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Electronic Theses, Treatises and Dissertations The Graduate School
2012 A Study of Rare-Earth Magnetism Through Spectroscopic Studies of Lanthanide-Based Single Crystals Sanhita Ghosh
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THE COLLEGE OF ARTS AND SCIENCES
A STUDY OF RARE-EARTH MAGNETISM THROUGH SPECTROSCOPIC STUDIES OF
LANTHANIDE-BASED SINGLE CRYSTALS
By
SANHITA GHOSH
A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Degree Awarded: Summer Semester, 2012 Sanhita Ghosh defended this dissertation on June, 28th, 2012.
The members of the supervisory committee were:
Stephen Hill Professor Directing Dissertation
Christopher Wiebe Professor Co-Directing Dissertation
Naresh Dalal University Representative
Pedro Schlottmann Committee Member
Grigory Rogachev Committee Member
The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements.
ii Dedicated to Ma and Baba
iii ACKNOWLEDGEMENTS
I express my sincere gratitude to all those, whose guidance and support have made this work possible. I would like to start by thanking my thesis supervisors, Dr. Stephen Hill and Dr. Christopher Wiebe, for having given me the opportunity to work on various interesting projects. I am grateful to them for having introduced me to the world of scientific research, and for helping me to develop the qualities required to work independently, while also providing me with their support and guidance at every step. They have been extremely patient with me, and have always helped me in learning new concepts and techniques. As a graduate student at Florida State University, I joined Dr. Wiebe’s group, where I worked for about two years, and was trained in single crystal synthesis and sample characterization techniques by Dr. Haidong Zhou and Dr. Benjamin Conner. Haidong has provided me with a number of samples that I have worked on over the last few years, and I thank him not only for the samples, but also for all his help and patience in training me in my first few months as a new graduate student. I joined Dr. Hill’s group in Fall 2009 when Dr. Wiebe relocated to Canada, and my sincerest thanks to every member of the Hill group for having made that transition as smooth as possible. I am grateful to Dr. Hill for having accepted me as his graduate student and for helping me identify new projects to work on, which would fit in with the studies that I had been involved in earlier. He has been a wonderful teacher and a source of support and guidance. Dr. Hill has supported me as a graduate research assistant and has also provided me with the opportunity to participate and present my work at various conferences in the past couple of years. That has given me the much needed exposure in current research trends, and has also been a major learning experience. I thank Dr. Hill for his guidance and encouragement. Dr. Saiti Datta, Dr. Alexey Kovalev, Dr. Christopher Beedle and Dr. Kyuil Cho have all been wonderful postdoctoral researchers in our lab from whom I have learnt a lot. A big thank you to Saiti for teaching me how to handle the equipment in the laboratory and for her valuable guidance in conducting experiments while I was new to the group. Thanks to Dr. Changhyun Koo, Dr. Junjie Liu, Mr. Muhandis Muhandis and Ms. Chelsey Morien for being the best lab-mates one could
iv hope for. Junjie has played the role of a teacher, an excellent co-worker and a friend, and I consider myself fortunate to have had him as a lab-mate. I am grateful to my PhD supervisory committee members, Dr. Pedro Schlottmann, Dr. Naresh Dalal and Dr. Grigory Rogachev for their time and support through the past four years. Their valuable suggestions and helpful discussions at committee meetings have made the successful completion of this work possible. My thanks to the scientists in CMS and EMR groups at the NHMFL for their support. Dr. Jurek Krzystek, Dr. Hans van Tol and Dr. Likai Song have helped me learn the operation of the Bruker spectrometer for pulsed EPR measurements. They have always been available to clarify my doubts and to help with the concepts of the pulsed technique. I would like to thank Dr. Naresh Dalal for allowing me to work on his Q-band EPR spectrometer at the Department of Chemistry, FSU and Dr. Vasanth Ramachandran for his help with conducting the experiment. I also express my thanks to the scientists and staff at the Machine Shop, the Electronics Shop, the Cryogenics Support, the Computer Support, the Receiving Department, the Safety Department and the DC Facility at the NHMFL. They have provided us with crucial support in various different aspects which has made it possible for us to carry on our work smoothly. It is impossible to recount the numerous direct and indirect ways in which various staff members at the lab have contributed to our work. I thank the machinists, especially Andy and Tom, who have been very helpful while working with me in building a probe for high frequency EPR measurements. Thanks to Russell Wood at the Electronics Shop for his help with soldering and electrical connections of equipment, Jim and Peter from Computer Support for solving various software glitches and Scott at the DC Facility for his support during our high-field experiments. I thank the administrative staff at the NHMFL and the Physics Department at FSU for always doing an amazing job in taking care of all required paperwork. Thanks to our former graduate coordinator Sherry Tointigh for the numerous ways in which she has helped new graduate students at the department. I would also like to thank Arshad Javed, Alice Hobbs, Mareen Redies and Miranda Hacker for their support with administrative stuff over the years. I would like to thank and acknowledge our collaborators from different institutions for their integral role in our work. Dr. Eugenio Coronado’s group from University of Valencia, Spain has provided us with the holmium polyoxometalate samples which constitute a major part of my thesis. I am grateful to Dr. Sylvain Bertaina from IM2NP-CNRS, France for the extremely
v valuable and instructive discussions on our pulsed EPR study on the holmium-polyoxometalate system. Dr. En-Che Yang’s group from Taiwan has sent us samples of tetra-nuclear lanthanide clusters which have also constituted part of my work. Dr. Enrique del Barco’s group from University of Central Florida, Orlando has contributed with the magnetic characterization of some of the samples. Neutron scattering experiments were carried out at NIST with support from Dr. J. S. Gardner. Dr. Michael Hoch at the NHMFL has been greatly involved in the work involving microwave excitations in kagomé systems, and has been instrumental in taking forward the project. I thank them all and consider myself fortunate to have been part of these collaborative projects. I would also like to thank my teachers whom I adore and respect. My first Physics teachers from Presidency College, Calcutta will always remain my most favorite teachers. I thank Dr. Dipanjan Ray Chowdhury, Dr. Debopiryo Shyam, Dr. Prodyot Sengupta and Dr. Pradeep Kumar Mukherjee amongst many others who taught us ‘why’ and ‘how’ to love Physics. I am also grateful to my teachers at IIT-Bombay, and would like to acknowledge Dr. Dibyendu Das, Dr. Indra Dasgupta and Dr. Tapanendu Kundu for their guidance. I have been fortunate to have been taught by a number of wonderful teachers at FSU, and I thank Dr. Pedro Schlottmann, Dr. Jorge Piekarewicz, Dr. Alexander Volya and Dr. Nicholas Bonsesteel for the amazing graduate courses they have taught. I acknowledge the various funding sources that have made it possible to carry out this work. I thank the NSF which has funded most of the projects that I have been involved in. Much of the work reported in this thesis has been carried out at the NHMFL which is funded by the NSF (Grant No. DMR – 0654118) and by the State of Florida. On the personal front, I thank my friends for their support in innumerable ways over these past five years. Thanks to Mukesh Saini, Siddhartha Adak, Krishanu Shome, Saurish Chakraborty, Rudresh Ghosh, Moumita Aich, Reshmi Das, Suman Dhayal and Naureen Ahsan for being friends I can always count on. Lots of thanks to Mukesh for his help in teaching me Matlab and for physics discussions in general. The company of a number of people has made my stay in Tallahassee bearable, and at times also enjoyable; I thank them all. It is almost impossible for me to say “thank you” to my parents, and I would like to end by saying that nothing that I have been able to learn or achieve would be possible without their love, guidance, support and encouragement.
vi TABLE OF CONTENTS
List of Tables ...... ix List of Figures...... x Abstract...... xiv 1. INTRODUCTION TO RARE-EARTH MAGNETISM...... 1 1.1 Magnetism...... 1 1.2 Rare Earth Magnetism ...... 6 1.3 Organization of the Thesis...... 13
2. EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION ...... 15 2.1 Electron Paramagnetic Resonance (EPR) Spectroscopy ...... 15 2.1.1 Instrumentation for High Frequency EPR Experiments ...... 19 2.2 Pulsed EPR Spectroscopy...... 34 2.2.1 Classical Description of the Theory of Pulsed EPR Spectroscopy...... 35 2.2.2 A Brief Overview of the Quantum Mechanical Description...... 43 2.3 Synthesis of Single Crystals: Floating Zone Method ...... 46 2.4 Sample Characterization Techniques...... 49 2.5 Neutron Diffraction...... 52
3. STUDY OF A MONONUCLEAR LANTHANIDE-BASED SINGLE-MOLECULE MAGNET...... 54 3.1 Introduction to Single-Molecule Magnets ...... 54 3.2 Mononuclear Lanthanide-based Polyoxometalate Single-Molecule Magnets ...... 59 3.2.1 Background and Motivation ...... 60 3.2.2 HoPOM: Structural Information and Synopsis of Previous Studies...... 63 9- 3.3 High Frequency Single Crystal EPR of [HoxY1-x(W5O18)2] ...... 66 9- 3.4 Low Frequency Single Crystal EPR of [HoxY1-x(W5O18)2] ...... 74 3.5 Discussion of EPR Results on HoPOM...... 84 9- 3.6 Study of [HoxY1-x(W5O18)2] in deuterated solvent ...... 96 3.7 Summary...... 98
4. FRUSTRATED MAGNETISM: STUDY OF SHORT RANGE ORDERING IN THE
MODIFIED HONEYCOMB LATTICE COMPOUNDS SrL2O4 (L = LANTHANIDE) ..100 4.1 Introduction to geometric frustration ...... 100 4.2 Motivation ...... 104 4.3 Description of the crystal structure of SrL2O4 ...... 106
vii 4.4 Synthesis and Characterization of SrHo2O4...... 106 4.5 Neutron Scattering ...... 110 4.5.1Elastic Neutron Scattering ...... 110 4.5.2 Inelastic Neutron Scattering...... 110 4.6 Discussion and Conclusion...... 114
5. FRUSTRATED MAGNETISM: MICROWAVE INDUCED EXCITATIONS IN THE
KAGOMÉ SYSTEMS Pr3Ga5SiO14 AND Nd3Ga5SiO14 ...... 116 5.1 Introduction and Motivation ...... 116 5.2 Experimental Results ...... 119 5.2.1 Magnetic Susceptibility ...... 121 5.2.2 High frequency EPR study...... 122 5.3 Discussion and Analysis ...... 126 5.4 Conclusion ...... 136
6. SUMMARY...... 137 REFERENCES ...... 140 BIOGRAPHICAL SKETCH ...... 150
viii LIST OF TABLES
1.1 Properties of Tripositive Rare Earth Ions [4, 7] ...... 9
1.2 Tabulation of the multiplicative factors of Stevens operators for ground states of trivalent rare earth elements [8] ...... 12
2.1 Tabulation of signal amplitudes for high frequency probe...... 32
2.2 Properties of the magnets used in carrying out EPR experiments reported in this dissertation...... 34
3.1 Tabulation of LF parameters and Stevens constants for Ho3+ in HoPOM [8, 65] ...... 66
3.2 Tabulation of T2 values for HoPOM single crystals of various dilutions ...... 78
ix LIST OF FIGURES
1.1 The angular dependence of the 4f orbitals...... 7
1.2 Square of the radial wavefunctions for the 4f, 5s, 5p and 6s orbitals in Gd3+ ...... 10
2.1 Schematic representation of Zeeman Effect showing the splitting of degenerate energy levels in the presence of an external magnetic field...... 16
2.2 Photograph showing the parts of a rotating cylindrical cavity with the fundamental resonance mode at 50.4 GHz...... 24
2.3 Schematic diagram showing the directions of rotation about two orthogonal axes in a two- axis rotation experiment ...... 25
2.4 Photograph of the transmission probe showing all the parts assembled ...... 26
2.5 Schematic representation of the construction of transmission probe ...... 30
2.6 High frequency probe components...... 31
2.7 The fixed laboratory frame showing the applied field B0 along the z-axis and the microwave field B1 along the x-axis ...... 32
2.8 The microwave field B1 in the laboratory frame and the rotating frame...... 36
2.9 Pictorial representation of the response of the magnetization to a π/2 pulse ...... 37
2.10 The nutation of the magnetization M about the effective field, Beff ...... 38
2.11 Hahn echo sequence ...... 41
2.12 Depiction of Rabi oscillations showing variation of echo amplitude as a function of time..45
2.13 Schematic depiction of the image furnace for floating-zone method of crystal growth ...... 47
2.14 Images of the stages of single crystal growth by the floating-zone method ...... 48
2.15 X-ray diffraction pattern of SrHo2O4 ...... 50
2.16 Schematic representation of time of flight measurement using disc chopper spectrometer at NIST ...... 51
3.1 The energy barrier for a Mn12Ac molecule in zero-field...... 56
x 9- 3.2 Different views of [Ln(W5O18)2] cluster in HoPOM ...... 63
3.3 A pictorial representation of the crystal field scheme for HoPOM as reported in Ref. [65] by AlDamen et al through calculation of LF parameters...... 67
3.4 Temperature dependence of EPR spectra for HoPOM (x = 0.25) at 50.4 GHz ...... 68
3.5 Energy level diagram showing the splitting of the mJ ± 4 states in the presence of hyperfine coupling with 165Ho nucleus...... 69
3.6 Schematic representation of the planes of rotation for the two-axis rotation capability ...... 70
3.7 EPR spectra for HoPOM (x = 0.25) single crystal at 50.4 GHz at T = 2 K ...... 71
3.8 This figure shows the minimum peak positions, extracted from the angle-dependent data for each end-plate orientation, with the hyperfine interaction ignored...... 72
3.9 Comparison of simulated and experimental peak positions corresponding to angle dependent EPR spectra shown in Fig. 3.7...... 73
3.10 Angle dependence of CW EPR spectra for HoPOM (x = 0.25) single crystal at 9.7 GHz ...75
3.11 Optimization of parameters for spin echo experiment ...... 75
3.12 Measurement of T2 for HoPOM single crystal...... 77
3+ 3.13 Log-log plot showing the variation of T2 with Ho concentration ...... 79
3.14 Plot of ED spectra and the variation of T2 with magnetic field for HoPOM single crystal (x = 0.25) at a frequency of 9.7 GHz at T = 4.5 K...... 80
3+ 3.15 Variation of T2 as a function of temperature for HoPOM single crystal with 25% Ho concentration ...... 81
3.16 Rabi oscillations observed for HoPOM single crystal (x = 0.25) at T = 4.5 K ...... 82
3.17 Energy level diagram showing the mJ ± 4 states in the presence of hyperfine coupling with 165Ho nucleus...... 83
4 3.18 Simulated energy level diagram with the inclusion of the transverse (B4 ) term in the LF Hamiltonian ...... 84
3.19 Comparison of experimental and simulated spectra for HoPOM at 9.7 GHz ...... 86
xi 3.20 Comparison of experimental and simulated angle dependent CW spectra for HoPOM at X- band ...... 87
3.21 Simulated energy level scheme for HoPOM with the red lines showing the expected field positions of the ΔmJ = ±1 transitions at f ~ 350 GHz, which should be the most intense transitions as per EPR selection rules...... 88
3.22 Simulated energy level scheme for HoPOM with the red lines showing the expected field positions of the ΔmJ = ±1 transitions at f ~ 350 GHz, for a crystal aligned with its quantization axis 10 degrees off from the direction of the magnetic field...... 89
3.23 Angle dependence of CW spectra of HoPOM at X-band in a dual-mode resonator (T~ 4.5 K)...... 91
3.24 Schematic representation of the relative orientations of the microwave field B1 with respect to the applied static field, B0 in parallel and perpendicular mode...... 91
3.25 Comparison of parallel-mode experimental and simulated CW spectra for HoPOM at X- band ...... 93
3.26 Comparison of CW and ED spectra for HoPOM ...... 93
3.27 Simulated frequency versus field dependence extracted from the energy level diagram shown in Fig. 3.17 ...... 94
3.28 Comparison of CW EPR spectra of protonated and deuterated HoPOM single crystals with 10% concentration of Ho3+...... 95
3.29 The ED spectrum for deuterated HoPOM single crystal with 10% concentration of Ho3+ at 9.7 GHz at T ~ 5K ...... 96
4.1 Geometries that exhibit spin-frustration...... 101
4.2 Common frustrated lattices...... 102
4.3 Characteristic behavior of inverse susceptibility versus temperature for paramagnets, ferromagnets and anti-ferromagnets...... 103
4.4 Crystal structure of SrL2O4 (Grey: Lanthanide; Red: Oxygen; Blue: Strontium)...... 105
4.5 Variation of inverse magnetic susceptibility with temperature for SrHo2O4 ...... 107
4.6 Variation of the specific heat capacity of SrHo2O4 and SrY2O4 with temperature ...... 107
xii 4.7 Inelastic neutron scattering data for SrHo2O4 single crystals at T = 1.5 K showing band structures corresponding to crystal field levels ...... 109
4.8 Intensity profile for inelastic neutron scattering for SrHo2O4 single crystal at T = 1.5 K ..111
4.9 Comparison of elastic neutron scattering data and simulation for SrHo2O4 single crystal at T = 1.5 K...... 112
4.10 Diffuse scattering fits for SrHo2O4 single crystal...... 113
5.1 Kagome lattice...... 117
5.2 Magnetic susceptibility data for NGS and PGS ...... 120
5.3 EPR spectra for PGS at a frequency of 127 GHz...... 123
5.4 EPR spectra for PGS at a frequency of 116 GHz...... 123
5.5 EPR spectra for NGS at a frequency of 116 GHz ...... 124
5.6 Frequency dependent EPR spectra for PGS at T ~ 3K...... 125
5.7 Plot of peak positions of EPR spectra for PGS at T ~ 3 K over a range of frequencies between 100 GHz – 250 GHz...... 127
5.8 Mechanism for the formation of induced magnetic moment through an admixture of ground state and excited state separated by a gap of Δ in the presence of sufficient exchange interactions between neighboring ions ...... 127
5.9 Energy level scheme presented by Lumanta et al [103] which reports the gap between the lowest two energy levels to be ~ 20 K for PGS...... 128
5.10 Spin-wave picture for a ferromagnetic system...... 130
5.11 Spin waves viewed in perspective (top) and viewed from above (bottom) showing one wavelength...... 130
5.12 Schematic representation of AFM spin waves...... 132
5.13 Plot depicting the resonance fields for the different modes shown in Fig. 5.4 for PGS at 116 GHz...... 135
xiii ABSTRACT
This dissertation presents a study of rare-earth magnetism using spectroscopic techniques. The features of strong spin-orbit coupling and the presence of hyperfine coupling which lead to coupled electro-nuclear crystal field states are typical of lanthanides, and contribute to distinct magnetic properties. In the work reported here electron paramagnetic resonance (EPR) and neutron scattering techniques have been employed for investigating the ground state magnetic properties of a few lanthanide-based single crystal samples. The two kinds of samples investigated in the course of this work include single molecule magnets (SMMs) and geometrically frustrated systems.
EPR studies on mononuclear lanthanide-based SMMs, which have attracted considerable attention due to their potential application in spintronic devices, form a major portion of the research reported in this dissertation. In these systems, the magnetization is associated with a single rare-earth ion (holmium) which facilitates mitigation of spin decoherence due to nuclear hyperfine and electron dipolar interactions by dilution and isotope purification. High frequency EPR studies on HoPOM indicates transverse spin orbit anisotropy, which considerably affects the magnetization relaxation properties by giving rise to a tunneling gap between excited electro- nuclear spin states. Electron spin echo measurements at 9 GHz demonstrate long relaxation times (~ 100 ns) for concentrated samples, with much longer values for diluted samples containing deuterated solvent. Besides a detailed study involving the evaluation of the spin Hamiltonian parameters and measurement of transverse relaxation times for HoPOM samples of various concentrations, we also attempt to investigate the mechanism leading to the observed long coherence times. We propose that there is mitigation of decoherence in this system due to the nature of the tunneling gap, which leads to an insensitivity of the spin dynamics to field fluctuations.
The other area of focus of the work presented in this dissertation involves studying spin frustrated lattices which give rise to novel ground state properties. The low temperature behavior of the modified honeycomb lattice compound SrHo2O4 has been characterized by dc magnetic
xiv susceptibility, heat capacity and neutron scattering experiments indicating lack of long range ordering down to 1.8 K. Elastic neutron scattering measurements show diffuse scattering indicative of short range ordering between nearest neighbor Ho3+ spins. Inelastic neutron scattering experiments carried out at multiple temperatures show the presence of five crystal field levels up to 80 K in energy, which is in agreement with the specific heat measurement on the system. The distorted kagomé lattice compounds, Pr3Ga5SiO14 and Nd3Ga5SiO14, were also studied as examples of spin frustrated systems. Magnetic resonance experiments on single crystals of these isostructural samples show complex multi-peak spectra with strong systematic temperature dependence. The nature of the observed excitations in high frequency EPR measurements indicate that they correspond to collective excitations akin to spin-wave resonances, caused by the formation of spin clusters whose correlation length depends on field and frequency. This study potentially provides an experimental basis for the investigation of antiferromagnetic spin wave – like resonances in the kagomé lattice.
The two kinds of samples discussed demonstrate interesting ground state properties, and have been probed using spectroscopic techniques for an insight into rare earth magnetism.
xv CHAPTER ONE
INTRODUCTION TO RARE-EARTH MAGNETISM
The study of magnetism and magnetic properties of materials has always been of great importance and interest in the field of condensed matter physics. Since the discovery of
“loadstone” (Fe3O4) in the ancient ages, our understanding of magnetism has evolved from long range interaction of ferromagnetic bodies into a study of magnetic ordering in the quantum mechanical regime. This dissertation is a study of rare-earth magnetism using spectroscopic techniques, and in this introductory chapter we shall discuss the underlying physics of magnetism in the context of the rare earth elements. In the subsequent chapters, we shall deal with compounds that have different rare earth ions as their magnetic cores, and this chapter aims to lay the groundwork for our understanding of the “spin-physics” of the systems we have studied using experimental techniques like electron paramagnetic resonance spectroscopy and neutron diffraction.
1.1 Magnetism
The origin of magnetism is the magnetic moment of an electron, and it is a combination of the orbital angular momentum and the intrinsic angular momentum [1-3]. The orbital motion of an electron about the nucleus gives rise to the orbital magnetic moment µorbital (or μL). In the simplest classical picture, we can describe this orbital motion as the revolution of a particle of mass m and charge e with a certain frequency in an orbit. As described by the laws of classical electrodynamics, the motion of a charged particle gives rise to the orbital magnetic moment which can be expressed as Eq. (1.1), where L denotes the orbital angular momentum. e L … (1.1) L 2m
1 Besides the orbital contribution, the magnetic moment of an electron also has a contribution from its intrinsic magnetic moment, µspin (or μS), which arises as a result of its intrinsic spin. Spin is a fundamental property of elementary particles, and is a quantum mechanical concept which is denoted by the spin quantum number S. Although the concept of electron spin does not exist in the classical regime, historically this intrinsic angular momentum was referred to as the ‘spin angular momentum’ by considering a classical analogy where an electron was treated like a sphere rotating about its own axis. The comparison of the motion of an electron with that of a spinning top helps us form an intuitive picture of the resulting angular momentum associated with its motion. However, this classical model fails to provide an accurate description of the essentially quantum mechanical degree of freedom which is referred to as the spin. The Stern - Gerlach experiment in 1922 provided experimental evidence of the quantization of spin, and acted as a turning point in our understanding of magnetism. In this experiment, a beam of electrically neutral silver atoms were passed through a non-homogeneous magnetic field, and instead of the classically expected continuous density distribution, a discrete spectrum was observed on the detector screen. This demonstrated that the projection of the spin magnetic moment in the presence of a magnetic field can have only discrete orientations in space. Using the classical analogy, the spin magnetic moment can be expressed as Eq. (1.2), and the total magnetic moment associated with an electron is the vector sum of the orbital and spin components, as shown in Eq. (1.3). The proportionality factor between μS and S is twice as large as that for μL and L. e S … (1.2) S m L S … (1.3) The classical analogy was introduced for an introductory intuitive idea, and hereafter in this chapter we follow the quantum mechanical formalism for our discussion. Since electrons possess magnetic moments, there is interaction with their local environment in an atom or a molecule. In case of certain materials, the electronic configuration is such that it causes the magnetic moments to line up in a certain preferred direction, giving rise to a net magnetization in the system. This process can occur either spontaneously or in the presence of an externally applied magnetic field. The response of the net magnetization (M) of a system in an applied magnetic field can be studied experimentally. Therefore, although we are dealing with an essentially quantum
2 mechanical system, we can apply the classical Bloch equations to study the evolution of the bulk magnetization of a system under the experimental conditions [1-3]. While the semi-classical approach helps us form an intuitive idea of the bulk properties, an understanding of magnetism requires a quantum-mechanical description of the electronic states. For a single electron in the presence of a central potential V, the time independent Schrödinger equation can be expressed as Eq. (1.4). 2 ˆ 2 H V E … (1.4) 2m In the above equation, the electronic wavefunction Ψ for a single electron can be expressed as a function of the quantum numbers n, l, ml and ms. n is the principal quantum number, and l is the orbital quantum number which characterizes the electronic subshell, with specific values for s-, p-, d- and f- electrons. The quantum numbers ml and ms represent the projections of l and s respectively. The quantum numbers can have discrete allowed values with n = 1, 2, 3, …, l = 0,
1, 2, …, n-1, ml = -l,-l+1, …, l and mS = ± ½.
ln ,, ml ,ms … (1.5) The single electron problem in the presence of a central potential due to the nucleus can be solved exactly using the Schrödinger equation. However, in order to describe the magnetic properties of an atom or an ion, we are faced with a complicated many-particle system consisting of the nucleus and the electrons. Since an exact solution is not possible, the Hartree approximation is the method chosen for solving the problem [4, 5]. The complete mathematical formalism of the solution is beyond the scope of this chapter, so we will outline the important steps in developing an understanding of the quantum mechanical description of electronic wave- functions. According to the Hartree approximation, an electron experiences a potential resulting from the nucleus and from the averaged charge density due to the other electrons. We start with the Schrödinger equation (Eq. 1.4), where the wavefunction is a function of the space (r) and spin (σ) co-ordinates, and can be expressed as Ψ(r1σ1, r2σ2, …, rZσZ) for a system with atomic number Z. The non-relativistic Hamiltonian can be expressed as Eq. (1.6). 2 Z Z 2 Z 2 ˆ 2 1 e Ze H i … (1.6) 2m i 2 ji ri rj i ri
In the above equation, the first term represents the kinetic energy, the second term accounts for the Coulomb repulsion between the electrons and the third term is the Coulomb potential due to 3 the nucleus. In order to solve for the wavefunction with this Hamiltonian, the problem is first treated as a system where each electron moves independently in the field of the nucleus and that of the averaged potential field caused by the charge density of the other electrons. The difference between this effective Hamiltonian and the actual Hamiltonian is then treated as the perturbation potential. Solving for the Schrödinger equation using the method of separation of variables, the wavefunction can be expressed as follows [4] : Z 1 r ,| where r R Yr , m , … (1.7) nlm msl i nlm msl nl lml S i r Due to the spherical symmetry, the eigenfunction can be written as a product of the radial part and the angular part as shown in Eq. (1.7), and is a function of the quantum numbers n, l, ml and ms. The spin co-ordinate, σ can have two values ± ½. Thus, there is a 2(2l+1) - fold degeneracy in the energy eigenvalues. Following the Pauli exclusion principle, linear combinations of the above solutions that are anti-symmetric with respect to the spatial and spin co-ordinates are allowed. The total wavefunction for the sub-shell, Ψ(LSMLMS) can be written in terms of the one-electron wavefunctions, ψ(mlms), and are conveniently expressed using the state vector notation shown in Eq. (1.8) [4, 5].
LSM L M S m msl LSM L M S m msl … (1.8) mm sl The coupling between the spin and the orbital angular momenta significantly affects the magnetic properties of rare-earth elements [4, 6, 7]. The Hamiltonian representing the spin-orbit interaction is expressed as Eq. (1.9), where the + and – signs in the term correspond to electronic configurations having lesser or greater than a half-filled subshell. The coefficient λ is the spin- orbit coupling constant and is a function of the 4f radial wavefunction for lanthanides. The difference in the strength of spin-orbit coupling in the case of f-electrons relative to the case of 3d electrons lead to significant differences between lanthanides and transition metals. Thus, while we have spin-orbit coupled states in the case of rare-earths, the 3d transition metal elements possess pure spin states with S as the good quantum number. It is important to note here that it is the interplay between the relative strengths of the spin-orbit coupling and the crystal field effect that is significant, and this will be discussed later in the chapter. ˆ H LS LS L S … (1.9)
4 Due to the strong spin-orbit coupling in rare-earths, the eigenstates can be represented in terms of the total angular momentum, J as shown in the state vector notation in Eq. (1.10) [4].
JM J LS LSM L M S JM J LS LSM L M S … (1.10) MM SL The ground state quantum numbers that represent an atom can be evaluated using a set of three rules known as Hund’s rules, which can be stated in the following order [1-4, 7] : (i) For a given electronic configuration, the term with the maximum spin multiplicity has the lowest energy. As a consequence of this rule, the total spin angular momentum of an atom, S, is maximized. The multiplicity for the state is (2S+1). (ii) For a given spin multiplicity, the term with the largest value of total orbital angular momentum, L lies lowest in energy. As a consequence of this rule, after the maximization of S, L is maximized. (iii) For an atom having a half-filled or less than half-filled outermost subshell, the level with the lowest value of total angular momentum, J lies lowest in energy. In the case of an outermost shell that is more than half-filled, the level with the highest value of J lies lowest in energy. Thus, the third rule considers the lifting of degeneracy due to spin-orbit coupling. As a result of this rule, J is |L – S| and |L + S| for atoms with less than half-filled and more than half-filled shells respectively. This rule is significant for heavy rare-earth elements with strong spin-obit coupling. However, for systems with weak L-S coupling, this rule is not valid, and the crystal- field effect has a dominant contribution.
The Zeeman Effect describes the interaction of the spin with an external magnetic field, B, and the corresponding Hamiltonian is as follows: ˆ H Zeeman B L 2S B … (1.11) e In the above equation, μB is called the Bohr magneton ( ). We can relate this to the B 2m classical expressions of orbital and spin angular momenta shown in Eq. 1.1 and 1.2 with the angular momenta expressed in units of . Using perturbation theory to evaluate the magnetic contribution to the energy eigenvalue, E1, we can arrive at Eq. (1.12), where |1> and |2> correspond to energy eigenstates with eigenvalues E1 and E2 [4, 5].
5 2 1 B B L 2S 2 E B B 1 L 2S 1 … (1.12) 12 E1 E2
Applying the Wigner-Eckert theorem in the |J MJ L S> basis, it follows that the matrix elements of (L + 2S) are proportional to those of J, and the proportionality constant is the Landé g factor, gJ. Mathematically, this can be expressed as shown in Eq. (1.13) [4]. ' ' JLSM J L 2S JLSM J g J JLS JLSM J J JLSM J … (1.13)
The Landé g factor is, JJ 1 L L 1 SS 1 g 1 … (1.14) J 2 JJ 1 Using the above definition of the g-factor, the effective magnetic moment of an atom can be expressed as Eq. (1.15). g J B J … (1.15) The magnetization of N atoms can be expressed as Eq. (1.16), where F is the free energy defined by Eq. (1.17) [1-4]. 1 M F … (1.16) V B E n F Nk BT ln exp … (1.17) n k BT The magnetization of a sample is defined as the magnetic moment per unit volume, and this is the experimental observable that can be measured with the application of a magnetic field. We discuss magnetization measurements on spin frustrated lanthanide-based crystals in Chapters 4 and 5. In the discussion so far, we have introduced the basic concept of magnetism, and have presented the formalism of the electronic wavefunctions in terms of the orbital and spin quantum numbers. In the next section, we will focus on magnetism as applicable to rare-earths.
1.2 Rare Earth Magnetism
The “rare earths” are a set of elements in the periodic table that are characterized by the progressive filling of the 4f- electronic shell. Although they are relatively plentiful in the Earth’s
6 Fig. 1.1 The angular dependence of the 4f orbitals. The shapes of the orbitals for ml = -3, -2, …, 2, 3 have been shown. The red and green colors represent the lobes with positive and negative amplitudes of the wavefunction. crust, these elements are typically not found in concentrated, easily extractable forms due to their geochemical properties. Hence they came to be known as the “rare earths”. The f-elements can be divided into two groups, the lanthanides and the actinides, each consisting of fourteen elements. The lanthanides are associated with the filling of the 4f- shell (Z = 58 to Z = 71) while the actinides are associated with the filling of the 5f- shell (Z = 90 to Z = 103) [7]. In this dissertation, we report studies on the spectroscopic characterization of the magnetic properties of a few lanthanide-based single crystals. As we have seen previously in this chapter, the magnetic properties of an element are governed by its electronic configuration. Hence, we need to take into consideration the features of the 4f- eigenfunction in order to account for the properties unique to these elements. The electronic configuration of the neutral lanthanides may be expressed as the closed shell Xenon configuration followed by the 4f-electrons and two or three outer electrons (6s2 or 5d16s2) [7]. Lanthanides usually exist in the +3 oxidation state, and the properties of the tripositive lanthanide ions have been tabulated in Table 1.1 for reference, since some of these values (shown in blue) have been referred to in subsequent chapters. It has been experimentally observed that the binding energy and the spatial extension of the wavefunction drop suddenly at the commencement of the lanthanide series. In case of the lanthanides, a deep potential well develops near the nucleus and the 4f electrons are drawn towards the interior [4, 6, 7]. This effect is called the “lanthanide contraction” and arises as a result of the imperfect shielding of one 4f electron by another [7]. A rapid contraction of the 4f eigenfunction is observed with an increase in the atomic number of the element, as one proceeds through the lanthanide series. This is reflected in the ionic and the atomic radii of the elements in the solid 7 state. No such effect occurs in the case of s, p or d shells, thereby leading to distinct properties observed in the case of the rare-earth elements. The angular dependences of the 4f-wavefunctions [4] are shown in Fig. 1.1. The f- electrons correspond to l = 3, and the figure shows the shapes of the orbitals for the different ml values = -3, -2, -1, 0, 1, 2 and 3. It can be seen from the figure that the charge clouds are highly anisotropic, and the anisotropy of the spin-orbit coupled states is clearly manifested in the magnetic properties exhibited by the systems.
In our discussion so far, we have considered the magnetic properties of free atoms or ions. However, when we study magnetic ions in crystal lattices, the crystal breaks down the isotropy of space and the effects of the crystal field have to be taken into consideration. Since transition metal elements (characterized by d-electrons) and rare earths (characterized by f-electrons) broadly form the two classes of materials usually investigated for studying magnetic properties, we show a comparison of how the crystal field typically affects the core ions in the two cases. The d and f ions behave differently in crystals, because the partially filled d-shell is on the outside, while the partially filled f-shell is buried inside the valence shell (lanthanide contraction). Fig. 1.2 shows the radial wavefunctions for the 4f, 5s, 5p and 6s orbitals in Gd3+, which demonstrates the effect of lanthanide contraction whereby the f orbitals lie well inside the ion. The d-shell has a strong overlap with the surrounding ions leading to strong crystal field effects. In this case, the crystal field effect is considered first, and the spin-orbit coupling is treated as a perturbation. In the case of lanthanides however, the spin-orbit coupling is dominant, and the crystal field effects, which are relatively weaker, are treated as a perturbation. The spin orbit coupling, λ in the case of lanthanides is typically about 20-100 times stronger than the crystal field as a result of the lanthanide contraction. In the case of transition metals, the situation is the other way around, and it is the relative strengths of both these effects that affect the anisotropy of the system. Due to the spin-orbit coupling, the eigenstates of rare earth elements are expressed in the |J MJ> basis, where J = L+S.
The crystal field effect accounts for the electric field due to the charge distribution around an ion, and acts on the 4f electrons to gives rise to the magnetic anisotropies which are characteristic of the lanthanides. The potential due to the crystal field may be expressed as follows [4] :
8 Table 1.1 Properties of tripositive rare earth ions [4, 7]. It tabulates the L, S and J quantum numbers and the calculated g-values for tripositive lanthanide ions. The ones that have been highlighted in blue are the ones we will refer to in the following chapters.
Atomic Ionic radii Lanthanide Ion+++ 4fn L S J g No. (pm) 57 Lanthanum La 103 0 0 0 0 --
58 Cerium Ce 102 1 3 1/2 5/2 6/7
59 Praseodymium Pr 99 2 5 1 4 4/5
60 Neodymium Nd 98.3 3 6 3/2 9/2 8/11
61 Promethium Pm 97 4 6 2 4 3/5
62 Samarium Sm 95.8 5 5 5/2 5/2 2/7
63 Europium Eu 94.7 6 3 3 0 --
64 Gadolinium Gd 93.8 7 0 7/2 7/2 2
65 Terbium Tb 92.3 8 3 3 6 3/2
66 Dysprosium Dy 91.2 9 5 5/2 15/2 4/3
67 Holmium Ho 90.1 10 6 2 8 5/4
68 Erbium Er 89 11 6 3/2 15/2 6/5
69 Thulium Tm 88 12 5 1 6 7/6
70 Ytterbium Yb 86.8 13 3 1/2 7/2 8/7
71 Lutecium Lu 86.1 14 0 0 0 --
9 Figure 1.2 Square of the radial wavefunctions for the 4f, 5s, 5p and 6s orbitals in Gd3+. This figure demonstrates the effect of lanthanide contraction which causes the f shell to be buried on the inside. This figure has been taken from Ref. [9].
e R vcf r Rd … (1.18) r R where ρ(R) is the charge density of the surrounding electrons and nuclei. The solution for the crystal field potential, vcf (r) can be expressed in terms of the spherical harmonics as a multipole expansion along with appropriate symmetry considerations (Eq. 1.19). It should be noted here that the standard notation for spherical harmonics are henceforth employed and therefore the expansion variables l and m used in the context spherical harmonics are different than the ones used earlier in the Hamiltonian formulation.
m l vcf r Al Yr lm rˆ … (1.19) lm
10 As we have mentioned earlier, the crystal field effect can be treated as a first order perturbation in the case of lanthanides, since the crystal field energy in these systems is usually small as compared to the spin-orbit splitting. It is known that f-electrons cannot have multipole distributions with l > 6; hence only matrix elements corresponding to l ≤ 6 need to be considered. The crystal field Hamiltonian was first formulated by Stevens in 1952 [8] and the Hamiltonian can be written in terms of the Steven’s operators as shown in Eq. (1.20) [4, 9]. ˆ m l ˆ m H cf Al l r Ol J i … (1.20) i lm ˆ m ˆ m H Bl Ol J i … (1.21) i lm m l In the above equations, Ôl is the Stevens operator, r is the radial factor corresponding to the 4f states and αl represents the Stevens factors. αl is usually written as α, β and γ for l = 2, 4 and 6 m respectively. The subscript l in the operator Ol denotes the order of the term, while the superscript m represents m-fold symmetry. The numerical values for the Stevens factors for rare- earths can be found in Ref. [8] and have been tabulated in Table 1.2. Conventionally, the crystal m l field Hamiltonian is expressed as Eq. (1.21) where Al , αl and r are combined together and m denoted by a single crystal field parameter Bl . The operator equivalent method significantly simplifies the representation of the crystal field potential, which would otherwise have involved the direct calculation of the charge distribution. The beauty of the Stevens operator formalism is that it is consistent with the symmetry of the molecule, and has the same transformation properties under rotation as the potential. Given the crystal field potential in Cartesian co- ordinates, the operator equivalent method can be applied to replace (x, y, z) by (Jx, Jy, Jz) while m taking into account the non-commutation of Jx, Jy and Jz. The Bl terms required to represent the crystal field Hamiltonian for a specific system is guided by symmetry [4, 8, 10]. The symmetry elements consistent with the co-ordination geometry of the magnetic core thus play a very significant role in determining the magnetic properties of a system. We shall discuss the relevance and importance of symmetry considerations in Chapter 3 where we show how the crystal field Hamiltonian strongly affects the magnetization relaxation properties. The crystal field Hamiltonian lifts the degeneracy of the pure ionic |JMJ> states, and can be diagonalized to yield the crystal field energies and wave-functions.
11 Table 1.2: Tabulation of the multiplicative factors of Stevens operators for ground states of trivalent rare earth elements [8]
Ion α β γ Ce3+ -2/35 2/7*45 0 Pr3+ -52/11*152 -4/55*33*3 17*16/7*112*13*5*34 Nd3+ -7/3333 -8*17/11*1113*297 -1719*5/132*113*33*7 Pm3+ 14/11*11*15 952/13*33*113*5 2584/112*132*3*63 Sm3+ 13/7*45 26/33*745 0 Eu3+ 0 0 0 Gd3+ 0 0 0 Tb3+ -1/99 2/11*1485 -1/13*33*2079 Dy3+ -2/9*35 -8/1145273 4/112*132*33*7 Ho3+ -1/30*15 -1/11*2730 -5/13*33*9009 Er3+ 4/45*35 2/11*15*273 8/132*112*33*7 Tu3+ 1/99 8/311*1485 -5/13*33*2079 Yb3+ 2/63 -2/77*15 4/13*33*63
Besides the effect of the crystal field, which acts independently at the individual ionic sites, the two-ion interactions also play a significant role in magnetic ordering. The most dominant form of this type of interaction in case of rare-earths is the indirect exchange coupling [4, 9] and it refers to the interaction between the magnetic moments of pairs of ions. The indirect exchange coupling can be expressed in the form of the Heisenberg interaction as shown in Eq. (1.22). ˆ 1 H ex J ij J i J j … (1.22) 2 ij The electronic magnetism in rare-earths is also commonly affected by the hyperfine interaction between electron and nuclear spins. ˆ H hf I i A J i … (1.23) i Eq. (1.23) represents the general form of the Hamiltonian representing the hyperfine interaction. The coupling between the electron and nuclear magnetic moments has negligible effect if the strength of the coupling constant, A is significantly lower than crystal field effects and exchange
12 interactions. In the case of rare-earths however, the lanthanide contraction often leads to strong hyperfine coupling, which is characteristic of these elements. It can be seen in our EPR studies on a holmium-based system reported in Chapter 3 that the degeneracy of the electronic energy levels is lifted via hyperfine interaction to give rise to electro-nuclear crystal field states. This feature is characteristic of a number of rare-earths, and elements with strong hyperfine coupling and large nuclear spins have gained attention as being potentially interesting systems for studying rare-earth qubits. We will return to this discussion in Chapter 3.
1.3 Organization of the Thesis
In this chapter, we have presented a theoretical discussion of rare-earth magnetism in the light of the properties of rare-earth elements in crystal lattices. In Chapter 2, we will discuss the experimental techniques used in our study of various lanthanide-based single crystals, and provide a description of the instrumentation employed for the experiments. Sample synthesis by the floating-zone method of crystal growth, sample characterization techniques like x-ray diffraction and magnetic susceptibility measurements, EPR spectroscopy and neutron scattering will be discussed. Since EPR spectroscopy forms a bulk of the studies carried out, emphasis will be given on the description of the technique of continuous wave and pulsed EPR measurements. Chapter 3 deals with a study of coherent spin manipulation in a holmium-polyoxometalate 9- system ([HoxY1-x(W5O18)2] ) using EPR spectroscopy. The effects of crystal field anisotropy and the correlation of the molecular symmetry with magnetization relaxation properties of this sample will be discussed through a comparison of experimental results with simulations. An account of the measurements of the relaxation times for samples of various dilutions of this system will also be presented in order to investigate the spin relaxation mechanisms and the factors contributing to mitigation of decoherence in this system. In Chapter 4, we discuss our studies on a spin-frustrated honeycomb lattice compound (SrHo2O4) using neutron scattering techniques to model the short-ranged magnetic ordering observed in the system. In Chapter 5, we will continue our discussion on frustrated magnetism, and present our EPR and magnetization studies on two distorted kagomé lattice compounds, Pr3Ga5SiO14 and Nd3Ga5SiO14. A qualitative discussion will be presented, based on our results from spectroscopic studies, which indicate the
13 presence of collective excitations akin to spin-wave resonances arising as a result of short-range ordered clusters. Chapter 6 will conclude this thesis with a synopsis and brief discussion of the studies carried out on various lanthanide-based single crystalline samples using spectroscopic techniques.
14 CHAPTER TWO
EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION
This chapter contains an outline of the various experimental techniques and a description of the instrumentation used in the course of this dissertation work, which involved studying the magnetic properties of lanthanide-based single crystals. Electron paramagnetic resonance (EPR) spectroscopy is the primary technique that was used to probe and study several different kinds of lanthanide-based materials that exhibit interesting behavioral patterns such as those of a single- molecule magnet, and novel ground state properties arising as a result of geometric spin frustration. In this chapter, we shall focus primarily on discussing the theory and technique of EPR spectroscopy and also provide a description of the instrumentation that was used in conducting the experiments. Thereafter, we shall move on to the other experimental techniques that were employed in sample synthesis and characterization, and for studies involving neutron scattering measurements.
2.1 Electron Paramagnetic Resonance (EPR) Spectroscopy
The term spectroscopy refers to the study of the discrete and the contiguous levels of a system, and is an extremely powerful technique that has evolved into various specialized branches. EPR is a specific branch of spectroscopy where microwave radiation is used in order to probe the magnetic energy levels of a system with unpaired electrons, and serves as a useful means for determining the structure and dynamics of paramagnetic species. It deals with a study of the interaction of the electron magnetic moment with its environment, and with an applied magnetic field. As we have discussed in Chapter 1, the magnetic moment of an electron arises as a result of its orbital motion and its intrinsic spin. This gives rise to an orbital and a spin angular momentum. In the presence of an external magnetic field, B0, the magnetic moment of an electron aligns itself either parallel or anti-parallel to the direction of the field, each orientation
15 Fig. 2.1 Schematic representation of Zeeman Effect showing the splitting of degenerate energy levels in the presence of an external magnetic field. corresponding to a specific energy. This effect of the lifting of the degeneracy of energy levels in the presence of a static magnetic field is known as the Zeeman Effect [11-14], and can be depicted by Fig. 2.1. The technique of EPR involves a study of the electronic transitions of unpaired electrons from a lower energy level (often the ground state) to a higher level. The process can be described through the following equation:
E E2 E1 h … (2.1) where, E1 and E2 are the two energy levels, E is the amount of energy required to cause the transition, h is Planck’s constant and ν denotes the microwave frequency. The statistical population distribution of electron spins between the two energy levels is given by the Maxwell-
Boltzmann distribution in which the ratio of the population of the upper state (N2) to that of the lower state (N1) can be expressed as Eq.2.2 where k is Boltzmann’s constant and T is the absolute temperature. N E 2 exp … (2.2) N1 kT In the simplest case, we can consider a two-level system for spin S = ½, where the levels are depicted by the projection of the spin quantum number, mS = ± ½, and an EPR transition can occur between these two energy levels. Calculation of the matrix elements corresponding to
16 possible transitions between different states gives rise to the formalism of selection rules so that transitions are allowed between spin levels when mS = ± 1. For higher spin systems, there are
(2S+1) electronic spin states that are characterized by the quantum number mS. In some cases, along with the electron spin, we also need to consider the interaction between the electron and the nucleus which leads to further lifting of degeneracy of the electro-nuclear energy levels giving rise to hyperfine structure. For a system with nuclear spin I, each of the Zeeman levels have a multiplicity of (2I + 1) and the allowed EPR transitions are usually governed by the selection rule mI = 0.
The behavior of a spin system can be described using the spin Hamiltonian which was first formulated by Abragam and Price in 1951, and is probably the most important mathematical tool in the description of EPR. In the simplest cases, the spin Hamiltonian can be expressed as a sum of terms representing the Zeeman interaction, the hyperfine interaction and the zero-field splitting term, as shown in Eq.2.3 ˆ ˆ ˆ ˆ ˆ ˆ H B .. SgB S D.. S .. SAI ... (2.3) In the above equation, Ŝ is the electronic spin operator, g is the Landé g-tensor, D is the zero- field splitting tensor, Î is the nuclear spin operator and A is the hyperfine coupling parameter. The first term represents the Zeeman interaction in the presence of an external field, where the Landé g-tensor can be expressed as Eq.2.4. For a free electron, the g-factor is isotropic and has a value of ~ 2. For an arbitrary orientation, the symmetrical g-factor tensor is represented as shown in the first part of Eq.2.4. However, it can be diagonalized to give it a simpler form when expressed in its quantization axis co-ordinates as shown in the second part of Eq.2.4.
g xx g xy g xz g x' 0 0 g g yx g yy g yz where gij g ji ; g 0 g y' 0 ... (2.4) g zx g zy g zz 0 0 g z' The angular dependence of the g-tensor is referred to as the g-anisotropy and this can be determined through EPR experiments. For certain symmetries (for example, a cubic symmetry), the g-tensor is isotropic, with gx = gy = gz. In the case of systems with axial symmetry (for example, trigonal or tetragonal crystal field symmetries), gz = g|| and gx = gy = g┴.
17 The second term in the spin Hamiltonian (Eq.2.3), which is the zero-field splitting term, is of great significance in EPR, and takes into account the sources of anisotropy that lift the degeneracy of a state in the absence of an external magnetic field. Spin-orbit coupling, which arises as a result of the interaction between the orbital angular momentum and the spin angular momentum of an electron, usually has a dominant contribution to this zero-field splitting term in the case of lanthanides. The zero-field splitting tensor also accounts for the field gradient arising as a result of the neighboring atoms in a crystal, and breaks the isotropy of space. This term can be represented as a sum of the axial and transverse components which can be evaluated based on the co-ordination symmetry of the crystal field. This will be discussed in detail in Chapter 3, where we show the contribution of this term in an EPR experiment and also demonstrate how the symmetry of the system affects the formulation of this part of the Hamiltonian. Considering a 2 simple case with axial symmetry, this zero field splitting term can be expressed as D[Sz – 1/3 S(S+1)], where the quantization axis has been assumed to be along z. In additional to this axial term, there is often a transverse zero-field splitting term in cases of lower symmetry systems. 2 2 The transverse operator is expressed as: E(Sx –Sy ). D and E are known as the second order axial and rhombic zero-field splitting parameters respectively. Here we have considered only terms up to the second order, and neglected higher order terms that may exist depending on the symmetry and the spin state of the system.
The third term in the spin Hamiltonian represents the hyperfine coupling between electron and nuclear spins. As with the g-tensor, the hyperfine coupling tensor A can also be expressed in its diagonal basis. Depending on the symmetry of the system, it can either be isotropic or anisotropic. Strong hyperfine couplings are commonly observed in the case of rare earth systems as a result of the lanthanide contraction effect discussed in Chapter 1. In our study on holmium polyoxometalates discussed in Chapter 3, hyperfine interactions were observed in the EPR spectra as a result of coupling between the electron spin and the Ho3+ nuclear spin.
The energy eigenvalues for a system can be evaluated by using the spin Hamiltonian described above in the Schrödinger equation. With the spin Hamiltonian operating on the eigenstates
|S, ms>, the (2S+1) dimension eigenmatrix is obtained which can be diagonalized to yield the eigenvalues. As we have already seen in Chapter 1, rare-earth elements exhibit strong spin-orbit
18 coupling. In such cases, the wavefunction is expressed in the |J, mJ> basis, where J is the total angular moment and mJ is the appropriate quantum number instead of mS. In our work involving lanthanides, it is the |J, mJ> basis that is relevant. The exact formalism of the spin Hamiltonian depends greatly on the system being studied, and only a simple case involving the terms that are most commonly relevant in an EPR experiment have been introduced above.
EPR spectroscopy offers several advantages allowing high spectral resolution and sensitivity. It enables a precise determination of the anisotropy of a sample which may arise as a result of various factors including spin dipolar interactions, exchange interaction, spin-orbit coupling and hyperfine coupling. Although we mention spin dipolar interactions and exchange interactions here for the sake of completeness, we have not dealt with the formalism of these terms in this chapter, since they are not relevant to the Spin Hamiltonian we will refer to in the subsequent chapters. The experimental technique of EPR allows the identification of the processes contributing to the magnetic properties of the sample being studied. It also enables a precise evaluation of the various parameters of the spin Hamiltonian, thereby providing an insight into the mechanism of spin interactions. The ability to precisely quantify the strength of the interactions discussed above is of great significance, because this also allows the potential for better control in synthesizing systems with desired properties. Therefore, this technique has found widespread applications in a number of fields.
2.1.1 Instrumentation for High Frequency EPR Experiments
The instrumentation discussed here allows high frequency (50 – 900 GHz) EPR measurements on single crystals in the 7 T, 15 T, 31 T and 35 T magnets at the NHMFL, used for conducting most of the experimental results reported in this thesis. The same experimental set-up may be used in other magnet systems too, but this involves adjusting the dimensions of the components of the probe to fit into respective magnet bores. It is significant to note that the availability of such high frequencies (up to 900 GHz) and high magnetic fields is a rather unique combination that offers several advantages over most commercial EPR spectrometers that operate either at X- band (~ 9 GHz) or Q-band (~ 34 GHz) frequencies. The features of this set-up which make it an extremely useful and powerful system for EPR experiments are as follows: 19 (i) Large and continuous frequency range which often makes it possible to study systems with considerable zero-field splitting. The zero-field splitting, which has been discussed in the previous section, accounts for the anisotropy that leads to the lifting of degeneracy between states in the absence of an applied magnetic field. In the simplest case for a system with only axial anisotropy (D), this term governs the separation between the lowest two energy states at zero field. In a large number of samples, this separation is significantly large, and EPR transitions can only be observed either by using high frequency sources, or by application of high magnetic fields. Thus, high frequency EPR allows the observation of a number of EPR ‘silent’ species, which do not have allowed transitions at X-band frequencies. The availability of a large frequency range also enables tracking an observed transition over a large frequency range for precise measurement of the spin Hamiltonian parameters.
(ii) High sensitivity which enables the study of small single crystal samples and also enables observation of low intensity transitions.
(iii) The rotating cavity used for most of our high frequency EPR experiments (discussed later in further details) provides a mechanism for precisely orienting a single crystal with the field applied along its quantization axis. In cases where an experiment is performed on a single crystal whose quantization axes and planes are not previously known, this rotation capability is the only factor that enables carrying out the measurements at the desired orientation.
The instrumental set-up for conducting the high frequency EPR experiments that are reported in the subsequent chapters consists of the following components: Probe
Magnet
Millimeter Vector Network Analyzer (MVNA)
Electronics for observation, measurement and recording of the EPR signal
20 In this section, we only focus on the description of the probes that have been used for the experiments performed. A detailed description of the working of the MVNA and the electronic components used for measuring and recording the signal has been discussed in Refs. [15-20].
2.1.1.1 Probes
The high frequency EPR experiments (frequency > 50 GHz) involve the use of two kinds of probes: (a) waveguide probe and (b) transmission probe.
(a) Waveguide Probe: The waveguide probe consists of an assembly of rectangular waveguides coupled to a resonant cavity. The dimensions of the waveguides used in constructing the probe and the shape and dimension of the cavity are chosen depending on the microwave frequency at which the fundamental mode is desired. Waveguide: A waveguide is a structure that transmits electromagnetic waves between its end- points with little loss. Waveguides can be cylindrical or rectangular, but for our discussion we will only consider rectangular waveguides. Maxwell’s equations (Eq. 2.5-2.6) lay the foundations of electromagnetism and are the starting point for understanding the propagation of guided waves [13, 21-23]. B E ... (2.5) t D H J ... (2.6) t Considering an electromagnetic wave propagating in the z-direction with a phase factor of exp(iωt – kz), the above equations can be simplified to obtain Eq.(2.7 -2.12) where μ and ε represent the permeability and the permittivity of the medium respectively [13]. E z kE iH ... (2.7) y y x E kE z iH ... (2.8) x x y E E y x iH ... (2.9) x y z
21 H z kH iE ... (2.10) y y x H kH z iE ... (2.11) x x y H H y x iE ... (2.12) x y z Electromagnetic waves propagating along transmission lines can be of three kinds: transverse electromagnetic (TEM for which Hz = Ez = 0), transverse electric (TE for which Ez = 0, Hz ≠ 0) and transverse magnetic (TM for which Hz = 0, Ez ≠ 0). The hollow rectangular waveguides do not transmit TEM modes and support only TE and TM modes. The configuration of the TE and
TM modes can be obtained by substituting Ez = 0 and Hz = 0 respectively in Eq. (2.7 – 2.12). Solving these equations with the appropriate boundary conditions yields the solution. The complete solution starting from the above equations is beyond the scope of this chapter and can be found in Ref. [13]. The dimensions of the rectangular cross-section of the waveguide determine the lowest cut-off frequency that can be transmitted through it.
The rectangular waveguides used for constructing the waveguide probes are made of copper and stainless steel. The choice of these materials is based on the property of thermal conductivity of these materials, so that a stable low temperature can be obtained. The waveguides are rigidly coupled to a resonant cylindrical cavity at the bottom of the probe by means of coupling holes or iris. A proper coupling of the waveguides to the cavity is crucial for minimizing microwave leak and thereby obtaining good resonance modes. To this end, the dimension of the coupling holes is given due consideration while designing a cavity. The diameter of the coupling hole is chosen so as to optimize the signal by ensuring good coupling and minimizing loss at the same time. A larger coupling hole ensures stronger coupling, but on the flip side, it also leads to greater loss and a reduction in the Q value (defined in the following section) of the cavity. Thus, an optimum condition has to be reached, and for a cavity designed for a fundamental resonance mode of 50 GHz, the coupling hole diameter has been determined to be λ/6 [17, 24]. A microwave leak occurs if a leak wave which passes directly through the two waveguides without passing through the cavity (which is contrary to the ideal situation, where the entire signal at the detector should come from the cavity). In the event that the leak signal is comparable to the signal coming from
22 the cavity, the resonance modes get distorted, thereby making it difficult to distinguish the desirable signal from the background [17]. Thus, efforts are made in the construction of the probe and development of the coupling mechanism to minimize the amount of leak signal. While some degree of microwave leak through the iris is inevitable, it is extremely important to keep it at a minimum. Careful consideration of the size of coupling holes, making staggered (rather than leveled) joints in the waveguides and ensuring a rigid coupling between the waveguides and the cavity are some of the steps taken in order to cut down the leak signal.
Microwave resonant cavity: A resonant cavity is a hollow conductor that is closed at both ends and is designed so as to support standing electromagnetic waves. At the resonance frequency, the cavity sustains microwave oscillations forming standing waves from superimposed microwaves that are reflected from the cavity walls. The shape and size of a cavity determines the modes that are supported by it. The resonance frequencies for a cavity of a given shape and dimension can be calculated by starting from Maxwell’s equations while applying boundary conditions consistent with the geometry [13]. For obtaining the solutions for the modes supported by a cylindrical waveguide or cavity, Laplace’s equation is expressed in cylindrical coordinates and the solutions are obtained in terms of the roots of the Bessel function. In the case of a cylindrical cavity of radius r and height h, the resonance frequencies for the TE and the TM modes are given by Eq.2.13 and Eq.2.14 respectively [13]:
2 2/1 1 x2 p mn |TE 2 ... (2.13) r h
2 2/1 1 dx 2 p mn |TM 2 ... (2.14) r h th th In Eq.(2.13 – 2.14) xmn is the n root of the m order Bessel function and dxmn represents its derivative. An important parameter that qualifies the performance of a cavity is the quality factor (Q) which is a dimensionless quantity defined as follows: 2 (energy stored ) Q ... (2.15) energy lost per cycle For the cavities discussed here, the higher the Q factor, the better its performance. Typical values for the Q factor of the cavities used in our experiments were in the range of 10,000 – 17,000 at
23 Fig. 2.2 Photograph showing the parts of a rotating cylindrical cavity with the fundamental resonance mode at 50.4 GHz. This shows the unassembled cavity so that all the parts can be seen. The measuring tape has been placed for an estimate of the size of the cavity parts.
low temperatures. Quantitatively, the Q factor represents a resonator’s bandwidth with respect to its central frequency. Losses leading to low Q factors may be due to dissipation and radiative loss through the coupling holes. Hence, the quality of the coupling of the cavity to the waveguides has a strong effect on the quality of signal observed in an experiment.
In our experimental set-up, a rotating cylindrical cavity made of copper was used. The cavities used were designed for the fundamental TE011 mode to be at approximately 50 GHz. Fig. 2.2 shows the parts of a rotating cavity. The sample may be placed either on the end-plate or on the quartz pillar. After closing the cavity with the end-plate, the assembly is sealed by attaching the end-cap. The spring is placed to apply uniform pressure through the end-cap on the cavity end- plate in order to minimize microwave leak and to ensure good coupling. The rotation capability
24 Fig. 2.3 Schematic diagram showing the directions of rotation about two orthogonal axes in a two-axis rotation experiment. The rotating capability of the cavity design allows the φ rotation and the stepper motor assembled to the magnet enables the θ rotation. The magnet has a transverse field which has been shown in the figure.
of this cavity allows the sample to be rotated with respect to the applied magnetic field and is a particularly useful feature for single crystal EPR. When the cavity is assembled, the teeth of the worm are in contact with the worm gear on the end-plate. The worm has a thin rod attached to it, which runs up to the top of the probe and can be rotated. As the worm is rotated, the end plate is also rotated since the gears are in contact. This makes it possible to rotate the sample during an experiment and is an extremely useful feature. The significant point to be noted with respect to this rotation capability is that this design enables sample rotation without affecting the coupling of the cavity. The rotation mechanism leaves all the mechanical connections between the cavity and the waveguides unaffected. Since the coupling is achieved through the side-walls of the cavity (Refer to Figs. 2 and 4 in Ref. [24]), and only the end-plate is rotated (in contrast to rotating the entire cavity), the cylindrical geometry is preserved and the sample remains in an identical environment of the electromagnetic field. This ensures that the coupling is not affected while carrying out an angle dependent EPR experiment on a sample. A detailed account on the design and the working of the cavity can be found in Ref. [24].
25 Fig. 2.4 Photograph of the transmission probe showing all the parts assembled. The blue dotted arrows depict the flow of the microwave through the probe.
This rotating cavity can be used in the 7 T Quantum Design PPMS magnet at the NHMFL, which is a transverse split-coil superconducting magnet, to carry out two-axis rotation experiments. This has been done in our work on the holmium-polyoxometalate discussed in Chapter 3. The experiment involves rotating the sample about two orthogonal axes. Rotating the end-plate with the cavity-rotation mechanism described above, rotates the sample in one plane, while a stepper motor assembled to the magnet allows rotation of the probe in the x-y plane. A schematic figure depicting the rotation axes is shown in Fig. 2.3. Two-axis rotation makes it possible to achieve orientations that are not restricted only to one plane, as in the case of a uniaxial rotation. This feature enables the identification of the quantization axis of a crystal, even if it does not lie on the planes defined by the crystal faces. Thus, it is an extremely useful technique, especially for samples such as the one discussed in Chapter 3, for which one has no
26 information about the relative orientation of the quantization axis with respect to the crystal axis or crystal face prior to conducting an experiment.
(b) Transmission Probe: The transmission probe consists of an assembly of two brass pipes fitted with a sample holder at the end which encloses two mirrors to reflect the microwave radiation. These probes are useful for high frequency (> 500 GHz) EPR experiments, and an attempt was made to design and build a transmission probe as a part of this dissertation. The reason for the construction of this high frequency probe was to carry out EPR experiments at very high frequencies (500 – 900 GHz) where it is not optimum to use the waveguide-cavity probe discussed in the previous section. The waveguide-cavity probe works on the principle of the cavity perturbation technique [24] and has specific resonant modes. As we go to higher and higher frequencies, these resonance modes become more and more closely spaced. Thus, in the over-moded regime, the waveguide-cavity probe essentially becomes similar to a transmission probe. As mentioned in the earlier discussion, the microwave signal in the waveguide probe is coupled to the cavity by means of tiny coupling holes (~ 0.51 mm in diameter for cavity with fundamental mode with frequency of 50 GHz [24]), and this design is best suited to minimize direct microwave leak between the waveguides. However, this severely restricts the amount of signal that is coupled through the cavity, and thereby at higher frequencies limits the throughput of the probe. Therefore, in order to obtain maximum signal throughput via a probe, a transmission probe with pipes of the widest usable diameter (the diameter of the pipes is limited by the bore of the magnet) is preferably used.
Unlike the waveguide probe which has a resonant cavity that supports specific modes, the transmission probe has a uniform signal output and is based on the simple idea of reflecting the incident microwave radiation by means of a mirror and then recording the transmitted output after it passes through the sample. A photograph of the transmission probe is shown in Fig. 2.4. Fig. 2.5 shows a schematic figure depicting the parts of the probe and the transmission of microwave through it. Fig. 2.6 shows photographs of the parts that were designed in constructing the probe for the Quantum Design and Oxford magnet systems used in our laboratory (listed in Section 2.1.1.2). The two brass pipes serve as the pathways for the incident and the transmitted microwave radiation. The incident radiation goes through one of the pipes and is reflected by a
27 45o mirror. It then passes through the sample placed in the removable sample holder, and is reflected by another 45o mirror so that it is transmitted upwards through the other brass pipe. The mirrors are inserted from the sides of the cylindrical base of the probe, and are held in position by two set screws for each mirror. This design was preferred over one in which the mirrors are rigidly attached (non-detachable) in the assembly due to its relative ease of machining. This also turns out to be advantageous because it offers the control of making slight changes to the mirror orientation for maximizing signal throughput. The dimension of the circular cross section of the mirrors was chosen such that they fit tightly into their slots in the base of the probe, with little room for movement, and can be held using set screws from both sides. The sample holder has a circular hole with a step on its boundary, where a mylar film can be placed for attaching the sample. The advantage of this design is that the sample holder can easily be detached without disturbing the rest of the set-up, thereby making it relatively quick and simple to change samples. The purpose of the two conical structures seen in the figure is to narrow down the microwave signal on to the mirrors and reduce stray reflection. In designing the cones, care was taken to ensure that the diameter of the top of the cone was matched exactly with the diameter of the brass pipe used. This was done to ensure a smooth transmission for the microwave without the presence of mismatched connections that might contribute to stray reflections. The diameter of the brass pipes used was selected to be the maximum diameter that would fit in the magnet bores to maximize the signal output. The materials used in making this probe were brass and copper. The transmission pipes, the cones and the mirrors were made of brass, and the other parts of the sample-holder assembly were made of copper. The choice of materials used was made with consideration of various factors like tensile strength and thermal conductivity. In order to make the probe compatible with two different magnet systems (with different bores and depths), two sets of pipes (one set soldered to the head of the probe, and another set soldered to the bottom) that can be joined with a clamp have been used. Although it is generally preferable to minimize the number of joints for maximum signal strength, this design offers the advantage of adding an extension to easily change the probe length. Hence, the same probe can be used in two magnets by either attaching or removing the extension. The probe head contains a fitting for a Fischer connector that allows a precise measurement of the temperature near the sample space. Using a thermometer attached on the surface of the copper sample holder, with wires running up to the probe head, the temperature can be measured by connecting the relevant electronics to the
28 Fischer connector port. Cernox resistance thermometers [25] are used for measuring the sample temperature. These sensors are built of materials with a negative temperature coefficient of resistance, and are calibrated so that the measured resistance can be converted to read the temperature. They allow measurement over a wide temperature range and have low magnetic field induced errors [25], making them a suitable choice for our experiments. The ends of the two brass pipes at the head of the probe are fitted with clamps for connection to the horns that attach to the source and the detector.
The probe was tested on the bench to check for signal intensity at high frequencies. It was found that we could achieve reasonable signal amplitude (40 – 62 dB) for several frequencies in the range 600 GHz – 850 GHz at room temperature. The values of the signal intensity (in units of decibels) obtained at various frequencies have been tabulated in Table 2.1 (a) and (b) for probe lengths corresponding to that of the Quantum Design and the Oxford magnets respectively. All the measurements tabulated correspond to the same receiver gain of 40 dB of the MVNA. In order to estimate the loss introduced as a result of transmission through the probe length, the signal amplitude was tested by connecting the source and detector directly to the mirror assembly at the bottom of the probe. In this set-up the signal amplitude was found to be ~ 70 dB for most frequencies in this range. It can be seen through a comparison of the signal strengths shown in Tables 2.1 (a) and (b), that the addition of the extension (~ 2.2 feet in length) in order to increase the length of the probe causes an appreciable reduction in the signal output. This is expected as a result of transmission losses introduced by two factors: the increase in length and the inclusion of an extra junction in the probe. However, we do obtain reasonable signal intensities that should allow measurements up to relatively high frequencies for both lengths. In order to maximize the signal output, the orientation of the mirrors was slightly adjusted for the first few measurements, and then fixed with set screws at their optimum positions. It was observed that a small change in the angle of the mirrors was sufficient to considerably reduce the signal throughput. Thus, care should be taken to test and maintain the desired orientation of the mirrors. Typical signal amplitudes at which measurements are carried out with the waveguide - cavity probe range between 55 – 80 dB for frequencies ranging from 50 GHz up to approximately 400 GHz. The receiver gain on the MVNA is adjustable (0 dB, 10 dB, 20 dB, 30 dB and 40 dB), and it is set at a gain value suitable for the strength of the signal observed corresponding to a specific mode.
29 Fig. 2.5 Schematic representation of the construction of transmission probe. (Parts not to scale). (a) Front view showing the parts of the probe and the direction of flow of the microwave signal (b) A rotated 3-dimensional perspective of the apparatus.
30 Fig. 2.6 High frequency probe components. (a) Parts of the probe head (b) Photograph showing the parts of the probe bottom including the sample holder, mirrors, cylindrical casing, cones and end-cap (c) Photograph showing how the parts are assembled.
31 Table 2.1. Tabulation of signal amplitudes for high frequency probe. They correspond to testing the transmission probe for (a) the Quantum Design magnet and (b) the Oxford magnet on the bench at room temperature. All the signal amplitudes listed in this table correspond to a receiver.
(a) Probe for Quantum Design Magnet (b) Probe for Oxford Magnet Frequency (GHz) Signal amplitude (dB) Frequency (GHz) Signal amplitude (dB)
647.445 49.4 570.644 52.2
671.7034 53.2 601.0312 50.7
705.4128 45.6 638.1574 51.8
719.9820 46.4 686.322 46.3
724.0158 58.5 706.7034 42.3
730.7658 50.2 720.7974 46.2
735.6960 60.1 763.7436 44.8
742.8510 51.8 786.051 40
756.8802 49.7 802.638 38.7
764.8452 50.3 811.544 41.4
773.8578 50.1 818.3682 40.8
785.0898 52.8
794.4426 49.8
804.9024 56.7
820.3788 48.6
835.6116 49.3
32 Thus, comparing the typically observed signal amplitudes in the two probes, we note that although the strength of the signal is lower at higher frequencies (> 400 GHz), we are still able to obtain reasonable throughput with the new transmission probe. The probe remains to be tested with a sample placed in the magnet, but it offers the possibility of carrying out high frequency measurements in the two magnet systems in our laboratory. This would then be a useful addition to the equipment used for high frequency EPR experiments, since the waveguide probes in current use in these two magnet systems do not have sufficient throughput at frequencies above ~ 450 GHz. The two types of probes that have been discussed have their own optimum frequency regimes, and complement each other, making EPR experiments possible over a very wide frequency range.
2.1.1.2 Magnets
The high frequency EPR experiments reported in this work were carried out in the following magnet systems: 7 T Quantum Design PPMS transverse split-coil superconducting magnet (with two-axis rotation capability)
15 T Oxford Instruments axial superconducting magnet
31 T axial resistive magnet at the NHMFL DC Facility
35 T axial resistive magnet at the NHMFL DC Facility
The key parameters characterizing the above mentioned magnet systems used for high frequency EPR have been tabulated in Table. 2.2. Low frequency EPR experiments were carried out in the X-band (9.7 GHz) Bruker ELEXSYS E 580 spectrometer at the NHMFL with capability for both continuous wave and pulse spectroscopy. Q-band (34 GHz) experiments were conducted using a Bruker ELEXSYS 500 spectrometer at the Department of Chemistry and Biochemistry, FSU.
33 Table 2.2: Properties of the magnets used in carrying out EPR experiments reported in this dissertation.
Maximum Magnet bore Probe Length Temperature Magnet Type field (T) (mm) (m) range (K) Transverse Quantum split-coil 7 T 26 1.15 2 – 300 K Design superconducting Axial Oxford 15 T 32 1.9 1.7 – 300 K superconducting 31 T Axial resistive 31 T 32 2 1.3 – 300 K 35 T Axial resistive 35 T 32 2 1.3 – 300 K
2.2 Pulsed EPR Spectroscopy
Although EPR spectroscopy had traditionally been used as a continuous wave (CW) technique, pulsed EPR has played a significant role in the study of electron spin dynamics in recent years. In a nutshell, the method of pulsed EPR involves the application of short (~ a few tens of nanoseconds) and intense microwave pulses, and then measuring the emitted microwave signal generated by the magnetization of the sample. The pioneering work in the field of pulsed EPR spectroscopy was carried out at Bell Laboratories during the 1960s, but the technique started undergoing rapid development since the 1980s which led to an increased availability of commercial spectrometers for pulsed measurements, which allows a direct and efficient means of studying spin dynamics. CW and pulsed EPR experiments both have their own advantages, and depending on the sample and the kind of properties we are interested in probing, the appropriate technique is selected. An important advantage of a CW EPR experiment is the higher sensitivity. Another advantage is that a CW experiment can often be conducted at relatively higher temperatures as compared to a pulsed experiment. Most pulsed experiments need to be carried out at low temperatures in order to measure the relaxation times which are usually too short at higher temperatures for samples of our interest. On the other hand, a pulsed EPR experiment 34 Fig. 2.7 The fixed laboratory frame showing the applied field B0 along the z-axis and the microwave field B1 along the x-axis. allows us to directly study the spin dynamics of a system and measure the magnetization relaxation properties of a spin ensemble. In systems with multiple entities having very different relaxation properties, the pulsed technique can be employed to selectively address a specific system of spins by controlling the microwave pulse sequence used to excite the sample. Thus, both these EPR techniques have significant applications in studying the magnetic properties of a system. In the work presented in Chapter 3, extensive use has been made of pulsed EPR spectroscopy to study the relaxation times of holmium-polyoxometalate samples of various concentrations. Pulsed EPR experiments were carried out at X-band (9.7 GHz) frequency using a commercial Bruker ELEXSYS E 580 spectrometer [26, 27].
2.2.1 Classical Description of the Theory of Pulsed EPR Spectroscopy
This treatment of the theory of pulsed EPR is based on Refs. [27, 28]. Although a rigorous theoretical discussion must involve quantum mechanics, here we present a classical description for an intuitive understanding of the phenomena. Since experiments are carried out on an ensemble of electron spins, the macroscopic magnetization of the sample may be described in terms of classical physics for understanding the basic aspects of the technique. In order to provide a description of the pulsed EPR technique, it is convenient to consider the magnetization
35 Fig. 2.8 The microwave field B1 in the laboratory frame and the rotating frame. In the stationary lab frame, the linearly polarized B1 can be considered to have two components in opposite directions with the same angular velocity ω0. In a frame rotating at the same velocity ω0, we need to only consider a stationary M0 along the z-axis and a stationary B1 along x-axis. The faster rotating component (2ω0) can be neglected [27]. of an electron spin in a rotating frame, rather than the stationary laboratory frame. This considerably simplifies the picture of the dynamics of the magnetization and is hence the chosen method. We first define our laboratory frame, and then move to the rotating frame to show how the two frames are related. Fig. 2.7 shows the laboratory frame defined by three orthogonal axes, where the external magnetic field, B0 is along the direction of the z-axis. The direction of the microwave field, B1 is parallel to the x-axis. The torque on an electron spin placed in an external magnetic field causes it to precess about the field. This phenomenon is called Larmor precession and may be expressed as Eq.2.16 where ωL is the Larmor frequency and γ is the proportionality constant.
L B0 ... (2.16) When a sample is placed in a magnetic field, there are a large number of electron spins which align themselves in either the “parallel” or the “anti-parallel” configuration following Boltzmann statistics. The electron spins precess about B0 and create a net stationary magnetization, M0 along
36 Fig. 2.9 Pictorial representation of the response of the magnetization to a π/2 pulse. Application of the π/2 pulse flips the magnetization by 90 degrees and aligns it along the x- axis. The pulses are named according to the angles by which they flip the magnetization vector [27]. the z-axis. In most EPR experiments, linearly polarized microwaves are used which create a microwave field B1, which is perpendicular to the applied field B0 (in our convention, B1 is along the x-axis). The strength of the microwave field is typically much weaker as compared to the applied field. It is useful to regard the linearly polarized B1 field as the sum of two magnetic fields rotating in opposite directions at the microwave frequency. In order to visualize the effect that the field B1 has on the magnetization, we now move to a rotating frame which is synchronized with one of the rotating components of B1. If the microwave frequency is ω0, then the resonance condition can be defined as: ωL = ω0.
In a coordinate frame that rotates at an angular velocity ω0, one of the components of B1 appears stationary, while the other appears to rotate at an angular velocity of 2ω0. This has been depicted in Fig. 2.8 which shows the components of B1 in both reference frames. The fast component rotating at 2ω0 can be neglected because it does not have a significant influence on the magnetization (since this frequency is far off resonance). Thus, in the rotating frame we
37 Fig. 2.10 The nutation of the magnetization M about the effective field, Beff. For a homogeneously broadened spectrum, each spin packet has a slightly different Larmor precession frequency, thereby leading to this “off-resonance” condition depicted in this figure [27].
effectively have a stationary B1 since we can do away with all the rotating components by transforming to the new co-ordinate system that rotates at the resonance frequency. The magnetization M0 will now precess about the stationary field B1 following Larmor precession.
This frequency of precession, ω1 is called the Rabi frequency. The magnetization M0, which was originally aligned along the z-axis, will therefore be rotated by an angle α with respect to the z- axis as a result of precession about B1. This happens as long as the microwave field is on, and the angle α is called the tip angle which depends on the magnitude of the microwave field and the length of the pulse, tp.
| B1 | t p … (2.17) Microwave pulses are therefore often referred to by the angle by which the magnetization is tipped as a result of the pulse. The effect of a π/2 pulse is pictorially shown in Fig. 2.9 which shows the tipping of the magnetization with respect to the pulse being either “on” or “off”.
38 In the discussion so far, only the resonance condition has been considered, with the Larmor frequency of precession being exactly equal to the microwave frequency. However, there is always an inhomogeneous broadening in the lineshape of an EPR spectrum which can be considered to be composed of a number of spin packets, each of which are homogeneously broadened, and have a small difference in frequency with respect to each other. Thus, not all parts of the spectrum can be exactly on resonance like the idealized condition we have considered so far. We therefore need to include the off-resonance effects to account for the true nature of evolution of a spin-system in the presence of a microwave pulse. In case of a transverse magnetization having a frequency ω = ω0 + ΩS, it will appear to rotate in the x-y plane, unlike the resonance condition (ΩS = 0) where M was aligned along the y-axis. Fig. 2.10 depicts the off- resonance situation which has two consequences that alter the picture in comparison to the simpler on-resonance case. Firstly, the magnetization rotates in the x-y plane instead of being aligned along y. Secondly, in this situation B0 does not disappear in the rotating frame, since the magnetization is no longer stationary. The magnetization is now tipped by a field Beff, which is the vector sum of B0 and B1, causing it to nutate at a frequency ωeff [28].
2 2 eff 1 S … (2.18)
1 1 The angle between Beff and the z-axis (direction of the static field) is tan . In this S discussion, we have seen how the magnetization responds to microwave pulses. Next, we apply this understanding to describe the features of a pulsed EPR experiment.
Relaxation Time: Relaxation times are a measure of the rate of decay of magnetization. In our discussion thus far, we have assumed that once a spin system has been excited, the precession of the magnetization continues the same way without any decay. This is clearly an unrealistic situation, and we need to take into account the decay of the magnetization vector. The electron spins interact with their surroundings causing the magnetization components in the x-y plane to decay. There are two kinds of relaxation processes which are characterized by the longitudinal relaxation time (T1) and the transverse relaxation time (T2). The longitudinal relaxation time is a measure of how quickly the magnetization returns to its original alignment along the z-axis. The transverse relaxation time measures how quickly the magnetization decays in the transverse (x-y) plane. The longitudinal relaxation is also referred to as the spin-lattice relaxation, and is a result 39 of the transfer of energy from the electron spin to the crystal lattice. The transverse relaxation time, also referred to as the spin-spin relaxation, is mainly a result of dipolar spin couplings. If a microwave pulse is applied at t = 0, then after a time t >> T1, T2, a steady state is reached. The spin dynamics in a two-level system are described by the Bloch equations as shown below. Although the relaxation processes cannot be exactly described in the classical picture, the Bloch equations provide a phenomenological description. It should be noted that although the classical treatment enables an understanding of the basic phenomena, this description fails in complex cases where there are more than two levels involved (i.e. S > ½). The longitudinal relaxation time
T1 can be expressed in terms of the time evolution of the z-component of the magnetization as shown in Eq.2.19. The transverse magnetization in the x-y plane follows an exponential decay and defines T2 as shown in Eq.2.20. dM (M M ) z z 0 … (2.19) dt T1 dM M , yx , yx … (2.20) dt T2 In the rotating frame, the Bloch equations can be represented as [28]:
dM x M x S M y … (2.21) dt T2
dM y M y S M x 1M z … (2.22) dt T2
dM z M z M 0 1M y … (2.23) dt T1 In the discussion that follows we describe the technique of estimating the relaxation time in a pulsed EPR experiment.
Echo: An ensemble of spins with different Larmor frequencies give rise to spin echoes when multiple pulses are applied. When a π/2 pulse is applied, it produces a Free Induction Decay (FID) signal which decays away after a finite time interval. However, for an EPR spectrum that is inhomogeneously broadened, this FID signal can be recovered by applying another refocusing microwave pulse. The refocused signal is known as the echo. The spin echo concept was first formulated by Erwin Hahn in 1950 for studying nuclear spin echoes [29]. Spin echo
40 Fig. 2.11 Hahn echo sequence. (a) The Hahn-echo pulse sequence showing the π/2 and π pulses with a separation τ (b) The time-evolution of the magnetization on the transverse plane in response to the pulse sequence shown in (a). The four stages shown in (b) correspond to the time denoted by 1, 2, 3 and 4 in (a). Thus, the Hahn echo sequence refocuses the signal to obtain an echo using multiple excitation pulses [28]. measurements are a powerful technique to study spin dynamics and have found widespread applications in fields like coherent optics and microwave spectroscopy. The Hahn echo sequence can be expressed as π/2 – τ - π – τ – echo, and has been depicted in Fig. 2.11(a). It consists of two pulses with tip angles π/2 and π with a separation of time τ. Fig. 2.11(b) shows the evolution of the magnetization vector with time for the Hahn echo pulse sequence shown. The π/2 pulse flips the magnetization along the – y-axis. After the application of the pulse, the different spin packets precess with their own individual Larmor frequencies causing the transverse magnetization to spread out on the x-y plane. When the second pulse (π pulse) is applied after a time τ, it causes the phase of the magnetization to be changed by 180o. Since the direction of rotation of the spin packets is not changed by this pulse, it causes the magnetization vectors to refocus along the y- axis, and creates the electron spin echo.
41 A pulsed EPR experiment is most commonly performed by applying a Hahn echo pulse sequence and recording the echo-detected spectrum. The pulse durations used in an experiment is determined by the linewidth of the spectrum. In the simple case, the linewidth of the transition is narrower than the excitation bandwidth. In a spin echo experiment, the echo height is proportional to the number of spins that are excited by the π/2 pulse. The process of transverse relaxation discussed above leads to an exponential decay in the echo height as a function of the delay time τ. Thus, for measuring T2, the separation between the pulses (τ) is successively increased over time, while measuring the corresponding echo intensity. This gives rise to an exponential decay curve which enables us to estimate T2. The longitudinal relaxation time, T1, is measured by the application of an inversion pulse which flips the magnetization vector from +z to the –z direction, followed by the Hahn echo pulse sequence for detection. In our pulsed EPR work on holmium-polyoxometalate crystals discussed in Chapter 3, this technique has been used to estimate and compare the relaxation times for samples of various concentrations.
ESEEM: Electron Spin Echo Envelope Modulation (ESEEM) is a technique in pulsed EPR to measure the interaction between the electron spin and neighboring nuclear spins [30]. This interaction is experimentally manifested as a periodic oscillation in the echo-height (shown in
Fig. 3.12 of Chapter 3) which is superimposed on the exponential decay curve typical for a T2 measurement. These oscillations carry signature of the nearby nuclei and give information about the local environment of the ion. In order to analyze the ESEEM data, the exponential decay is first subtracted, and then the Fourier transform of the remaining oscillations is computed to extract the ESEEM frequency. We observed ESEEM effects in case of dilute holmium polyoxometalate samples (discussed in Chapter 3), and a similar analysis was carried out to identify the nuclei giving rise to the interactions that manifest themselves in the form of the oscillations on the decay curve. Observation of the ESEEM effect in a T2 measurement provides information about the mechanisms contributing to the spin relaxation process, and is therefore significant in pulsed EPR experiments.
42 2.2.2 A Brief Overview of the Quantum Mechanical Description
The classical picture we have discussed so far is very useful in forming an intuitive understanding of the effect of a microwave pulse sequence on the bulk magnetization of a simple two level system. However, for a proper quantitative treatment of the phenomena for more complex systems, a quantum mechanical description is required. The goal of this discussion is to provide a brief introduction to the quantum mechanical formalism based on the treatment presented in Chapter 4 of Ref. [28]. The time evolution of an ensemble of spins can be expressed using the density operator formalism which is analogous to the Bloch equations. The evolution of a wavefunction can be described by Schrödinger equation: d | iH t |)( … (2.24) dt
The density operator (σ) is defined as the weighted sum of the wave functions |ψk> with a time- independent probability pk as shown in Eq.2.25 and the time evolution of σ can be expressed as Eq.2.26.
t)( pk | k k | … (2.25) k d i p k (H | k k | | k k | H ) i(H H ) … (2.26) dt k The above equation can be written as a commutator to obtain the Liouville-von Neumann equation as follows (Eq.2.27): d [Hi t),( t)]( … (2.27) dt The time evolution of the density operator σ can be mathematically expressed using the product operator formalism, which decomposes σ into a linear combination of orthogonal basis operators. A particularly appealing feature of this treatment is the fact that the operators have a clear physical meaning, and that the effects of pulses and delays can be interpreted as geometric rotations. The spin operators are represented as Ŝx, Ŝy and Ŝz. For a spin ½ system, these spin operators are depicted as the Pauli matrices as shown in Eq.2.28. 1 0 1 1 0 i 1 1 0 ˆ ˆ ˆ S x S y S z … (2.28) 2 1 0 2 i 0 2 0 1
43 For expressing the time evolution of σ, we first introduce the notation that is usually used in the product operator formalism. For a product operator A evolving under another product operator B, Eq.2.29 can be written to express the evolution. A shorthand notation that is commonly used to express Eq.2.29 is shown in Eq.2.30. eBi eA Bi C … (2.29)
A B C … (2.30) In the above equations, the rotation angle corresponds to the angle by which the spin is flipped as a result of the microwave pulse. The general solution for Eq.2.30 is shown in Eq.2.31. A B Acos Bi , A sin for B A … (2.31) A B A for B A We use the spin operators in this product operator formalism to describe the two-pulse Hahn echo sequence that we have already described in Section 2.2.1. The Hamiltonian for a free precession, Ĥfree, (corresponding to a delay in the absence of an applied pulse) and that for a pulse about x, Ĥpulse, can be expressed as Eq.2.32 and Eq.2.33 respectively. ˆ ˆ H free S z … (2.32) ˆ ˆ Hpulse,x 1Sx … (2.33) The Hahn echo sequence is expressed as: π/2 – τ - π – τ – echo. Using the operator notation, the effect of this pulse sequence can be expressed quantitatively as follows[28, 31]:
S 2 x S z S y SzS cos S S y sin S S x … (2.34)
Sz cos S S y sin S S x
tSzS sin S t S x cos S t S y The four expressions shown in Eq.2.34 respectively represent the density operator σ (t) at times
1, 2, 3 and 4 of the pulse sequence, as shown in Fig. 2.11(a). In the above equations, ΩS is the frequency off-set arising as a result of inhomogeneous broadening of a spectrum which consists of a number of spin packets, each with its own Larmor frequency. Here, we have provided a brief preliminary description of the theoretical formalism that is used for studying the effect of microwave pulses on the time evolution of the electron spin. A complete treatment of this theory can be found in Ref. [28] on which we have based the discussion presented here.
44 Fig. 2.12 Depiction of Rabi oscillations showing variation of echo amplitude as a function of time. The oscillations are damped and decay exponentially with a characteristic time of ~ 2T2.
Rabi Oscillations: Rabi oscillations represent the cyclic behavior of a two-level quantum system in the presence of an oscillating driving field [32, 33]. Qualitatively, this can be understood by considering a two-level system where the population of each of the levels oscillates with a characteristic frequency known as the Rabi frequency. The observation of Rabi oscillations, which are experimentally manifested as oscillations with decaying echo amplitude, provide evidence for the existence of long-lived coherence in a quantum system. In the absence of external excitation, the Hamiltonian of a two-level quantum system with states |0> and |1> having energies E0 and E1 respectively can be expressed as: ˆ H 0 E0 0 0 E1 1 1 … (2.35) In the presence of the external oscillating field, the Hamiltonian for the system becomes [28]: Hˆ Hˆ R expi t 0 1 R exp i t 1 0 … (2.36) 0 2 R 2 R
45 If at time t = 0, the spin is assumed to be in state |0>, then after time t, the probability of it existing in state |1> is given by p1(t) as shown in Eq.2.37. 1 p t)( 1 cos t … (2.37) 1 2 R This periodic oscillation in population between the spin states gives rise to Rabi oscillations and the frequency ωR is called the Rabi frequency. For a spin system characterized by spin S, the
Rabi frequency for a transition from state mS to mS+1 can be expressed as Eq.2.38. For a spin ½ system this translates to Eq.2.39, which leads to a linear relationship between ωR and the microwave field B1 [28]. g B (m ,m ) B 1 (SS )1 m (m )1 … (2.38) R S S1 S S g B B 1 … (2.39) R
Spin echo experiments for observation of Rabi oscillations and studying the variation of ωR with
B1 were carried out in our work on holmium polyoxometalate reported in Chapter 3. As we will see in the next chapter, Rabi oscillations were observed for this sample as an experimental confirmation of long-lived spin coherence which makes the system an interesting candidate for potential applications. In a typical experiment involving Rabi oscillations, the amplitude of the oscillations decays exponentially with time (Fig. 2.12). This damping, which is primarily due to the inhomogenity of the microwave field, causes the oscillations to die down after a certain time interval.
2.3 Synthesis of Single Crystals: Floating Zone Method
The synthesis of some of the samples studied in the course of this dissertation was carried out at the National High Magnetic Field Laboratory (NHMFL) using the floating zone method of crystal growth [34, 35]. Single crystal growth by this method involves a two-step procedure: (1) synthesis of polycrystalline seed rods and (2) single crystal growth using the image furnace. These polycrystalline rods were synthesized by solid state reaction carried out by mixing stoichiometric proportions of the relevant chemicals, baking them in a furnace and then pressing them into rods. The reactants were ground together and fired at ~ 950o C for 20 hours. This step
46 Fig. 2.13 Schematic depiction of the image furnace for floating-zone method of crystal growth. The molten zone created by sharply focused light from two ellipsoidal mirrors can be seen in the figure. Figure taken from Ref. [35]. was repeated 2-3 times, and thereafter the reactants were pressed into thin rods by application of pressure. The pressed rod was fired for 24 hours at a temperature of ~ 1050o C in order to obtain the polycrystalline feed and seed rods required for single crystal growth. While pressing the powder in order to obtain the polycrystalline rods, care was taken to ensure that smooth, symmetric cylindrical rods were formed with a uniform cross section. This is significant for them to be used as feed rods for good quality single crystal growth using the floating zone method. Attempts were also made to form long rods that would enhance the chances of obtaining a large single crystal. Since neutron scattering experiments form one of the major techniques used in studying the samples synthesized, large crystal size was desirable.
The image furnace is used for high quality crystal growth using the floating zone method. The furnace has a bulb and mirror arrangement which can be adjusted to focus the light from the halogen bulbs in order to create the “zone” which can be heated to very high temperatures. Fig. 2.13 shows a schematic diagram of the floating zone image furnace which consists of two
47 Fig. 2.14 Images of the stages of single crystal growth by the floating-zone method. In the first figure, the two polycrystalline rods can be seen with their tips approaching the molten zone. In the next stage, the tip of the seed rod starts melting as it enters the “zone”. The third figure sows the final stage with a stable molten zone. Figure taken from Ref. [35].
concave ellipsoidal mirrors, two halogen bulbs, an upper and a lower shaft, and a quartz tube. In order to grow a single crystal, one starts with two polycrystalline feed and seed rods which are attached to the upper and lower shafts of the image furnace respectively. The shafts to which the feed rods are attached are movable and can move up and down, as well as rotated about the vertical axis. The upper shaft has a hook attached at the bottom for suspending the feed rod, while the lower shaft has a ceramic holder for the seed rod. The shafts with the polycrystalline rods are enclosed within a quartz tube. The drive shafts are rotated in opposite directions and the upper one is lowered into the ‘zone’ to start melting the bottom tip of the sample slowly.
The ‘zone’ is a tiny volume (less than one cubic centimeter) which is heated to extremely high temperatures (up to 2000o C) by the sharply focused light from the 1500 Watt bulbs. Once the tip of the upper rod has melted properly, the adjacent tips of the feed and seed rods are carefully joined. This creates a molten region at the ‘zone’ which needs to be monitored carefully until a stable molten zone is obtained. Once stable, both the upper and the lower rods are pulled down at slow speeds, so that the liquid gradually solidifies to form a single crystal. The speed at which the rods are pulled is an important parameter which often determines the quality of the crystal grown. The furnace at the NHMFL allows pulling speeds ranging from 0.5 mm per hour to 50 mm per hour and, although it is generally advisable to keep the speed slow, one might sometimes 48 need to pull at a faster rate depending on the stability of the zone. The success of a crystal growth depends greatly on careful monitoring in order to maintain a stable zone. A CCD camera is fitted on the furnace which enables the experimentalist to constantly view the image of the feed and seed rods on a computer. Fig. 2.14 shows a set of photographs at different stages of a typical floating zone method crystal growth.
The image furnace is designed such that the growth environment can be controlled depending on the conditions best suited for the sample being grown. The quartz tube which encloses the sample is fitted with an inlet and outlet through which gas flow can be controlled. Some crystals are grown in air, while others may require an atmosphere of a specific gas like oxygen, helium or argon. This is determined by the chemical composition of the sample and the reactivity with the environment. Proper selection of the atmosphere is crucial for obtaining crystals with the desired composition. The gas pressure inside the chamber can also be controlled through valve regulators fitted at the gas outlet pipes. Samples which are volatile often need to be grown under pressure in order to maintain the correct composition. The floating zone furnace at the NHMFL sustains crystal growths at pressures of up to 10 atmospheres.
Single crystals belonging to the family SrL2O4 (L = Er, Ho, Yb) discussed in Chapter 4 and single crystals of the kagomé-like langasites Pr3Ga5SiO14 and Nd3Ga5SiO14, which have been discussed in Chapter 5 were synthesized using the floating zone method of crystal growth.
2.4 Sample Characterization Techniques
The crystals synthesized were characterized by measurements involving the following techniques which shall be discussed briefly:
X-ray Diffraction Magnetic susceptibility measurement
49 Fig. 2.15 X-ray diffraction pattern of SrHo2O4. The plot shows the Bragg peaks as a function of the diffraction angle which is characteristic of the lattice spacing of a crystal and is therefore used to characterize samples.
X-ray Diffraction: X-ray diffraction (XRD) is one of the most widely used non-destructive techniques for studying crystal structures by measuring the intensity of the scattered x-ray beam as a function of the scattering angle, . The lattice spacing, and hence the crystal structure can be determined by application of Bragg’s Law which is expressed as Eq.2.40 where d = crystal lattice spacing, = scattering angle, = wavelength [1, 3]. 2d sin n … (2.40)
Powder XRD measurements on the samples synthesized were carried out using CuK1 radiation (1.54059 Å) with a Ge monochromator and recorded by an imaging plate Guinier camera. Fig.
2.15 shows the XRD pattern obtained for SrHo2O4 which was used to confirm that the sample has the correct composition and phase.
Magnetic Susceptibility: Magnetic susceptibility is a measure of the degree of magnetization of a material in the presence of a magnetic field and can be defined as [1]:
50 Fig. 2.16 Schematic representation of time of flight measurement using disc chopper spectrometer at NIST. Figure based on Ref. [37].
M T )( … (2.41) H In this expression, M is the magnetization and H is the applied field. DC magnetic susceptibility measurements were carried out for sample characterization using a Quantum Design superconducting quantum interference device (SQUID). A SQUID is a highly sensitive magnetometer containing a superconducting loop containing Josephson junctions and is used for measuring weak magnetic fields. At high temperatures, the Curie-Weiss Law defines magnetic susceptibility () through the following expression [1]: C T )( … (2.42) T CW
In Eq.2.42, C is the Curie constant and θCW is the Curie-Weiss temperature. Plotting the variation of inverse magnetic susceptibility as a function of temperature allows the evaluation of the Curie
51 constant and Weiss temperature for a sample. Magnetic susceptibility measurements were carried out on a number of samples for our work on frustrated spin systems including SrHo2O4
(discussed in Chapter 4), Pr3Ga5SiO14 and Nd3Ga5SiO14 (discussed in Chapter 5).
2.5 Neutron Diffraction
Neutron diffraction is increasingly becoming an important tool in crystallographic studies. It is similar to X-ray diffraction, but gives complementary information about the structure of a solid due to the difference in the radiation source used for probing. X-rays typically have energies of the order of a few thousand electron-volts, while neutrons have thermal energies of the order of a few tens of milli-electron volts (meV). Thus, while x-ray diffraction techniques give useful information about the crystal structure, it is possible to extract information about lattice and spin excitations from neutron diffraction studies. The interaction between the magnetic spins of the neutrons with the magnetic dipole moment of the sample being probed gives valuable information about the microscopic magnetic structure of the material. It is therefore a very powerful technique for studying the dynamics of solids [36].
Neutron scattering can either be elastic or inelastic in nature. Elastic neutron scattering, as the name suggests, occurs when the neutrons collide with the atoms of the sample being probed, and are deflected in direction without undergoing a change in energy. This gives rise to diffraction pattern (similar to ones obtained in a Bragg x-ray diffraction experiment) providing information about long-ranged as well as short-ranged spin interactions. Experimental data and analysis of elastic neutron scattering has been shown in Chapter 4. Inelastic neutron scattering, on the other hand, is accompanied by a change in energy of the deflected neutrons and occurs when the collision of the incident neutrons with the sample results in the creation of phonons or other excitations which absorb part of the energy. This energy change can be interpreted as the energy difference between quantized levels, and can therefore be used as a means of describing the crystal field levels of a sample. In the study of SrHo2O4 discussed in Chapter 4, experimental results and analysis of neutron scattering data have been shown.
52 The neutron scattering measurements were carried out at the Disc Chopper Spectrometer (DCS) facility at National Institute of Standards and Technology (NIST) [37]. This spectrometer uses the principle of time of flight measurement, and has been schematically depicted in Fig. 2.16
[30]. A neutron beam with an energy Ei is incident on the sample after passing through the pulser and the monochromator. The time of arrival of the beam at the sample can be computed to yield the value tS. After passing through the sample, the neutrons are scattered either elastically (without any change in energy) or inelastically (with gain or loss in energy), and reach the detector at time tD. In the course of the experiment, the distribution of the times of flight (t = tD – tS) of the scattered neutrons is obtained. With the knowledge of the incident and scattered energies of the beam, and the scattering angle, the energy exchanged between the neutrons and the sample can be computed. The distribution of the scattered signal intensity as a function of scattering angle and time of flight (I(2θ, t)) is thus obtained in a neutron scattering experiment conducted using the DCS spectrometer. The combination of long flight paths, accurately known spectrometer distances, and chopper and timing electronics enables precise measurement of neutron velocities and energy changes[37].
53 CHAPTER THREE
STUDY OF A MONONUCLEAR LANTHANIDE-BASED SINGLE MOLECULE MAGNET
In this chapter, we present our study of the spin dynamics of a mononuclear holmium-based single-molecule magnet (SMM) using the technique of pulsed electron paramagnetic resonance (EPR) (discussed in Section 2.1 in Chapter 2) in order to measure the spin relaxation time for this system. Before we proceed to discuss our specific system of interest, we shall briefly outline the basic features of SMMs and their relevance in potential applications involving information storage and processing using the dynamics of an electron spin. We shall also introduce the ideas of magnetic anisotropy, quantum tunneling and electron spin coherence in this section since these are key concepts that are relevant to this study.
3.1 Introduction to Single-Molecule Magnets
SMMs, as the name suggests, are molecule-based magnets which are distinct from conventional bulk magnets. A SMM may be defined as a molecule that exhibits slow relaxation of magnetization that is purely of molecular origin [38]. This means that below a certain critical temperature which is defined as the blocking temperature, once the molecule is magnetized by an external magnetic field, it retains its magnetization even after the removal of the external field. SMMs can be regarded as mesoscopic materials that act as an intermediate between the macroscopic and the microscopic worlds. Investigation of these materials gives us an insight into how quantum properties evolve in the classical regime [38-40]. Since the discovery of the slow magnetization relaxation property of [Mn12O12(CH3COO)16(H2O)4] (Mn12Ac) in the 1990’s [41], the study of SMMs has evolved into a major branch of research involving theoretical and experimental efforts from physicists and chemists [42-50].
54 Magnetic anisotropy is a key feature of a SMM and is the property of a magnetic system to have a preferred orientation of the spin. This is the crucial feature that enables the application of magnetic particles or molecules in information storage by representing the ‘spin up’ and ‘spin down’ states as bits. In order to understand the properties of SMMs, we introduce the general spin Hamiltonian, and briefly describe each component as follows: ˆ ˆ ˆ ˆ H H Zeeman H ZFS H hf … (3.1) ˆ ˆ H Zeeman B .. SgB (Zeeman term) … (3.2) ˆ ˆ 2 S(S )1 ˆ 2 ˆ 2 k ˆ k H ZFS DS z E(S x S y Bn On (Zero field splitting) … (3.3) 3 ,kn ˆ ˆ ˆ Hhf .SIA (Hyperfineterm) … (3.4) In the above equations, µB is the Bohr magneton, g is the Landé g-tensor and Ŝ is the spin operator. The first term in Eq. 3.1 represents the Zeeman interaction, which describes the splitting of the energy levels in the presence of an external magnetic field B. The zero-field splitting term in the Hamiltonian (second term in Eq. 3.1) introduces the zero-field splitting tensor, D , which represents the anisotropy in the absence of an external field. The zero-field splitting predominantly arises as a result of spin-orbit coupling and spin-spin coupling. In EPR spectroscopy, the D tensor is typically expressed in terms of the axial anisotropy (D) and the transverse anisotropy (E) terms as shown in Eq. 3.3. In case of SMMs, the axial anisotropy is negative (D < 0) and leads to an energy barrier between the lowest lying mS (mS represents the spin projection of a state defined by spin S) states which gives rise to the slow relaxation of magnetization (Fig. 3.1). Fig. 3.1 shows the magnetization relaxation barrier for Mn12Ac which has a spin state, S = 10, and this has been used as an example to demonstrate typical SMM behavior. The axial part of the Hamiltonian is diagonal in the |S, mS> basis. The transverse term,
E, causes mixing of states which differ by mS = ± 2 and can lead to quantum mechanical tunneling. The last term in Eq. 3.3 represents higher order terms that need to be considered for k systems with S > 3/2 where Ôn are the Stevens operators. The third term in Eq. 3.1 is the hyperfine interaction which represents the coupling between the electron spin (S) and the nuclear spin (I) through a hyperfine coupling parameter, A. The above approach in describing the spin Hamiltonian is known as the giant-spin approximation. In this representation, only the spin ground state is considered, and it is assumed that the first excited state is much higher in energy.
55 Fig. 3.1 The energy barrier for a Mn12Ac molecule in zero-field. This figure shows the typical energy barrier (ΔE) that prevents reversal of magnetization in a SMM. This figure has been adapted from the following source: J. D. Lawrence, Comprehensive High Frequency Electron Paramagnetic Resonance Studies of Single Molecule Magnets, PhD Dissertation, The University of Florida (2007) [16].
The spin of the system is considered to be the vector sum of the individual spins in the cluster
(for example, the ground state spin is calculated to be S = 10 for Mn12Ac, which has eight ferromagnetically coupled S = 2 Mn3+ centers, all of which are antiferromagnetically coupled to the four S = 3/2 Mn4+ centers in the molecule). This approximation is valid in the strong exchange limit (J >> D), where the exchange interaction (J) between the spins is dominant, allowing them to be represented as a single “giant spin”. This implies that information of the local spin interactions is lost in this description. However, this is often the chosen method for large clusters containing multiple spin centers, and is a useful technique for providing a quantitative description for such spin systems. An alternative representation for the spin Hamiltonian is the multi-spin Hamiltonian, which accounts for the exchange interactions between the spin centers in a molecule. This is the preferred representation for smaller molecules, but would become computationally extremely complicated (since the dimension of the Hilbert space would be (2S+1)(2I+1)) for larger molecules like Mn12Ac.
56 In most conventional SMMs, the intramolecular exchange interactions are strong enough for the ground state spin multiplet to be sufficiently separated from the higher energy levels. The exchange Hamiltonian describes the interactions between unpaired electrons, where Ji,j denotes the isotropic exchange coupling constant. The exchange Hamiltonian for an N-spin cluster can be expressed as Eq. 3.5. Intermolecular exchange interactions are usually very weak in the case of SMMs due to the presence of bulky ligand groups, and there is usually no long-ranged ordering.
N N ˆ ˆˆ H ex J SS jiij … (3.5) i ij As we can see from the pictorial depiction in Fig. 3.1, the height of the energy barrier depends on the magnitude of D and the magnitude of the ground state spin, S. SMMs are thus characterized by the following features: High spin ground state Large negative uniaxial anisotropy Sufficiently strong exchange interaction
These are the typical requirements in building a SMM for it to exhibit slow relaxation of magnetization. As a result of the isolation of the ground state spin multiplet, it can be regarded as a state with a perfectly rigid magnetic moment. For an axial system (E = 0), to the simplest approximation, the zero-field splitting term can be expressed as Eq. 3.6 ˆ 2 H ZFS,axial DS z … (3.6)
As shown through the example of Mn12Ac (Fig. 3.1), a negative axial anisotropy term (D < 0) makes the ground state of the system correspond to the largest value of the spin projection, mS 2 (since Energy = DmS ). Such a barrier thus leads to the bistability of the magnetic moment since it prefers to orient itself either parallel or anti-parallel to the quantization (-z) axis. The property of magnetic bistability, demonstrated by SMMs, make them potential candidates for applications which aim to use electron spin states as bits for information storage and processing. There are two mechanisms that allow this spin state to change: thermal activation and quantum tunneling. Thermally assisted quantum tunneling may also occur as a combination of the two processes. One of the major challenges in synthesizing SMMs is the design of molecules with a large anisotropy barrier and high blocking temperatures.
57 Quantum tunneling is one of the fascinating features exhibited by SMMs where a particle tunnels through a potential barrier, contrary to the laws of classical physics [40]. Let us consider a molecular system with an axial anisotropy barrier. At low temperatures, all the spins populate only the ground state of the system and the barrier prevents the change of the spin projection from the ‘spin up’ to the ‘spin down’ state. However, the introduction of a transverse anisotropy leads to the mixing of the otherwise degenerate states on each side of the barrier and opens up the possibility of quantum tunneling across the barrier. The mixing of states is determined by the 2 matrix elements (
Pure quantum mechanical tunneling occurs at very low temperatures, when only the ground state is populated. In this scenario, the tunnel-splitting gives rise to symmetric and anti-symmetric combinations of the ground state wavefunction, which allows the spin to tunnel through the barrier and reverse itself. Thermally assisted quantum tunneling occurs at higher temperatures when some of the excited spin states are also populated. In the absence of an external magnetic field, this can lead to tunneling between higher states that are at resonance. The presence of magnetic field biases the potential barrier and puts states with different spin projections at resonance with each other. In this situation, quantum tunneling is followed by phonon emission as the spin relaxes back to the ground state. Experimental evidence for quantum tunneling can be observed through the occurrence of steps in the magnetization in a hysteresis curve [51, 52].
EPR spectroscopy is one of the most powerful techniques used in studying SMMs since it allows a direct mechanism for studying spin levels, rather than the approach of thermodynamic measurements [38]. Although commercial EPR instrumentation is largely dominated by spectrometers suited for low frequency (X-band) studies, high-frequency high-field EPR (HF- EPR) has been essential for investigating SMMs with high spin ground states and large zero-field splitting. This technique enables the evaluation of the anisotropy parameters, thereby giving an insight into understanding the magnetic properties of molecular magnets. HF-EPR offers the
58 advantages of high resolution, high sensitivity and the ability of observing the spectra for species that are “EPR-silent” at lower frequencies. Pulsed EPR has also emerged as a preferred technique for probing spin dynamics through spin-echo measurements which give a measure of spin relaxation times of a system. In the work discussed in this chapter, both high-frequency and pulsed EPR techniques have been employed.
Before we conclude our introductory discussion on SMMs, we visit the concept of electron spin coherence. Quantum coherence implies a fixed phase relationship between two wave functions and is as a critical property that is required for a material to be a suitable candidate for quantum computation. In case of molecular magnets, long coherence times were not expected due to the presence of strong dipolar fluctuations in concentrated samples. However, SMMs have been shown to exhibit long spin-spin relaxation times in cases where the degree of decoherence can be suppressed through dilution and cooling down to very low temperatures, so that all spins are polarized. In such systems, with sufficiently long relaxation times, coherent manipulation of spin is possible. As we discuss later in the chapter, the significant feature which makes the study of coherent spin manipulation in holmium-polyoxometalate interesting, is the fact that relatively long coherence times (~ few hundred nanoseconds) were observed even for fairly concentrated samples at temperatures of ~ 4.5 K. This feature not only makes the system an attractive candidate for potential applications, but also offers the opportunity for probing the basic properties which lead to a mechanism for mitigation of spin decoherence. In this chapter, we discuss our EPR studies on this system and propose a possible explanation for the observed relaxation times.
3.2 Mononuclear Lanthanide-based Polyoxometalate Single-Molecule Magnets
In this work, we study a lanthanide-based SMM which is a recent addition to the class of materials exhibiting SMM properties. This section involves a discussion of lanthanides in the context of SMMs and aims to describe the motivation behind studying these systems.
59 3.2.1 Background and Motivation
The study of SMMs has attracted considerable attention due to their potential application in molecular spintronic devices and quantum computation. The basic building blocks for quantum computation [53] are quantum bits (qubits), and recent work by Bertaina et al reported studies on a new family of spin qubits based on rare earth ions which demonstrated remarkably long relaxation times (~ 50 μs) at 2.5 K, thereby suggesting that rare earth qubits might potentially be applicable for quantum information processing at 4He temperatures [54]. Electron spin coherence studies have also been carried out on a number of systems based on transition metal elements, including Fe4 [55] and Fe8 [56, 57]. However, the situation is significantly different in the case of rare-earth ions in crystals, which exhibit strong magnetic anisotropy due to spin-orbit coupling, as we shall see in this discussion. It is significant to mention a few key features that are generally applicable to rare earth systems in order to develop the background for the work that we shall discuss subsequently in the chapter. In case of rare earth systems, the total spin (S) is not a good quantum number due to the presence of strong spin (S) – orbit (L) coupling. This makes the total angular momentum J (J = L + S) the preferred quantum number, which in turn is coupled to the crystal field in these systems. The rare earth elements often have strong hyperfine coupling as a result of interaction between the nuclear spin, I, and the electron spin, S, leading to electro- nuclear crystal field states. This strong hyperfine coupling is an outcome of the “lanthanide- contraction” which causes the f-electron shell to be drawn inwards, towards the nucleus, as a result of an imperfect shielding of f-electrons. This feature has been discussed in Chapter 1.
The properties of rare earth qubits offer certain distinct advantages and also lead to interesting physics, which has made these systems attractive, both from the perspective of gaining a fundamental understanding of the spin systems, as well as for potential applications in spintronic devices. Some of the features which make the study of rare earth qubits interesting are as follows [54] : (i) The crystal field symmetry has a pronounced effect on the energy levels of the system, thereby offering new possibilities for scaling. Although the crystal field effect in lanthanides is weaker than in the case of most transition metal ions, and is usually treated as a perturbation, it nonetheless significantly affects the magnetization 60 relaxation properties. It has been shown in the work by Bertaina et al [58] that the direction of the applied magnetic field and the electric field gradient due to the surrounding ligands affect the relaxation time and the Rabi frequency in case of rare- earth qubit systems. This feature is unlike usual spin-qubits and is unique to rare earth systems due to the magnetic anisotropy associated with strong spin-orbit coupling. Thus, these systems offer the potential for scaling due to their dependence on the crystal field. (ii) Most rare earths have large magnetic moments making spin manipulation possible at relatively low driving magnetic fields [54]. The large value of the total angular
momentum due to spin-orbit coupling gives rise to large magnetic moments (~ 10 μB for Ho3+). (iii) The strong hyperfine coupling produces a large number of states with slightly differing resonance frequencies. Although there have been studies demonstrating that nuclear hyperfine coupling increases the degree of decoherence, thereby making coherent manipulation of spin difficult [59, 60], the presence of the hyperfine coupling actually turns out to be an advantage in certain cases. This is due to the fact that in the case of rare-earths, the strong hyperfine interaction leads to coherent coupling between I and S giving rise to electro-nuclear states, as opposed to the situation in the case of transition metals. The hyperfine interaction produces a large number of qubit states (2I+1) with close resonance frequencies, making it relatively simple to address them selectively for potential applications involving quantum computation. In addition, there have been reports in literature which propose that systems with high A and I show EPR “cancellation resonances” that might contribute to mitigation of decoherence [61]. This is relevant to our study on holmium-polyoxometalate and has been discussed in further details at the end of the chapter.
The use of rare earths in SMMs has therefore become popular due to the novel and interesting spin physics demonstrated by these systems. The properties of rare-earths discussed above make these systems distinct from transition metal elements. Traditionally, most SMMs were based on polynuclear complexes formed by clusters of exchange-coupled transition metal elements
(examples: Mn12Ac, Fe8). However, recent work by Ishikawa et al showed that a SMM can also
61 be obtained by sandwiching a single trivalent lanthanide ion between two phthalocyanine dianion ligands [62]. This led to efforts in synthesizing SMMs based on a single lanthanide ion rather than the traditional approach of building multi-nuclear transition metal complexes. One significant advantage of mononuclear systems is that they are relatively simple in terms of the control we have over their co-ordination chemistry. Smaller molecules ensure better control in manipulating ligands and obtaining the desired properties. Another feature that makes small molecules attractive is the relative ease of understanding the interactions within the molecule. Polynuclear systems with a large number of spin centers can only be treated quantitatively through the giant-spin approximation, whereby all excited spin-states and local exchange interactions are ignored. Smaller molecules, however, can be studied using the multi-spin model for a more detailed understanding of the spin interactions within the molecule. The sample we discuss in this chapter is a mononuclear system with a trivalent holmium ion as the magnetic core. Besides computational ease, these systems also offer another significant advantage. The magnetic properties in these mononuclear systems are associated with only a single ion, thereby making it simpler to carry out dilution and isotope purification for reduction in spin decoherence due to dipolar couplings. AlDamen and co-workers successfully synthesized SMMs with a lanthanide ion encapsulated by a polyoxometalate (POM) cage [63], where the lanthanide has similar co-ordination chemistry as that of the bis(phthalocyaninato) lanthanide complex reported by Ishikawa. A detailed account of the molecular structure has been provided in the following section. The encapsulation by the POM cage in these complexes offers the advantage of preserving bulk SMM properties outside of the crystal, which makes them scalable and addressable on surfaces by techniques like that of a scanning tunneling microscope [64]. These complexes were synthesized with various lanthanide cores including Tb, Dy, Ho, Er, Tm and Yb. Extensive magnetostructural characterization has been reported on all the members of this family, and it was observed that the Tb, Dy, Ho and Er derivatives of the series showed characteristic SMM behavior [65]. Our system of interest in this work is the holmium-based polyoxometalate SMM: Na9[HoxY1-x(W5O18)2].xH2O, hereafter referred to as HoPOM. We carried out continuous wave (CW) EPR measurements on single crystals of the complex at multiple frequencies. Pulsed EPR spectroscopy was also performed, and the measurement of relaxation times on samples with various concentrations of Ho3+ spins showed fairly large coherence times even for concentrated samples. Simulations of the ground state energy levels
62 9- 9- Fig. 3.2 Different views of [Ln(W5O18)2] cluster in HoPOM (a) Structure of [Ln(W5O18)2] cluster in HoPOM (b) & (c) Representation showing the co-ordination of the central Ln3+ ion in the cluster. Purple: Lanthanide, Grey: Tungsten, Red: Oxygen. Fig. taken from Ref. [65]. and EPR spectra were carried out in an effort to understand these interesting results, and also to evaluate the Hamiltonian parameters by comparison with the experimental spectra.
3.2.2 HoPOM: Structural Information and Synopsis of Previous Studies
The HoPOM cluster can be described as a “double-decker” structure, where the holmium ion is 6- sandwiched by two anionic [W5O18] groups [65]. The bulky POM ligand provides good isolation of the anisotropic Ho3+ ions. The anionic clusters are surrounded by sodium cations that are octahedrally coordinated with oxygen atoms. The Na+ cations interact with the highly 9- charged [Ho(W5O18)2] anions for electroneutrality, and thereby introduce a large number of 9- water molecules in the lattice [65]. The structure of the [Ho(W5O18)2] cluster can be seen in Fig. 3.2. We pay close attention to the co-ordination geometry of the central Ho3+ ion since this determines the symmetry, and thereby the properties of the system. The angle Φ marked in Fig. 3.2 (c) can be defined as the offset between the two squares defined by the planes through the coordinating oxygen atoms. As shown in the Fig. 3.2 (c), the angle Φ corresponds to 44.2o [65], o which is very close to the value of Φ = 45 that is expected for an ideal D4d symmetry [66]. The
63 system therefore corresponds to a geometry that gives rise to an approximate D4d ligand field
(LF) symmetry. The deviation from the ideal D4d configuration, although seemingly small, is sufficient to completely change the magnetic relaxation properties of the system. In this work, we show how this symmetry configuration strongly affects the properties of HoPOM through an analysis of our EPR studies on the system.
In this class of nanomagnets, the magnetic anisotropy required for slow relaxation of the magnetization exhibited by SMMs, arises as a result of the splitting of the J ground state of the Ho3+ ion in the presence of a LF [67]. As we have discussed in Section 3.2.1, the strong spin- orbit coupling in these systems makes the total angular momentum, J, the good quantum number. For certain LF symmetries, the splitting of the J ground state can stabilize sublevels with large
|mJ| values, and thus lead to a quantization axis of magnetization. The work done by AlDamen and co-workers reports dc magnetic susceptibility, specific heat and NMR measurements on the III 9- 3+ 3+ 3+ 3+ members of the [Ln (W5O18)2] (Ln = Tb , Dy , Ho and Er ) family, and an evaluation of the LF parameters based on these measurements [65]. Ishikawa et al developed a procedure for determining the LF parameters for pthalocyaninatolanthanide complexes through least-squares fitting of the magnetic susceptibility data [68]. This procedure was followed in the work by AlDamen et al, where the LF parameters, and thereby the energy level scheme of the ground state multiplet for HoPOM has been reported [65].
The Spin Hamiltonian for the system can be expressed as follows: Hˆ Zeeman term Hyperfine coupling term LF Hamiltonian … (3.7) In this discussion, we focus on the LF term which describes the effect of the field due to the surrounding ligands on the central paramagnetic core. Molecular symmetry is the key factor that determines the ligand field parameters of a system and is therefore crucial for an understanding of the spectroscopic transitions. From symmetry considerations, the LF Hamiltonian for HoPOM may be expressed as Eq. 3.8, following C4 point group symmetry [65]. A point group is a set of symmetry operations for which at least one point remains fixed for all operations in the relevant mathematical group. C4 denotes 4-fold rotational symmetry about an axis, and the LF Hamiltonian consistent with this symmetry is shown below: ˆ 20 ˆ 0 40 ˆ 0 44 ˆ 4 60 ˆ 0 64 ˆ 4 H A2 r O2 A4 r O4 A4 r O4 A6 r O6 A6 r O6 … (3.8)
64 3+ k In the above equation, α, β and γ are Stevens constants for Ho [8], Ôn are the Stevens k k operators, r are radial factors and An are numeric parameters. In the above equation we have considered terms up to the 6th order since it is not possible to have multipole distributions with n k > 6 for the f-shell. The Stevens operators Ôn relevant to our system are: ˆ 0 2 O2 3Jz (JJ )1 ˆ 0 4 2 2 O4 35Jz 30( (JJ )1 )25 Jz ((3( JJ ))1 (6 JJ ))1 ˆ 0 6 4 2 2 O6 231Jz (315 (JJ )1 735)Jz (105 (( JJ ))1 525 (JJ )1 294)Jz ((5( JJ ))1 3 ((40 JJ ))1 2 60 (JJ ))1 … (3.9)
ˆ 4 1 4 4 O J J 4 2 ˆ 4 1 2 4 4 O 11J (JJ )1 ,38 J J 6 2 z AB BA where J J iJ ,J J iJ andA,B x y x y 2 These operators can be obtained by using the operator equivalent method where the angular momentum operators have the same properties as the spherical harmonics used in the expansion k of electrostatic potentials of the appropriate symmetry [8, 10]. The operators Ôn with k = 0 correspond to the axial terms and the ones with k ≠ 0 represent the transverse terms. Given that the geometry of HoPOM corresponds to an approximate D4d symmetry, the LF Hamiltonian was further simplified by AlDamen et al [65] to include only the axial terms. This serves as a reasonably good approximation for fitting thermodynamic measurements which are not sensitive to the higher order transverse terms that lead to a mixing of states, but do not considerably affect the overall energy level picture. Thus, following this approximation, the two transverse terms with k ≠ 0 vanish to give the LF Hamiltonian shown in Eq. 3.10. ˆ 20 ˆ 0 40 ˆ 0 60 ˆ 0 0 ˆ 0 0 ˆ 0 0 ˆ 0 H A2 r O2 A4 r O4 A6 r O6 B2 O2 B4 O4 B6 O6 … (3.10)
The above axial Hamiltonian corresponds to D∞h symmetry which can be expressed as follows:
D∞h ≡ C∞ X Ci [66]. This is a high symmetry group where C∞ represents cylindrical symmetry k k and Ci represents inversion symmetry. The authors report the values of the LF parameters (An r ) k k 9- consistent with this simplified LF Hamiltonian. The parameters (An r ) for the [LnW10] [Ln = Tb, Dy, Er and Ho] series have been tabulated by AlDamen et al [65]. Table 3.1 shows the LF parameters [65] and Stevens constants [8] for Ho3+, from which we evaluated the coefficients 0 0 0 B2 , B4 and B6 for the HoPOM system. Following the procedure developed by Ishikawa et al,
65 Table 3.1: Tabulation of LF parameters and Stevens constants for Ho3+ in HoPOM [8, 65]
0 2 0 4 0 6 0 0 0 A2 r A4 r A6 r B2 B4 B6 α β γ (cm-1) (cm-1) (cm-1) (MHz) (MHz) (MHz)
1 1 5 -270.4 -202.0 37.7 18027 202 -1.4 450 30030 3864861
the LF parameters were calculated by AlDamen et al [65] from simultaneous fits to χmT values over a temperature range of 2 K to 300 K. It can be seen from Ref. [65] that good agreement was obtained between the experimental data and the theoretical expectation based on the calculated parameters. The energy level scheme of the HoPOM system has also been reported by carrying out the diagonalization of the (2J + 1) dimension eigenmatrix that can be constructed from the Hamiltonian of the system using the calculated LF parameters. Ho3+ has the quantum numbers S = 2, L = 6, J = 8 and I = 7/2. Since Ho3+ is an integer spin system, it is a Kramers ion. Eq. 3.11 shows the Hund’s rule of coupling, from which gJ was evaluated to be 1.25 JJ 1 L L 1 SS 1 g 1 25.1 … (3.11) J 2 JJ 1 3+ The ground state doublet for Ho has been reported to be mJ = ± 4 [65]. The first excited state corresponding to mJ = ± 5 is well separated from the ground state by an energy gap of approximately 16 cm-1 [65]. Fig. 3.3 shows a pictorial depiction of the energy level scheme worked out by AlDamen et al [65]. Given the large separation between the ground state and the excited states, we only focus on the mJ = ± 4 states in our EPR experiments.
9- 3.3 High Frequency Single Crystal EPR of [HoxY1-x(W5O18)2]
EPR experiments were performed on the mononuclear holmium polyoxometalate system, 9- 3+ 3+ [HoxY1-x(W5O18)2] , where x denotes the concentration of the Ho ion. Samples with Ho concentration corresponding to 100%, 25%, 10%, 0.65% and 0.1% were used for our experiments. The non-magnetic element, Yttrium (Y), was added in required proportions to
66 Fig. 3.3 A pictorial representation of the crystal field scheme for HoPOM as reported in Ref. [65] by AlDamen et al through calculation of LF parameters. We can see from this figure that the ground state doublet for HoPOM is mJ = ±4, which is sufficiently separated from the higher energy levels. achieve the diluted samples of various Ho3+ concentrations. High frequency EPR studies were carried out on dilute single crystals of HoPOM with 25% concentration of Ho3+ (x = 0.25), at frequencies ranging from 50 to 150 GHz in a 7 T superconducting Quantum Design magnet. The experiments were conducted using the rotating cavity technique (described in Section 2.1 of Chapter 2) while employing a millimeter vector network analyzer associated with several different sources and multipliers as a spectrometer [17, 24]. The continuous wave (CW) spectra at 50.4 GHz show eight spectral lines due to hyperfine coupling to the 165Ho nucleus (I = 7/2)
(Fig. 3.4). Fig. 3.5 shows a schematic representation of the energy level diagram for mJ = ± 4 states in the presence of hyperfine interactions, which lift the degeneracy of each mJ level, splitting them into (2I +1) levels corresponding to the nuclear spin projections, mI. The arrows in the figure represent the expected transitions between the mI multiplets, following the selection rule mI = 0. This energy level diagram has been constructed by assuming a simple two-level 0 0 0 picture for the system including only axial terms (B2 , B4 and B6 ) in the LF Hamiltonian. This 67 Fig. 3.4 Temperature dependence of EPR spectra for HoPOM (x = 0.25) at 50.4 GHz. The eight lines seen in the spectra represent the hyperfine transitions due to strong coupling with the 165Ho nucleus. was done following the work of AlDamen et al [65] using the same parameters as those reported in Ref. [65]. In the discussion that follows, we show that this assumption provides a good fit to our experimental data at high frequencies. However, it is significant to note that this picture corresponds to ΔmJ = 8 transitions, which are normally forbidden by EPR selection rules. Therefore, the observation of the spectra shown in Fig. 3.4 indicates that a transverse term is required in the Hamiltonian to allow these strongly forbidden transitions through mixing of states. As we shall see later in our discussion, the high frequency EPR peak positions are insensitive to this transverse term, although it is essential to explain and understand the EPR spectra both at high and low frequencies. Fig. 3.4 shows the EPR spectra at a frequency of 50.4 GHz over a temperature range of 2 K – 10 K, and it can be seen that the expected eight hyperfine transitions consistent with the energy level diagram shown in Fig. 3.5 were observed. The data shows that the intensity of the signal reduces as the temperature is increased. Since the population of energy levels is guided by Boltzmann statistics, the temperature dependence of
68 Fig. 3.5 Energy level diagram showing the splitting of the mJ ± 4 states in the presence of hyperfine coupling with 165Ho nucleus. This energy level diagram corresponds to a LF Hamiltonian with only axial terms. The blue arrows represent the expected EPR transitions at 50 GHz.
EPR spectra indicates whether the observed transitions are ground state transitions or not. This is due to the fact that only the ground state is populated at very low temperatures, while higher levels start to get populated as the temperature is increased. The variation of the signal intensity in the temperature dependent spectra shown in Fig. 3.4 indicates that they are ground state transitions.
The spectral lines show clear angle dependence at 50.4 GHz as a result of strong anisotropic ligand field associated with the Ho3+ ions. An angle-dependent experiment was carried out while rotating the crystal about two orthogonal axes, in order to determine the orientation of the quantization axis of the sample. The physical appearance of the single crystal resembled that of a rectangular parallelepiped, and studying the angle dependence of the EPR transitions showed that the easy-axis of the crystal does not fall on the crystal faces. The procedure of two-axis rotation enabled the exact identification of the quantization axis of magnetization. A preliminary
69 Fig. 3.6 Schematic representation of the planes of rotation for the two-axis rotation capability. The figure shows the sample placed on the quartz pillar in the rotating cavity in the presence of a transverse magnetic field. This figure has been taken from Ref. [24] and adapted. description of the two-axis rotation technique using a cylindrical rotating cavity in the Quantum Design PPMS magnet has been provided in Chapter 2. For this experiment, the sample was placed on the quartz pillar mounted on the end-plate of the resonant cavity (Refer to Fig. 2.2 in Chapter 2 for a description of the parts of the rotating cavity). The worm - gear assembly of the cavity allows rotation of the sample about one axis (φ-rotation), while a motor attached to the magnet system enables rotation of the probe about an orthogonal axis (θ-rotation). Fig. 3.6 shows a schematic depiction of the planes of rotation of the sample in a two-axis rotation experiment. Fig. 3.7(a) shows one set of representative angle dependent spectra at 50.4 GHz and at a temperature T = 2 K. Ten such data sets were collected for different cavity end-plate orientations, and the minimum peak position of the observed transitions was identified to locate the quantization axis. It was observed that the lines show considerable broadening as we move away from the quantization-axis towards the hard plane. The plot of the peak positions from Fig 3.7(a) as a function of the orientation has been shown in Fig. 3.7(b). In order to identify the orientation corresponding to the minima of the peak positions, the hyperfine splitting was first
70 Fig. 3.7 EPR spectra for HoPOM (x = 0.25) single crystal at 50.4 GHz at T = 2 K. (a) Angle dependent EPR spectra for HoPOM single crystal at 50.4 GHz showing clear angle dependence of the hyperfine transitions. (b) Plot of the peak positions showing the 8 mI transitions corresponding to the spectra shown in (a). Inset: Polynomial fit to the data after collapsing the corresponding mI transitions. This figure shows one set of representative angle dependent spectra for the two-axis rotation experiment carried out to identify the easy-axis orientation.
71 Fig. 3.8 This figure shows the minimum peak positions, extracted from the angle-dependent data for each end-plate orientation, with the hyperfine interaction ignored. The blue arrow points to the end-plate orientation at which the minima for the transitions was observed. The red line shows the polynomial fit to the data points.
ignored and all the mI levels were collapsed into one, as shown in the inset of Fig. 3.7(b).
Ignoring higher order contributions, the energy difference corresponding to the observed ΔmI = 0 hyperfine transitions between mJ = ±4 levels can be expressed as Eq. 3.12.
EmJ ,mI 2g J B BmJ 2Am mIJ … (3.12)
In the above equation, mI can take values ranging from +I to –I in steps of 1, which in this case ranges from +7/2 to -7/2. The set of eight curves shown in Fig. 3.7(b) differ only in their mI values, and therefore these lines can be collapsed into one (as shown in the inset of Fig. 3.7(b)) by taking an average. The polynomial fit to the data was used to determine the minimum field position of the transition for a given orientation of the cavity end-plate. Similar plots were made for each of the ten end-plate orientations at which the angle-dependent experiment was carried out. After identifying the minimum point from each set, they were plotted as a function of the end-plate orientation angle. Fig. 3.8 shows the minimum field positions of the transitions, corresponding to the various end-plate orientations. The end-plate position marked ‘5’ in Fig. 3.8
72 Fig. 3.9 Comparison of simulated and experimental peak positions corresponding to angle dependent EPR spectra shown in Fig. 3.7. The symbols represent the experimental data points and the lines are simulated. It can be seen from this figure that a reasonable degree of correspondence was observed between the experimental and simulated spectra. The g-value and the hyperfine coupling constant were evaluated through a comparison of the experimental and simulated spectra at 50.4 GHz, corresponding to the orientation where the quantization axis of the crystal is parallel to the external magnetic field. corresponds to the configuration for which experimental data has been shown in Fig. 3.7, and represents the true minima of the angle-dependent EPR spectra. Thus, the two-axis rotation experiment enabled us to identify the crystal orientation for which the observed transitions were at the lowest field positions. A single-axis rotation, which is more commonly performed, is not sufficient for experimentally determining the easy-axis of a sample if there is no way to identify the correct plane of rotation. This work provides one such example where the ability to rotate the sample about two orthogonal axes is extremely valuable, making it possible to achieve the correct orientation in the absence of any prior knowledge of the crystallographic planes and axes.
The g-value and the hyperfine constant, A, were assumed to be isotropic and were evaluated by fitting the experimental spectra. The program ‘Easyspin’ [69, 70] which is based on the Matlab platform was used for all simulations reported in this chapter. The ‘Easyspin’ program ‘pepper’
73 was used for simulation of CW EPR spectra by defining the system parameters and the 0 experimental conditions. The parameters used for the simulation are as follows: B2 = 18027 0 0 0 0 0 MHz, B4 = 202 MHz and B6 = -1.46 MHz. The values of B2 , B4 and B6 used in these simulations are the same as those reported by AlDamen et al [65]. We started with an approximate estimate for the values of isotropic g (~ 1.26) and A (~ 850 MHz), and then varied these parameters to obtain a good correspondence with the experimental spectra at 50.4 GHz. Fig. 3.9 shows a comparison between the experimental and simulated spectra for HoPOM. The values of g and A as determined by simulating the EPR spectra at 50 GHz are as follows: g = 1.246, A = 871 MHz.
The g-value estimated experimentally agrees well with the value of gJ (1.25) predicted by Hund’s rule. While we cannot rule out the possibility of some degree of anisotropy in g and A, we note from our simulations, that we can obtain a reasonable degree of agreement using isotropic parameters. Thus, in our high frequency EPR experiments, the expected eight hyperfine transitions were observed, and the experimental data was analyzed to yield the values of the parameters g and A for HoPOM.
9- 3.4 Low Frequency Single Crystal EPR of [HoxY1-x(W5O18)2]
Low frequency EPR studies on the sample were carried out in a commercial X-band Bruker ELEXSYS spectrometer at a frequency of 9.7 GHz. Fig. 3.10 shows the angle dependence of the 9- CW EPR spectrum of [HoxY1-x(W5O18)2] (x = 0.25) at T = 4.5 K. Electron spin echo measurements were also carried out on the sample using a Hahn echo sequence of /2−−, with two microwave pulses of duration 12 ns and 24 ns respectively with a separation of 120 ns. The technique of pulsed EPR spectroscopy and the mechanism of obtaining a Hahn echo by applying multiple pulses have been discussed in Chapter 2. The echo-detected (ED) spectra for the sample with 25% concentration of Ho3+ show the presence of three strong peaks. Additional features, which were comparatively lower in intensity, were also observed in between the main features in the ED spectra. We note here, that the experimental set-up for the X-band measurements did not have the mechanism to orient the crystal with its quantization axis exactly parallel to the external
74 Fig. 3.10 Angle dependence of CW EPR spectra for HoPOM (x = 0.25) single crystal at 9.7 GHz.
Fig. 3.11 Optimization of parameters for spin echo experiment. The above figures show the ED spectra for HoPOM (x = 0.25) at 9.7 GHz at T = 4.5 K. (a) The dependence of the ED spectra on the pulse sequence at a constant power attenuation of 10 dB. (b) The dependence of the ED spectra on the microwave power attenuation using the pulse sequence: 12-120-24 ns.
75 magnetic field. Hence, although we carried out angle dependent studies, the orientation corresponding to the minimum field values for the observed transitions does not necessarily represent the exact easy-axis position. However, from our high-frequency angle dependent measurements, we have an estimate of the degree of deviation, and can therefore still interpret our low frequency results and draw significant conclusions from them. The pulse sequence and the microwave power attenuation were varied in order to determine the parameters optimum for the experiment. This optimization experiment was carried out in order to maximize the signal strength of the echo, and also to check if any of the observed spectral features changed significantly as a result of altering the pulse-sequence or the microwave power. To accomplish this, the ED spectra was first recorded for various pulse sequences keeping the microwave power attenuation at a constant value of 10 dB. Next, the ED spectra were also recorded for various power attenuations with the same pulse sequence. The optimized parameters of 12-120-24 ns for the pulse sequence and an attenuation of 6 dB for the microwave power were henceforth used for the spin-echo measurements throughout the experiment. Besides the pulse sequence and the microwave power, other experimental parameters that were optimized include receiver gain, modulation amplitude, number of averages and sweep rate. The variations of the ED spectra with pulse sequence and power attenuation are shown in Fig. 3.11. We observe from the figure that each of the more intense peaks exhibit splitting. This splitting is probably related to instantaneous spin diffusion effects. Spin diffusion denotes the spread of magnetization between spins through dipolar coupling, and occurs when spins with the same resonance frequency before a pulse have different resonance frequencies after the application of the pulse [28]. This effect is more likely to occur in concentrated samples as a result of a change in the local fields for individual spins due to spin-spin interactions. We note from the first four spectra shown in Fig. 3.11(a) that the degree of splitting in the peaks changes as a function of the length of the second pulse. This indicates that instantaneous spin diffusion is the most likely cause of the observed splitting in the intense ED peaks. A thorough test can be performed to confirm this explanation by studying the spectra for a larger number of pulse sequences, which would help establish the effect of the length of the second pulse on the ED spectra [71, 72]. Another possible factor that could lead to the observed splitting is the non-linear and complex nature of the energy levels at low excitation frequencies. However, the effect of the pulse lengths on the spectra suggests that instantaneous spin diffusion is the most likely cause of this observed effect. The CW and ED
76 Fig. 3.12 Measurement of T2 for HoPOM single crystal. (a) T2 for HoPOM single crystal (x = 0.25) at T = 4.5 K at B ~ 0.2 T. The exponential fit to the decay curve for determining T2 (~ 92 ns) has been shown in red. Inset: Measurement of T2 for HoPOM single crystal (x = 0.0065) at T = 4.5 K at B ~ 0.2 T. The oscillations seen in the data correspond to the observation of ESEEM effects for samples of lower dilutions. The exponential fit to the decay curve for determining T2 (~ 455 ns) has been shown in red.
EPR experiments described above for the sample with 25% concentration of Ho3+, were repeated for the more dilute crystals in order to carry out a comparative study of our results while varying the concentration of Ho spins. An ED spectrum could not be observed for the pure (100% concentration) sample.
3+ The transverse relaxation time (T2) was measured for samples with Ho concentrations of 25%,
10%, 0.65% and 0.1%. For measuring T2, the separation (τ) between the π/2 and π pulses was successively increased over time, while measuring the echo amplitude. Thus, the decay in the echo-height as a function of time was recorded. This can be fit using an exponential decay function in order to estimate the value of T2. Fig. 3.12 shows the experimental data for a T2 measurement for a single crystal with 25% Ho3+ concentration, and the exponential fit to the
77 Table 3.2: Tabulation of T2 values for HoPOM single crystals of various dilutions.
Concentration 25% 10% 0.65% 0.1% (% of Ho3+)
T2 (ns) 92 120 450 600
decay curve for determination of the relaxation time. The value of T2 was found to consistently increase with an increase in dilution of the sample. This is expected since spin dipolar couplings, which are a major source of spin dephasing, are reduced with an increase in the degree of sample dilution. The values of T2 at magnetic field values corresponding to the most intense ED peaks have been tabulated in Table 3.2 for samples of various concentrations.
Electron spin echo measurements have been reported on a number of transition metal as well as rare earth systems [54-58, 73, 74]. T2 values measured for some of these systems are as follows: 3+ ~ 0.63 μs for Fe8 [56], ~ 630 ns for Fe4 [55], ~ 50 μs for Er :CaWO4 [54] and 3 μs for deuterated samples of Cr7Ni [73]. In the works cited here, sample dilution (for example: Er -4 -6 3+ concentration of 10 to 10 in Er :CaWO4; concentration of 0.5 mg/mL for Fe4 where the sample was diluted into a frozen solvent matrix) [54, 55, 58], cooling down to very low temperatures [74] and application of a high magnetic field ~ 10 T to polarize the spins in Fe8 [56] have been demonstrated to be effective means for increasing the coherence time. It is known that dipolar interactions, hyperfine coupling and spin-phonon interactions contribute to decoherence in a system [59, 60, 75]. Hence, the rather long relaxation times (~ 100 ns) observed in the case of fairly concentrated (25% and 10%) samples of HoPOM at T ~ 4.5 K are striking, and suggest that there is a mechanism facilitating mitigation of spin decoherence due to nuclear hyperfine and electron dipolar interactions in these systems. This will be discussed further in the following section of this chapter where we try to correlate the nature of the energy levels in the presence of the transverse LF contribution, with the observation of long coherence times in our spin echo experiments. This observation is of great interest since it presents an example of a system where there is a controllable means of mitigating decoherence without employing any of the above- mentioned known techniques.
78 3+ Fig. 3.13 Log-log plot showing the variation of T2 with Ho concentration..
In case of diluted samples with 0.65% and 0.1% concentrations of Ho3+, Electron Spin Echo Envelope Modulation (ESEEM) effects were observed while measuring the spin relaxation times (shown as an inset in Fig. 3.12). ESEEM effects are caused by the interaction between the electron spin and neighboring nuclear spins. It is experimentally manifested through the occurrence of periodic oscillations in the echo-intensity and carries signatures of the nearby nuclei and the local environment. It can be seen in the inset of Fig. 3.12 that ESEEM modulations are observed superimposed on the decay curve typical for a T2 measurement. In order to extract the frequency of the modulations, the exponential decay was first subtracted from the data. Next, a Fourier transform of the subtracted signal was performed to obtain the value of the frequency. The nuclear Larmor frequency for protons (1H) is 42.5 MHz/T. Thus, the frequency for protons at a field of 0.1605 T (the value of the magnetic field corresponding to which the ESEEM data has been shown in Fig. 3.12) is 6.8 MHz. An estimate of the ESEEM frequency ~ 6.6 MHz from the experimental data, therefore, suggests that they are due to the protons present in the solvent of the crystals.
79 Fig. 3.14 Plot of ED spectra and the variation of T2 with magnetic field for HoPOM single crystal (x = 0.25) at a frequency of 9.7 GHz at T = 4.5 K. This plot demonstrates the T2 weighted nature of the ED spectrum.
Fig. 3.13 shows the variation of the T2 time for samples of different concentrations. The log-log plot, based on the four dilutions studied, shows a non-linear variation of T2 with concentration, which is indicative of the fact that dipolar coupling is not the only mechanism leading to spin dephasing. Our preliminary findings based on comparison of the T2 values of HoPOM single crystals in protonated and deuterated solvents (discussed later in Section 3.6) indicate that there are two factors affecting the relaxation mechanism: (i) dipolar spin coupling and (ii) the coupling between the Ho electron spin and the solvent nuclei. Therefore, in order to accurately establish the true nature of the variation of T2 with sample concentration, the evaluation of relaxation times of samples of a larger number of dilutions is required. For the sample with 25% Ho3+ concentration, the spin relaxation time was measured over the entire field range (0 T – 0.35 T) of the spectrum, while increasing the magnetic field values in small steps of 50 G. The experimental data for this measurement has been shown in Fig. 3.14 which clearly demonstrates
80 Fig. 3.15 Variation of T2 as a function of temperature for HoPOM single crystal with 25% Ho3+ concentration. Measurements were attempted at higher temperatures, but the decay time was too small to be estimated with a reasonable degree of accuracy.
the T2 weighted nature of the ED spectrum. It can be seen from the figure that the variation of T2 follows a very similar pattern as that demonstrated by the ED spectrum.
The temperature dependence of T2 has been shown in Fig. 3.15. This measurement could only be carried out over a small temperature range of 4.5 K – 10 K. At higher temperatures, the signal intensity was very weak making the data too noisy for an accurate estimate of the relaxation times. Due to instrumental limitations of a flow cryostat, stable temperatures lower than 4.5 K could not be achieved. Therefore, the pulsed measurements reported were carried out at the lowest achievable stable temperature of ~ 4.5 K. The longitudinal relaxation time (T1) was also measured, and it was estimated to be approximately of the order of 1 µs. The T1 measurement was carried out by the application of an inversion pulse (π pulse), followed by a Hahn echo sequence for detection. As defined in Chapter 2, T1 is a measure of the rate of relaxation along the longitudinal direction, and occurs as a result of the dissipation of energy into the lattice.
81 Fig. 3.16 Rabi oscillations observed for HoPOM single crystal (x = 0.25) at T = 4.5 K. The plot shows the variation of the observed Rabi oscillations with power attenuation. Inset: The linear variation of the Rabi frequency with the microwave field.
Thus, T1 estimates the spin-lattice interaction and acts as one of the factors contributing to decoherence. T1 and T2 can be mathematically extracted by solving for the time evolution of magnetization as expressed by the Bloch equations.
Clear coherent electron-spin oscillations were observed by measuring the variation in echo amplitude as a function of the first pulse. The pulse sequence for the Rabi measurements was as follows: variable pulse – T- /2 − − - τ - echo, where the length of the variable pulse was successively increased in steps of 4 ns. Fig. 3.16 shows Rabi oscillations (quantum oscillations resulting from the coherent absorption and emission of photons driven by an electromagnetic wave [32, 33]; discussed in Chapter 2) observed in case of the sample with 25% concentration of Ho3+. Rabi oscillation measurements were carried out for various power attenuations of the microwave field and the linear dependence of the Rabi frequency with the microwave field B1 is shown as an inset in the figure. Similar experiments for observing Rabi oscillations were also
82 Fig. 3.17 Energy level diagram showing the mJ ± 4 states in the presence of hyperfine coupling with 165Ho nucleus. This energy level diagram corresponds to a LF Hamiltonian with only axial terms. The blue arrows represent the expected EPR transitions at 50 GHz. The red arrows represent the expected EPR transitions at 9.7 GHz. The green arrows represent the field positions of the observed EPR transitions at 9.7 GHz. This schematic representation shows the difference between the observed transitions at low frequency and that expected from this energy level picture. The low field regime on this figure has been highlighted to show the contrast between the nature of the levels with (shown in Fig. 3.18) and without a transverse term in the LF Hamiltonian. successfully performed on the more dilute HoPOM samples. In the context of potential application of molecular magnets in quantum computation, the observation of Rabi oscillations in V15 cluster was recently reported by Bertaina et al [76]. This work established that quantum oscillations, indicative of long-lived quantum coherence, can be achieved in molecular magnets if the decoherence due to dipolar fluctuations can be significantly reduced. The observation of Rabi oscillations in HoPOM shows that long-lived coherent spin manipulation is possible in these SMMs. The long T2 relaxation times and observation of Rabi oscillations make the system an interesting candidate for potential applications.
83 4 Fig. 3.18 Simulated energy level diagram with the inclusion of the transverse (B4 ~ 91.5 MHz) term in the LF Hamiltonian. It can be seen that the energy level diagram in the low field regime changes significantly from that shown in Fig. 3.17 and opens up a gap of approximately 9 GHz that allows us to observe the EPR transitions at X-band.
3.5 Discussion of EPR Results on HoPOM
In order to interpret and understand our experimental results on HoPOM, let us go back to the discussion of the LF Hamiltonian we introduced in Section 3.2.2 earlier in the chapter. In the treatment followed by AlDamen et al, an exact D4d symmetry of the molecule had been assumed, and a good fit to the thermodynamic data was obtained [65]. However, we note that although we could understand our high frequency (50.4 GHz) EPR results based on this Hamiltonian (Fig. 3.9), it does not agree with the low frequency (9.7 GHz) experimental spectra. Fig. 3.17 shows the energy level diagram for the system with only the axial terms in the LF Hamiltonian. This representation is the same as the picture that has already been shown in Fig. 3.5, but also includes the field positions of the expected and the observed transitions at 9.7 GHz. It can be seen from the figure (through the mismatch between the positions of the red and green arrows), that our experimentally observed transitions do not correspond to what was expected from this
84 energy level diagram. This shows that our representation of the system as shown in Fig. 3.17 is incorrect, at least in the low field region. By introducing a transverse term in the LF Hamiltonian, we observed that the presence of this term, which leads to a mixing of states, is required in order to explain the spectra observed at low frequencies. We note here that the observation of the strongly forbidden ΔmJ = 8 EPR transitions at 50 GHz was the first indication of the presence of transverse anisotropy. However, at higher frequencies the positions of the EPR peaks are relatively insensitive to the contribution of this transverse term, thereby allowing us to obtain good agreement between experiment and simulation based on an axial LF Hamiltonian. Fig. 3.18 shows the simulated energy level diagram corresponding to the LF Hamiltonian with a 4 B4 term, as shown in the figure. In this simulation, the parameters reported by AlDamen et al 4 have been used for the axial terms [65], and a transverse term, B4 (~ 91.5 MHz), has been included. It can be seen from the figure that the addition of this transverse term transforms the energy level diagram and opens up a gap of approximately 9 GHz which just allows us to observe the low frequency spectra at X-band. The inclusion of this transverse term in the Hamiltonian is also consistent with symmetry considerations, since the co-ordination geometry 4 of the holmium ion does not actually conform to an exact D4d symmetry. The value of the B4 term was determined through attempts at matching our experimental spectra at 9.7 GHz with the CW spectra simulated using the ‘Easyspin’ program [69]. Fig. 3.19 shows a comparison between 4 the experimental and simulated spectra, and we note that after the inclusion of the B4 term in the LF Hamiltonian, we were able to obtain reasonably good correspondence between the two spectra. It was observed that the low frequency spectrum was extremely sensitive to the frequency as well as the crystal orientation, and better agreement between the simulated and the experimental spectra at 9.7 GHz was not possible due to uncertainties in the crystal orientation.
The simulations presented thus far were carried out using exactly the same axial LF parameters as reported by AlDamen et al [65], while varying only the transverse term to observe the closest correspondence. The parameters of the Hamiltonian presented here seem to be a reasonably good estimate; however, in the presence of a large number of adjustable variables, it is difficult to well constrain all the parameters simultaneously. Thus, while we obtain reasonable correspondence by using the axial LF parameters exactly as reported in Ref. [65], we note that there is room for slightly varying the axial LF terms too. Although the axial parameters that were evaluated by
85 Fig. 3.19 Comparison of experimental and simulated spectra for HoPOM at 9.7 GHz. The figure shows good correspondence between the two spectra with the transverse term in the LF Hamiltonian.
AlDamen et al [65] by fitting the χmT data agree very well with the thermodynamic measurements, it is important to realize that these measurements are not as sensitive to the anisotropy parameters as the EPR spectra. Our attempt at simulating the angle dependent EPR spectra at X-band is shown in Fig. 3.20. Comparing the experimental and simulated spectra shown in Fig. 3.20 in bold and thin lines respectively, we see that we are able to reproduce most of the features observed in the angle dependent experiment. The values of the LF parameters 0 0 0 used for these simulations are as follows: B2 = 18027 MHz, B4 = 209 MHz, B6 = -1.53 MHz 4 0 0 and B4 = 94 MHz. This set of parameters varies slightly from the B4 and B6 terms obtained 0 0 0 from Ref. [65] (earlier values from Ref. [65] are: B2 = 18027 MHz, B4 = 202 MHz and B6 = - 1.46 MHz). The high frequency EPR spectra are not too sensitive to this small change, but they have a much stronger effect on the low frequency spectra at X-band. The spectra at 9.7 GHz are extremely sensitive even to small changes in the frequency as well as the transverse LF term, so it is not possible to have an accurate evaluation of a unique set of parameters with a high degree of certainty. However, we can deduce from the comparison of experiment and simulation (Fig.
86 Fig. 3.20 Comparison of experimental and simulated angle dependent CW spectra for HoPOM at X-band. The bold lines show the experimental spectra, while the thin lines are the simulated spectra. The experimental spectra shown in this figure are the same as those shown in Fig. 3.10. The values of the LF parameters used for the simulation are as follows: 0 0 0 4 B2 = 18027 MHz, B4 = 209 MHz, B6 = -1.53 MHz and B4 = 94 MHz. For the simulations shown in this figure, the axial LF parameters have been tweaked slightly as compared to those reported in Ref. [65] to obtain better correspondence. The orientations for the four simulated spectra shown above are 19, 30, 55 and 65 degrees respectively. These angles represent the projections corresponding to the rotations on the plane containing the quantization axis. It can be observed from this comparison, that most of the observed spectral features in the angle-dependent study have been reproduced through our simulations, thereby providing confirmation of a reasonable degree of accuracy in our estimated parameters.
87 Fig. 3.21 Simulated energy level scheme for HoPOM with the red lines showing the expected field positions of the ΔmJ = ±1 transitions at f ~ 350 GHz, which should be the most intense transitions as per EPR selection rules. This simulation has been carried out for a crystal oriented with its quantization axis parallel to the static magnetic field. It can be seen from this figure, that the ΔmJ = ±1 transitions are expected to be ~ 9.5 T for a frequency of 350 GHz.
3.20) that we have a fairly good hold on the parameters in the Hamiltonian describing the system. Since we can reproduce the features of the angle dependent spectra reasonably well, we 0 conclude that the set of parameters providing the best fit to our data are as follows: B2 = 18027 0 0 4 MHz, B4 = 209 MHz, B6 = -1.53 MHz and B4 = 94 MHz.
We demonstrate through this study that the deviation of the LF symmetry from the exact D4d configuration, although small, has an extremely pronounced effect on the properties of the system. We also conclude that the presence of the transverse anisotropy in the LF Hamiltonian is essential for explaining the observed EPR transitions. Since the higher frequency transitions are not sensitive to this term, we were able to observe correspondence between our experimental and simulated spectra at 50 GHz even with just the axial terms. However, our low frequency
88 Fig. 3.22 Simulated energy level scheme for HoPOM with the red lines showing the expected field positions of the ΔmJ = ±1 transitions at f ~ 350 GHz, for a crystal aligned with its quantization axis 10 degrees off from the direction of the magnetic field. It can be seen by comparing this figure with Fig. 3.21, that the misalignment of 10 degrees is sufficient to push the ground state transitions out of the frequency - field range investigated.
4 experiment shows the necessity of the B4 term in the Hamiltonian, the inclusion of which enabled us to successfully explain our spectra.
Our discussion so far has been restricted to the mJ = ± 4 ground state. The separation between the ground state and the first excited state (mJ = ± 5) as calculated from the energy level scheme reported by AlDamen et al [65] is approximately 16 cm-1. Therefore, given this crystal field scheme, we expect to be able to observe the most intense EPR transitions (ΔmJ = ± 1) if we go up to high frequencies and high magnetic fields. However, although single crystal EPR experiments were conducted at frequencies up to ~ 400 GHz in a 15 T superconducting magnet, we did not observe the ΔmJ = ± 1 transitions. Fig. 3.21 shows the simulated energy level diagram consistent with the Hamiltonian parameters discussed above, with the red lines denoting the most
89 intense transitions at f = 350 GHz. The peak positions of the expected transitions shown in this figure correspond to the orientation of the quantization axis parallel to the external magnetic field. We can see from the field positions of the transitions in this simulation, that we should have been able to observe these transitions within the frequency and field range available to us. In order to probe why this was not the case in our experiments, we refer to Fig. 3.22 which shows the same energy level picture, but simulated for a case where there is a misalignment of the quantization axis of the crystal with respect to the external field by 10 degrees. We observe from this figure, that the small misalignment is sufficient to push these transitions out of the frequency – field range of the experiments conducted. We are aware from the two-axis angle dependent experiment (discussed in Section 3.3), that there was a sufficient degree of misalignment in our initial experiments when attempts were made to observe the ΔmJ = ± 1 transition. Given that the quantization axis of the system does not lie along the physical crystal planes, it was not possible to orient the sample accurately without a two-axis rotation experiment. Thus, it appears from the simulations shown in Fig. 3.22 that the misalignment of the sample in our high-frequency experiments could be the reason behind not observing the expected ΔmJ = ±1 transition. The simulations of the energy levels and the expected transitions were carried out by using the ‘levels’ and ‘levelsplot’ functions in the Easyspin program [69].
X-band CW experiments on HoPOM single crystals were also carried out in a dual mode resonator which allows us to record the EPR spectra for microwave polarization parallel and perpendicular to the applied static magnetic field. This experiment was carried out in an attempt to observe whether normally forbidden transitions following different selection rules are allowed in parallel-mode EPR for HoPOM. The angle dependent CW spectra for parallel and perpendicular modes at X-band have been shown in Fig. 3.23. The differences in the spectra for the two cases probably correspond to the excitation of transitions arising from different selection rules. The microwave frequencies at which CW measurements were conducted in the dual mode resonator are as follows: 9.42 GHz for parallel mode and 9.64 for perpendicular mode. Parallel mode EPR is frequently employed as a sensitive experimental technique in case of systems with high integer spin ground states. In the conventional perpendicular mode EPR spectroscopy, the oscillating microwave field (B1) applied to induce the spin transitions is aligned perpendicular to the static magnetic field (B0). In case of parallel mode EPR on the other hand, the polarization of
90 Fig. 3.23 Angle dependence of CW spectra of HoPOM at X-band in a dual-mode resonator (T~ 4.5 K). (a) Spectra for perpendicular mode (b) Spectra for parallel mode.
Fig. 3.24 Schematic representation of the relative orientations of the microwave field B1 with respect to the applied static field, B0 in parallel and perpendicular mode.
91 B1 is oriented parallel to B0. A schematic representation of the relative orientations of the fields in the two cases is shown in Fig. 3.24. Perpendicular mode EPR is the more widely used technique since most samples that were conventionally studied using EPR spectroscopy were half-integer spin systems, for which this mode gives rise to transition probabilities that are largest for the selection rule ΔmS = ± 1. However, in the case of integer spin systems with S ≥ 1, parallel mode EPR is often the preferred technique, especially at low frequencies, since it gives rise to large transition probabilities for ΔmS = 0 transitions. A number of studies have been reported on high integer spin systems where parallel mode EPR at X-band has been successfully used to investigate the transitions between high spin states that are otherwise very weak or silent in the perpendicular mode [77-79]. In our study on the HoPOM system we were able to observe parallel mode transitions at X-band frequency. Fig. 3.25 shows a comparison between the experimental and simulated CW spectra for the parallel mode. The system parameters used for obtaining the simulated spectrum are the same as those used for the perpendicular mode simulations presented previously in the chapter. It can be seen from the figure that reasonably good correspondence was observed between the experimental result and the simulation.
An important feature noted from our experimental results, is that the comparison of the experimental CW and ED spectra at 9.7 GHz shows significant differences. In our study on various HoPOM systems we consistently observed that the CW and ED spectra did not exactly match each other. Fig. 3.26 shows a comparison of the CW and ED spectra for a HoPOM single crystal with 25% Ho3+ concentration. We observe here that the more intense peaks in the ED spectra do not correspond to the peaks in the CW spectrum. In order to understand this seemingly anomalous feature, we note that the ED spectrum is weighted by T2. Thus, the more intense peaks in the ED spectra correspond to longer relaxation times. We have shown earlier that the inclusion of the transverse term in the LF Hamiltonian opens up a gap of approximately 9 GHz between pairs of levels with the same nuclear spin projection. Effectively, this translates into the creation of a gap between the electro-nuclear spin states, where the frequency of the EPR transitions becomes insensitive to the external field. Fig. 3.27 shows the variation of the frequency with the applied magnetic field. This frequency versus field variation has been plotted by taking the difference between the energy levels corresponding to ΔmI = 0 and ΔmI = ± 1 transitions. The curvature of the energy levels in the low field regime (non-linear regime) is such
92 Fig. 3.25 Comparison of parallel-mode experimental and simulated CW spectra for HoPOM at X-band.
Fig. 3.26 Comparison of CW and ED spectra for HoPOM. The two spectra were collected at the same sample orientation at T ~ 5 K. It can be seen from this figure that the more intense peaks in the ED spectrum do not correspond to the CW peaks.
93 Fig. 3.27 Simulated frequency versus field dependence extracted from the energy level diagram shown in Fig. 3.17. The blue line corresponds to 9.7 GHz, the frequency at which the experiment was performed. The black solid lines and the red dashed lines represent the ΔmI = 0 and ΔmI = ± 1 transitions respectively. The red arrows show the positions of the CW peaks as shown in Fig. 3.26. It can be noted that the field positions of the ED peaks (Fig. 3.26) correspond to the minima positions in between the CW peaks and lie near the df/dB ~ 0 positions. that the tunneling gap between the electro-nuclear spin states makes the spin dynamics insensitive to the dipolar field fluctuations. In order to measure the sensitivity of the transition frequency (f) to a change in the magnetic field (B), a parameter called the effective gyromagnetic df ratio can be defined as: [80]. It has been shown in Ref. [61, 80] that a reduction in eff dB eff is associated with a significant reduction in the decoherence rate in the case of bismuth qubits in df silicon. Following these examples, the positions of frequency minima corresponding to ~ 0, dB is expected to reduce decoherence. Interestingly, it is observed from Figs. 3.26 and 3.27 that the peak positions of the more intense ED peaks correspond with these minima positions. Hence, we
94 Fig. 3.28 Comparison of CW EPR spectra of protonated and deuterated HoPOM single crystals with 10% concentration of Ho3+. The spectra were collected at 9.7 GHz at T ~ 5 K. The sharp feature at 0.17 tesla is an impurity signal from the resonator. propose that decoherence arising due to sensitivity to magnetic fluctuations is considerably reduced leading to unexpectedly long T2 relaxation times in HoPOM. A thorough theoretical treatment is required to test this suggestion, but our experimental findings seem to be consistent df with the idea that the long coherence times observed in this system correspond to the low dB ratio in certain parts of the EPR spectra. As we have discussed earlier, most spin echo experiments that report considerably long T2 values [54-57], make use of conditions like extremely low concentrations, very low temperature or application of high magnetic field in order to reduce dipolar decoherence. Hence, our observation of T2 times of the order of a few hundred nanoseconds, even in fairly concentrated samples of HoPOM, is significant from the point of view of probing the mechanism of the reduction in decoherence, and also for potential applications.
95 Fig. 3.29 The ED spectrum for deuterated HoPOM single crystal with 10% concentration of Ho3+ at 9.7 GHz at T ~ 5K. The four strong peaks in the spectrum have been labeled.
9- 3.6 Study of [HoxY1-x(W5O18)2] in deuterated solvent
As a continuation of our work on HoPOM systems, a comparative study was carried out between
Na9[HoxY1-x(W5O18)2].xH2O (H) and Na9[HoxY1-x(W5O18)2].xD2O (D) in order to observe any potential effects of the solvent deuteration on the relaxation times exhibited by the samples. CW EPR experiments were performed both on powder and single crystal samples at X-band (9.7 GHz) and Q-band (34 GHz) using commercial Bruker spectrometers. The powder experiments were carried out in order to ensure whether any small differences observed in the spectra were intrinsic properties of the system, or caused by slight changes in experimental parameters like sample orientation. Comparison of the CW EPR spectra obtained at a temperature of 5 K at X- band and at Q-band frequencies showed no significant difference between the deuterated and non-deuterated (protonated) sets of HoPOM samples. Fig. 3.28 shows a comparison between the
96 X-band CW spectra of systems H and D with 10% concentration of Ho3+. The protonated crystals used for obtaining the data shown in this comparative plot belonged to a new batch samples that were received along with the deuterated set. The signal intensity for the deuterated set was consistently observed to be lower in comparison, and this could possibly be due to sample size or crystal quality. Powder samples of both sets with 100%, 50% and 10% concentration of Ho3+ spins were studied. Pulsed EPR experiment at 9.7 GHz was also performed as part of this comparative study on single crystals of both sets. Fig. 3.29 shows the ED spectrum for deuterated HoPOM single crystal with 10% concentration of Ho3+. The intensity of the spectrum was much weaker as compared to the protonated samples of various concentrations studied earlier. In order to obtain relatively clean data where the peaks could be well distinguished from the background noise, averaging was carried out by repeating the measurement a number of times. Smaller peaks in between the more intense ED peaks were clearly observed in case of the protonated samples, as can be seen from the data that has been shown earlier in Fig. 3.26. It is likely that these features are present for the deuterated samples too, but were too weak to be observed. The spectrum shown corresponds to T ~ 5K. The spin- spin relaxation time was also measured, and it was observed that the T2 values for the samples with the deuterated solvent were significantly larger than those of the non-deuterated ones. 3+ Longer T2 values were observed for the sample with 10% Ho of the deuterated set (T2 ~ 506 ±
22 ns for P1 and T2 ~ 342 ± 16 ns for P2) as compared to the corresponding non-deuterated counterpart (T2 ~ 120 ns). The difference in the T2 value observed at field positions corresponding to the ED peaks P1 and P2 is of significance. Although we do not have a clear understanding of the cause of this difference, it might indicate a difference in the spin relaxation mechanisms. Our experimental findings therefore show that longer coherence times can be obtained for HoPOM by preparing samples with deuterated solvents. It has been discussed in Ref. [59] that a strong coupling of the electron spin to the solvent protons can lead to faster phase decoherence. The use of deuterated solvent as opposed to a protonated one has been shown to be 2 effective in achieving longer T2 times in [55, 73]. The gyromagnetic ratio of deuterium ( D) is about 6 times smaller than that of hydrogen (1H), which means that the coupling strength to 2D is much lower. Thus a deuterated solvent can lead to a significant reduction in the decoherence rate as demonstrated in the work by Ardavan et al [73]. We observe this to be true in case of our HoPOM samples as well, which indicates that the coupling to the solvent protons is one of the
97 dominant mechanisms leading to spin dephasing in this system. Since achieving longer relaxation times would make a system more attractive from the perspective of potential applications, this is a significantly impactful result. It should, however, be noted that extracting the T2 values from the decay of the spin echo was considerably difficult for the deuterated crystal due to the very low intensity of the ED spectrum. Again, multiple averaging was done while measuring the decay in the echo height in order to clearly observe the signal. Thus, there is scope of some degree of error in our measurements on the deuterated sample. Also, given the current availability of samples, we have been able to carry out spin-echo measurement on only one concentration (10%) for the deuterated set. Therefore, further detailed study remains to be continued for deuterated HoPOM crystals of different dilutions to establish how the relaxation times are affected as a result of the change in solvent. Given the challenge in carrying out comparative measurements due to the difficulty in obtaining identical sample orientation, it is crucial to estimate T2 for a number of concentrations in order to have a quantitative analysis regarding the effect of solvent deuteration. Studying deuterated samples of lower concentrations (< 1%) would be especially interesting since we expect to observe ESEEM effects as a result of coupling to 2D. This would then enable us to have a better insight into the mechanisms contributing to spin decoherence, thereby allowing better control in obtaining desired effects.
3.7 Summary
To summarize, we have carried out a comprehensive study on HoPOM single crystals using EPR spectroscopy. We show that the LF symmetry plays a crucial role in understanding the spin dynamics of the system. Although a LF Hamiltonian with only axial anisotropy was sufficient to explain the thermodynamic properties, we note through our simulations that the inclusion of a transverse term is essential for the mixing of states that gives rise to the EPR transitions observed in our experiments. Good agreement was observed between the experimental and simulated spectra at 50.4 GHz as well as 9.7 GHz, and the energy level diagram for the system in the low field regime has been presented. Spin echo EPR experiments enabled the estimation of the transverse relaxation time, and their variation with Ho3+ concentration was also studied. The surprisingly long T2 times (~ 100 ns) in concentrated samples point towards a mechanism that
98 facilitates the lowering of the degree of decoherence that is expected to arise as a result of hyperfine and dipolar interactions. It is suggested that the transverse anisotropy creates a tunneling gap in the low field regime that leads to a relative insensitivity of the spin dynamics to dipolar field fluctuations. This allows the long spin relaxation time in HoPOM. A comparative study between dilute HoPOM crystals with deuterated and non-deuterated solvents demonstrated that the deuteration considerably increased the relaxation time of the system. Rabi oscillations were observed in dilute HoPOM samples providing experimental evidence of long-lived coherent spin manipulation.
Although we have demonstrated a number of interesting properties of HoPOM through this study, a few questions based on our experimental observations still remain unanswered. The strong dependence of the low frequency EPR spectra on sample orientation and the difficulty in aligning the sample along its quantization axis make it complicated to carry out a comparative study between different samples. Therefore, a theoretical treatment that formulates the mechanisms leading to suppression of decoherence in the HoPOM system would enable the verification of our suggestions based on experimental findings. In addition, there is scope for further investigation to test the existence of g-anisotropy, hyperfine anisotropy, higher order 4 transverse LF term (B6 ) and quadrupolar effects [81]. Our simulations indicate that we can explain our EPR spectra to a reasonably good degree of correspondence with isotropic g and A 4 values, and the presence of a B4 term in the LF Hamiltonian. However, it remains to be investigated whether any of these other effects offer a better fit to the data. Besides allowing an insight into the interesting properties of HoPOM, this work also demonstrates the use of CW and pulsed EPR spectroscopy as extremely powerful complementary tools for probing the spin dynamics of a system.
99 CHAPTER FOUR
FRUSTRATED MAGNETISM: STUDY OF SHORT RANGE ORDERING IN THE MODIFIED HONEYCOMB LATTICE
COMPOUNDS SRL2O4 (L = LANTHANIDE)
The results presented in this chapter can be found in the article: Short range ordering in the modified honeycomb lattice compound SrHo2O4, S. Ghosh, H. D. Zhou, L. Balicas, S. Hill, J. S. Gardner, Y. Qiu and C. R. Wiebe, J. Phys.: Condens. Matter 23 (2011).
4.1 Introduction to geometric frustration
Geometrically frustrated magnets have held the attention of the solid state community as testing grounds for our current understanding of magnetism [82]. The term frustration refers to the inability of a system to exist in a unique ground state, and when this arises solely as a result of the geometric properties of an atomic lattice, it is called geometric frustration. In recent years, the study of systems exhibiting geometric magnetic frustration has become a topic of active research for physicists due to the novel ground state properties possessed by them. Geometric frustration arises as a result of the incompatibility of the local antiferromagnetic interactions with the global symmetry imposed by the crystal structure leading to degeneracy in ground states, and is usually observed in crystal lattices based on a triangular geometry [82]. Frustration also occurs commonly in disordered systems, but our interest in this present study is restricted to the realm of geometric magnetic frustration. It was earlier believed that most systems of interest have a very small number of available ground states and exhibit long range ordering at low temperatures. This assumption made it relatively simple to probe such materials by applying the technique of mean field theory. However, this does not hold for systems exhibiting strong
100 Fig. 4.1 Geometries that exhibit spin-frustration. The square lattice shown in (c) is a frustrated topology only when nearest amd next-nearest neighbor interactions are comparable.
geometric magnetic frustration which are highly degenerate and cannot be explained by mean field theory. Pauling’s work on ice in 1945 first introduced the idea that a system could possess a thermodynamically large number of accessible ground states [82]. In the early 1930s, a discrepancy was observed between the values of entropy of water measured spectroscopically (188.7 J/mole K at a temperature of 298 K) and that determined by the integration of specific heat divided by temperature (185.3 J/mole K with temperature varying from 10 K to 298 K). Pauling attributed this difference to the intrinsic disorder in the structure of ice, and showed through the application of statistical mechanics calculations that the difference matches the finite entropy possessed by ice at zero temperature. Later, in the 1970s, Toulouse and Anderson introduced the concept of frustration in order to explain spin glass behavior [82]. Since then, a plethora of new systems have been discovered which have strong frustration effects. Geometrically frustrated systems show several interesting properties such as those of spin liquid, spin ice and spin glass. The degeneracy in ground states leads to interesting physics of frustrated systems which is yet to be explored and explained to a great degree. Understanding the properties of such materials requires a blend of successful experimental efforts in synthesis and characterization, as well as the development of theoretical models to explain the observed features.
101 Fig. 4.2 Common frustrated lattices. The lattice geometries shown above represent the most commonly studied examples of frustrated lattices.
The canonical example for demonstrating geometric frustration is that of a lattice based on an equilateral triangular plaquette [83]. Fig. 4.1(a) shows that in case of a lattice based on a triangular geometry only two of the three nearest neighbor spins can satisfy the condition of being aligned anti-parallel to each other to fulfill the minimum energy constraint.
The Hamiltonian for nearest neighbor two spin interactions can be written as a scalar product of the spin operators ˆ ˆ ˆ H ex 2 SJ 1 S 2 … (4.1) where S1 and S2 are the two spin operators and J is the exchange constant. In case of an antiferromagnetic interaction (J < 0), the energy is minimized for anti-parallel spin alignments and Fig. 4.1(a) makes it clear that a triangular arrangement of spins is geometrically frustrated since the minimum energy condition cannot be satisfied simultaneously by all three spins for nearest neighbor interactions. Geometric frustration also occurs for tetrahedral geometry which may be considered as a combination of four edge-shared equilateral triangles. In the case of such an arrangement, two of the four spins are frustrated as shown in Fig. 4.1(b). In contrast to the cases of triangular and tetrahedral configurations, the square geometry is not a frustrated one for nearest neighbor interactions (Fig. 4.1(c)). However, if the nearest and next nearest neighbor interactions become comparable in magnitude, a square planar arrangement of spins also becomes a frustrated system for J < 0. Fig. 4.2 shows some common examples of frustrated lattices which are based on corner and edge sharing triangular and tetrahedral geometries.
102 Fig. 4.3 Characteristic behavior of inverse susceptibility versus temperature for paramagnets, ferromagnets and anti-ferromagnets.
In this chapter, we shall present a comprehensive study of the magnetic properties of a frustrated system carried out through elastic and inelastic neutron scattering experiments. In recent developments in the field of the study of spin frustrated systems, model materials have been sought after in the kagomé [84], pyrochlore [85] and fcc lattice [86, 87] structures with varying degrees of success. In many of the compounds, systems find ways to reduce the ground state degeneracy through lattice distortions, complex routes to magnetic ordering, or through energetic compromises using orbital degrees of freedom. Hence, the search continues for compounds which display low temperature ground state properties similar to those of a spin liquid as predicted by Anderson [88].
The presence of magnetic frustration in a system can be studied using experimental techniques involving measurement of magnetic susceptibility, specific heat and neutron scattering, since such studies carry signatures of frustration in a material. The suppression of ordering temperature is used as an empirical measure of the degree of frustration in a system. Theoretically, there is no order at any finite temperature for a triangular antiferromagnet. However, in actual cases, anisotropy and long range interactions come into play and overcome frustration to produce long ranged order. Our focus in the present study is concentrated on systems where the effect of
103 frustration dominates and considerably lowers the ordering temperature. In order to gauge the degree of frustration in a material, a frustration parameter (f) may be defined as [82] :
expected ordering temperature (w) f … (4.2) observed ordering temperature (Tc) A system with f > 10 is empirically considered to be strongly frustrated. Fig. 4.3 shows the general behavior of the variation of the inverse magnetic susceptibility as a function of temperature for ferromagnets, paramagnets and antiferromagnets. For ferromagnets,
θW is positive and TC = θW. In case of antiferromagnets, θW is negative and TC (which is equal to the Neel temperature TNeel) is less than |θW|. Geometrically frustrated magnets can be considered as an extreme case of antiferromagnets where f = |θW| / TC > 10.
As in the case of magnetic susceptibility, specific heat measurements also carry signatures of the presence of magnetic frustration in a material. Geometrically frustrated magnets exhibit residual entropy at low temperatures as compared to an ordered system. In some frustrated systems large residual entropy is present, and such behavior can be experimentally studied through heat capacity measurements. Thus, magnetic susceptibility and heat capacity measurements are chosen as some of the basic techniques employed in studying frustrated magnetism.
4.2 Motivation
In this chapter, the study of short-ranged magnetic ordering through magnetic characterization and neutron scattering measurements on the modified honeycomb lattice compounds, SrL2O4, will be discussed. The honeycomb lattice, in principle, should not be a frustrated topology [86].
However, recent work on materials of the series SrL2O4 has shown that the compounds show promise for exploring disordered magnetism in honeycomb sublattices [89]. These compounds exhibit lack of long-ranged ordering at low temperatures due to the interplane connectivity of the honeycomb layers through an edge-shared triangular network. This unique topology has motivated recent experimental as well as theoretical studies of this series of materials [89-91]. In this study, we carried out single crystal synthesis, sample characterization and neutron scattering experiments on a few members of the SrL2O4 series in order to gain an insight into the properties of these geometrically frustrated magnetic materials. Although synthesis and preliminary 104 Fig. 4.4 Crystal structure of SrL2O4 (Grey: Lanthanide; Red: Oxygen; Blue: Strontium) [86].
characterization was carried out on SrHo2O4, SrEr2O4, SrYb2O4 and SrDy2O4 members of this series, only SrHo2O4 will be discussed in this chapter since neutron scattering studies were performed on this system.
We shall outline the synthesis and characterization of single crystalline SrHo2O4 followed by a discussion of its magnetic properties based on neutron scattering experiments. It has been reported from previous experimental work on powder samples of SrHo2O4 that this material does not show any sign of long-ranged ordering down to 1.8 K [89]. Our work shows that the system exhibits short-ranged ordering below 4K which is demonstrated through elastic neutron scattering experiments. A crystal field scheme was also worked out based on inelastic neutron scattering data, which has been shown to be consistent with specific heat measurements, giving useful insight into the properties of the system.
105 4.3 Description of the crystal structure of SrL2O4
We shall start with a general description of the structure of SrL2O4 crystals where L = Gd, Dy, Ho, Er, Tm and Yb. In this work we have focused on studying only a few members of this group, but nonetheless the crystal structure description holds true for these isostructural compounds.
The SrL2O4 crystals have a tetragonal configuration. They are made up of LO6 octahedra running along the crystallographic c direction with Sr-O polyhedra in between them. The lanthanide sublattice in these compounds form chains of edge-shared triangles arranged in the shape of a honeycomb [89]. Fig. 4.4(a) shows the crystal structure of SrL2O4 as viewed along the c-axis of the crystal with the bold lines outlining the honeycomb shaped arrangement of lanthanides in the a-b plane. There are double rows of the LO6 polyhedra that form the walls of the honeycomb.
The Sr atoms are located in the cavities formed by the LO6 octahedra. It is interesting to note that the LO6 polyhedra in the SrL2O4 crystals show slight irregularities in terms of the bond lengths and bond angles, i.e., all the L-O bond lengths are not equal. The difference in the L-O bond lengths is significant because it might possibly lead to Ising behavior as seen in SrDy2O4 [89]. The magnetic properties of the system are determined by the arrangement of the lanthanides and hence we pay special attention to them. As seen in Fig. 4.4(b), the lanthanide atoms form chains of edge-shared triangles along the c-axis which cause inter-plane connectivity between the honeycomb layers, and hence lead to geometric frustration in these crystals.
4.4 Synthesis and Characterization of SrHo2O4
Polycrystalline SrHo2O4 was made by solid state reaction in order to act as the feed and seed rods for single crystal growth. A mixture of SrCO3 and Ho2O3 in stoichiometric proportions was ground together and fired at 950oC for 40 hours. The mixture was then pressed into 4 mm diameter 60 mm rods under 400 bars of hydrostatic pressure and fired at 1300oC for 24 hours. The chemical reaction is shown in Eq. 4.3.
SrCO3 Ho O32 SrHo O42 CO2 … (4.3)
A single crystal of SrHo2O4 was synthesized by the floating zone technique discussed in Chapter 2. The crystal growth was carried out in an image furnace with the feed and seed rods rotating in
106 Fig. 4.5 Variation of inverse magnetic susceptibility with temperature for SrHo2O4. Inset: low temperature variation of magnetic susceptibility with temperature at fields of 0.1 and 1 T.
Fig. 4.6 Variation of the specific heat capacity of SrHo2O4 and SrY2O4 with temperature. The red line with circles represents SrHo2O4 and the blue solid line represents SrY2O4. Lower inset: low temperature variation of specific heat capacity for SrHo2O4 showing a broad peak at 3.5 K. Upper inset: variation of ∆S with temperature obtained by integration of the area under the heat capacity curve for SrHo2O4.
107 opposite directions at 25 rpm at a rate of 5 mm hour-1. Small pieces of the single crystal were ground into a fine powder for x-ray diffraction (XRD) measurements and the results confirmed that that the sample had the correct composition and was a single phase. The X-ray Laue diffraction was used to orient the crystal.
The dc magnetic susceptibility measurements were carried out with a superconducting quantum interference device (SQUID) magnetometer with the magnetic field oriented along the [001] axis of the crystal. The magnetic susceptibility of the SrHo2O4 single crystal was measured over a temperature range of 2 – 300 K. The variation of inverse susceptibility with temperature is shown in Fig. 4.5. The sample follows Curie-Weiss behavior between 30 and 300 K and a straight line fit was obtained for the linear variation. The Weiss temperature θW was fitted for the system and a value of -15K was obtained, which is close to the value of -16 K obtained by Karunadasa et al [89]. Magnetic susceptibility measurements were carried out up to temperatures much lower than the Curie-Weiss temperature, and yet no feature indicating the establishment of long ranged ordering was observed down to 2 K. This is typical for a frustrated system and hence the result of the susceptibility measurement provides experimental evidence of geometric
W 15 K frustration in SrHo2O4. The frustration index can be estimated as f 10 . The TN 3.0 K graph inset in Fig. 4.5 shows the variation of magnetic susceptibility carried out at low temperatures. A broad peak indicative of short ranged ordering in the single crystal was observed at a temperature of about 4 K in a field of 0.1 T. An application of a field of 1 T was enough to suppress the weak ordering of the Ho3+ spins. Similar measurements were also carried out on
SrEr2O4 single crystals and similar behavior was observed. The work by Karunadasa et al [89] reports magnetic susceptibility measurements of powder samples of various members of the
SrL2O4 series. It is significant to note that none of their samples showed any sharp features that are associated with long range ordering down to 1.8 K.
The specific heat measurement on the single crystal was performed using a physical property measurement system (PPMS) with the [001] crystal orientation aligned parallel to the magnetic field. The variation of specific heat capacity of SrHo2O4 with temperature is shown in Fig. 4.6. Two interesting features were observed in the specific heat data for the sample at low
108 Fig. 4.7 Inelastic neutron scattering data for SrHo2O4 single crystals at T = 1.5 K showing band structures corresponding to crystal field levels. (The intensity is depicted by the color according to the scale on the right.) temperatures as shown in the inset of Fig. 4.6. A sharp upturn at a temperature below 1.8 K and a broad feature at around 3.5 K were observed. The broad peak at low temperature signifies the presence of short ranged ordering. The reason for the sharp feature is not clearly understood; it could possibly be due to a magnetic transition, but there is no conclusive evidence to support this at present. Another possibility is the augmentation of the nuclear contribution to the specific heat due to the local field of the Ho3+ spins. Beyond 10 K the specific heat of the sample showed the expected T3 variation in accordance with the Debye formalism for specific heat. The area under the curve was integrated after carrying out a background subtraction of using specific heat data of a polycrystalline sample of SrY2O4 which is non-magnetic in nature. The change in entropy, ΔS, as denoted by this area was plotted as a function of temperature and can be seen as an inset in Fig. 4.6. The graph reaches saturation beyond 100 K and the saturation value was observed to be 13.87 which is near the value for Rln5.
109 4.5 Neutron Scattering
Neutron scattering experiments were carried out on SrHo2O4 single crystals at the Disc Chopper Spectrometer (DCS) facility at National Institute of Standards and Technology (NIST). Neutron beams with a wavelength of 2.9 Å were used and the experiment was carried out at temperatures of 1.5 K, 6 K, 10 K and 100 K. In order to visualize and analyze the data obtained, the software DAVE [92] was used.
4.5.1 Inelastic Neutron Scattering
The Fig. 4.7 shows the pictorial representation of inelastic neutron scattering data on SrHo2O4 at 1.5 K. The flat dispersion typical of quantized crystal field levels was observed. The intensity profile with the variation of the intensity as a function of the difference between the incident and scattered neutron energies has been plotted in Fig. 4.8. The figure shows the presence of one strong elastic peak and three relatively weaker inelastic peaks at corresponding to ΔE values of 0.80 meV, 1.64 meV and 3.61 meV at a temperature of 1.5 K. At temperatures of 10 K and 100 K, an additional peak was observed at around 7 meV as can be seen as the inset (a) in Fig. 4.8. This additional feature corresponds to a higher temperature transition which is not thermally populated at 1.5 K. Thus, the data for inelastic neutron scattering point to the presence of five crystal field levels which have been depicted in inset (b) of Fig. 4.8 after carrying out a conversion from the energy to the temperature scale (11 K ≈ 1 meV).
4.5.2 Elastic Neutron Scattering
Elastic neutron scattering experiments have been conducted on powder samples on various members of the SrL2O4 series by Karunadasa et al [89]. Their work reports the presence of a broad peak indicative of short range ordering in the case of Er, Ho and Dy members of the series.
The authors mention that no such evidence was observed in the case of SrYb2O4 or SrTm2O4, possibly due to the low magnetic moments of these lanthanides. Neutron scattering work done on
SrEr2O4 single crystals by Petrenko et al also show evidence of short ranged ordering at low
110 Fig. 4.8 Intensity profile for inelastic neutron scattering for SrHo2O4 single crystal at T = 1.5 K. The observed inelastic peaks at 0.80, 1.64 and 3.61 meV are marked in the figure. Insets: (a) An additional peak at 7 meV observed at T = 10 K (b) Schematic depiction of the five crystal field levels suggested by the inelastic neutron scattering data. temperatures [91]. In our work, we present an analysis based on our neutron scattering experiments on single crystals of SrHo2O4. The intensity profile diagram for elastic neutron scattering on SrHo2O4 single crystal has been shown in Fig. 4.9(a). The color scale shown alongside the figure represents the intensity of the scattered signal. Low intensity diffuse scattering indicative of short range ordering of Ho3+ spins is clearly seen from the figure. The chemical Bragg peaks of the single crystal can be seen as an array of high intensity sharp peaks.
The diffuse scattering is comparatively much lower in intensity (as seen in the color plot in Fig. 4.9(a)), but is clearly observable in the elastic neutron scattering measurement. The data shows the scattering distribution in the [H, K, 0] plane. Attempts were made to extract and model this diffuse scattering by taking several cuts along the H and K axes. The intensity profile plots thus obtained by taking cuts, were rescaled to plot the variation of intensity as a function of the
111 Fig. 4.9 Comparison of elastic neutron scattering data and simulation for SrHo2O4 single crystal at T = 1.5 K. (a) Elastic neutron scattering data for SrHo2O4 single crystal at T = 1.5 K. The green patches represent diffuse scattering due to short range ordering of Ho spins. (b) Neutron scattering simulation as a result of nearest neighbor Ho spin interactions in SrHo2O4. In the simulated figure, the darker colors (black) represent low intensity, while the lighter colors (white) represent the higher intensities in scattering.
112 (a)
(b)
Fig. 4.10 Diffuse scattering fits for SrHo2O4 single crystal. (a) Fit to diffuse scattering along the [H 0 0] direction. (b) Fit to diffuse scattering along the [0 K 0] direction. (T = 1.5 K) The error bars are a measure of the standard deviation of the intensity.
113 reciprocal lattice vector Q. Using standard data for the magnetic form factor of Ho3+, the intensity was rescaled and the structure factor S(Q) was plotted as a function of Q, where S(Q) is proportional to I/F2 (I = intensity, F = form factor). Following the example of neutron scattering analysis on Tb2Ti2O7 by Gardner et al [83], the structure factor for the crystal was calculated assuming nearest neighbor spin correlations. In the study on Tb2Ti2O7 [83], the form of the structure factor was given by Eq. 4.4, where Q = 2π(h, k, l) and a represents the spin-spin correlation function for nearest neighbor interactions. h k h l k l S(Q) 1 acos cos cos cos cos cos … (4.4) 2 2 2 2 2 2
In a similar fashion the expression for the structure factor for SrHo2O4 using a model based on nearest neighbor correlations can be expressed as Eq. 4.5, where di represents the Ho nearest neighbor distances in the a-b plane and a denotes the spin correlation function.
S(Q) = 1 + a[cos (hd1) + cos (kd2)] …(4.5) Comparison between the simulated graph (Fig. 4.9(b)) based on this correlation function and the experimental neutron scattering data show close resemblance, indicating that the short range ordering arises as a result of nearest neighbor interactions only. The experimental data were successfully fit using a simple cosine function (Fig. 4.10 (a) and (b)) which is consistent with the idea that nearest neighbor interactions between the Ho spins give rise to the observed diffuse scattering spectrum.
4.6 Discussion and Conclusion
The neutron scattering and heat capacity measurements carried out in this work on SrHo2O4 single crystals show clear experimental evidence of the existence of short range ordering of Ho spins. Modeling the observed diffuse scattering spectra shows that it arises as a result of nearest neighbor Ho spin interactions. Our simulation based on calculations for nearest neighbor correlations is in good agreement with the experimental elastic neutron scattering data. A correlation was also observed between the results of inelastic neutron scattering and specific heat capacity measurements on the sample. The neutron scattering data indicates the presence of five crystal field levels, transitions between which were observed in terms of the peaks in the
114 graphical representation of the data (Fig. 4.8). This conclusion is in accordance with the plot of the variation of ΔS with temperature (shown as inset in Fig. 4.6) which reaches a saturation value of approximately Rln5, which would be expected if there were five available energy levels per Ho3+ ion in this energy range. Although we theoretically expect this saturation value to be Rln8 for Ho3+, this is not the observed value possibly because some of the levels are not accessible from the ground state. However, the low temperature behavior of specific heat for the sample has not yet been satisfactorily explained. The upturn in heat capacity could either be indicative of a low temperature transition, or it could be due to a nuclear Schottky-like effect. This leaves the scope for further investigation involving neutron scattering and heat capacity measurements at lower temperatures in order to better understand the properties of the system.
There has been some recent speculation in the literature that spin-ice like orderings could exist in frustrated magnetic systems [93]. One of the key signatures for these ground states is the presence of characteristic ‘pinch-points’ within the diffuse magnetic neutron scattering profile [94]. We observe through our measurements that such ‘pinch-points’ are not present in the low temperature diffuse scattering spectra for SrHo2O4, and our fit to the experimental data using a simple nearest neighbor correlation is adequate to explain the results. With our current data, we cannot rule out the possibility that the short ranged ordering between Ho3+ spins persists in the limit of 0 K. Thus, further neutron scattering experiments are required at mK temperatures to get a clearer picture, and also to establish the explanation for the low temperature heat capacity behavior.
115 CHAPTER FIVE
FRUSTRATED MAGNETISM: MICROWAVE INDUCED
EXCITATIONS IN THE KAGOMÉ SYSTEMS PR3GA5SIO14 AND
ND3GA5SIO14
Some of the results presented in this chapter can be found in the article: Electron magnetic resonance studies of the Pr3Ga5SiO14 and Nd3Ga5SiO14 kagomé systems, Sanhita Ghosh, Saiti Datta, Haidong Zhou, Michael Hoch, Christopher Wiebe and Stephen Hill, J. Appl. Phys. 109, 07E137 (2011).
5.1 Introduction and Motivation
Geometric frustration effects in a variety of antiferromagnetic (AFM) materials have attracted considerable attention due to the interesting ground state properties that these systems exhibit. In Chapter 4 we have provided an introduction to geometric frustration which arises as a result of an incompatibility of the local AFM interactions with the global symmetry imposed by the crystal structure, giving rise to ground state degeneracy [82, 95]. Various novel magnetic phases such as those of spin-glass, spin-ice and spin-liquid are found in frustrated systems depending on the lattice structure. These features have made the study of strongly frustrated systems interesting and challenging. In this work, we probe the ground state properties of two kagomé systems,
Pr3Ga5SiO14 (PGS) and Nd3Ga5SiO14 (NGS), using microwave radiation.
The kagomé lattice consists of a 2-dimensional network of corner-sharing triangles and is one of the most sought after examples of geometrically frustrated magnetism, since few such examples exist. Fig. 5.1(a) shows the geometry of a perfect kagomé lattice. The ground state for a kagomé system of classical Heisenberg spins, which interact antiferromagnetically, is highly degenerate
116 Fig. 5.1 Kagomé lattice. (a) An ideal kagomé lattice (b) A distorted kagomé lattice (c) Structure of PGS / NGS (isostructural) showing the kagomé planes in the crystal. (Red = Pr / Nd, Blue and Green: Ga). Figure taken from Ref. [103].
[82, 95]. The degeneracy gives rise to a large number of low energy excitations. Considerable theoretical interest has been focused on the S = 1/2 kagomé lattice which is predicted to show spin liquid behavior at very low temperatures [96]. Experimental work has also been carried out quite extensively on a number of systems which include spin 1/2 systems, specifically herbertsmithite (ZnCu3(OH)6Cl2), and several higher spin systems [97-100]. The recently discovered NGS and PGS compounds, which belong to the langasite family of materials, have trigonal crystal structures with space group P321 where Ln3+ [Ln = Lanthanide (Nd, Pr)] magnetic moments occupy a network of corner-sharing equilateral triangles in the a-b plane that form a distorted kagomé lattice [26, 101, 102] (Fig. 5.1 (b)). This topology is equivalent to that of isolated kagomé planes stacked in the crystallographic c-direction, if only nearest neighbor interactions are considered. Hence, to a reasonably good approximation, NGS and PGS may be treated as perfect kagomé systems with the condition that the next-nearest neighbor interactions are much weaker compared to the nearest neighbor interactions. Fig. 5.1(c) shows the crystal structure of NGS and PGS which are isostructural compounds. Much attention has recently been focused on the study of these rare-earth kagomé systems [26, 101-105]. NGS and PGS both possess large single ion anisotropies and theoretical calculations predict a semi-classical spin liquid phase at low temperatures [106, 107]. An important distinction between the two compounds is that NGS with total angular momentum, J = 9/2 is a Kramers ion, while PGS (J = 4) is not [104]. Kramers theorem states that the energy states of ions with an odd number of electrons are at least doubly degenerate in the presence of purely electrostatic fields [108]. Following this theorem, rare-earth ions have been classified as Kramers ions or non-Kramers
117 ions depending on whether the number of 4f-electrons is odd or even. Thus, in the presence of a crystal field, the levels of a Kramers ion are at least doublets. For a non-Kramers ion on the other hand, the ground state degeneracy can be totally lifted. Kramers theorem plays a crucial role in determining the magnetic properties of an ion. In the absence of a magnetic field a permanent magnet must have a degenerate ground state. Therefore, for a non Kramers ion (example: Pr3+, Ho3+, Tb3+), if the ground state due to crystal field effects is a singlet, then it is non-magnetic. In the case of NGS, the ground state is known to be a doublet since it is a Kramers system. In the case of PGS, the ground state could potentially be either a singlet or a doublet, which has to be determined through a study of its magnetic properties. Experimental evidence in support of potential spin liquid type behavior in Pr3Ga5SiO14 (PSG) is provided by neutron scattering and other measurements [103], but the ground state magnetic properties of this system have been the subject of some controversy [105].
Low temperature magnetic susceptibility, neutron scattering, nuclear magnetic resonance (NMR), and muon spin relaxation measurements (µSR) have so far been reported on these materials as a means of probing their ground state configuration. Magnetization studies have also revealed the presence of strong magnetic anisotropy associated with crystal field effects [101]. Recent reports of neutron scattering measurements down to sub-Kelvin temperatures (~ 30 mK) do not show the presence of any new Bragg peaks indicative of long-range ordering in either NGS or PGS [26, 103]. However diffuse scattering indicative of short range ordering was observed in both systems at milli-Kelvin temperatures. The absence of long-range order coupled with the strong frustration makes these systems good candidates for spin-liquid ground states [26].
In the case of NGS, Zhou et al. postulated a spin-liquid ground state on the basis of neutron diffraction studies [26]. In the same system, a persistent disordered state which exhibits rather slow magnetic fluctuations indicative of a collective magnetic state down to 60 mK in zero-field was predicted in the work by Zorko et al [102]. These fluctuations were shown to be significantly suppressed via the application of a relatively weak applied magnetic field (0.5 T). In the case of PGS however, the true nature of the ground state remains controversial: while zero- field work by Zorko et al. suggests a nonmagnetic ground state [105], reports based on NMR
118 indicate the presence of a field-dependent magnetic ground state [103]. In-field experiments on PGS, which include neutron scattering and nuclear magnetic resonance (NMR), show magnetic properties at temperatures of 1 K and below [103]. However, zero-field nuclear quadrupole resonance (NQR) and μSR experiments suggest that that the ground state is a non-magnetic singlet [105]. In the present study involving the technique of high frequency electron paramagnetic resonance (EPR) measurements carried out over a range of applied magnetic fields, we aim to shed more light on the ground state properties of both systems.
Very little EPR work appears to have been carried out on kagomé materials at low temperatures [109]. Recently, high field EPR measurements have been made on the quasi-1D AFM system
BaCo2V2O5 [110]. Over a range of applied magnetic fields this material has a spin liquid phase, and it has been claimed that low energy excitations are induced by the microwave radiation. Theoretical calculations of the dynamical structure factor S q, , obtained as the Fourier transform of the 1D correlation function, have been used to support the conclusions. However, efforts in characterizing the magnetic properties by applying the EPR technique to kagomé lattices does not appear to have been undertaken in any previous work. We carried out magnetic susceptibility measurements and multi-frequency EPR studies on both NGS and PGS single crystals in order to probe the interesting ground state properties exhibited by these frustrated AFM systems in a kagomé-like lattice.
5.2 Experimental Results
Magnetic susceptibility and EPR experiments were performed on oriented single crystals of NGS and PGS. The single crystals used for our measurements were grown from polycrystalline seed rods in an image furnace using the floating-zone method of crystal growth (discussed in Chapter 2). The crystals were characterized and face-indexed by means of X-ray diffraction in order to identify the crystallographic axes and planes. The crystals studied had an easily identifiable flat plane which was determined to be the ab-plane with the crystallographic c-axis perpendicular to it.
119 Fig. 5.2 Magnetic susceptibility data for NGS and PGS. (a) The variation of inverse magnetic susceptibility for NGS with temperature for two crystal orientations at an external magnetic field of 0.1 T. Inset: The variation of magnetization of NGS with external magnetic field at T = 2 K. (b) The variation of inverse magnetic susceptibility for PGS with temperature for two crystal orientations at an external magnetic field of 0.1 T. Inset: The variation of magnetization of PGS with external magnetic field at T = 2 K. The green lines show the straight line fits to the linear variation in the high temperature limit.
120 5.2.1 Magnetic Susceptibility
The DC magnetic susceptibility of NGS and PGS single crystals were measured over a temperature range of 2-300 K in a superconducting quantum interference device (SQUID) magnetometer for two crystal orientations with respect to the external magnetic field. Fig. 5.2 shows the variation of inverse susceptibility 1/χ with temperature for a magnetic field of 0.1 T applied along the [001] axis and in the (110) plane for the two systems. Comparison of the data for the two crystal orientations shows the existence of strong magnetic anisotropy in accordance with what had been reported by Bordet et al [101]. The 1/χ plot shows Curie-Weiss behavior and straight line fits to the two sets of data give values of ~ -16 K and -35 K as the Curie-Weiss temperatures (θCW) for NGS and PGS respectively. Linear fits were obtained for the high temperature (150 K – 300 K) linear variation of χ-1 as a function of T (Fig. 5.2). The Curie-Weiss
Law (Eq. 5.1) was used to evaluate the value of θCW from the slope and y-intercept of the straight line fits to the χ-1 vs. T plots. C T )( T CW … (5.1) 1 In Fig. 5.2, the slope of the linear fit corresponds to , while the value of the y-intercept C corresponds to CW . Thus, application of Curie-Weiss Law allows the evaluation of the Curie- C
Weiss constant that is characteristic of the sample. Although we attempted to estimate θCW through linear fits to the high temperature susceptibility data, it should be noted that it is not possible to evaluate this reliably in these systems due to crystal field effects [101]. It is, however, clear from Fig. 5.2 (b) that in the case of PGS, at temperatures down to 2 K, there is no evidence for temperature independent Van Vleck paramagnetism [3] that would have been expected if the ground state was a singlet in the single ion description. The effective magnetic moment, μeff, for the trivalent lanthanide ions in the two systems can be calculated from the experimentally 3+ determined value of C. The Nd effective magnetic moment in NGS is 3.63 µB for the crystal 3+ oriented with its c-axis parallel to the magnetic field, and that for Pr in PGS is 3.38 µB for the same crystal orientation. The theoretically calculated values for the magnetic moments for Nd3+ and Pr3+ are 3.62 and 3.54 respectively [1]. The insets in Fig. 5.2 show the magnetization (M) for
121 these materials at 2 K as a function of applied field μ0H up to 5 T. In case of NGS, the magnetization saturates above a field of ~ 3.5 T. In the case of PGS however, the magnetization does not saturate in this range. Previous measurements on PGS have shown no saturation of M at 1.6 K in fields up to 10 T [103]. From our magnetization data, a magnetic ground state is suggested for both NGS and PGS. However, later in the chapter, we pay attention to previous studies on PGS in order to understand the reasons for the conflicting reports on its ground state nature, and potentially contribute to the picture through arguments based on our experimental observations from EPR measurements.
5.2.2 High Frequency Magnetic Resonance Study
Continuous wave (CW) high–field (0 – 30 T) magnetic resonance experiments were conducted on oriented NGS and PGS single crystals over the range of 50 – 500 GHz using a waveguide probe with a resonant cavity while employing a Millimeter Vector Network Analyzer (MVNA) associated with several different sources and detectors as spectrometer [17, 24]. Higher frequency (500 – 800 GHz) experiments were also carried out on the samples using a transmission probe. A description of the probes used for the measurements has been provided in Chapter 2. The high frequency EPR measurements on both systems reveal rich spectra with a large number of peaks over the entire frequency range studied. Fig. 5.3(a) shows a representative set of EPR spectra obtained as a function of magnetic field (up to 15 T) at a frequency of 127 GHz for a PGS single crystal, oriented with its c-axis parallel to the applied field, for temperatures in the range 2 – 10 K. A complex multi-peak spectrum was found at temperatures below ~ 10 K. Fig. 5.3(b) shows a plot of the variation of the peak positions of the observed transitions with respect to temperature, and strong systematic shifts with temperature can be observed. Similar systematic temperature dependence of the spectra was observed for a number of frequencies. Fig. 5.4 shows the EPR spectra obtained at a frequency of 116 GHz over a temperature range of 1.3 – 20 K up to a field of 31 T. This data shows that the complex-multi peak spectrum observed in Fig. 5.3 persists to higher fields and up to a temperature of ~ 15 K. The temperature dependence of the spectral peak positions reveals a plateau region for each frequency, followed by a sharp decrease in field position of the peaks at higher temperatures. We note here that the EPR spectra and their evolution with temperature are very different from the 122 (a) (b)
Frequency ~ 127 GHz
Fig. 5.3 EPR spectra for PGS at a frequency of 127 GHz. (a) Temperature dependence of the EPR spectra at f ~127 GHz from 2 K to 10 K. The strong shifts in the spectral peak positions can be seen from this figure. (b) Plot of the peak positions of the transitions shown in Fig. 5.3 (a).
(a) (b) 25 K 20 K 15 K 10 K 16 12 K 8 K 6 K 4.3 K 3.1 K 2.2 K 1.3 K 14 12
10 P8 P7 8 P6 P5 P4 6 P3
P (tesla) field Magnetic P2 P P P P 9 P 2 3 4 P 6 P P 10 4 P1 P 5 7 8 1 Frequency ~ 116 GHz
Cavity Transmission (arb. units-offset) (arb. Transmission Cavity 0 5 10 15 20 25 30 2 0 5 10 15 20 25 30 35 Magnetic field (tesla) Temperature (K)
Fig. 5.4 EPR spectra for PGS at a frequency of 116 GHz. (a) Temperature dependence of the EPR spectra at f ~116 GHz from 1.3 K to 25 K. More transitions are observed as we go up to higher magnetic fields. (b) Plot of the peak positions of the transitions shown in Fig. 5.4 (a).
123 Fig. 5.5 EPR spectra for NGS at a frequency of 116 GHz. (a) Temperature dependence of the EPR spectra at f ~ 116 GHz from 1.4 K to 44 K. We observe that the observed excitations persist up to higher temperatures in case of NGS as compared to PGS. (b) Plot of the peak positions of the transitions shown in Fig. 5.5 (a). typical behavior seen in isolated paramagnetic spins in which the peaks do not show any significant temperature dependent shifts. This unusual feature points to the possibility of the existence of some form of collective excitation of the electron spins. Similar experiments as those described above were carried out on single crystals of NGS, and a very similar behavior was observed as shown in Fig. 5.5. The figure shows the temperature dependence of the EPR spectra at a frequency of 116 GHz up to a field of 31 T over a temperature range of 1.4 K – 44 K. It can be seen from the spectra that the observed resonances persist up to the highest fields investigated. We also note that the observed multi-peak spectra for NGS persist up to temperatures ~ 35 K. Our experimental observations thus seem to suggest similar physics in the two systems. The only significant difference between the results obtained from the two compounds is that the resonances persist up to relatively higher temperatures in the case of NGS.
The shifts in the peak positions observed in our EPR experiments on NGS and PGS are reminiscent of AFM resonances, where the mode frequencies depend directly on the sub-lattice magnetization [3]. The striking similarity between the nature of the variation of peak positions observed in the temperature dependent measurements for PGS and the variation of the AFM resonance frequency for MnF2 as a function of temperature can be seen from Fig. 18 of Chapter
124 Fig. 5.6 Frequency dependent EPR spectra for PGS at T ~ 3K. We can see from the figure that the spectra do not show the systematic frequency – field dependence as expected in the case of conventional independent particle EMR spectra.
16 in Ref. [3]. A discussion on AFM resonance frequencies as a function of the exchange field and the anisotropy field can be found in Ref. [3].
Although our measurements reveal a strong and systematic temperature dependence of the spectra, no such systematic dependence was observed while varying the microwave frequency. No clear frequency versus field correspondence was observed for the resonances, and higher frequency measurements showed a larger number of peaks (Fig. 5.6). Fig. 5.6 shows the EPR spectra of PGS for a few different frequencies at a temperature of ~ 3 K. Fig. 5.7 shows a plot of the field positions of the observed transitions over a frequency range of 100 GHz – 250 GHz at T ~ 3 K. The absence of a systematic frequency versus field dependence, as well as the observation of larger number of resonances at higher excitation frequencies can be seen from this figure. These features point to the fact that the resonances observed in the spectra do not correspond to conventional independent particle EPR transitions, possibly arising instead from collective excitations associated with antiferromagnetically ordered clusters. 125 5.3 Discussion and Analysis
In this section, we shall focus mainly on PGS since the true nature of its ground state has not yet been established in the presence of conflicting suggestions from different studies. We shall however also refer to our studies on NGS in order to compare the similarities and the dissimilarities between the two systems. The evidence from the zero field experiments [105], together with the magnetization results and the in-field NMR and neutron scattering findings [103], suggest the following description for the Pr3+ ions in PGS. In zero field, the ground state is a non-magnetic singlet. However, in an applied field, interaction with the field, together with the exchange interactions between neighboring Pr3+ ions, might lead to an induced moment through the admixture of low lying crystal field states (at an energy Δ above the ground state), with the ground state. The induced moment could then give rise to the low temperature magnetic properties that are observed in various in-field measurements. Such excited state-ground state admixture behavior has been shown to occur in a number of systems in which the exchange interaction is sufficiently large and Δ is sufficiently small [111-115]. Recent inelastic neutron scattering experiments on the spin glass PrAu2Si2 have shown the presence of induced moments of this type [114]. In the case of PrAu2Si2, the ground state has been reported to be a singlet which is separated from the excited doublet state by a gap Δ ~ 0.7 meV, and the interionic 3+ exchange interaction between neighboring Pr ions, Jex, causes an admixture of these states, thereby giving rise to an induced moment [114]. The suggested mechanism of the formation of the induced moment in PrAu2Si2 through mixing of states is shown in Fig. 5.8 [114]. There are examples of theoretical and experimental work on similar induced-moment systems [111, 113, 115] which show that for systems having sufficiently strong exchange interaction and small Δ, there can be formation of ground state magnetic moment through admixture of states below a certain critical temperature. For a two-level system, this critical temperature can be defined as:
1 J 2 n ex Tc ln 2 where α represents the matrix element coupling the two states and n is J ex the degeneracy of the excited state [114]. We note here, that in the case of PGS, the gap Δ between the lowest two levels corresponds to ~1.5 meV (18 K) [105]. Given this picture that we have from previous reported studies on PGS, we try to formulate arguments about possible
126 Fig. 5.7 Plot of peak positions of EPR spectra for PGS at T ~ 3 K over a range of frequencies between 100 GHz – 250 GHz. It can be seen from the figure that a large number of resonances were observed at higher frequencies giving rise to complex multi-peak spectra indicative of collective excitations.
Fig. 5.8 Mechanism for the formation of induced magnetic moment through an admixture of ground state and excited state separated by a gap of Δ in the presence of sufficient exchange interactions between neighboring ions. This scheme has been reported in Ref. [114] for spin glass order induced in PrAu2Si2.
127 Fig. 5.9 Energy level scheme presented by Lumanta et al [103] which reports the gap between the lowest two energy levels to be ~ 20 K for PGS. scenarios that might explain our experimental findings on this system. In the absence of a well- developed mathematical model to explain these complex systems, our analysis will only be a qualitative one which aims to examine potential models by comparison with various studies along with our understanding of the spectra observed in the EPR experiments.
Our observation of excitations at low fields and at low temperatures suggests that magnetic properties are associated with the ground state of PGS. Lumanta et al.[103] present a schematic crystal field level diagram (Fig. 5.9) based on an analysis of specific heat data which shows the separation between the ground state and the first excited state for PGS to be ∆ = 20 K [103]. The NQR relaxation rate measurements give a slightly lower value for ∆, which corresponds to 18 K [105]. Given this estimate of the gap, it follows that the excitations observed in the EPR spectra for PGS at low temperatures with T << 18 K, and for fields B < 15 T has to essentially arise from the ground state. As discussed earlier, the behavior of the spectra with field and temperature does not resemble that of a conventional paramagnetic system which typically exhibits little variation in resonance field with temperature, and for which Zeeman interaction gives rise to a spectrum whose resonance field positions scale with the excitation frequency. Evidence that the features in the spectra do not correspond to conventional independent-spin EPR transitions is provided by the frequency dependent EPR measurements carried out at T ~ 3 K. It was found that the spectral peaks did not vary in a predictable, systematic way with the variation of frequency. As originally suggested by Zorko et al. for NGS [102], the present magnetic resonance studies seem to confirm a collective magnetic ground state for both compounds. Since the EPR studies are necessarily conducted in the presence of a magnetic field it is not yet clear if these properties persist to zero field. It is likely that the applied field suppresses quantum fluctuations, thereby 128 destroying any supposed spin-liquid state and stabilizing finite sized ordered regions or clusters that are long-lived relative to the high-frequency EPR time scale (10 – 100 ps). In order to account for the observed multiple-peak features in the EPR spectra, it is suggested that excitation of spin wave-like modes occurs in short-range AFM ordered regions of the sample where the correlation length depends both on field and temperature. This mode excitation model can potentially explain the strong spectral shifts observed in the temperature dependent measurements as well as the lack of systematic variation with microwave frequency. The presented data thus suggest that the observed transitions arise as a result of the development of spin correlations, either upon cooling or upon increasing the magnetic field. Although our EPR results for PGS are consistent with the existence of finite-sized ordered clusters giving rise to standing waves that depend on the geometry of the clusters, the exact nature of these ordered clusters remains to be established.
Discussion of spin waves:
We shall briefly digress from our discussion of PGS and NGS to introduce the concept of spin- waves since we propose that spin-wave like modes might correspond to the observed transitions in our EPR experiments. Spin waves can be qualitatively described as a propagating collective excitation in a magnetic lattice [3]. FM and AFM spin waves are very distinct in nature, and while there are well-developed theoretical models and a number of examples for FM spin wave resonances, the phenomenon of AFM spin waves is not as widely observed. In the case of a ferromagnet, all the spins are aligned parallel to each other in the ground state (Fig. 5.10(a)). Considering the semi-classical picture discussed in Ref. [3], an excited state may be obtained either by reversing just one spin as shown in Fig. 5.10(b) or by letting all the spins share the reversal as depicted in Fig. 5.10(c). It can be shown that an excitation of much lower energy can be achieved in the situation shown in Fig. 5.10(c) as compared to the one shown in Fig. 5.10(b) [3]. These low energy excitations of a spin-system give rise to a wave-like form and are called magnons. Magnons are analogous to lattice vibrations or phonons with the distinction that magnons are oscillations in the spin orientations, while phonons are oscillations in the atomic positions on a lattice [3]. Fig. 5.11 shows a schematic depiction of a ferromagnetic spin wave viewed from the top and in perspective.
129 Fig. 5.10 Spin-wave picture for a ferromagnetic system. (a) Pictorial representation of the ground state of a ferromagnet with all spins aligned parallel. (b) A possible excitation with one spin reversed. (c) Low-lying excitation forming spin-wave. The spin vectors precess on the surfaces of cones, with successive periodic advances. This figure has been drawn following the example given in Ref. [3].
Fig.5.11 Spin waves viewed in perspective (top) and viewed from above (bottom) showing one wavelength.
Let us consider a linear N-spin system having only nearest neighbor ferromagnetic interaction (J). The ground state wavefunction for the system, when all the spins are aligned parallel to each other can be denoted as: 0 321 ... N , where αi represents the eigenfunction for the parallel configuration. As the system is excited out of its ground state, we can consider the next state to be represented by means of a linear combination of wavefunctions, with each member of the combination representing a spin reversal at a different lattice site [116]. This can be mathematically stated as shown in Eq. 5.2.
130 k k c j j where j 21 ... j 1 jj 1... N … (5.2) j
The wavefunction ψk corresponds to a spin wave with wave number k, and the allowed values of k can be evaluated by solving the Schrödinger equation with the appropriate periodic boundary conditions. The detailed solution is beyond the scope of this chapter, and can be found in Ref. [116]. The expression relating the energy with the wave number can be expressed as Eq. 5.3, which can be approximated to Eq. 5.4 for small values of k. In the following equations, 1 E NJ is the ground state energy and a is the lattice spacing. 0 2 1 E N 4 J 2J cos(ka) 0 … (5.3) 2
22 E E0 Jk a … (5.4)
(E – E0) corresponds to the energy of the spin wave and, in the quantized notation, the dispersion
2 relation for a ferromagnetic spin wave can be expressed as Eq. 5.5 showing E k k dependence.
22 Ek k const.Ja k … (5.5) In the semi-classical representation [3, 116], we can consider a lattice consisting of N precessing spins (Fig. 5.10(c)), where each spin differs in phase from the preceding one by θ, thereby forming a continuous wave travelling through the lattice.
In the case of an anti-ferromagnet, J < 0 and the neighboring spins tend to align themselves anti- parallel to each other. An antiferromagnet can be represented by the two-sublattice picture, where all spins of one sublattice point up, while all spins of the other sublattice point down in the ground state configuration. One might naively expect that although the net magnetization in an antiferromagnet would be zero in the absence of an applied field, the precessions and phase relationships of an AFM spin wave excitation would be the same as in the case of a ferromagnet. However, it was first shown by Hulthén [117] that the dispersion relation for an anti- ferromagnetic spin wave follows the relation: E k k … (5.6) Later, the work by Anderson [118] provided a formal treatment of the spin wave theory for anti- ferromagnets, and confirmed the above dispersion relation. The distinct difference in the
131 Fig. 5.12 Schematic representation of AFM spin waves. (a) Spin wave in an AFM as seen in perspective at a given time. The figure shows the two sublattice representation with spins 1, 3, 5, … pointing up and spins labeled 2, 4, 6, … pointing down. The up spins precess in a circle of radius R while the down spins precess in a circle of radius P (R>P). (b) Top view collapsed on circles of radius R and P showing the phase relationship between the precessing spins. The arrows in the figure represent the direction of precession. The positions marked -1, -2, -3, -4 are those of imaginary vectors pointing opposite to spins 1,2,3,4 [116]. dispersion relation between the two cases implies that it takes much more energy to excite an antiferromagnetic spin wave as compared to a ferromagnetic one. A pictorial model for spin- wave excitations and a detailed account of FM and AFM modes for spin-waves can be found in
132 Ref. [116]. In this discussion, we follow the treatment of AFM spin waves presented by Keffer et al in Ref. [116] for an intuitive understanding of the phenomena through a simple analysis of the semi-classical picture. Fig. 5.12 shows the semi-classical picture of spin waves in an antiferromagnet, where the system has been divided into two sublattices, with spins 1,3,5,… pointing up, and spins 2,4,6,… pointing down. It is important to remember that the two sublattices cannot be treated independently due to the exchange interactions between nearest neighbor spins. All spins must either precess clockwise (mode 1) or anti-clockwise (mode 2) in order to maintain the required phase relationship. This can be achieved if the up spins precess in a circle of larger radius (mode 1) or of a smaller radius (mode 2) as compared to the radius of precession of the down spins. This has been shown pictorially in Fig. 5.12(a) where the up spins precess in a circle of radius R which is greater than the radius P for the down spins. Fig. 5.12(b) shows the collapsed top view showing the precessions of the spins in different phase positions in the ‘up’ and ‘down’ sublattices. Following the method of Herring and Kittel [119], and starting
dSi from the equation of motion for Si: i Si , H , we can arrive at Eq. 5.7 and Eq. 5.8 for the dt magnitudes of the torque acting on an up spin and a down spin respectively in the presence of an external magnetic field H0. The complete mathematical derivation involving the solution of the equation of motion can be found in Ref. [116]. x and y have been marked in Fig. 5.12(b) and R and P are dimensionless quantities.
2J Tu RH 0 2yS0 u R … (5.7) g
2J Td PH 0 2xS0 d P … (5.8) g Using geometry, one can arrive at the following relationship between x, y, R and P as shown in Eq. 5.9 [116]. y x kaP R … (5.9) R P 8R 2 8P 2 For small k, the above equation can be approximated to Eq. 5.10. y x ka … (5.10) R P 2 The combination of Eq. 5.7, 5.8 and 5.10 yields the dispersion relation for antiferromagnetic spin
133 waves as shown in Eq. 5.10 and 5.11, thereby leading to the k dependence. 2J H S ak (Mode )1 … (5.10) 0 0 2J H S ak (Mode )2 … (5.10) 0 0 This concludes our discussion of FM and AFM spin waves, and we return to our study of the observed microwave excitations of the PGS and NGS distorted kagomé lattice systems.
Hypothesis suggesting spin wave-like excitations in PGS:
Spin wave modes have been observed in a number of kagomé systems [99] and spin-liquid pyrochlores [120-122]. Most theoretical models developed in order to describe frustrated 2D systems focus on the very low temperature properties of the kagomé lattice and involve dynamical models which predict excitations such as spinons [123-126]. For the presented spectra, the temperature dependent shifts in the field values at which the observed excitations occur, are similar to the behavior found in AFM systems, where the mode frequencies depend on the sub-lattice magnetization. FM and AFM standing spin wave resonances have been reported in a number of thin film systems [127, 128] and sub-micron sized magnetized discs [129, 130] with well-defined geometries. In Ref. [127] Golosovsky et al. demonstrate that a standing wave solution for a ferromagnetic thin-film of a specific thickness leads to a linear relationship between the resonance field Bn and the square of the mode number n. At present, there is no available theory that predicts AFM spin wave resonances similar to those that have been observed in our EPR experiments in the case of a kagomé lattice. Neutron scattering results on PGS clearly show that regions of short-range order appear in the system at low temperatures and in applied magnetic fields [103]. It is however not clear what nature of interactions lead to these short range ordered clusters. Although neutron scattering measurements indicate the presence of spin clusters of the size of a few lattice spacings [103], a well-defined estimate of the correlation length under given temperature and magnetic field conditions is not known at present. Nevertheless, a model that supports the formation of spin wave type excitations is a potential candidate that could explain the observed EPR spectra in these kagomé systems.
134 1.6 15
1.4 10
(T)
n
1.2 B 5
n 0 1.0 0 2 4n 6 8
log B log 0.8 Slope = 0.6157 +/- 0.04329 0.6 0.0 0.2 0.4 0.6 0.8 1.0 log n
Fig. 5.13 Plot depicting the resonance fields for the different modes shown in Fig. 5.4 for PGS at 116 GHz. The logarithmic plot of the resonance field positions as a function of mode number suggests an approximate Bn ∞ √n relationship. The two highest field transitions have been omitted in obtaining this fit since it is likely that at such high fields other interactions might come into play causing the observed deviation. Inset: Plot of Bn vs. mode number (n).
In order to carry out an analysis of the EPR spectra of PGS, it might be useful to adopt an empirical relationship between the absorption mode number n, as determined from the position
of peaks in the spectra, and the corresponding magnetic field Bn. We adopt the form Bn n , with the exponent to be determined from a fit to the experimental results. Fig. 5.13 is based on the 2.2 K spectrum for PGS at a frequency of 116 GHz (shown in Fig. 5.4) and shows a plot of log Bn versus log n which yields a straight line fit with a value for close to 0.6. The inset in
Fig. 5.13 shows a plot of Bn versus n. In this plot, the two highest field transitions have not been considered in obtaining the linear fit since it is likely that at such high fields other interactions might come into play causing the observed deviation. It should be noted here that the numbering of the modes of the observed resonances shown in this figure is arbitrary, and is only intended to attempt to observe any systematic variation of the resonance field positions consistent with the
135 idea of excitation of modes in the PGS kagomé system. Further theoretical work is needed to test this suggestion and establish the nature of the low-energy excitations observed in PGS and NGS.
5.4 Conclusion
In conclusion, we note that our magnetic susceptibility and magnetic resonance studies on PGS and NGS provide experimental evidence for a collective magnetic ground state in both systems. The magnetic resonance spectra that have been obtained provide evidence of low energy collective spin excitations in the PGS system. The striking similarity in the experimental observations in cases of PGS and NGS suggest similar physics in both systems. An empirical approach in analyzing the spectra suggests a systematic relationship between the excitation mode number and the corresponding magnetic field at which microwave absorption is observed. However, this analysis is a purely qualitative one. A model that explains the observed resonances and allows a thorough quantitative analysis of our experimental results is yet to be developed. This work provides an experimental basis for the investigation of spin wave-like resonances that are observed in these kagomé systems, and will hopefully act as a stimulus for the development of a theoretical model that can explain the novel and complex physics not only in NGS and PGS, but also in other similar systems.
136 CHAPTER SIX
SUMMARY
In this chapter we present a summary of the work that has been discussed in the previous chapters of the thesis. The work presented in this thesis involves a study of rare earth magnetism carried out on lanthanide-based single crystalline samples using spectroscopic techniques like EPR and neutron scattering.
In Chapter 1 we have provided an introduction in order to build up an understanding of magnetism as applied to lanthanide ions which are characterized by the filling of the f-shell. The concepts of spin-orbit coupling and crystal field effects have been introduced in this chapter, since the interplay between the relative strengths of these two effects play a prominent role in contributing to the anisotropy of the system. The strong spin-orbit coupling and the presence of hyperfine coupling between electron and nuclear spins are typical of lanthanides, and significantly affect the magnetic properties of rare-earth systems. Therefore, these features have been discussed in the introductory chapter to provide a brief theoretical background behind the spin physics of molecules with lanthanides as their magnetic core. Chapter 2 provides a description of the experimental techniques involved in carrying out the work reported in the thesis. It also includes a discussion of some of the aspects of the instrumentation employed in carrying out the experiments. Various experimental techniques including sample synthesis (floating zone method), x-ray diffraction, measurement of magnetic susceptibility, EPR spectroscopy and neutron scattering have been used for the various studies conducted. We provide a brief outline of all these different techniques in Chapter 2, while focusing mainly on EPR spectroscopy since a major bulk of the work done has involved the use of this technique. Both continuous wave and pulsed EPR have been discussed, along with a description of the probes used for conducting the EPR experiments reported in this thesis. A
137 transmission probe for carrying out high frequency EPR was designed as part of this work, and a description of this probe has been provided in the chapter.
In Chapter 3 we have discussed our study on a mononuclear holmium polyoxometalate (HoPOM) single-molecule magnet (SMM). This system forms an example of a lanthanide-based SMM and is an interesting candidate, both for studying spin relaxation properties, and also from the point of view of potential spintronic applications. Continuous wave and pulsed EPR techniques were employed for a detailed study on dilute single crystals of this system. In Chapter 3, we have presented a discussion of the motivation behind studying these systems, followed by a comparison between our experimental results and simulation for evaluating the parameters of the spin Hamiltonian describing the system. The study demonstrates the strong correlation between molecular symmetry and magnetization relaxation properties of the sample. Pulsed EPR results on samples of various dilutions have been reported in the chapter which demonstrates the effect of sample dilution on the transverse relaxation time due to a reduction in dipolar couplings which form a major source of decoherence. Through an analysis of our EPR spectra and simulations of the energy level scheme, we have attempted to propose an explanation for the mechanism leading to reduction in decoherence and enabling the observation of T2 times of ~ 100 ns for concentrated (25% and 10%) samples.
Chapter 4 deals with a study of frustrated magnetism in a modified honeycomb lattice compound. SrHo2O4 single crystals were investigated as a part of the study which involved magnetic characterization of systems of the SrL2O4 family (L = lanthanide). A description of the sample synthesis by the floating-zone method of crystal growth, and sample characterization studies have been reported in the chapter which demonstrate the lack of long ranged ordering in this frustrated system. We have shown through an analysis of elastic neutron scattering measurements on oriented single crystals that short range diffuse scattering consistent with nearest neighbor interactions between Ho3+ spins is present. Inelastic neutron scattering experiments have also been carried out, and an energy level scheme for the crystal field levels have been presented on the basis of the neutron scattering spectra.
138 Chapter 5 is a continuation of the discussion on frustrated magnetism where we report our study on Pr3Ga5SiO14 (PGS) and Nd3Ga5SiO14 (NGS) kagomé lattice compounds. In this work we have carried out high frequency magnetic resonance experiments on these isostructural systems in order to probe the nature of the ground state excitations. The measurements reveal complex multi-peak spectra for both the compounds which are characterized by strong, systematic temperature dependence and a lack of the expected systematic variation with microwave frequency. This feature, which is unlike conventional independent particle EPR, leads us to speculate that it is likely that the observed resonances are collective spin-wave-like excitations associated with the formation of clusters whose correlation length depend on temperature and magnetic field. In Chapter 5 we have presented our experimental results along with a qualitative discussion of a possible model that might explain the observations. However, a theoretical analysis is required to test this suggestion, and the reported EPR results potentially provide an experimental basis for the investigation of collective excitations in the kagomé lattice.
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149 BIOGRAPHICAL SKETCH
Sanhita Ghosh was born in Calcutta, India. She went to school at Our Lady Queen of the Missions School and La Martiniere for Girls in Calcutta. She earned her Bachelor of Science degree in physics with honours from Presidency College, Calcutta, and obtained her Master of Science in Physics from the Indian Institute of Technology-Bombay (IIT-B). She joined Florida State University to pursue her doctoral studies in the field of Experimental Condensed Matter Physics, and worked under the supervision of Dr. Stephen Hill and Dr. Christopher Wiebe at the National High Magnetic Field Laboratory, Tallahassee, Florida. She wishes to pursue a career involving research in material science and applications of magnetic resonance techniques.
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