).  A. L. Barra, D. Gatteschi, and R. Sessoli, C hemistry A European Journal 6 1608 (2000).  C. Kirman, J. Lawrence, S. Hill, E. C. Yang, and D. N. Hendrickson, J. Appl. Phys. 97 10M501 (2005).
168  S. Carretta, E. Liviotti, N. Magnani, P. Santini, and G. Amoretti, Phys. Rev. Lett. 92 207205 ( 2004).  A. Wilson, J. Lawrence, E. C. Yang, M. Nakano, D. N. Hendrickson, and S. Hill, Phys. Rev. B 74 140403 (2006).  C. M. Ramsey, E. del Barco, S. Hill, S. J. Shah, C. C. Beedle, and D. N. Hendrickson, Nat Phys 4 277 (2008).  S. Hill, S Datta, J. Liu, R. Inglis, C. J. Milios, P. L. Feng, J. J. Henderson, E. del Barco, E. K. Brechin, and D. N. Hendrickson, Dalton Trans. 39 4693 (2010).  O. Pieper, T. Guidi, S. Carretta, J. van Slageren, F. El Hallak, B. Lake, P. Santini, G. Amorett i, H. Mutka, M. Koza, M. Russina, A. Schnegg, C. J. Milios, E. K. Brechin, A. Juli, and J. Tejada, Phys. Rev. B 81 174420 (2010).  W. Wernsdorfer and R. Sessoli, Science 284 133 (1999).  J. Liu, B. Wu, L. Fu, R. B. Diener, and Q. Niu, Phys. Re v. B 65 224401 (2002).  W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia, and D. Mailly, J. Appl. Phys. 87 5481 (2000).  L. D. Landau, Phys. Z. Sowjetunion 2 46 (1932).  E. C. G. Stueckelberg, Helv. Phys. Acta 5 369 (1932 ).  C. Zener, Proceedings of the Royal Society of London. Series A 137 696 (1932).  S. Miyashita, J. Phys. Soc. Jpn. 64 3207 (1995).  G. Rose and P. C. E. Stamp, J. Low Temp. Phys. 113 1153 (1998).  hys. Rev. B 67 134403 (2003).  A. Vijayaraghavan and A. Garg, Phys. Rev. B 79 104423 (2009).  D. A. Garanin and R. Schilling, Phys. Rev. B 71 184414 (2005).  E. M. Chudnovsky and D. A. Garanin, Phys. Rev. Lett. 87 187203 (2001).  K. M. Mertes, Y. Suzuki, M. P. Sarachik, Y. Paltiel, H. Shtrikman, E. Zeldov, E. Rumberger, D. N. Hendrickson, and G. Christou, Phys. Rev. Lett. 87 227205 (2001).  A. Garg, EPL (Europhysics Letters) 22 205 (1993).
169  D. Loss, D. P. DiVincenzo, and G. Grinstein, Phys. Rev. Lett. 69 3232 (1992).  J. von Delft and C. L. Henley, Phys. Rev. Lett. 69 3236 (1992).  P. Bruno, Phys. Rev. Lett. 96 117208 (2006).  M. V. Berry and M. Wilkinson, Proceedings of the Royal Society of London. A. Ma thematical and Physical Sciences 392 15 (1984).  A. Caneschi, D. Gatteschi, N. Lalioti, C. Sangregorio, R. Sessoli, G. Venturi, A. Vindigni, A. Rettori, M. G. Pini, and M. A. Novak, Angew. Chem. Int. Ed. 40 1760 (2001).  R. Clrac, H. Miyasaka, M. Yamashita, and C. Coulon, J. Am. Chem. Soc. 124 12837 (2002).  P. Gambardella, A. Dallmeyer, K. Maiti, M. C. Malagoli, W. Eberhardt, K. Kern, and C. Carbone, Nature 416 301 (2002).  M. Ferbinteanu, H. Miyasaka, W. Wernsdorfer, K. Nakata, K. i. Sugiura, M. Yamashita, C. Coulon, and R. Clrac, J. Am. Chem. Soc. 127 3090 (2005).  T. Kajiwara, M. Nakano, Y. Kaneko, S. Takaishi, T. Ito, M. Yamashita, A. Igashira Kamiyama, H. Nojiri, Y. Ono, and N. Kojima, J. Am. Chem. Soc. 127 10150 (2005)  H. Miyasaka, T. Madanbashi, K. Sugimoto, Y. Nakazawa, W. Wernsdorfer, K. i. Sugiura, M. Yamashita, C. Coulon, and R. Clrac, Chemistry A European Journal 12 7028 (2006).  T. D. Harris, M. V. Bennett, R. Clerac, and J. R. Long, J. Am. Chem. Soc. 132 3980 (2010).  H. Miyasaka, M. Julve, M. Yamashita, and R. Clerac, Inorg. Chem. 48 3420 (2009).  R. J. Glauber, Journal of Mathematical Physics 4 294 (1963).  B. Barbara, Journal De Physique 34 1039 (1973).  J. H. Luscombe, M. Luban, and J. P. Reynolds, Phys. Rev. E 53 5852 (1996).  H. Miyasaka, T. Madanbashi, A. Saitoh, N. Motokawa, R. Ishikawa, M. Yamashita, S. Bahr, W. Wernsdorfer, and R. Clrac, Chemistry A European Journal 18 3942 (2012).  R. Inglis, E. Hou ton, J. Liu, A. Prescimone, J. Cano, S. Piligkos, S. Hill, L. F. Jones, and E. K. Brechin, Dalton Trans. 40 9999 (2011).
170  G. Arom, P. Gamez, O. Roubeau, P. C. Berzal, H. Kooijman, A. L. Spek, W. L. Driessen, and J. Reedijk, Inorg. Chem. 41 3673 (20 02).  M. Mola, S. Hill, P. Goy, and M. Gross, Rev. Sci. Instrum. 71 186 (2000).  S. Takahashi and S. Hill, Rev. Sci. Instrum. 76 023114 (2005).  S. Hill, R. S. Edwards, S. I. Jones, N. S. Dalal, and J. M. North, Phys. Rev. Lett. 90 217204 (2003).  S. Hill, N. Anderson, A. Wilson, S. Takahashi, N. E. Chakov, M. Murugesu, J. M. North, N. S. Dalal, and G. Christou, J. Appl. Phys. 97 10M510 (2005).  J. Lawrence, E. C. Yang, R. Edwards, M. M. Olmstead, C. Ramsey, N. S. Dalal, P. K. G antzel, S. Hill, and D. N. Hendrickson, Inorg. Chem. 47 1965 (2008).  J. Lawrence, E. C. Yang, D. N. Hendrickson, and S. Hill, PCCP 11 6743 (2009).  K. Park, M. A. Novotny, N. S. Dalal, S. Hill, and P. A. Rikvold, Phys. Rev. B 65 014426 (2001)  S. Hill, S. Maccagnano, K. Park, R. M. Achey, J. M. North, and N. S. Dalal, Phys. Rev. B 65 224410 (2002).  D. E. Graf, R. L. Stillwell, K. M. Purcell, and S. W. Tozer, High Pressure Research 31 533 (2011).  E. del Barco, S. Hill, C. C. Beedle, D. N. Hendrickson, I. S. Tupitsyn, and P. C. E. Stamp, Phys. Rev. B 82 104426 (2010).  A. K. Hassan, L. A. Pardi, J. Krzystek, A. Sienkiewicz, P. Goy, M. Rohrer, and L. C. Brunel, Journal of Magnetic Resonance 142 300 (2000).  J. Krzys tek, A. Ozarowski, and J. Telser, Coord. Chem. Rev. 250 2308 (2006).  A. Garg, Phys. Rev. Lett. 83 4385 (1999).  F. Li and A. Garg, Phys. Rev. B 83 132401 (2011).  C. S. Park and A. Garg, Phys. Rev. B 65 064411 (2002).  u and A. Garg, Phys. Rev. Lett. 88 237205 (2002).  E. del Barco, A. D. Kent, S. Hill, J. M. North, N. S. Dalal, E. M. Rumberger, D. N. Hendrickson, N. Chakov, and G. Christou, J. Low Temp. Phys. 140 119 (2005).
171  M. S. Foss Feig and R. F. Jonath an, EPL (Europhysics Letters) 86 27002 (2009).  W. Wernsdorfer, S. Bhaduri, C. Boskovic, G. Christou, and D. N. Hendrickson, Phys. Rev. B 65 180403 (2002).  P. L. Feng, C. Koo, J. J. Henderson, M. Nakano, S. Hill, E. del Barco, and D. N. Hendri ckson, Inorg. Chem. 47 8610 (2008).  J. J. Henderson, C. Koo, P. L. Feng, E. del Barco, S. Hill, I. S. Tupitsyn, P. C. E. Stamp, and D. N. Hendrickson, Phys. Rev. Lett. 103 017202 (2009).  E. C. Yang, W. Wernsdorfer, S. Hill, R. S. Edwards, M. Nakano, S. Maccagnano, L. N. Zakharov, A. L. Rheingold, G. Christou, and D. N. Hendrickson, Polyhedron 22 1727 (2003).  E. C. Yang, W. Wernsdorfer, L. N. Zakharov, Y. Karaki, A. Yamaguchi, R. M. Isidro, G. D. Lu, S. A. Wilson, A. L. Rheingold, H. Ish imoto, and D. N. Hendrickson, Inorg. Chem. 45 529 (2005).  G. Redler, C. Lampropoulos, S. Datta, C. Koo, T. C. Stamatatos, N. E. Chakov, G. Christou, and S. Hill, Phys. Rev. B 80 094408 (2009).  J. O. D. George F. Koster, Pobert G. Wheeler and Hermann Statz, Properties of the Thirty Two Point Groups (M.I.T. Press, Cambridge, MA, 1963).  F. A. Cotton, Chemical Applications of Group Theory (A Wiley Interscience Publication, 1990).  J. van Slageren, S. Vongtragool, B. Gorshunov, A. Mukhin and M. Dressel, Phys. Rev. B 79 224406 (2009).  E. Liviotti, S. Carretta, and G. Amoretti, The Journal of Chemical Physics 117 3361 (2002).  A. L. Barra, A. Caneschi, A. Cornia, D. Gatteschi, L. Gorini, L. P. Heiniger, R. Sessoli, and L. Sora ce, J. Am. Chem. Soc. 129 10754 (2007).  R. Maurice, eacute, mi, C. de Graaf, Guih, and N. ry, Phys. Rev. B 81 214427 (2010).  J. Liu, C. Koo, A. Amjad, P. L. Feng, E. S. Choi, E. del Barco, D. N. Hendrickson, and S. Hill, Phys. Rev. B 84 09 4443 (2011).  H. M. Quddusi, J. Liu, S. Singh, K. J. Heroux, E. del Barco, S. Hill, and D. N. Hendrickson, Phys. Rev. Lett. 106 227201 (2011).
172  D. N. Hen drickson, Inorg. Chem. 50 7367 (2011).  J. Liu, C. C. Beedle, H. M. Quddusi, E. d. Barco, D. N. Hendrickson, and S. Hill, Polyhedron 30 2965 (2011).  C. C. Beedle, University of Californian, San Diego, 2010.  J. C. Hempel, The Journal o f Chemical Physics 64 4307 (1976).  W. B. Lynch, R. S. Boorse, and J. H. Freed, J. Am. Chem. Soc. 115 10909 (1993).  K. Hijii and S. Miyashita, Phys. Rev. B 78 214434 (2008).  J. Liu, E. del Barco, and S. Hill, Phys. Rev. B 85 012406 (2012).  S. Bahr, C. J. Milios, L. F. Jones, E. K. Brechin, V. Mosser, and W. Wernsdorfer, Phys. Rev. B 78 132401 (2008).  W. Wernsdorfer, T. C. Stamatatos, and G. Christou, Phys. Rev. Lett. 101 237204 (2008).  O. Waldmann, A. M. Ako, H and A. K. Powell, Inorg. Chem. 47 3486 (2008).  Y. Sanakis, M. Pissas, J. Krzystek, J. Telser, and R. G. Raptis, Chem. Phys. Lett. 493 185 (2010).  J. M. Zadrozny, J. Liu, N. A. Piro, C. J. Chang, S. Hill, and J. R. Long, Chem. C ommun. (2012).  C. Coulon, Cl, eacute, R. rac, L. Lecren, W. Wernsdorfer, and H. Miyasaka, Phys. Rev. B 69 132408 (2004).  L. Bogani, A. Vindigni, R. Sessoli, and D. Gatteschi, J. Mater. Chem. 18 4750 (2008).  A. Tomkiewicz, F. Villain and J. Mrozinski, J. Mol. Struct. 555 383 (2000).  595 225 (2001).  J. Martnez Lillo, D. Armentano, G. De Munno, F. Lloret, M. Julve, and J. Faus, Inorg. Chim. Acta 359 4343 (2006).  A. Bencini and D. Gatte schi, Electron Paramagnetic Resonance of Exchange Coupled Clusters (Springer, Berlin, 1990).
173 BIOGRAPHICAL SKETCH Junjie Liu was born in Changsha, the capital city of Hunan province in China. He spent the early part of his education in Changsha. In the summer of 2003, he graduated from high school and joined the Tsinghua University in Beijing, China. During his undergraduate studies, Junjie Liu joined the program called fundamental science, mathematics and physics which emphasizes fundamental educatio n in both mathematics and physics In 2007, he received his bachelor degree in physics from Tsinghua University and then joined the University of Florida in Gainesville, Florida, in the same year. After finished his core courses, Junjie Liu joined Dr. Step hen Hill's lab in the summer of 200 8. Later in 2009, he moved with Dr. Hill s lab to the National High Magnetic Field Laboratory at Tallahassee, where he kept working on his research projects. His PhD project mainly focuses on using high frequency/high fie ld electron paramagnetic resonance to study the spin dynamics in molecule based magnets. These projects include wide range collaborative interdisciplinary research projects with condensed matter physics and inorganic chemistry groups. Meanwhile, he also ca rried out many theoretical studies to understand the correlations between the symmetry and quantum tunneling of magnetizations in molecule based magnets.