1 PHASE TRANSITIONS, T HERMODYNAMICS, AND M AGNETISM OF THE LOW DIMENSIONAL ANTIFERROMAGNETS C r ( DIETHYLENETRIAMINE)(O2)2 H2O AND (CH3)2CHNH3CuCl3 By Y OUNGHAK KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSIT Y OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011
2 2011 Younghak Kim
3 To my parents and sister
4 ACKNOWLEDGMENTS First of all I sincerely thank my advisor Professor Yasumasa Takano, for his patient guidance, inspiration, and encouragement with deep insight based on broad experiences He is always full of energy to explore nature and ready to discuss it with m e Under his guidance, I have been able to make new contributions to science. I woul d also like to thank my supervisory committee members Prof essors Pradeep Kumar, Bohdan Andraka, Yoonseok Lee, and Daniel Talham for their mentoring. Through this Ph. D. project, I have had a privilege of working with many people to whom I would like to say thanks Dr. Hiro yuki Ts uj ii had developed the calorimeter used in t his work and co llaborat ed with me in some of the IPA CuCl3 experiments described in sections 6.3.1 and 6.3.2. He has been always willing to help m e understand physics involved in these experiments Dr. Ju Hyun Park, Tim Murphy, and G lover Jones have give n great help at the N ational H igh M agnetic Field L aboratory (NHMFL) during experiments answering technical questions and solving problems Prof Naresh Dalal, Dr. Arneil Reyes, Dr. Phillip Kuhns, Dr. Sarita Nellutla, Dr. Narpinder Kaur, and Tiglet Besara contributed greatly to the success of the Cr(dien) experiments. Dr. Yasuo Yoshida, Dr. Tao Hong, Travis Miller, and Brad Atkins helped me in various parts of this project I would like to give special thanks to Dr. Robert DeSeri o and Charles Parks for their help and advice in teaching undergraduate labs A lthough I do not mention their names because of the limited space, I would also like to thank many other people in the department, the university, and the NHMFL for help. Last but not least, I would like to say love heartily to m y parents and sister and appreciate their support and encouragement Without their love and devotion, I could not have finish ed this long journey.
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 7 LIST OF FIGURES .......................................................................................................... 8 ABSTRACT ................................................................................................................... 12 CHAPTER 1 OVERVIEW ............................................................................................................ 14 2 THEORETICAL BACKGROUND ............................................................................ 17 2.1 Quantum Critical Points .................................................................................. 17 2.2 Magnons ........................................................................................................ 19 2.3 Haldane Gap .................................................................................................. 23 2.4 Bose Einstein Condensation .......................................................................... 24 3 EXPERIMENTAL TECHIQUES .............................................................................. 31 3.1 Specific Heat Measurements ......................................................................... 31 3.1.1 Calorimeter Design .............................................................................. 31 3.1.2 Calorimeter Electronics and Thermometer Calibration ........................ 33 3.1.3 Relaxation Calorimetry ........................................................................ 35 3.2 Magnetocaloric Effect Measurements ............................................................ 38 3.3 Ma gnetic Torque Measurements ................................................................... 41 3.4 Nuclear Magnetic Resonance ........................................................................ 42 4 THERMODYNAMICS OF THE S=1 ANTIFERROMAGNET Cr(DIETHYLENETRIAMINE)(O2)2H2O .................................................................. 54 4.1 Basic Properties of Cr(diethylenetriamine)(O2)2H2O ..................................... 54 4.2 Experimental .................................................................................................. 55 4.2.1 Specific Heat and Magnetocaloric Effect ............................................. 55 4.2.2 Heat Release from Proton Spins near the QCP .................................. 56 4.3 Analysis of the Results ................................................................................... 57 4.3.1 Specific Heat and Magnetocaloric Effect ............................................. 57 4.3.2 Heat Release from Proton Spins near the QCP .................................. 58 4.4 Discussion ...................................................................................................... 62
6 5 PROTON NMR IN THE S=1 SQUARELATTICE ANTIFERROMAGNET Cr(DIETHYLENETRIAMINE)(O2)2H2O NEAR THE QUANT UM CRITICAL POINT ..................................................................................................................... 76 5.1 Motivation ....................................................................................................... 76 5.2 Results and Discussions ................................................................................ 76 5.2.1 Field Sweeps ....................................................................................... 76 5.2.2 Frequency Sweeps .............................................................................. 7 8 5.2.3 Site Dependence of the SpinLattice Relaxation Time T1 .................... 80 5.2.4 Dependence of T1 on Temperature and Magnetic Field ...................... 81 5.3 Summary ........................................................................................................ 83 6 (CH3)2CHNH3CuCl3 ................................................................................................. 97 6.1 Previous Work on (CH3)2CHNH3CuCl3 ........................................................... 97 6.2 Experimental .................................................................................................. 99 6.3 Results and Analysis .................................................................................... 100 6.3.1 Specific Heat and Magentocaloric Effect ........................................... 100 6.3.2 Magnetic Torque ................................................................................ 101 6.3.3 Heat Release from Proton Spins near the QCP ................................ 102 6.4 Conclusions .................................................................................................. 105 7 SUMMARY ........................................................................................................... 123 LIST OF REFERENCES ............................................................................................. 125 BIOGRAPHICAL SKETCH .......................................................................................... 129
7 LIST OF TABLES Table page 4 1 T0, Tq, and Hq of the data points demarcating a quantum critical region in the H T diagram shown in Figure 410. .................................................................... 75
8 LIST OF FIGURES Figure page 2 1 Typical quantum critical phase diagram. ............................................................ 28 2 2 Spin wave. .......................................................................................................... 28 2 3 Dispersion relations of ferromagnetic (left) and antiferromagnetic (right) magnons. ............................................................................................................ 29 2 4 Dispersion relations of low lying excitations in 1D HAF for S = 1/2 (left) and S = 1 (right). ........................................................................................................ 29 2 5 Zeeman splitting of the triplet state of a spin dimer. ........................................... 30 3 1 3He/4He dilution refrigerator at the NHMFL Millikelvin Laboratory. ..................... 46 3 2 Calorimeter for specific heat and magnetocaloric effect measurements. ........... 47 3 3 Schematic of the calorimeter electronics setup for the specific heat and magnetocaloric effect measurements. ................................................................ 48 3 4 Models for the analysis of a relaxation of the platform temperature in a specific heat measurement. ................................................................................ 49 3 5 Illustration of the principle of the magnetocaloric effect due to a field sweep. .... 49 3 6 Schematic view of the cantilever magnetometer. ............................................... 50 3 7 Schematic arrangement for NMR. ...................................................................... 51 3 8 Energy diagram of a nuclear spin ( I = 1/2) in an applied field H ........................ 51 3 9 Principle of the spinecho technique. .................................................................. 52 3 10 NMR pulse sequence for T1 measurements. ...................................................... 53 4 1 Crystal structure and basic unit of Cr(diethylenetriamine)(O2)2 and H2O molecules. .......................................................................................................... 64 4 2 Image of the Cr(dien) sample. ............................................................................ 65 4 3 P rocedure of the heat release experiment. ........................................................ 65 4 4 Specific heat as a function of temperature at different magnetic fields. .............. 66 4 5 Magnetocaloric effect curves of Cr(dien) at different reservoir temperatures ..... 67
9 4 6 Magnetic phase diagram of Cr(dien), determined by specific heat and magnetocal oric effect measurements. ................................................................ 68 4 7 T between the sample and the thermal reservoir as a function of the magnetic field during field sweeps. .......................................... 69 4 8 T between the sample and the thermal reservoir during field sweeps at 0.1 T/min. ........................................................................ 70 4 9 Amount of heat released at (a) 181 mK and (b) 96 mK as a function of Tq, the temperature from which the sample was rapidly quenched. ............................... 71 4 10 T curves (squares), marking a quantum critical region of Cr(dien) delimited by two straight lines through the data points. ...................................................................................... 72 4 11 T the t emperature difference between the sample and the thermal reservoir, after a field sweep at 0.1 T/min is stopped at t = 0. ............... 73 4 12 .................... 74 4 13 Amount of heat released vs. the field at which field sweep was stopped. .......... 74 5 1 NMR spectra at 540.7 MHz, showing the spin echo intensity as a function of magnetic field. .................................................................................................... 85 5 2 Frequency sweep NMR spectra at 12.7 T at 1.30 K and 0.36 K. ........................ 86 5 3 Frequency sweep NMR spectra at 11.0 T at 2.50 K, 1.55 K, and 0.50 K. .......... 87 5 4 Scaled plots of the data shown in Figure 53. ..................................................... 88 5 5 Two examples of proton NMR relaxation curves, both taken at 12.7 T. ............. 89 5 6 Proton NMR T1 as a function of frequency at 1.31 K and 0.36 K at 12.7 T. ........ 90 5 7 NMR spectra obtained at 1.27 K 1.50 K from integrated spin echoes at nine frequencies. ........................................................................................................ 91 5 8 Proton NMR T1 in Cr(dien) as a function of temperature in fields ranging from 12.0 T to 12.7 T. ................................................................................................. 92 5 9 Proton NMR T1 in Cr(dien) as a function of temperature in fields ranging from 12.0 T to 12.7 T. ................................................................................................. 92 5 10 Stretching parameter as a function of temperature at various magnetic fields. .................................................................................................................. 93
10 5 11 2D contour plot and 3D plot of proton T1, determined from stretchedexponential fits, as a function of temperature and magnet ic field. ...................... 94 5 12 2D contour plot and 3D plot of 1/ T1T as a function of temperature and magnetic field. .................................................................................................... 95 5 13 2D contour plot and 3D plot of the stretching parameter as a function of temperature and magnetic field. ......................................................................... 96 6 1 Schematic diagram and the crystal structure of IPA CuCl3. .............................. 106 6 2 Specific heat of IPA CuCl3 in magnetic fields normal to the B plane. ............... 107 6 3 Specific heat of IPA CuCl3 in magnetic fields normal to the C plane. ............... 108 6 4 Magnetocaloric effect curves at various temperatures. .................................... 109 6 5 Magnetic phase diagram of IPA CuCl3. ............................................................ 110 6 6 Exponent of the power law fit as a function of the highest transition temperature, for H B plane and H C plane. ................................................. 111 6 7 Transition field scaled with the g factor for the two applied field directions. ...... 112 6 8 Magnetic torque divided by magnetic field at various temperatures, for field applied perpendicular to the B plane. ............................................................... 113 6 9 Magnetic torque for H C plane. ...................................................................... 114 6 10 Derivatives of magnetic torque divided by magnetic field for H B plane at various temperatures. ....................................................................................... 115 6 11 Comparison of the magnetic torque data shown in Figure 6 10 with the phase boundary determined from speci fic heat (solid squares) and the magnetocaloric effect (open squares), shown in Figure 6 5. ............................ 116 6 12 Comparison of the magnetic torque data for H C plane, partly shown in the inset to Figure 6 9, with the phase boundary determined from the magnetocaloric effect (open circles), shown in Figure 6 5. ............................... 117 6 13 T between the sample and the thermal reservoir as a function of the magnetic field during field sweeps at 0.1 T/min. ..................... 118 6 14 Amount of heat released at (a) 180 mK and (b) 99.7 mK as a function of Tq, the temperature from which the sample was quenched. .................................. 119 6 15 Crystal structure of IPA CuCl3. ......................................................................... 120
11 6 16 T curves (triangles and circles), marking a quantum critical region of IPA CuCl3 delimited by two lines through the data points. ........................................................................... 121 6 17 Relaxation rate 1/ vs. the field at which field sweep was stopped. .................. 122 6 18 Amount of heat released at T0 = 180 mK as a function of the field at which field sweep ended. ............................................................................................ 122
12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PHASE TRANSITIONS, T HERMODYNAMICS, AND M AGNETISM OF THE LOW DIMENSIONAL ANTIFERROMAGNETS Cr( DIETHYLENETRIAMINE)(O2)2 H2O AND (CH3)2CHNH3CuCl3 By You nghak Kim May 2011 Chair: Yasumasa Takano Major: Physics A quantum critical point (QCP), a zerotemperature singularity at which a line of conti nuous phase transition ends, gives rise to novel phenomena, some of which can be observed at nonzero temperatures. In this dissertation, we focus on the QCP s of two dissimilar quantum antiferromagnet s, Cr(diethylenetriamine)(O2)2H2O i n which a transition between an antiferromagntically ordered phase and the fully polarized state occurs at the QCP, and (CH3)2CHNH3CuCl3 whose QCP is between a quantum mechanically disordered phase and an antiferromagnetically ordered phase. Using these two compounds, Cr(dien) and IPACu Cl3 for short, w e show that nuclear spins frozen in a highenergy nonequilibrium state by temperature quenching are annealed by quantum fluctuations near a QCP. This new phenomenon, observed in a spin 1 squarelattice antiferromagnet and a spin1/2 ladder antiferromagnet having di ssimilar QCPs, provides a powerful general tool with which to map a quantum critical region near a QCP and to investigate the dynamics of the quantum fluctuations that underlie quantum criticality.
13 The experiment relies extensivel y on a novel application of calorimetry. In Cr(dien), we have also used proton NMR to probe the quantum fluctuations of Cr4+ spins at magnetic fields very near the QCP a special part of a quantum critical region th at is inaccessible to our calorimetric me thod. As a prerequisite for this research, we have also mapped the magnetic phase diagrams of Cr(dien) and IPA CuCl3 with a particular emphasis on finding the exact locations of the QCP s. For this purpose, w e have employed specific heat, magnetocaloric ef fect and in the case of IPA CuCl3 magnetic torque measurements W e find power law dependences of the critical field of the longrange antiferromagnetic order on tem perature in both compounds. In Cr(dien), the power law exponent is that of the threedi mensional Ising universality, for the magnetic field direction we have investigated. In IPA CuCl3, the power law exponent s for the two magnetic field directions we have studied indicate a Bose Einste in condensation of magnons at the c losing of the Haldane gap of the magnons by the magnetic field.
14 CHAPTER 1 OVERVIEW Q uantum criticality is the behavior of a macroscopic system at a continuous phase transition in th e limit of zero temperature [ 1 ] Such a transition is produced by controlling a tunable parameter such as the magnetic field, pressure, and chemical composition. Quantum criticality is believed to underlie nonFermi liquid behavior in some conductors and nonconve ntional superconductivity in cup rates, pnictides, and some heavy fermion metals [ 2 3 ] Dynamics in the vicinity of a quantum critical point ( QCP) is particularly interesting. On the one hand, crit ical slowing down occur s at a QCP, with a diverging equilibr ation time. On the other, quantum fluctuations near the QCP should enhance the energy transfer rate between different degrees of freedom. This dissertation examines the properties of two quantum magnets, Cr (diethylenetriamine)(O2)2H2O and (CH3)2CHNH3CuCl3, near their QCPs. The first compound, Cr(dien) for short, is a novel Cr4+ compound synthesized by Naresh Dalals group in the Chemistry Department of Florida State University [ 4 5 ] It is a quasi two dimensional antiferromagnet in which squarelattice layers of the S = 1 Cr4+ spins are weakly coupled. The saturation field, 12.392 0.003 T, in which the line of continuous phase transition between the paramagnetic phase and the antiferromagnetic phase terminates, is the QCP of this material. The second material, also known as IPA CuCl3, is a quasi onedimensional antiferromagnet comprising the S = 1/2 spins of Cu2+. These spins form two leg ladders with antifer romagnetic leg interactions and as a result, possess a Haldane gap. The QCP of this material is at 9.995 0.002 T for one field direction and 9.399 0.004 T for another, perpendicular direction. At these critical
15 fields, the Haldane gap is completely suppressed by magnetic field, and inter ladder interactions lead to antiferromagnetic ordering. For both compounds, we first characterized the temperature dependence of the critical field near the QCP. We have found that Cr(dien) exhibits an Ising criticality, whereas IPA CuCl3 shows a Heisenberg criticality described as a BoseEinstein condensation of magnons. We also discovered a new quantum critical phenomenon, quantum fluctuationdriven release of heat from hydrogen nuclear spins, in both compounds. To observ e this phenomenon, the sample i s temperature quenched in a magnetic field, an operation which leaves the hydrogen nuclear spins frozen in a highenergy nonequilibrium state. Subsequently, as the magnetic field is swept towards the QCP, the divergent quantum fluctuations of the Cr4+ or Cu2+ spins quickly anneal the nuclear spins. W e employ a calorimetric technique to examine the dynamics of the annealing and concomitant heat release from the nuclear spins. Our discovery of t his phenomenon in two dissimilar antiferromagnets, differing in dimensionality and spin quantum number, sugges ts that it is a generic property of a variety of QCP. The results for Cr(dien) have been published in Physical Review Letters [ 6 ] In addition, we used proton NMR to investigate the dynamics of the Cr4+ spins in Cr(dien) near the QCP We have found that the proton spinlattice relaxation time T1 becomes drastically short near the QCP caused by divergent quantum fluctuations of the Cr4+ spins At the same time, the relaxation becomes strongly nonexponential near the QCP. We characterize the nonexponential behavior and briefly suggest a possible link between this phenomenon and the dynamic critical behavior of an Ising system.
16 The outline of the dissertation is as follows Chapter 2 reviews the theoretical backgr ound of this work Chapter 3 descri bes our experimental techniques : specific heat measurement s, magnetocaloric effect measurement s, magnetic torque measurement s, and NMR. Chapter s 4 t hrough 6 present the experimental results on Cr(dien) and IPA CuCl3. Finally, Chapter 7 summarizes the dissertation.
17 CHAPTER 2 THEORE TICAL BACKGROUND In this chapter, we review four theoretical concepts that f o r m a basis of this work. They are quantum critical points, magnons, Haldane gap, and BoseEinstein condensation. 2.1 Quantum Critical Points Since Hertz introduced the idea of quantum c riticality in 1976 [ 7 ] many people have been studying quantum critical behavior of condensed matter, theoretical ly and experimentally A QCP is a point, in a phase diagram, at which a zerotemperature phase transition takes place. Such a transition is produced by tuning nonthermal parameters such as the magnetic field, pressure, and chemical composition, as illustrated in Figure 2 1. Among thes e, magnetic field is the most convenient parameter, which can be changed with ease with out in terrupti ng an experiment In general there are two kinds of phase transitions: first order transitions and continuous transitions, also known as higher order tra nsitions A firstorder transition is accompanied by an abrupt change in first derivative s of the free energy. In contrast, at a continuous phase transition, an anomaly occurs only in second or higher order derivatives of the free energy while the first d erivatives remain continuous. For example, a transition from a paramagnet to a ferromagnet or an antiferromagnet is usually continuous, accompanied by a peak anomaly in specific heat Only when a continuous phase transition occurs a t zero temperature is the critical point a QCP. Most often, a QCP appears in a phase diagram as the zerotemperature end point of a line of continuous transitions.
18 A continuous phase transition at a nonzerotemperature, classical critical point is caused by thermal fluctuations. Thermal fluctuations are absent at zero temperature, where a quantum phase transition takes place. Instead, t he fluctuations at a QCP are caused purely by a quantum mechanical effect due to the Heisenberg uncertainty At a QCP, both spatial and temporal extents of quantum fluctuations diverge, as does the sp atial extent of thermal fluctuations at a classical critical point The divergent quantum fluctuations at a QCP are believed to be the driving mechanism behind a variety of novel phenomena, which chal lenge conventional theoretical approaches to collective behavior of many body systems and may also lead to technological applications. They are responsible for the breakdown of Fermi liquid behavior in some heavy fermion metals [ 8 9 ] and for the emergence of exotic states of matter such as unconventional superconductors [ 10, 11] nematic electron fluids in two dimensional electron gases and in Sr3Ru2O7 [ 12 14] and reentrant hiddenorder states in URu2Si2 [ 15] Presently, many theoretical and experimental studies in condensed matter physics are devoted to the challenging task of elucidating such phenomena. Particularly important are investigations of dynamic phenomena near a QCP, such as quantum fluctuationdriven relaxation and tunneling, because dynamic and static properties near a QCP are thought to be inseparably linked. Antiferromagnets are particularly attractive candidates f or such studies, because their QCPs can be reached by applying an e xternal magnetic field instead of varying the chemical composition of the material, a parameter difficult to control precisely as has been found in studies involving conduction electrons However, not every antiferromagnet fills the bill. The antiferromagnet under
19 study must be low dimensional and must consist of ions of a small quantum number re quirements which enhance quantum fluctuations 2.2 Magnons Similar to phonons, which are collective excitations of a crystal lattice, magnons are collective excit ations of electron spins in a magnetic solid [ 16 18 ] Whereas phonons are quantized sound waves, magnons are quantized spin waves. There are important differences bet ween spin waves in a ferromagnet, illustrated in Figure 2 2, and those in an antiferromagnet. We will first describe spin waves in a ferromagnet, then those in an aniferromagnet, followed by a description of special kinds of magnons in onedimension a l quantum magnet s, antiferromagnet s of small spin quantum number s. For simplicity, we consider a onedimensional ferromagnet described by a Heisenberg Hamiltonian: i i iS S J H1 (2 1) where J is the strength of the exchange interaction, and Si the spin operator at site i The exchange interaction J is negative; later, for an antiferromagnet it will be positive It is easy to derive a dispersion relation for spin waves in a ferromagnet [ 19] The Heisenberg equation of motion for the spins is H S i dt S dj j, 1 (2 2A) 1 1, j j j j j jS S S SS S i J (2 2B) j j j j j j j j j j j jS S S S S S S S S S S S i J 1 1 1 1 (2 2C)
20 1 1 j j jS S S J (2 2D) The components of this equation are y j y j y j x jS S S JS dt dS1 12 (2 3A) x j x j x j y jS S S JS dt dS1 12 (2 3B) 0 dt dSz j (2 3C) We look for a solution of the form ) ( t kja i x jue S, (2 4A) ) ( t kja i y jve S (2 4B) where u and v are constants, k the wave vector, and a the lattice constant Substituting these into Eqs. 2 3A and B results in ) ( ) (] 2 [t kja i ika ika t kja ive e e JS ue i (2 5A) ) ( ) (] 2 [t kja i ika ika t kja iue e e JS ve i (2 5B) That is, 0 ) cos 1 ( ) cos 1 ( v u i ka JS ka JS i (2 6) We can obtain the solution by using the determinant as follows: 0 ) cos 1 ( 2 ) cos 1 ( 2 i ka JS ka JS i (2 7) Thus, ) cos 1 ( 2 ka S J (2 8)
21 This is the dispersion relation for spin waves, i.e. magnons, in a ferromagnet. For ka Eq. 2 8 becomes 2 2k Sa J (2 9) An antiferromagnet has two sublattices, say sites A with spins pointing to the let and sites B with spins pointing to the right We assign even indices j to sites A and odd indices j to sites B [ 19] For even j y j y j y j x jS S S JS dt dS1 12 (2 10A) x j x j x j y jS S S JS dt dS1 12 (2 10B) For odd j y j y j y j x jS S S JS dt dS1 12 (2 11A) x j x j x j y jS S S JS dt dS1 12 (2 11B) Using y xiS S S (2 12) we rewrite Eqs. 2 10 and 211 as ] 2 [1 1 j j j jS S S iJS dt dS for even j ( 2 13A) a nd ] 2 [1 1 j j j jS S S iJS dt dS for odd j (2 13B) As in the ferromagnetic case, we look for a solution of the form
22 ) ( t kja i jue S for even j (2 14A) ) ( t kja i jve S for odd j (2 14B) Substituting Eqs. 2 14A and B into 213A and B results in ] 2 [ v e e u JS u ika ika for even j (2 15A) ] 2 [ u e e v JS v ika ika for odd j (2 15B) That is, v u ka ka JS v u 1 cos cos 1 2 (2 16) So, 0 2 cos 2 cos 2 2 JS ka JS qka JS JS (2 17) which is 0 ) cos 1 ( 42 2 2 2 ka S J (2 18) Therefore, ka JS sin 2 (2 19) This is the dispersion relation for spin waves, i.e. magnons, in an antiferromagnet. For small k, the dispersion has a linear dependence on k ) (k in contrast to the k2 dependence ) (2k for a ferro magnet as shown in Figure 2 3. In either case, magnons have no energy gap at k = The quantum numbers of a magnon in a ferromagnet are known to be S = 1 and ms = 1 whereas for a magnon in an antiferromagnet they are known to be S = 1 and ms =
23 To derive the magnon dispersion for an antiferromagnet, we have assumed that the spins are ordered in a state comprising two sublattices. Magnetic order is often absent, however, in quantum magnets. As a result, the magnon dispersion in such a quantum magnet can greatly differ from Eq. 2 19. In the next section, we describe the case of onedimensional antiferromagnets, a quantum magnet consisting of socalled spin dimers. 2.3 Haldane Gap In the previous section, we have considered a onedimensional (1D) Heisenberg antiferromagnet (HAF) given by Eq. 2 1 only as a model for an antiferromagnet that is spontaneousl y ordered with two sublattices. True 1D HAFs do not order, however, even at zero temperature, because a spin in 1D does not have enough neighbor spins and th us quantum fluctuations become overwhelming. As a result the magnon dispersion is not given by Eq. 2 19 and depends on whether the quantum number S of spins is a full integer or a half of an odd integer. For the Hamiltonian given by Eq. 2 1 in the S = 1/2 case Bethe [ 20] has given the e xact wave functions, which lead to the following dispersion for low lying excitations : ka J k sin 2 ) ( (2 20) This dispersion may not appear dramatically different from Eq. 2 19. However, the spin quantum number S of the excitations is 1/2 inst ead of 1. For this and other reasons, they are called spinons instead of magnons. Moreover, immediately above the dispersion given by Eq. 2 20 lies a continuum of twospinon excitations as shown in Figure 2 4 [ 2 1 ]
24 When the spin quantum number S is an integer, the Schr dinger equation for the Hamiltonian given by Eq. 2 1 cannot be exactly solved. As was predicted by Haldane [ 22] however, there exists in this case an energy gap, called the Haldane gap, between the quantum mechanically disordered ground state and low lying excit ations as shown in Figure 2 4. In particular, if S = 1, the Haldane gap is [ 23] J 41 0 (2 21) U nlike in the S = 1/2 case, the spin quantum number S of the excitations is 1, but ms can be 1 or 0 instead of only 1. For this reason, these excitations are sometimes called triplons rather than magnons. Haldanes prediction has been confirmed by a number of experiments on 1D S = 1 HAFs. One of the best examples is Ni(C2H8N2)2N O2( ClO4) also known as N ENP [ 24] It has been shown [ 25] that twoleg S = 1/2 H eisenberg spin ladders with an antiferromagnetic leg exchange also has a Haldane gap. One such antiferromagnet is (CH3)2CHNH3CuCl3, the subject of Chapter 6. The presence of the Haldane gap leads to physical properties of integer spin 1D HAFs, and two leg S = 1/2 Heisenberg spin ladders such as (CH3)2CHNH3CuCl3, fundamentally different from those of half oddinteger spin 1D HAFs For instance, the spatial correlation function of the integer case has a n exponential decay as opposed to a power law decay in t he half oddinteger case [ 26] Moreover as the Haldane gap closes at a sufficiently high magnetic field, a QCP appears in the magnetic phase diagram, as explained in the next section. 2. 4 BoseEinstein Condensation Bose Einstein condensation (BEC) was initially found in liquid 4He [ 27, 28] as had been proposed as the explanation of its superfluidity [ 29, 30] and more recently in
25 ultracold atomic gases [ 31] It can also occur in a system of other bosons such as gapped magnons. To explain this we first consider the simplest quantum magnet with gapped excitations, a collection of antiferromagnetically coupled pairs of S = 1/2 spins with weak antiferromagnetic interpair exchanges. Such a pair is called a spin dimer. Of this system consisting of spin dimers, the ground state is a collection of singlet dimers 2 1 0 0sm S (2 22) and low lying excitations are mixtures of singlet dimers and triplet dimers 1 1sm S (2 23) 2 1 0 1sm S (2 24) 1 1sm S (2 25) In a magnetic field H the Zeeman effect splits the triplet states, causing the energy of the ms = H with a slope B, as illustrated in Fig ure 2 5. Here g is the g factor and B the Bohr magneton. At a critical field B cg H (2 26) level separation between the singlet and triplets of a dim er at zero field, analogous to the Haldane gap between the quantum mechanically disordered ground state of an integer spin 1D HAF and magnons Because of the antiferromagnetic interdimer interactions, however, the ground state does not immediately become a collection of ms = which all the spins are aligned with the magnetic field. Instead, the triplets are
2 6 delocalized they are magnons in a general sense and only an infinitesimal number of them will condense into the ground state at Hc. As the fie ld is increased further, an increasing number of triplets will condense into the ground state, resulting in increasing magnetization. The ground state is now antiferromagnetically ordered owing to the interdimer interactions. Therefore, Hc is a QCP that separates a macroscopic singlet state from an antiferromagnetically ordered state. Because delocalized triplets are bosons, the magnetic ordering can be described as a BEC of magnons [ 25, 32] As this handwaving argument for a BEC suggests, the magnetic field corresponds to the chemical potential of an ideal Bose gas or an atomic gas. For magnons in integer spin 1D HAFs or two leg S = 1/2 Heisenberg spin ladders, the wave functions are far more complicated than Eqs. 2 22 through 225 and are in fact unknown. However, the effect of a magnetic field on them will be exactly the same. The only key difference is that the interactions responsible for antiferromagnetic ordering are 2 26 is the Haldane gap. One of the most important consequences of an antiferromagnetic ordering as a BEC of magnons is the universal power law [ 1 3 ] T H Hc (2 27) where H is the transition field at nonzero temperature T and Hc the cri tical field at zero temperature. The exponent depends on spatial dimension d : = d /2 (2 28) For threedimensional ordering, = 3/2. The first evidence for a BEC of magnons was found in TlCuCl3 [ 32 33] an antiferromagnet consisting of dimers of S = 1/2 Cu2+ spins. In this material, the ordering
27 temperature T was determined at each H from specific heat and agreement with Eq. 2 27 was found with = 1 .67 0.07 close to the theoretical value 3/2. More recently, additional evidence for BEC has been found in a few other spindimer compounds, including BaCuSi2O6 [ 34] in which evidence for crossover has been observed between a threedimensional BEC and a twodimensional BEC due to geometric frustration. In principle, a BEC of magnons can occur also near the saturation field of a Heisenberg antiferromagnet, since an antiferromagnetically ordered state at a field just below the saturation field can be viewed as a dilute system of ms = 1 magnons condensed in the background of a fully polarized vacuum [ 35 ]
28 Tuning Parameter, gT Quantum Critical Region 0 Phase B Phase A gc Figure 21. Typical quantum critical phase diagram. g is a tuning parameter such as the magnetic field, pressure, and chemical composition. At a critical value, gc, a quantum phase transition occurs at T = 0 The solid line is a phase boundary between two phases A and B, and the dash lines mark a crossover to the quantum critical region. Figure 22. Spin wave. Top: spin wave state of a ferromagnet. a is the lattice constant. Bottom: top view of the spin wave, showing a wavelike behavior.
29 0 ak ak0 Figure 23. Dispersion relations of ferromagnetic (left) and antiferromagnetic (right) magnons. For small k, the ferromagnetic case is 2k whereas the anti ferromagnetic case is k k0 k0 Figure 24. Dispersion relations of low lying excitations in 1D HAF for S = 1/2 (left) and S = 1 (right). In the left panel, t he shaded region is the two spinon continuum. In the right panel, a has been taken to be 1.
30 1 1sm S 2 1 0 1sm S 1 1sm S 2 1 0 0sm S HcS=0S=1Energy Magnetic Field Figure 25 Zeeman splitting of the triplet state of a spin dimer. The critical field Hc at which the ms = triplet state reaches the ground state is / B.
31 C HAPTER 3 EXPERIMENTAL TECHIQUES Specific heat, the magnetocaloric effect, magnetic torque, and nuclear magnetic resonance (NMR) are valuable tools for the study of quantum critical behavior of magnetic materials, and often superconductors, near a QCP. Thes e tools provide useful information on the static and dynamic properties of materials under study In this chapter, I will describe their principles and how they were used in this work. Most of the results presented in this dissertation were obtained in an Oxford Instruments 3He/4He dilution refrigerator with a 20T superconducting magnet at the Millikelvin Laboratory of the NHMFL in Tallahassee. Figure 31 shows another, similar refrigerator at the NHMFL. The operating temperature of the refrigerator used f or this work is typically between 20 mK and 900 mK. It takes about 6 hours for the refrigerator to reach the base temperature after a toploading probe, on which an experiment is attached, is inserted in the mixing chamber 3.1 Specific Heat Measurements 3. 1.1 Calorimeter Design Specific heat provides an unambiguous signature at a phase transitions driven by temperature or magnetic field. As illustrated in Figure 3 2, the calorimeter used for specific heat measurements for this dissertation consists of a sam ple platform, a silver block, a silver ring, and a vacuum can. This calorimeter was designed and built by Hiroyuki Tsujii, a former postdoc of our group, with components produced by our machine shop [ 36] The calorimeter attaches to the bottom of the toploading probe for the dilution refrigerat or. The 6.3 mm thick silver block and the 1.9 cm diameter silver ring serve as a thermal reservoir. Silver is used here because, in magnetic fields, its
32 nuclear specific heat is one of the smallest of all metals, ensuring a short equilibration time. The sample platform is a 6.4 mm diameter and 0.13 mmthick sapphire disk, on which a thin film of a Ti Cr alloy has been evaporated as a heater and a small slice of a 220 Spe er resistor has been attached with EPO TEK 417 silver epoxy as a thermometer. The electrical leads for the heater and thermometer serve as a weak thermal link between the platform and the reservoir, as well as a mechanical support of the platform. The 24. 6 mm diameter vacuum can isolates the reservoir and platform thermally from the liquid helium in the mixing chamber, allowing the temperature of the experiment to be easily set anywhere between 20 mK and 7 K with a heater on the silver block. The can is made of brass in order to reduce eddy current heating when the magnetic field is swept for a magnetocaloric effect measurement A grease joint with a taper of 7tight vacuum seal [ 37] The v acuum feedthroughs for the electrical leads consist of two rows of six goldplated pins taken from a singlein line pin (SIP) socket and glued with Stycast 2850FT epoxy into individual holes drilled on the lid of the vacuum can. The original spacing of the pins of the SIP socket is maintained in the feedthroughs, so that a pair of SIP sockets for the external leads and another pair for the internal leads directly plug into them from above and below, respectively. T he calorimeter is evacuated at room temper ature through a 1.3 mm diameter CuNi tube and a 1/16 inch diameter stainless steel tube, which run in series along the length of the toploading probe. The silver block is suspended by a rod of Vespel SP 22, a thermal insulator, attached to the lid of the vacuum can. A weak thermal contact for the block is provided by twelve 79 m diameter silver leads that run to the block from the feedthroughs. The
33 block has sixteen silver pins for providing heat sinks to the leads; the twelve leads are attached to twelv e of the pins with EPO TEK 417 silver epoxy. The thermometer for 79 m diameter Karmaalloy wire is wound around a silver post on the block; Karma alloy comprises Ni 7 4%, Cr 20%, Al 3%, and Fe 3%. The leads for the platform thermometer and platform heater are 76 m diameter wires of 9 0% Pt 10% Rh. Each lead is soft soldered to a heat sink made of a small piece of 0.1 mm thick silver foil that has been glued with Stycast 2850FT to the silver ring. 3.1.2 Calorimeter Electronics and Thermometer Calibration The electronics for the calorimeter consists of a PAR 124A lock in amplifier, a decade resistor, three dc current sources, and a multimeter, as shown in Figure 3 3. To m easure a specific heat with the relaxation method, which wi ll be described in section 3.1.3, first the temperature of the reservoir is set at a desired value by applying a dc current to the block heater T he resistance of the block thermometer is read with the four wire method employing one of the current sources and the multimeter used as a voltmeter T he Wheatstone bridge for the sampleplatform thermometer is balanced by manually matching the decade resistor with the resistance of the thermometer, while using the lock in amplifier as a null detector. Subsequently, the current to the platform heater is turned on to produce a temperature difference between the platform and the reservoir, resulting in an off balance voltage across the Wheatstone bridge. The bridge is rebalanced by adjusting the decade resistor. Finally, the platform heater current is turned off, and the ensuing relaxation of the off balance voltage is measured by the lock in amplifier, with the dc analog output of the lock in amplifier recorded by a data-
34 acquisition board installed on a computer The bridge is driven by the internal oscillator of the lock in amplifier, set at 441 Hz. T he excitation level is frequently checked t o ensure that the platform thermometer is not overheated while the off balance bridge is maximized. During an experiment, the decaderesistor values that balance the Wheatstone bridge with the platform heater off and on are converted to temperatures by using a calibration function embedded in a LabView program on the com puter Since the resistance of the platform thermometer varies slightly from run to run, however, more accurate conversions are carried out after the experiment by comparing the platform thermometer resistance with the block thermometer resistance, which i s stable. The resistance of the block thermometer depends not only on temperature but also on the magnetic field. Therefore, the thermometer has been calibrated at several fields in situ by measuring the relaxation times of the calorimeter with highpurit y standard samples of silver, platinum, and indium In the field range of the present experiment, calibration uncertainties are about 1.0% at 0.8 K and about 1.8% at 0.1 K. The principle of the calibration method is as follows. W e choose two materials whos e specific heats have quite different temperature dependences at a given magnetic field. We then measure the relaxation times of the samples of those two materials of known masses in our calorimeter, at a given t emperature. The ratio of the two relaxation times will yield the temperature uniquely provided that the phonon and electron specific heats of the samples are accurately known and their nuclear spin specific heat and quadrupolar specific heat if any, can be calculated. The three highpurity metals satisfy these requirements Three instead of two are needed to cover different field and
35 temperature regions. The method in effect measures the heat capacities of the standard samples in magnetic field but eliminates a propagated uncertainty due to the unc ertainty of the thermal conductance of the weak link by taking the ratio of relaxation times. 3.1.3 Relaxation Calorimetry The calorimeter described in Sec. 3.1.1 is used to measure the specific heat of a sample with the relaxation method [ 38 40] In this method, the temperature of the sam ple platform is raised by about 1 % to 5% above the reservoir temperature by applying a dc current to the platform heater. After the platform thermometer has reached a constant temperature, the heater is turned off and the ensuing relaxation of the platform temperature is recorded. In reality, we record the off balance signal of the Wheatstone bridge for the thermometer. The time const ant of the relaxation yields the specific heat of the sample as follows. The heat flow Q from the sample platform to the reservoir is given by T Q (3 1) where is the thermal conductance of the weak link between the sample platform and the reservoir, and T the temperature difference between them ( Figure 3 4 ( a ) ) According to the definition of heat capacity, CdT dQ (3 2) where C is the total heat capacity of the sample and t he sample platform, and T the sample platform temperature. Thus, dt T d C Q ) ( (3 3) since the reservoir temperature is held constant. From Eqs. 3 1 and 3 3, we obtain
36 dt T d C T ) ( (3 4) which yields C ( 3 5) where is the time constant of the exponential relaxation of T Here, the temperature dependences of C and have been ignored. This approximation is valid, since T with the heat er on is kept small, between 1% and 5% of T E quation 35 is the basis of the relaxation calorimetry. Using this equation, the total heat capacity C is obtained from whose temperature dependence has been determined, and the measured For we take its value at the average between the initial and final tem peratures. As a result, C is the value at that average temperature. From C the known heat capacity of the sample platform, Cadd, is subtracted to obtain the heat capacity Csample, and thus the specific heat, of the sample. At low temperatures, simple exponential relaxation of the sampleplatform temperature is not always observed. Often, the temperature instead drops rapidly first, then decays more slowly. This rapid initial drop, the socalled 2 effect, arises from either a poor thermal contact between the sample and the sample platform, as schematically shown in Figure 3 4 ( b ) or a poor thermal conductance of the sample, as shown in Figure 3 4 ( c) In the situation modeled by Figure 3 4 ( b ) th e temperature of the sample is uniform, but the thermal contact between the platform and the sample is poor so that a temperature gradient develops across the interface between the two. This causes the temperature of the platform to drop rapidly, the soca lled lumped 2 effect, until the temperature gradient at the interface leads to a heat flow from the sample to the
37 platform and thus a relaxation of the sample temperature. As a result, T follows the equation [ 39 ] 2 1 1 1 0/ 1 / t e a t e a T T (3 6) where T0 is T at t = 0 the time at which the sampleplatform heater is turned off and the parameters a1, 1, and 2 depend on the individual heat capacities and conductances shown in Figure 3 4 ( b ) The specific heat of the sample is given by 1 1 2 1 1 11 1 C a C a Cadd add sample (3 7) where 1 is the time constant of the slow relaxation and a1 its weight. Cadd is the heat capacity of the sample platform In the case modeled by Figure 3 4 (c) the thermal conductance of the sample is poor, leading to a temperature gradient across the sample, the socalled distributed 2 effect. When the heater is turned off at t = 0 a temperature gradient develops within the sample as well as between the sample and the sample platform, where sam ple is the thermal conductance of the sample. In this case, the temperature in the sample is taken to be constant in planes parallel to the sampleto platform boundary and to vary only with the distance from the boundary. Though the heat flow equation may be solved analytically in terms of C s, and extraction of these parameters from the relaxation data is difficult [ 38, 41 ] However, application of conservation of energy to the problem gives the heat capacity directly, since the total energy removed from the sample and
38 platform must equal the total amount of heat that flows through the thermal link. This simple reasoning leads to 0 ) ( =0dt t T T C (3 8) Since the equation is based on a fundamental principle, it can be applied to all cases regardless of the source of the poor conductance [ 41] In practice, however, it should be used only when the bundled2effect model is inadequate, since the numerical integration of T is susceptible to errors due to a noise. 3.2 MagnetocaloricEffect Measurements A peak in s pecific heat is an unambiguous indicator of a phase transition. However, mapping out a phase diagram with specific heat measurements requires a great amount of time. Also, the specific heat anomaly usually becomes weak, as a phase boundary becomes flat, meaning that the temperature dependence of the transition field becomes weak as the temperature approaches zero. To supplement specific heat measurements which suffer from these limitations, the magnetocaloric effect provides a very convenient, fast probe that works over a wide range of temperatures. Often, magnetocaloric effect measurements are made before specific heat measurements to make a quick survey of the phase diagram of a new magnetic material, followed by specific heat measurements for close inspection of temperature and field regions of particular interest with a higher accuracy. M agnetocaloric effect measurements employ the same relaxation calorimeter and ancillary electronics used in specific heat measurements. However, the principle of magnetocaloric effect measurements is different from specific heat measurements as explained below
39 The magnetocaloric effect is the response of the temperature or entropy of a magnetic material to a change in the applied magnetic field. This effect finds applications in the technique of adiabatic nuclear demagnetization that provides ultralow temperatures below 10 mK, as well as in the conventional magnetic refrigeration technology [ 42, 43] When a magnetic field applied to a magnetic material placed on a thermally is olated sample platform is changed by dH the platform temperature will change by dH C T M T dH T S H S dH H T dTH H H T S (3 9) Here, S is the entropy, M the magnetization, i.e., the total magnetic dipole moment of the sample. In deriving Eq. 3 9, Maxwells relation H TT M H S (3 10) and the expression for the constant field heat capacity H HT S T C (3 11) have been used. For the relaxation calorimeter described in Sec. 3.1.1, Eq. 3 9 does not strictly hold, since the sample platform of the calorimeter is weakly coupled to the thermal reservoir rather than thermally isolated, as depicted in Figure 3 5. Equation 31 again governs the heat flow through the weak link from the platform to the reservoir. On the other hand,
40 dH H S dT T S T TdS dQT H (3 12) From Eqs. 3 1 and 312, we obtain dt dH T M T dt T d C T H H ) ( (3 13) for T due to the magnetocaloric effect during a field sweep at the rate dH/dt Here, Eqs. 3 10 and 3 11 have been used along with the fact that the reservoi r temperature is held constant. By rearranging the terms in the above equation, the evolution of the temperature deviation is given by dt dH T M T dt T d TH ) ( (3 14) where = CH/ is the thermal relaxation time of the platform. As this equation shows, the magnetocaloric effect during a field sweep results in a temperature difference between the sample platform and the reservoir, a temperature difference whose field dependence reflects the dependence of the entropy on the magnetic field or, equivalently the temperature dependence of the magnetization. Thus, the magnetocaloric effect is a sensitive probe of a magnetic phase transition, at which the entropy and magnetization exhibit anomalies. Furthermore, this effect can di stinguish a firstorder transit ion from a continuous transition. In practice, it is found that the sweep rate of the applied field affects the reservoir temperature, which in turn affects the sampleplatform temperature. Since the reservoir is made of silver, eddy current generated by a field sweep causes heating, proportional t o the square of the sweep rate. To account for the heating effects, the field must be swept both upward and
41 down ward, and the resulting temperature variations referenced to the average background temperature. 3.3 MagneticTorque Measurements Magnetic torque is another tool that allows quick detection of a magnetic phase transition, although it is often a blunt probe that does not give the precise location of the transition. To measure this quantity, we have used a capacitive cantilever magnetometer built by Timothy Murphy at the NHMFL. The measurements were made in collaboration with him. Figure 3 6 illustrates his magnetometer, comprising a 12.7 m thick CuBe beam (the cantilever), a fixed electrode, and a header. A thermometer and heater that read and regulate the temperature of the sample are located on the mounting stage (not shown) of the magnetometer. The sample is attached to the cantilever with GE varnish to prevent it from falling off in response to an appli ed field and the gravity. The position of the sample should be close to the tip of the cantilever, as far from the fulcrum as possible, to maximize the magnetometer sensitivity, and on the symmetry line of the cantilever to minimize the effect of an unwant ed torque that twists instead of bends the cantilever. The cantilever technique was developed by Brooks et al. [ 44] among others. When a uniform magnetic field is applied to a magnetic sample, the direction of the sample magnetization usually differs slightly from that of the magnetic field. This will result in a torque HM (3 15) where M is the magnetization, and H the applied field. This torque on the sample bends the cantilever, whose deflection changes the capacitance between the cantilever
42 and the fixed electrode [ 45] For a small deflection, the capacitance change is proportional to the torque. The misalignment of the magnetization with respect to the magnetic field, a necessary condition for torque magnetometry, can arise from a number of sources The magnetic ions in the sample may have an anisotropic g tensor or the socalled singleion ani sotropy due to spinorbit interaction. The exchange interaction between the magnetic ions may contain a Dzyaloshinskii Moriya term [ 46, 47] which is anisotropic. Eve n in a sample that lacks such sources of anisotropy, a nonuniform demagnetization field will cause a misalignment of the magnetization, unless the sample is an ellipsoid. The capacitance of the cantilever magnetometer is measured with an Andeen Hagerling 2700A automatic capacitance bridge, operated at 1 k Hz with a 1 Vrms excitation. Typical measured capacitances of the magnetometer at the NHMFL are 0.9 to 1.1 pF. Typical sensitivities range from 10 7 emu to 10 9 emu for measurements at 20 T. 3.4 Nuclear Magnetic Resonance Nuclear magnetic resonance (NMR) is used in the work reported in Chapter 5, in order to measure the spinlattice relaxation time T1 of hydrogen nuclear spins in a magnetic sample placed in a magnetic field. In this section, I will describe the principle of the spinecho method, and its use in T1 measurements and NMR spectra measurements. Figure 37 is a schematic representation of the experimental geometry, in which a sample is placed inside an rf coil inserted in the bore of a superconduct ing magnet. The magnet produces a static magnetic field along the z axis. The z component of the angular momentum loosely called the spin of a n atomic nucleus in the sample is
43 then quantized, with allowed values of the quantum number, m ranging from I to I where I is the spi n quantum number of the nucleus. For example, 1H has I = 1/2 so that m can be either +1/2 or 1/2. When there is no applied field, the spin states of different m are degenerate. In applied magnetic field H h owever, the Zeeman effect will split the energy of these states as shown in Figure 3 8 for I = 1/2, with the energy of each state given by H E (3 16) where is the nuclear magnetic moment This quantity is written as pm I ge 2 (3 17) where g is the nuclear g factor, e the fundamental charge, the Planck constant, and mp the proton mass. The energy is quantized a ccording to H m E (3 18) where is the gyromagnetic ratio g e /2 mp. Consequently H (3 19) is the frequency of a photon that is absorbed or emitted during a transition between t he t wo states. This frequency is typically 1 MHz to 1 GHz, in the range of radio frequency [ 48] The photons that are absorbed or stimulate an emission are produced by an rf coil shown in Figure 3 7 This coil also det ects the photons emitted by the nuclear spins. Macroscopically, the detection occurs as an induced emf across the coil due to precession of nuclear magnetization around the z axis, the direction of the applied magnetic field H.
44 The dynamics of nuclear spins in a sample are characterized by two relaxation times, the spinlattice relaxation time T1 and the spinspin relaxation time T2. The spinlattice relaxation is a longitudinal magnetic relaxation, the recovery of the longitudinal component of the nuclear ma gnetization, Mz, to equilibrium after tipping of the nuclear spins by an rf pulse. T1 characterizes this relaxation as ) / 1 )( 0 ( ) (1T t e z M t z M (3 22) The spinspin relaxation, on the other hand, is a transverse magnetic relaxation of the xy component of the nuclear magnetization, Mxy. T2 characterizes this process as ) / 1 )( 0 ( ) (2T t e xy M t xy M (3 23) i n a frame rotating around the z axis at frequency given by Eq. 321. The transverse relaxation is caused by interactions between nuclear spins as well as by an inhomogeneity of the static field H providing slightly different precession frequencies to local nuclear magnetizations in different parts of the sample. To remove this second, uninteresting effect, the spinecho method is used. Figure 39 explains this method [ 49] First, a 90 xy plane, in the rotating frame Immediately after this pulse, all spins point in the same direction in the xy plane, but after a while their directions in the rotating f rame will sp read out because a slight inhomogeneity in H causes the spins in different parts of the sample to precess at different rates. The 180 pulse, flips all the spins by 180 wer spins ahead of faster spins, causing all the spins to line up once again in the xy plane at time 2 At this point, all the spins precess in phase in the laboratory frame, inducing an rf
45 emf across the coil that has generated the tipping pulses. This s ignal is the spin echo. The intensity of the spinecho signal plotted against 2 decays only with intrinsic T2 due to time dependent fluctuating random fields arising from interactions between nuclear spins, not with a shortened time constant that includes the uninteresting effect of the inhomogeneous magnetic field. Spin echoes are not only used for T2 measurements. Since an echo signal appears with a delay after the last rf pulse, it does not suffer from a slow recovery of the amplifier after a pulse, unl ike a freeinduction signal which immediately follows an rf pulse. For this reason, spin echoes are widely used for NMR spectra measurements as well as T1 measurements. Figure 310 shows the basic pulse sequence for T1 measurements with the spinecho technique [ 48] First, several 90 pulses are applied in order to saturate Mz(0) of all spins to zero. After a delay, tR, the recovery of Mz is monitored by a 90 180 pulse pair that produce a spin echo, whose intensity is proportional to the recovering Mz. The sequence is repeated many times, by varying tR while holding constant and T1 is determined from the recovering intensity of the spin echo as a function of tR. In our experiment, to be described in Chapter 5, tR ranged from 10 s to 10 s and was 45 s.
46 Figure 31. 3He/4He dilution refrigerator at the NHMFL Millikelvin Laboratory.
47 Platform thermometer Silver ring Silver block Silver pins V acuu m feedthrough Sapphire platform Pumping line T apere d grease seal Block thermometer V espel suppor t Brass support Block heater Platform heater Platform heater Platform thermometer Figure 32. Calorimeter for specific heat and magnetocaloric effect measurements The bottom view shows the sample platform made of a thin sapphire disk at center, attached to the silver ring via the leads for the heater and thermometer on it. The photograph to the right shows the actual calorimeter, with the vacuum can removed. Adapted from Ref. [ 36]
48 Figure 33. Schem atic of the calorimeter electronics setup for the specific heat and magnetocaloric effect measurements. A Wheatstone bridge is used to measure the resistance of the sampleplatform thermometer. Data is obtained by a data acquisition board (DAQ) on a computer. Via a GP I B bus, the computer also controls the current sources for the block and platform heaters and for the block thermometer, and reads the voltage across the block thermometer.
49 Figure 34. Models for the analysis of a relaxation of the platform temperature in a specific he at measurement [ 50] (a ) Simple onedimensional heat flow model w ith only one relaxation time. (b ) Model for the lumped 2 effects, with two relaxation times, 1 and 2, arising from a poor therm al contact between the sample and the platform, represented by the limited thermal conductance in series with the thermal conductance of the weak link between the platform and the reservoir. (c ) Model for the distributed 2 effects. The two time const ants in this case arise from the limited thermal conductance of the sample. [Adapted from Sherline, T. E. 2006. Antiferromagnetism in cesium Tetrabromocuprate(II) and Body CenteredCubic Solid Helium Three. Ph.D. dissertation ( Page 23, Figure 2.4). Universi ty of Florida, Gainesville, Florida.] Figure 35. Illustration of the principle of the magnetocaloric effect due to a field sweep [ 50] When the magnetic field on a sample is changed by dH over an infinitesimal time dt the temperature of the sample changes by d ( ) [Adapted from Sherline, T. E. 2006. Antiferromagnetism in cesium Tetrabromocuprate(II) and Body CenteredCubic Solid Helium Three. Ph.D. dissertation (Page 3 0 Figure 2.6). University of Florida, Gainesville, Florida.]
50 Figure 36. Schematic view of the cantilever magnetometer. A sample (a) is glued with GE varnish on the flexible CuBe beam (b). The beam is separated from the fi xed electrode (c), which is embedded in the base plate (d), by the sp acer (e). As explained in the text, when the magnetization ( M ) of the sample is misaligned with respect to the external magnetic field ( H ), a magnetic torque ( ) is exerted on the sample, bend ing the cantilever beam and thus leading to a change in the capacitance between b and c.
51 Figure 37 Schematic arrangement for NMR. Figure 38. Energy diagram of a nuclear spin ( I = 1/2) in an applied field H The energy difference E between the two spin states is proportional to H and the gyromagnetic ratio of the spin.
52 Figure 39. Principle of the spinecho technique. The precession of magnetization, shown with arrows, is depicted in the rotating frame. (a ) At t = 0, the equilibrium magnetization points in the z direction parallel to the applied field H (b ) A 90 xy plane. (c) Magnetizations at different parts of the sample rotate at slightly different rates due to an inhomogeneity of the applied field, causing the magnetizations to spread out in the rotating frame. (d) This spreading becomes more extensive with passing time. (e) A 180 x axis flips the magnetizations by 180) Finally, a faster magnetization catches up with a slower magnetizati on, so that all magnetizations become once again in phase and produce an induced emf (the spin echo) across the rf coil
53 Figure 310. NMR p ulse sequence for T1 measurements Five 90 pluses ( only t wo are shown for clarity ) saturate the magnetization, followed by a 90 180o pulse pair that moni tors magnetization recovery by spin echo. tR, known as a recovery time, is varied whereas the time between the 90 and 180o pulses for the spin echo is fixed Freeinductiondecay signal s immediately after the 90 pulses ha ve been omitted for clarity
54 CHAPTER 4 THERMODYNAMICS OF THE S=1 ANTIFERROMAGNET Cr( DIETHYLENETRIAMINE)(O2)2H2O 4. 1 Basic Properties of C r (diethylenetriamine)(O2)2H2O At a classical critical point, of a continuous phase transition occurring at a nonzero temperature, only the spatial extent of order parameter fluctuations diverges [ 51] In contrast, the divergence occurs both spatially and temporally at a QCP, in extricably linking dynamic and static properties [ 52, 53] A spectacular demonstration of this linkage is the quantum annealing the quantum fluctuationdriven relaxation of quenched disorder of a n Ising spin glass near its QCP, a phenomenon with broad implications in efficient algorithms for solving optimization problems i n circuit designs and logistics [ 54, 55] Despite the importance of the dynamics of quantum fluctuations near QCPs, experimental studi es of such dynamics have been scar ce largely limited to inelastic neutron scattering to which not all materials that possess QCPs are amenable. In our work, we have observed heat release from hydrogen nuclear spins caused by the quantum fluctuations of Cr4+ ionic spins near the magnetic f ield driven QCP. This phenomenon o pens up a unique avenue to investigate the dynamics of quantum fluctuations that underlie quantum criticality. For this study, we have chosen the inorganic coordination compound Cr(diethylenetriamine)(O2)2H2O hereafter ref erred to as Cr(dien), because it contains a large number of hydrogen nuclear spins [ 4 56] Cr(diethylenetriamine)(O2)2, the key building block of this c ompound, i s an oblate, elongated disk shaped m olecule, in which Cr4+ is located on the mirror symmetry plane, as shown in Figure 41 In the monoclinic crystal structure of Cr(dien), also shown in Figure 41, the S = 1 spins of Cr4+ form a square lattice parallel to the crystallographic ac plane, with an exchange energy J of
55 2.71 2.88 K according to magnetic susceptibility at temperatures between 1.8 K and 300 K [ 4 ] .1[ 4 ] The spins order antiferromagnetically at TN = 2 55 K in zero magnetic field Application of a high magnetic field depresses the ordering temperature, driving it to zero at a critical fi eld, which we have found to be 12.392 0.003 T. Heat release from hydrogen n uclear spins occurs near this QCP. 4.2 Experimental The singlecrystal sample of Cr(die n), shown in Figure 42, was grown by Narpinder Kaur at the Chemistry Department of Florida State University, by chemical synthesis in an ice bath, followed by slow evaporation of water, as described in Ref. [ 4 ] The crystal weighed 1.02 mg The experiment, done primarily at the NHMFL in Tallahassee, used the relaxation calorimeter described in section 3.1.1. In this calorimeter the sample temperature can be raised with the heater with ease and allowed to drop rapidly to the reservoir temperature by turning off the heater. Heat released in the sample, if any, can be readily detected as a spontaneous temperature diff erence bet we en the sample and the reservoir. The calorimeter was inserted in the mixing chamber of a toploading dilution refrigerator equipped with a 20tesla superconducting magnet. 4.2.1 Specific Heat and Magnetocaloric Effect Prior to heat release measurements, we first determined the magnetic phase diagram of Cr(dien) by means of specific heat and magnetocaloric effect measurements 1 The values of J differ from those given in Ref. [ 4 ] by a factor of 2, because we write unlike the authors of that reference the spin hamiltonian as j j i iS S J H where each nearest neighbor spin pair j i appears only once in the sum
56 at temperatures ranging from 0.2 K to 3 K in magnetic fields up to 12.5 T. T o minimize the addenda heat capacity, the sample was dir ectly glued on the calorimeter platform with Wakefield compound. The field was applied parallel to the crystallographic b axis, perpendicular to the ac plane of the monoclinic crystal. For the magnetocaloric effect measurements, the magnetic field was swept at the rate of 0.2 T/min. 4.2.2 Heat Release from Proton Spins near the QCP Figure 43 describes the procedure of the experiment First, the sample is heated from the thermal reservoir temperature, T0, to temperature Tq r anging from 266 mK to 1.52 K, in magnetic field Hq p arallel to the b axis the same direction as in the specific heat and magnetocaloric effect measurements. After 1.4 min at Tq, the sample is rapidly cooled back to T0 in 1.8 s by turning off the heater. This temperature quenching leaves the hydrogen nuclear spins the proton spins fr ozen in a highenergy nonequilibrium state corresponding to Tq. Subsequently, the magnetic field is swept at 0.2 T/min or 0.1 T/min through the critical field while the temperature difference T between the sample and the thermal reservoir is continuously measured to detect heat release from the proton spins The field sweeps were made at four different T0 ranging from 96 mK to 261 mK. Each sweep was immediately repeated, and T of the second sweep in which heat release from the proton spins no more occurs was subtracted from T of the first sweep in order to remove u ninteresting contributions of the magnetocaloric effect (at most 1.1 mK) and eddy current heating ( 3 mK at T0 = 96 mK and 8 mK at T0 = 181 mK ) We have also measured the relaxation time of the proton spins during the heat release. This was done by stopping the field sweep in the middle, at field H and recording the subsequent relaxation of the sample temperature toward T0. Again, the
57 sweep w a s immediately repeated to the same field, and the ensuing relaxation of T from the magnetocaloric effect and eddy current heating alone was recorded and subtracted from the data. The measurements were carried out only at T0 = 181 mK after the sample had been quenched from Tq = 796 mK at Hq = 13.5 T or from Tq = 761 mK at Hq = 10 T. measured in this manner is approximately T1 + Cn/ [ 57] where T1 is the nuclear s pinlattice relaxation time and Cn the nuclear heat capacity. At 181 mK, where the measurements were made, Cn/ = 11.1 s at 12.1 T. 4.3 Analysis of the Results 4.3.1 Specific Heat and Magnetocaloric Effect Figure 44 show s the result of specif ic heat meas urements. Peaks indicate an antiferromagnetic transition. A s the magnetic field is raised, both the transition temperature and peak height decreases. Above 12 T, the peak becomes too small to detect. To overcome this limitation, we made magnetocaloric effe ct m eas urements from 10.5 T to 12.5 T. As described in section 3 .2 the magnetocaloric effect is a sensitive, convenient tool for detecting a magnetic phase transition. However, it is usually unclear which feature of a T curve during a field sweep stands for a transition, if the transition is continuous as opposed to first order. This ambiguity in the exact location of a continuous magnetic transition is a problem common to all techniques that measure, directly or indir ectly, a magnetization. As in those cases, one needs to rely on specific heat results as a guide in choosing the correct feature. In the T curves for Cr(dien), a peak or a dip appears depending on whether the field is swept upward or downward, as shown in Figure 4 5. It turns out that the average temperature and field of these two
58 features in a pair of curves for upward and downward field sweeps for a fixed reservoir temperature give the correct transition point From the specific heat and magnetocaloric e ffect data, we have determined the magnetic phase diagram of Cr(dien) in terms of magnetic field and temperature, as shown in Figure 4 6. If Cr(dien) is a Heisenberg antiferromagnet the n the boundary of the ordered phase should obey a power law, Eq. 2 27, with the critical exponent = 3/2. Fitting the data up to 0.838 K, we obtain Hc = 12.392 0.003 T with = 2.01 0.02. This exponent i s in close agreement with = 2 predicted for the 3D Ising universality class [ 2 ] ruling out a BEC of magnons in Cr(dien). To identify the origin of the Ising anisotropy in this compound, further studies are required. 4.3.2 Heat Release from Proton Spins near the QCP Figures 4T between the sample and the thermal reservoir during field sweeps through the critical field Hc, when the thermal reservoir was held at T0 = 181 or 96 mK. Curves obtained at T0 = 261 and 219 mK are shown in Figure 4 8. As the field approaches Hc, heat is released in t he sample, T The heat release occur red only during the first field sweep after the temperature quenching of the sample from Tq, not during subsequent sweeps. The amount of released heat Q is obtained from the data via H TdH Q (4 1 ) where is the thermal conductance of the weak link between the sample and the thermal reservoir, and H the fieldsweep rate. As shown in Figure 4 9, Q depends on
59 both Tq and Hq. These dependences indicate unambiguously that the heat is indeed released from the proton spins as the following analysis shows The energy of nuclear spins per mole at temperature T in magnetic field H is T H Ra H T U2, (4 2 ) where the constant a is 2 23 ) 1 ( ) (Bk I I a (4 3 ) R is the gas constant, and kB the Boltzmann constant For the proton, I = 1/2. In the second step of the experiment, the sample is cooled rapidly in magnetic field Hq from temperature Tq, at which the proton spins are thermalized with the lattice, to T0, at which the nuclear spinlattice relaxation time is long. This causes the proton spins to freeze in a nonequilibrium state determined by Tq, At this point, the energy of the proton spins is U ( Tq, Hq). In the third step, where the magnetic field is swept to H the energy changes to U ( TqH / Hq, Hq) because the entropy of the proton spins stays constant and, as a result the t emperature that characterizes the nonequilibrium state of the proton spins changes from Tq to TqH / Hq. (N ote that the entropy of nuclear spins is a function of only H / T .) If this field is close to the QCP, then rapidly fluctuating dipolar fields and transferred hyperfine fields due to quantum fluctuations of the Cr4+ spins will cause the proton spins t o therm alize with the lattice. The heat released by the proton spins to the lattice will be then
60 3 ) 1 ( 1 ) ( ) / (0 2 0 2 0H T H T H I I k R T HT H RaH H T U H H H T U Qq q B q q q q (4 4 ) In n moles of sample, if the number of proton spins that participate in such heat release is nH per formula unit, then Eq. 4 4 must be multiplied by nHn : H T H T H I I k nR n Qq q B H 0 23 ) 1 ( (4 5 ) At T0 = 261 and 219 mK, the temperatures of the data shown in Figure 4 8 the agreement between the amount of released heat and Eq. 4 5 is poor, suggesting that the proton spins had partly equilibrated w ith the lattice before the field sweeps. For T0 = 181 mK, good agreement between experiment and Eq. 4 5 is obtained with nH = 10, as shown in Figure 4 9(a), except for a few points for which Hq is 18 T or 16 T. For those points, 1.4 min of waiting wa s probably insuffi cient to thermalize the proton spins at Tq. A t T0 = 96 mK the peak that appears at 12.76 T in Figure 4 7(b) during the downward field sweeps from 13.5 T is very sharp while T > 36 mK i.e. while the sample temperature T0 + T is higher than 132 mK. This indicat es that proton spins whose relaxation times are too short when T0 > 132 mK now participate d in heat release. When T0 = 181 mK, they had evidently reached thermal equilibrium with the lattice before each field sweep started and therefore did not participate in heat release. A s shown in Figure 4 9(b), good agreement between Eq. 4 5 and the T0 = 96 mK data for Hq = 13.5 T is obtained with nH = 1 5 of which five whose are very short at T0 > 132 mK are assumed to freeze at a highenergy nonequilibrium state at 132 mK instead of at Tq.
61 Evidently, those five proton spins do not participate in heat release even at T0 = 96 mK when quenched at Hq = 10 T, in the antiferromagnetic phase. Cr(dien) contains fifteen hydrogen atoms per formula unit, as shown in Figure 4 1(b). Among the thirteen in t he Cr(diethylenetriamine)(O2)2 molecule, the f ive bonded to nitrogens are closer to the Cr4+ ion than eight that are bonded to carbons The five hydrogens bonded to the nitrogens are located at 2.40 2.52 from the Cr4+, whereas the eight that are bonded to the carbons fall into two groups of four, each at 3.08 3.29 and 3.77 3.79 from the Cr4+. The water hydrogens are at 3.29 and 3.48 from the closest Cr4+ ions It is very likely that the five pr oton spins with short nuclear spin are of the hydrogens bonded to the nitrogens and thus experience stronger fluctuating dipolar field and transferred hyperfine field of the Cr4+ ion, whereas the ten proton spins with longer are of the eight hydrogens bonded to the carbons and the two i n the water molecule. The temperatures and fields of the peaks in Figures 4 7 and 48 are shown in the diagram given in Figure 4 10, along with the phase boundary between the highly polarized antiferromagnetic phase and similarly highly polarized paramagnetic phase. Since t he peak position depends slightly on the quenching conditions Tq and Hq, the peaks chosen for the figure are for similar Tq and Hq as much as possible for consistency, as listed in Table 41. This diagram suggests that the loci of the peaks converge on the QCP in the zerotemperature limit These loci delimit the region in which is shorter than the timescale of the experiment, a quantum critical region.
62 Figure 411 shows the relaxation of T at T0 = 181 mK at various H Six of the ele ven curves exhibit exponential relaxation whereas five exhibit stretched exponential relaxation: ) / ( 0 t e T t T (4 6 ) where t is time. The stretching exponent ranges from 0.61 to 0.78. The relaxation rate 1/ diverges as the field approaches Hc, as shown in Figure 4 12, indicating that divergent quantum fluctuations of the Cr4+ spins near the QCP drive the relaxation of the nuclear spins. The divergence is asymmetric around Hc, faster for H > Hc than for H < Hc. This asymmetry is also seen in Figure 4 1 0 as a wider quantum critical region for H > Hc than for H < Hc. The amount of heat released during the relaxation measurements is shown as a function of | H Hc| in Figure 4 13. The result indicates that the ten proton spins that release heat at T0 = 181 mK contain two groups, each comprising four to six spins per formula unit. Above Hc, one group relaxes at | H Hc| of about 0.8 T, whereas the other group relax es at fields closer to Hc. Similarly, below Hc, the first group relaxes at | H Hc| of about 0.6 T, whereas the second group relaxes at fields closer to Hc. The result is consistent with the molecular structure of Cr(diethylenetriamine)(O2)2 shown in Figure 4 1(b): the eight hydrogens bonded to the four carbons with longer protonspin fall into two groups, each consisting of four hydrogens, with distinct ranges of distances from the Cr4+. 4.4 Discussion Our results provide unambiguous evidence that temperature quenching of Cr(dien) leaves the proton spins frozen in a highenergy nonequilibrium state and, as the
63 magnetic field is then brought close to the QCP, quantum fluctuations of the Cr4+ ionic spins quickly anneal them to reach thermal equilibrium with the lattice. These results imply that the quantum fluctuationdriven heat rel ease from nuclear spins is a generic phenomenon to be found near a variety of QCP Because of the inextricable link between dynamic and static properties in quantum criticality, quantum critical systems are predicted to exhibit interesting, nontrivial rel axation phenomena during and after a sweep of a control parameter such as magnetic field and pressure through a QCP [ 58] and also after temperature quenching near the QCP [ 59] Our results warn, however, that the response of nuclear spins which are nearly ubiquitous to those changing parameters and to quantum fluctuations must be carefully taken into account in real solids. At the same time, heat release from nuclear spins promises to be a useful probe for the dynamics of quantum fluctuations that underlie quantum criticality in a variety of systems
64 Figure 41. Crystal structure and basic unit of Cr(diethylenetriamine)(O2)2 and H2O molecules (a) (top panel) Crystal structure of Cr(dien), with H2O molecules and H atoms omitted for clarity [ 60] The blue arrows indicate nearest neighbor distances between Cr4+ ions, 5.64 on average, within a squarelattice layer The interlayer spacing is approximately 6.9 (b) (bottom panel) B asic unit of the crystal : a pair of Cr(diethylenetriamine)(O2)2 and H2O molecules Yellow is Cr, red O, blue N, black C, and gray H. Drawn by Ronald Clark. [ Top panel r eprinted with permission from Ramsey, C. M. 2004. Thermomagnetic and EPR Probing of Magnetism in Low Dimensional Lattices and SingleMolecule Magnets. Ph.D. dissertation (Page 77, Figure 5. 3 ). Florida State University Tallahassee, Florida.]
65 Figure 42. Image of the Cr(die n) sample. P2' P1' P2P1 QCP Field Temperature Figure 43. P rocedure of the heat release e xperiment. The sample is heated from point P1 to P2 in the nearly completely polarized paramagnetic phase, or from P1 to P2 in the antiferromagetic phase, whose boundary (broken line) terminates in a QCP at T = 0. After rapid temperature quenching, the magnetic field is swept through the critical field, and the evolution of the sample temperature is recorded.
66 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 2 4 6 8 10 C (J/K mol)Temperature (K) 0 T 5 T 6T 7T 8T 9T 9.5T 10T 10.5T 11T 11.5T 12T Figure 44 Specific heat as a function of temperature at different magnetic fields [ 5 ] Peaks indicate clearly the transition at fields between 0 T and 11.5 T. At 12 T, the peak is barely visible. Magnetocaloric effect measurements were used at high fields, where specific heat peaks are small or barely visible. [Adapted from Kaur, N. 2010. Magnetic and Thermodynamics Studies on Spin 1 Compounds. Ph.D. dissertation (Page 89, Figure 5. 11). Florida State University, Tallahassee, Florida.]
67 11.0 11.5 12.0 12.5 13.0 13.5 0.474 0.476 0.478 0.480 0.482 0.484 0.486 T (K)H (T)0.456 K 11.0 11.5 12.0 12.5 13.0 13.5 0.832 0.834 0.836 0.838 0.840 0.842 0.844 0.846 0.797 K T (K)H (T) 10.5 11.0 11.5 12.0 12.5 13.0 0.95 0.96 0.97 0.98 T (K)H (T)0.908 K 10.0 10.5 11.0 11.5 12.0 12.5 1.384 1.388 1.392 1.396 1.400 1.404 1.408 T (K)H (T)1.322 K 10.0 10.5 11.0 11.5 12.0 12.5 1.54 1.55 1.56 1.57 1.58 1.479 K T (K)H (T) 9.0 9.5 10.0 10.5 11.0 11.5 1.80 1.81 1.82 1.83 1.84 1.85 1.734 K T (K)H (T) Figure 45. Magnetocaloric effect curves of Cr(dien) at different reservoir temperatures [ 5 ] Red and blue curves represent, respectively, the sample temperature during upward and downward field sweeps. Specific heat m easurements guarantee that the peaks and dips in the curves are transitions. [Adapted from Kaur, N. 2010. Magnetic and Thermodynamics Studies on Spin 1 Compounds. Ph.D. dissertation (Page 86, Figure 5.10). Florida State University, Tallahassee, Florida.]
68 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 0 2 4 6 8 10 12 14 Paramagnetic phase Antiferromagnetic phase Heat capacity Magnetocaloric effect Power-law fit H (T)T (K) Figure 46. Magnetic phase diagram of Cr(dien), determined by specific heat and magnetocaloric effect measurements [ 5 ] The critical field, Hc, is 12.392 0.003 T. [Adapted from Kaur, N. 2010. Magnetic and Thermodynamics Studies on Spin 1 Compounds. Ph.D. dissertation (Page 95, Figure 5. 17). Florida State University, Tallahassee, Florida.]
69 11.5 12.0 12.5 13.0 13.5 0 5 10 15 20 (a) T (mK)H (T) 11.5 12.0 12.5 13.0 13.5 0 20 40 60 80 (b) T (mK)H (T) Figure 47. Temperatur T between the sample and the thermal reservoir as a function of the magnetic field during field sweep s. (a ) The thermal reservoir was held at 181 mK as the field was swept at 0.2 T/min T emperature Tq from which the sample has been quenched ranges from 266 mK to 1.52 K. The quenching fields were 6 T (black), 9 T (purple), 11.5 T (green), 13.5 T (orange), 16 T (magenta), and 18 T (r ed). (b) The thermal reservoir was held at 96 mK, as the field wa s swept at 0.1 T/min. Tq ranged from 300 mK t o 797 mK. The quenching fields were 10 T (blue) and 13.5 T (red). In both frames, t he peaks to the right were observed during downward field sweeps and those to the left during upward sweeps. The critical field, Hc, is 12.392 0.003 T, as shown in Figure 4 6
70 11.0 11.5 12.0 12.5 13.0 13.5 0 2 4 6 (c) T (mK)H (T) (b) (a) T (mK) T (mK)11.0 11.5 12.0 12.5 13.0 13.5 -0.5 0.0 0.5 1.0 1.5 2.0 11.0 11.5 12.0 12.5 13.0 13.5 0 2 4 6 Figure 48. Temperature difference T between the sample and the thermal reservoir during field sweeps at 0.1 T/min. For ( a ) the thermal reservoir wa s held at T0 = 261 mK during the field sweeps as well as during temperature quenching of the sample. Blue and green: upward field sweeps after temperature quenching from Tq = 764 mK (blue) and 1.01 K (green) at Hq = 10.5 T; red: downward field sweep after temperature quenching from Tq = 796 mK at Hq = 13.5 T. For (b) the thermal reservoir is held at T0 = 219 mK. Blue: upward field sweep after temperature quenching from Tq = 1.00 K at Hq = 10 T; red: downward field sweep after temperature quenching from Tq = 1.04 K at Hq = 13.5 T.
71 0.0 0.5 1.0 1.5 0.0 0.1 0.2 0.3 (a)Q ( J)Tq (K) 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (b)Q ( J )Tq (K) Figure 49 Amount of heat released at (a ) 181 mK and (b ) 96 mK as a fu nction of Tq, the temperature from which the sample was rapidly quenched. In (a ) the sample was quenched at 6 T (black), 9 T (purple), 11.5 T (green), 13.5 T (orange), 16 T (magenta), and 18 T (red). T he lines represent Eq. 4 5 with nH = 10, each corresponding to the data points of the same color. In (b ) the sample was quenched at 10 T (blue) and 13.5 T (red). T he lines of corresponding colors represent Eq. 4 5 nH = 10 for the 10 T line; nH = 15, of which five are assumed to freeze only at 132 mK for th e 13.5 T line.
72 Figure 4T curves (squares), marking a quantum critical region of Cr(dien) delimited by two straight lines through the data points. Solid squares are for peaks at a sweep rate of 0 1 T/min open squares at 0 2 T/min. Circles represent the phase boundary detected by the magnetocaloric effect between the antiferromagnetic (AF) and nearly completely polarized paramagnetic (P) phases, with the broken line from a power law fit of da ta points up to 0.84 K.
73 0 200 400 600 800 0 5 T (mK)(b) (a) T (mK)Time (sec) 0 200 400 600 800 0 5 10 Fig ure 411 Relaxation of T the temperature difference between the sample and the thermal reservoir, after a field sweep at 0.1 T/min is stopped at t = 0. The reservoir is held at T0 = 181 mK The sample has been prepared by temperature quenching (a) from Tq = 796 mK at Hq = 13.5 T and (b) from Tq = 761 mK at Hq = 10 T. In (a), the curves were taken at 13.1 T, 13.0 T, 12.9 T, 12.85 T, 12.8 T, and 12.7 T (from top to bottom), after a downward field sweep from Hq toward Hc = 12.392 0.003 T stopped. In (b), the fields at which th e curves we re taken are 12.1 T, 12.0 T, 11.9 T, 11.8 T, and 11.7 T (from top to bottom), after an upward field sweep from Hq toward Hc stopped. For clarity, successive curves have been shifted by 1 mK.
74 Figure 4ld at which field sweep was stopped. Solid symbols are data from curves showing exponential relaxation, and open symbols from curves showing stretchedexponential relaxation. Lines are guides to the eye. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 |H-Hc| (T)Q ( J ) Figure 4 13. A mount of heat released vs. the field at which field sweep was stopped. Horizontal lines indicate the amounts of heat released during complete field sweeps through Hc. B lue and red squares are for Tq = 761 m K at Hq = 10 T and Tq = 7 96 m K at Hq = 13. 5 T, respectively.
75 Table 41. T0, Tq, and Hq of the data points demarcating a quantum critical region in the H T diagram shown in Figure 410. T 0 (mK) T q (K) H q (T) 96 0.797 13.5 0.762 10 181 0.796 13.5 0.500 9 219 1.04 13.5 1.00 10 261 0.796 1 3.5 0.764 10.5
76 CHAPTER 5 PROTON NMR IN THE S=1 SQUARELATTICE ANTIFERROMAGNET Cr( DIETHYLENETRIAMINE)(O2)2H2O NEAR THE QUANTUM CRITICAL POINT 5.1 Motivation In Chapter 4, we studied heat release from proton spins in temperaturequenched C r(dien) near the QCP between the antiferromagnetically ordered phase and the fully polarized state. This study provided insight to quantum fluctuations of the Cr4+ spins, demarcating a quantum critical region near the QCP However, our calorimetric method involving a field sweep could not measure a protonspin relaxation time shorter than about 27 s. This limit ed our studies to a magnetic field region at least 0.2 T away from the QCP. T o investigate the spin dynamics of Cr4+ at magnetic fields closer to the QCP, a regi on in which the protons relax faster, we have us ed proton NMR, which directly probes the proton spins and is able to measure fast protonspin relaxation. The experiment was done at the NHMFL with Tiglet Basala, Phillip Kuhns, and Arneil Reyes at temperatu res between 1.35 K and 300 mK in a 3He cryostat with a 17 T superconducting magnet. Using a spinecho technique at frequencies between 511.0 MHz and 540.7 MHz we have primarily focused on measuring T1 in a very narrow field range around the QCP, 12.0 T 12.7 T. In addition, w e have taken NMR spectra in this field range, as well as at 11.0 T, partly to study the effect of magnetic ordering on NMR peak positions In all measurements, the magnetic field was applied parallel to the crystallogra phic b axis, th e same as in all the experiment s described in Chapter 4. 5.2 Results and Discussions 5.2.1 Field S weeps W e first took proton NMR spectra of Cr(dien) at 1.30 K in the nearly completely polarized paramagnetic phase, by sweeping the magnet ic field at several fixed
77 frequencies The field was swept upward, typically over a 0.4 T wide range, at a rate of 0.25 T/min to ensure thermal equilibrium. We observed six peaks as shown in Figure 5 1 which shows the spinecho intensity at 540.7 MHz as a function of the fi eld. As described in section 4.3.2, in Cr(dien) there are thirteen hydrogen sites contained in a Cr(diethylenetriamine)(O2)2 molecule and two sites in a water molecule. Within the Cr(diethylenetriamine)(O2)2 molecule, the distances to the hydrogen sites from Cr4+ are 2.40 2.52 to the five hydrogen sites (H2, H1C, H1D, H3C, and H3D) attached to the three nitrogen atoms and 3.08 3.79 to the eight hydrogen sites attached to the four carbon atoms ( Ref. [ 4 ] for the labeling of the hydrogen sites.) The water hydrogen sites are at 3.29 and 3.48 from the closest Cr4+. The hydrogen sites attached to the nitrogen atoms are much closer to Cr4+ than are all the other hydrogen sites, and thus experience the st rongest dipolar fields and transferred hyperfine fields For our magnetic field direction, H || b the dipolar field of the Cr4+ spin at H2 attached to the middle nitrogen raises the total field, whereas the dipolar fields on the four sites attached to the t wo corner nitrogen atoms lower the total fields. As a result, the small isolated peak in Figure 51, appearing at the lowest field, comes from H2, and the peak at the highest field probably comes from the four hydrogen sites attached to the corner nitrogen atoms. To assign peaks to the water hydrogen sites and the hydrogen sites attached to carbon atoms, at larger distances from the nearest Cr4+, we need to consider the contribution of more than one Cr4+ spin to the dipolar fields. Our calculations of the d ipolar fields due to all Cr4+ spins that are within 12 of each hydrogen site suggest that the small peak at the second lowest field in Figure 51 comes from one of the water
78 hydrogen sites (H5B) and one of the two hydrogen sites (H4A) attached to one of the two corner carbon atoms, the third peak from the left comes from the remaining water hydrogen (H5A) and two of the hydrogen sites (H2A and H3B) attached to two noncorner carbon atoms, the second tallest peak comes from three hydrogen sites (H1A, H1B, and H4B) attached to the two corner carbon atoms, and the tallest peak comes from the two remaining hydrogen sites (H2B and H3A). In these peak assignment s, we have ignored transferred hyperfine fields, whose computation is beyond the scope of this dissert ation. 5.2. 2 Frequency Sweeps In NMR spectra taken with the fieldsweep technique, different points in the spectra correspond to different points in the magnetic phase diagram. To examine the effect of magnetic order ing on the NMR spectra, a frequency swe ep technique in which the magnetic field is fixed is therefore superior, although it requires frequent tuning of the tank circuit for the rf coil and thus takes longer to perform. We have taken frequency sweep spectra at 12.7 T and 11.0 T, as shown in Figures 5 2 and 53, which present integrated spinecho signals as well as summedup fast Fourier transforms (FFTs) of the signals. At 12.7 T, the frequency was changed in 0.1 MHz steps, which were decreased to 0.05 MHz at 11.0 T At 12.7 T, above Hc and thus in the nearly completely polarized paramagnetic phase, the spectra were taken at 1.30 K and 0.36 K. The 1.30 K spectra agree very well with the field sweep spectra, Figure 51, taken at the same temperature. Moreover, the spectral shape and widths show no change upon cooling to 0.36 K, as expected for a nearly completely polarized paramagnetic phase.
79 At 11.0 T, the spectra were taken at three temperatures 2.50 K, 1.55 K, and 0.50 K corresponding to the paramagnetic phase, near the phase boundary, and the antiferromagnetically ordered phase, respectively To facilitate better comparison of the spectra at the three temperat ures Figure 54 shows each spectra scaled with the ar ea under the curve. Comparison of the sweeps at 0.50 K and 2.50 K reveals that all the peaks in the paramagnetic phase split in two in the ordered phase, as marked in Figure 5 3, except the small isolated peak at the highest frequency. (We will call this peak, which does not split, the baby peak.) This splitting indicates that the Cr4+ spins in the ordered phase have two orientations as expected for a squarelattice antiferromagnet in which spins form two magnetic sublattices one with the spins tilted slightly to the left and the other with the spins tilted slightly to the right with respect to the magnetic field. Why does the baby peak remain unsplit in the ordered phase? Because this peak comes from a single proton site and because that site, attached to the middle nitrogen of the Cr(diethylenetriamine)(O2)2 molecule and thus the closest (2.40 ) to Cr4+, is dominated by the dipolar field produced by just one Cr4+ spin Moreover, along our field direction, H || b this site happens to be almost directly above or below the Cr4+. Consequently, the dipolar field there does not depend, to the first order, on the direction to which the Cr4+ spin tilts in the antiferromagnetically ordered phase and thus to which of the two sublattices the Cr4+ spin belongs. Incidentally, except for the effect of the dipolar field and transferred hyper fine field of the Cr4+ ions, the NMR peak heights should obey Curies law for the proton spin s and, as a result, should be inversely proportional to temperature. However, the peak
80 heights of the spectra at different temperatures shown in Figures 5 2 and 53 cannot be compared with each other, since we tuned the tank circuit at each temperature and thus the quality factor Q of the rf tank circuit varied slightly from spectra to spectra. 5.2.3 Site Dependence of the Spin Lattice Relaxation Time T1 To further confirm the assignment of the resonance peaks to proton sites, T1 is a powerful tool, since it is strongly influenced by fluctuations in the dipolar field and transferred hyperfine field, fluctuations whose strength varies from proton site to proton site For this purpose, 12.7 T is a convenient field since the temperature dependence of T1 is weak at this field as will be discussed in the next subsection, and thus we do not need to worry about a temperature drift during measurements. The measurements were made at 1.31 K and 0.3 6 K near the highest and lowest temperatures of our experiment ; within the temperature range of our experiment, the relaxation is closest to being exponential in the high temperature end, becoming increasingly nonexponential as the temperature de creases as will be discussed in the next subsection. Examples of nearly exponential and strongly nonexponential relaxations are shown in Figure 55. Figure 56 shows T1 at 12.7 T as a function of frequency. At this field, 540.7 MHz is the unshifted frequency. The figure shows that the more shifted is the resonance, the shorter is T1, indicating that the peaks with the largest shifts com e from proton sites closest to Cr4+ as discussed in section 5.2.1. The baby peak and the low frequency shoulder have the shortes t T1, as expected, since they come from the proton sites closest to Cr4+, one attached to the middle nitrogen and four attached to the two corner nitrogen atoms of the Cr (diethylenetriamine)(O2)2 molecule. In section 4.3.2, we argued on the basis of heat r elease results that it is very likely that the five proton sites that are attached to nitrogen atoms in Cr(dien) have shortest
81 relaxation times. The NMR T1 data verify that interpretation. Moreover, the heat release measurements in which field sweeps were stopped before reaching Hc indicated that the longer relaxation times of the eight proton sites attached to carbon atoms and the two sites within the water molecule comprise two groups, each containing four to six sites. The longer T1's in Figure 56(b) do divide into two such groups. T1's are shorter for t wo peaks, at 539.7 MHz and 543.8 MHz, which according to our dipolar field calculation come from four proton sites, H5B H4A, H2B, and H3A. 5.2. 4 Dependence of T1 on Temperature and Magnetic Field For the T1 measurements to investigate Cr2+ spin dynamics near the QCP we chose the second highest peak in the NMR spectra, since at 1.3 K this peak has almost no shift due to the dipolar field and transferred hyperfine field ( Figure 51) At 540.7 MHz, the fre quency to field r atio of the peak is 540.7 MHz/12.7 T = 42.575 MHz/T, very close to the gyromagnetic ratio 42.577468 MHz/T of the proton. We therefore expected that this peak would split by the smallest amount in the antiferromagnetically ordered phase near the QCP More important, this peak will not shift in the ordered phase, to the lowest order Therefore, the frequency can be held constant at a given magnetic field, as the temperature is varied. To estimate the frequencies of t his peak at eight other fi eld value s we have chosen for T1 measurements, we used the frequency to field r atio determined at 540.7 MHz For example, for 12.1 T, we estimated the frequency of the peak to be 12.1 T 42.575 MHz/T = 515.2 MHz. To confirm our estimates we then made a f ield sweep at each calculated frequency at 1.27 K 1.50 K, as shown in Figure 5 7 The field was again swept upward at a rate of 0.25 T/min to gua rantee thermal equilibrium These spectra ensured that the frequency estimate for each field is correct
82 As d escribed in section 3.4, the spinecho technique was used to monitor the proton magnetiz ation recovery after saturation and thus to measure T1. Two typical relaxation curves are shown in Figure 5 5 as mentioned before, one of them approximately exponential ( Eq. 3 22) and the other strongly nonexponential, exhibiting a stretched exponential behavior given by ) / ( 1 01T t e M t Mz z (5 1) Figures 58 and 59 show as a function of temperature, the relaxation time T1 obtained from such data by fitting them to exponential curves and stretchedexponential curves respectively. At 12.7 T, in the nearly completely polarized paramagnetic phase above the QCP, T1 i ncreases gradually and monotonically as the temperature is lowered. In contrast, a t fields smaller t han 12.4 T, there is a shallow minimum near the transition temperature, Tc, followed by a steep increase with decreasing temperature in the antiferromagnetically ordered phase. The data points at and very near 0.3 K, the lowest temperature of the 3He cryos tat, may suffer from temperature instability. However, this does not affect the majority of the data, which were taken at higher temperatures Figure 510 shows the stretching parameter of Eq. 5 1 as a function of temperature. The data points at different fields fall on a common curve labeled empirical line decreasing with decreasing temperature except near Tc, where a downward deviation from the common curve occurs. This means that the relaxation becomes less exponential as temperature decreases and also near Tc. Combining these two general trends suggests that the relaxation will be most strongly non-
83 exponential as Tc approaches zero, that is, at the QCP. At an Ising critical point, the autocorrelation function that characterizes fluctuations decays with a stretchedexponential time dependenc e, with the stretching exponent (d 1)/(d + 1), where d is the system dimensionality [ 61] It is very likely that our observation is related to this behavior of fluctuations but more theoretical work is needed to connect the two dots together. To present T1 and the str etching parameter as a function of temperature and magnetic field, two dimensional (2D) contour plot s and threedimensional (3D) graphs are useful, as sh own in Figures 5 11, 5 12, and 513. In particular, a s shown in Figure 5 12 1/ T1T exhibits a sharp peak, indicating the transition from the nearly completely polarized paramagnetic phase to the antiferromagnetically ordered phase. In the 2D contour plots, the red line and the black circles, taken from Figure 46, mark the phase boundary determined from specific heat and the magnetocaloric effect, and shown in black is the same line shifted by 0.14 T to indicate the probable phase boundary for the NMR sampl e. The difference between the two curves probably comes from different remanent fields of the magnets used in the two sets of measurements and, possibly, a slight difference in the crystal orientation. In Figure 5 9(b), the stretching parameter exhibits a dip near the QCP, suggesting that there is indeed an intrinsic stretchedexponential relaxation driven by quantum fluctuations. 5.3 Summary We have probed the spin dynamics of Cr4+ ions in Cr(dien) by means of proton NMR near the QCP The results reveal increasingly stretchedexponential behavior of proton spin lattice relaxation as the QCP is approached. How the quantum fluctuations of ionic spins near a QCP leads to such a behavior of nuclear spins is an interesting
84 question whose answer will require t heoretical investigation. Contrary to our expectation based on th e calorimetric results described in section 4.3.2, the relaxation rate, 1/ T1, does not diverge as the QCP is approached, although T1 does reach a minimum at Tc.
85 12.4 12.5 12.6 12.7 12.8 12.9 0 5000 10000 15000 20000 Intensity (a.u.)H (T) Figure 51. NMR spectra at 540.7 MHz showing the spin echo intensity as a function of magnetic field The arrows indicate the six peaks discussed in the text The vertical line mark the position of the unshifted peak, at 12.7 T. The spinlattice relaxation of this peak was studied as a function of temperature and magnetic field near the QCP
86 530 535 540 545 550 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 (a) Spectra at H=12.7 T (FFT)Intensity (a.u.)Frequency (MHz) 0.36 K 1.30 K 530 535 540 545 550 0 10000 20000 30000 40000 50000 (b) Spectra at H=12.7 T (Integration)Intensity (a.u.)Frequency (MHz) 0.36 K 1.30 K Figure 52. Frequency sweep NMR spectra at 12.7 T at 1.3 0 K and 0.36 K. Shown are (a) FFT sums and (b) integrated intensity, of spinecho signals
87 460 464 468 472 476 480 0 20000 40000 60000 80000 100000 120000 140000 160000 B A Intenstity (a.u.)Frequency (MHz) 0.50 K 1.55 K 2.50 K Spectra at H=11 T (FFT Sum) (a) 460 464 468 472 476 480 0 5000 10000 15000 20000 25000 30000 35000 Intenstity (a.u.)Frequency (MHz) 0.50 K 1.55 K 2.50 KSpectra at H=11 T (Integration)(b) Figure 53. Frequency sweep NMR spectra at 11.0 T at 2.50 K, 1.55 K, and 0.50 K Shown are (a) FFT sums and (b) integrated intensity, of spinecho signals The arrows indicate splitting of peaks, as the sample is cooled from nearly completely polarized paramagnetic phase to the antiferr omagnetically ordered phase. In (a), A represents the main group of peaks, which s plit in the ordered phase, and B represents the baby peak which does not split
88 460 464 468 472 476 480 0.0000 0.0004 0.0008 0.0012 0.0016 0.0020 0.0024 0.0028 0.0032 Area Scaled Intensity (a.u.) Spectra at H=11 T (FFT Sum) Frequency (MHz) 0.50 K 1.55 K 2.50 K (a) 460 464 468 472 476 480 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Spectra at H=11 T (Integration)Area Scaled Intensity (a.u.)Frequency (MHz) 0.50 K 1.55 K 2.50 K (b) Figure 54. Scaled plots of the data shown in Figure 53. The data have been scaled with the area under the curve at each temperature, for comparison.
89 19000 4000 5000 7500 10000 12500 15000 17500 10.00M 15.00 100.00 1.00k 10.00k 100.00k 1.00M 60000 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 10.00M 15.00 100.00 1.00k 10.00k 100.00k 1.00M Figure 55 Two examples of proton NMR relaxation curves, both taken at 12. 7 T Circles are integrated spinecho signals in arbitrary units; the horizontal axis, which is in a logarithm scale, is time in microseconds. (a) (top panel) At 1.30 K the deviation from an exponential fit (blue line) is small. (b) (bottom panel) At 320 mK, a stretched exponential fit (blue line) with = 0.50 is required to describe the relaxation.
90 538 540 542 544 546 548 10 20 30 40 50 60 0.36 K 1.31 KT 1 (ms)Frequency (MHz) (a) 538 540 542 544 546 548 10 20 30 40 50 60 70 80 0.36 K 1.31 KT 1 (ms)Frequency (MHz) (b) Figure 56. Proton NMR T1 as a function of frequency at 1.31 K and 0.36 K at 12.7 T. T1 has been extracted from data by (a) an exponential fit and (b) a stretchedexponential fit.
91 11.8 11.9 12.0 12.1 12.2 0 10000 20000 30000 511.0 MHz Intensity (a. u.) (a) 11.9 12.0 12.1 12.2 12.3 0 5000 10000 15000 20000 25000 515.2 MHz ( b) 12.0 12.1 12.2 12.3 12.4 0 4000 8000 12000 16000 519.4 MHz ( c) 12.1 12.2 12.3 12.4 12.5 0 5000 10000 15000 20000 523.7 MHz ( d) 12.2 12.3 12.4 12.5 12.6 0 5000 10000 15000 20000 25000 528.0 MHz ( e ) 12.3 12.4 12.5 12.6 12.7 0 4000 8000 12000 16000 532.2 MHz ( f ) 12.4 12.5 12.6 12.7 12.8 0 20000 40000 60000 536.5 MHz ( g ) 12.5 12.6 12.7 12.8 0 5000 10000 15000 20000 538.6 MHz ( h ) 12.5 12.6 12.7 12.8 12.9 0 5000 10000 15000 20000 540.7 MHz H (T)( i ) Figure 57 NMR s pectra obtained at 1.27 K 1.50 K from integrated spin echo es at nine frequencies. The arrow in each frame indicates the unshifted peak chosen for T1 measurements. These spectra were taken to confirm the estimated frequency at each chosen field of T1 meas urements. The f ield ranged from 12.0 T to 12.7 T, from (a) to (i) in 0.1 T increments except for (h), which is for 12.65 T.
92 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 10 100 T (K) 12.7 T 12.65 T 12.6 T 12.5 T 12.4 T 12.3 T 12.2 T 12.1 T 12.0 T 12.7 T (again) 12.5 T (again) 12.0 T (again) T1 (ms) Figure 58 Proton NMR T1 in Cr(dien) as a function of temperature in fields ranging from 12.0 T to 12.7 T. T1 has been determi ned from exponential fits of relaxation curves. Lines are guides to the eye. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 10 100 12.7 T 12.6 T 12.5 T 12.4 T 12.3 T 12.2 T 12.1 T 12.0 T T1 (ms)T (K) Figure 59 Proton NMR T1 in Cr(dien) as a function of temperature in fields ranging from 12.0 T to 12.7 T. T1 has been determined from stretched exponential fits of relaxation curves. Lines are guides to the eye.
93 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.3 0.4 0.5 0.6 0.7 0.8 12.7 T 12.6 T 12.5 T 12.4 T 12.3 T 12.2 T 12.1 T 12.0 T Stretch ParameterT (K) Empirical line Figure 510 Stretching parameter as a function of temperature at various magnetic fields. An empirical line (black solid line) indicates a general trend, from which the data deviate downward near the phase transition.
94 0.0 0.2 0.4 0.6 0.8 1.0 1.2 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 (a) SH&MCE Phase boundary1 Phase boundary2T1 (ms) T (K)H (T) 0 11.9 23.9 35.8 47.8 59.7 71.6 80.0 0.4 0.6 0.8 1.0 1.2 0 10 20 30 40 50 60 70 80 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 (b) T1(ms)T1 (ms)H (T)T (K) 0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 Figure 511 2D contour plot and 3D plot of proton T1, determin ed from stretchedexponential fits, as a function of temperature and magnetic field. In (a) r ed line and black circles mark the phase boundary between the antiferromagnetically o rdered phase and the nearly completely polarized paramagnetic phase, determined from specific heat and the magnetocaloric effect in section 4.3.1. T he black line is the same as the red line, but shifted to indicate a probable phase boundary for the NMR sample.
95 0.0 0.2 0.4 0.6 0.8 1.0 1.2 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 (a) SH&MCE Phase boundary1 Phase boundary2 1/T1T(ms-1K-1)T (K)H (T) 0 0.050 0.10 0.15 0.20 0.25 0.28 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 0.4 0.6 0.8 1.0 1.2 0.00 0.05 0.10 0.15 0.20 0.25 (b) H (T) T (K) 1/T1T ( ms-1K-1) 0 0.050 0.10 0.15 0.20 0.25 0.28 Figure 512 2D contour plot and 3D plot of 1/ T1T as a function of temperature and magnetic field. As shown i n (a) the m axima in 1/ T1T indicate the phase transition. Black circles, red line, and black line are the same as in Figure 5 11.
96 0.0 0.2 0.4 0.6 0.8 1.0 1.2 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 SH&MCE Phase boundary1 Phase boundary2T (K)H (T) 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 Strech parameter (a) 0.4 0.6 0.8 1.0 1.2 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 0.3 0.4 0.5 0.6 0.7 (b)H (T) Strech parameterT (K) 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 Figure 513 2D contour plot and 3D plot of the stretching parameter as a funct ion of t emperature and magnetic field. Black ci rcles, red line, and black line are the same as in Figure 5 11 A dark blue dip a region of small stretching parameter, appears near the QCP.
97 CHAPTER 6 (CH3)2CHNH3CuCl3 6.1 Previous Work on (CH3)2CHNH3CuCl3 (CH3)2CHNH3CuCl3, abbreviated IPACuCl3, forms a triclinic lattice (the 1 P space group) with lattice constants a = 11.692 0.008 b = 7.804 0.00 4 c = 6.106 0. 003 = 7 9.00 0.04 = 122.60 0.04, and = 116.47 0.04 [ 62] The convention among researchers who study the magnetism of IPA CuCl3 is to specify the direction of the applied magnetic field with respect to the so called A B and C planes the three orthogonal fac es of a single crystal rather than with respect to the crystallographic axes. The A plane is perpendicular to the c axis, and the C plane is the bc plane. The B plane is perpendicular to the A and C planes, thus perpendicular to the bc plane and parallel to the c axis [ 62 64] Manaka and coworkers [ 63, 64] who have extensively studied the magnetism of this compound for more than a decade, choose the unit cell differently from Roberts et al. [ 62 ] who were the first to determine the crystal structure. In this dissertation, we instead refer to the crystallographic axes defined by the latter authors. The magnetism of IPACuCl3 arises from S = 1/2 Cu2+ ions. In early experim ent s, the compound was thought, naively on a structural ground, to comprise ferromagnetic antiferromagnetic alternating chains running along the c axis [ 63] Instead, it was found later by inelastic neutron scattering [ 65] to consist of two leg spin ladder s, which r un parallel to the b axis as shown in Figure 6 1(b). A simple spin ladder has only two kinds of exchange interactions: a rung exchange and leg exchange. But the ladders in IPACuCl3 have four exchanges including an interladder exchange, as depicted in Figure 6 1(a). The rung exchange J1, res ponsible for the formation of S = 1 composite spins is
98 ferromagnetic and the leg exchange J2 is antiferromagnetic. The diagonal exchange J3 is also antiferromagnetic whereas the inter ladder exchange J4 is ferromagnetic. According t o inelastic neutron scattering and magnetic susceptibility [ 63, 65] J1 = 2 7 7 K J2 = 14 2 K J3 = 34 1 K and J4 = 3 .39 0 .01 K [ 66] The negative signs for J1 and J4 indicate ferromagnetic exchange. Theoretically, S = 1/2 two leg Heisenberg spin ladders with an antiferromagnetic leg exchange belong to the same universality class as S = 1 linear chain Heisenberg antiferromagnet s and should therefore have a Haldane gap between the quantum mechanically disordered ground state and the triplet magnon excitations as pointed out in section 2.3. As a result, the ground state below a critical field Hc will be a spin liqu id, with no longrange order. Above Hc, which is a QCP, the composite S = 1 spins will order antiferromagnetically owing to interladder interactions, and this ordering can be described as a BEC of magnons as explained in section 2.4. These theoretical exp ectations have been borne out by a number of neutron experiments on IPA CuCl3 [ 65, 67, 68] However, only a small number of data points were taken in these experiments to determine the critical exponent of Eq. 227. To determine the magnetic phase diagram of IPA CuCl3 in detail, particularly near the QCP, we have measured the specific heat and the magnetocaloric effect the method used for Cr(dien), as described in section 4.3.1. Preliminary results have been published as a short report [ 69] In addition, we have measured the magnetic torque in an attempt to extend the phase diagr am to about 24 mK, considerably lower than possible with the calorimetric method. We find a power law dependence of the critical field on temperature, with an exponent indicative of a BEC of magnons. Near the QCP,
99 we also find heat release from proton spin s after temperature quenching, demonstrating that the phenomenon we have first observed in Cr(dien) is probably a generic property of a variety of QCP. 6.2 Experimental All measurements were performed in a dilution refrigerator with a 20tesla superconduct ing magnet at the NHM FL in Tallahassee. The specific heat and magnetocaloric effect measurements were carried out in the relaxation calorimeter described in section 3.1, at temperatures down to 49.9 mK in magnetic fields up to 18 T. In order to minimize the addenda heat capacity, the sample was directly glued on the calorimeter platform with Wakefield compound. The applied magnetic field was perpendicular to the B plane or C plane of the crystal Magnetic torque was measured by a cantilever beam magnetometer des cribed in section 3.3. The sample was glued on the flexible CuBe beam with GE varnish and covered with a stretched piece of Teflon tape for protection. Using a rotating probe at the NHMFL, we oriented the sample so that the magnetic field was perpendicular to either the B plane or C plane, as in the specific heat and magnetocaloric effect measurements. The calorimeter for the specific heat and magnetocaloric effects measurements was also used for the heat release experiment. In this experiment, the magnetic field was applied only perpendicular to the B plane. The procedure of the experiment was the same as in the heat release experiment on Cr(dien) described in section 4.2.2. The singlecrystal samples of IPA CuCl3 used in these experiments were grow n by Akira Oosawas group in Tokyo by dissolving (CH3)2CHNH2HCl and CuCl2 in isopropano l and allowing the solvent to evaporate slowly at 30 [ 63]
100 6.3 Results and Analysis 6.3.1 Specific Heat and Magentocalori c Effect Figures 62 and 63 show the specific heat in the two field directions. The peak at each magnetic field indicate s an antiferromagnetic transition. The peak height, as well as the transition temperature, decreases with decreasing magnetic field. The peak disappears completely at 10 T, when the applied field is perpendicular to the B plane, and at 9.5 T when H C plane, indicating that these fields are close to the QCP at which the antiferromagnetic order disappears The rapid upt urn below about 0.4 K at these fields is due to the nuclear specific heat of copper and hydrogen, which makes it difficult to detect a transition peak at a temperature less than about 0.4 K. To overcome this limitation of the specific heat technique, we u sed the magnetocaloric effect at temperatures below 0.8 K as a com p lementary tool, as shown in Figure 6 4. As in Cr(dien), whose results were described in section 4 .3.1 comparison of the magnetocaloric effect data and specific heat peak positions indicated that the correct transition point is given by t he average temperature and field of the peak and dip in a pair of curves for upward and downward field sweeps at a fixed reservoir temperature. From specific heat and the magnetocaloric effect, we have deter mined the phase diagram in terms of magnetic field and temperature for each appl ied field direction. As shown in Figure 6 5, there is no magnetic ordering below a critical field Hc; near zero temperature, this field region is presumably a spinliquid phase, in which magnon excitations are gapped. Above Hc, 3D antiferromagnetic longrange order appears due to weak interladder exchange.
101 If IPACuCl3 is an ideal Heisenberg antiferromagnet, then its antiferromagntic ordering is expected to be a BEC of magnons, near the QCP as described in section 2.4. The universal power law for a BEC is given by Eq. 2 27 w ith an exponent To determine for IPACuCl3, we have fitted the data shown in Figure 6 5 to Eq. 227 by letting Hc and to be fitting parameters and by limiting the maximum temperature to which the fit is made. Figure 66 shows the exponent value versus the maximum t emperature of the fit. From this analysis, we find Hc = 9.99 5 0.002 T and = 1.47 0.06 for H B plane and Hc = 9.399 0.004 T and = 1. 50 0.06 for H C plane, by fitting the data up to 0.6 K and 0.8 K, respectively [ 69] The critical exponents are in excellent agreement with = 3/2 expected for 3D BEC. is related to the critical field Hc by Eq. 2 26. The g factor of IPA CuCl3 is slig htly anisotropic, with gA = 2.11 2. 06 < gB < 2.11, and gC = 2.26, according to ESR [ 63, 64] With these and from our Hc, we find 13.83 1 0.003 K < < 14.166 0.003 K for H B plane and = 14.268 0.006 K for H C plane, in good agreement with the value directly obtained by zerofield inelastic neutron scattering, = 13.58 9 K [ 65] Moreover, Figure 6 7, which shows the magnetic phase diagrams for the two field directions scaled with the g factor, indicates that the anisotropy of the phase diagram comes from the g factor anis otropy alone within our experimental uncertainties. Such a scaling was noted earlier by Manaka et al. [ 70] who compared the phase diagrams for H A plane and H C plane. Their phase diagrams extended, however, only down to about 0.7 K 6.3.2 Magnetic Torque Magnetic torque is a convenient tool to qualitatively measure magnetization as a function of magnetic field and to thereby determine a magnetic phase diagram, albeit
102 not a quantitatively accurate tool Figures 68 and 69 sh ow the magnetic torque, divided by the magnetic field, on IPA CuCl3 for H B plane and H C plane. This quantity, / H is roughly proportional to the sample magnetization, as can be understood from Eq. 3 15. For both field directions, / H is nearly zero in the quantum mechanically disordered phase below Hc. Since magnon excitations are gapped in this phase, the magnetization is zero at low temperatures. A bove Hc, / H increases rapidly as the magnetization increases w ith the magnetic field. In an attempt to precisely determine the critical field at each temperature from the torque data, we examine the derivatives dH H d / ) / ( and 2 2/ ) / ( dH H d which are shown as a function of magnetic field in Figures 6 10 and 61 1 for H B plane, and in the inset to Figures 6 9 and in Figure 6 1 2 for H C plane. The peak in the second derivative, 2 2/ ) / ( dH H d occurs systematically about 0.1 T below the transition deter mined from the magnetocaloric effect, whereas the shoulder in the first derivative closely follows the transition. However, we have not yet found a quantitatively u nique way to define this feature in the first derivative and thus cannot use the torque to e xtend the phase diagram from 0.17 K to about 24 mK. 6.3.3 Heat Release from Proton Spins near the QCP Like in Cr(dien), results for which were discussed in section 4 .3.2, we have found that proton spins in temperaturequenched IPA CuCl3 release heat near the QCP. In this experiment, the magnetic field was applied perpendicular to the B plane of the crystal For t he procedure of the experiment which w as similar to the Cr(dien ) experiment, refer to section 4.2 .2. The quenching t emperature Tq rang ed from 22 5 mK to 1.33 K The waiting time for thermal equilibrium at Tq was typically 2 min for high Tq
103 and 30 min for low Tq, a fter which the sample was rapidly cooled to the reservoir temperature T0 in less than 1 min The field sweeps through the critical field Hc were made at seven different T0 ranging from 56.2 mK to 271 mK, at the rate of 0.1 T/min or 0.2 T/ min. As the field is swept toward Hc, a pronounced peak appears in the temperature T between the sample and the thermal reservoir, as shown in Figure 6 13, indicating heat being released in the sample. As was found in Cr(dien), the heat release occurs only during the first field sweep after the temperature quenching of the sample from Tq. The amount of released heat Q is obtained from the data by using Eq. 4 1 As shown in Figure 6 14, Q depends on both Tq and Hq, the quenching field, in a manner very similar to the results for Cr(dien) discussed in section 4.3.2, clearly indicat ing that the heat is released from the proton spins As has been deri ved in that section, Q is given by Eq. 45 For the thermal reservoir temperature T0 = 180 mK, good agreement between experiment and the equation is obtained with nH = 7 as shown in Figure 6 14(a), except for a few points for which Hq is 6 T As shown in Figure 6 14(b), good agreement between the equation and the T0 = 99.7 mK data for Hq = 9 T is obtained with nH = 5 whereas nH = 4 gives the best fit for Hq = 11 T. IPACuCl3 contains ten hydrogen atoms per formula unit, as shown in Figure 6 15. Among them seven are significantly closer to a Cu2+ ion at 3.37 3.57 3 .72 3. 80 3.99 4.09 and 4.30 than other three, which are at 4.71 4.94 and 5.20 from the closest Cu2+ ion It is very likely that these seven hydrogen atoms which e xperience stronger fluctuating dipolar field and transferred hyperfine field of the
104 Cu2+ spins than other three do, are those whose nuclear spins release heat at T0 = 180 mK Among them, only four or five that are closest to a Cu2+ ion evidently release heat at T0 = 99.7 mK The positions of the heat release peaks are shown in Figure 6 16 along with the phase boundary between the spinliquid phase and the antiferromagnetically ordered phase. As in the case of Cr(dien), the diagram suggests that the loci of the peaks converge on the QCP in the zerotemperature limit. These loci delimit the quantum critical region in which is shorter than the timescale of the experiment. As in Cr(dien), we have also investigated the variation of the relaxation time of pr oton spins, within the quantum critical region. The measurements were carried out at T0 = 180 mK after the sample had been quenched from Tq = 779 mK at Hq = 9 T or from Tq = 826 mK at Hq = 12.5 T. The relaxation rate 1/ diverges as the field approaches Hc, as shown in Figure 6 17, indicating that divergent quantum fluctuations of the Cu2+ spins near the QCP drive the relaxation of the proton spins. The divergence is asymmetric around Hc, faster for H > Hc than H < Hc. This asymmetry is also seen in Figure 6 16 as a wider quantum critical region for H > Hc than for H < Hc. The amount of heat released during the relaxationtime measurements is shown as a function of | H Hc| in Figure 6 18. The result indicates that the seven hydrogen atoms whose nuclear spins release heat at this temperature consist of two groups, each comprising three or four hydrogens within our uncertainties Below Hc, one group rel e a se s heat at | H Hc| larger than about 0.45 T, whereas the other group does so at fields closer to Hc. This res ult is consistent with the distribution of hydrogencopper distances discussed earlier to explain the T0 dependence of the number of proton spins
105 that participate in heat release. Above Hc, there is no evidence for grouping in the field region of our study 6.4 Conclusions We have determined the magnetic phase diagram of IPA CuCl3 for two magnetic field directions, H B plane and H C plane, from specific heat and the magnetocaloric effect. The phase boundaries, H ( T ), between the gapped, spinliquid phase and the antiferromagnetically ordered phase scale with the anisotropic g factor, indicating that IPACuCl3 is an ideal Heisenberg antiferromagnet, with negligible singleion anisotropy or Dzyaloshinskii Moriya interac tion. As a result, the antiferromagnetic ordering near zero temperature can be described as a 3D BE C of magnons, as testified by the critical exp onents for the phase boundaries. Like in Cr(dien), temperature quenching of IPA CuCl3 leaves the hydrogen nuclear spins frozen in a highenergy nonequilibrium state and, as the magnetic field is then brought close to the QCP quantum fluctuations of the Cu2+ ionic spins quickly anneal them to reach thermal equilibrium with the lattice. But unlike in Cr(dien), where the QCP is between an antiferromagnetically ordered phase and a state completely polarized by the magnetic field, the QCP in IPA CuCl3 occurs between a gapped, spinliquid phase and a fieldinduced, antiferromagnetically ordered phase. Taken together with our result for Cr(dien), our result for IPA CuCl3 strongly suggests that the heat release from nuclear spins driven by quantum fluctuations is a generic phenomenon near a wide variety of QCPs.
106 Figure 61. Schematic diagram and the crystal structur e of IPA CuCl3. (a) (top panel) Schematic representation of spin ladders in IPACuCl3 with four exchanges: ferromagnetic rung exchange J1 (black solid lines) responsible for the formation of composite S = 1 spins, antiferro m agnetic leg exchange J2 (blue dot lines), antiferromagnetic diagonal exchange J3 (orange dot line), and ferromagnetic inter ladder exchange J4 (long dash line). B rown circles represent the S = 1/2 spins of Cu2+ ions (b) (bottom panel) Crystal structure of IPA CuCl3, with the exchanges indicated Brown spheres and green spheres represent Cu2+ ions and Cl, respectively. Ladders extend along the b axis. For clarity, IPA ions are not shown. The structure has been drawn with Balls & Sticks [ 71]
107 0.0 0.5 1.0 1.5 0 100 200 300 400 C (mJ/K mol)T (K) IPA-CuCl3, H B-planeC (mJ/K mol)T (K) 0 2 4 0 500 1000 1500 2000 2500 0 T 9.8 T 10 T 10.3 T 10.4 T 10.5 T 10.6 T 10.7 T 10.8 T 10.9 T 11 T 12 T 13 T 14 T 15 T 16 T 17 T 18 T Figure 62. Specific heat of IPA CuCl3 in magnetic fields normal to the B plane. From 10.3 T to 11.0 T, near the QCP, the data were taken in small field increments Bottom panel shows the low temperature region below 1.5 K. Lines are guides to the eye.
108 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 500 1000 1500 2000 IPA-CuCl3, H C-plane 9.5 T 9.8 T 10 T 18 TC (mJ/K mol)T (K) 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 9.5 T 9.8 T 10 TC (mJ/K mol)T (K) Figure 63. Specific heat of IPA CuCl3 in magnetic fields normal to the C plane. Bottom panel shows the low temperature region below 1.0 K. Lines are guides to the eye.
109 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.1 0.2 0.3 0.4 0.5 0.6 IPA-CuCl3 H_|_B planeT(H = 9 T)+2T (K)H (T) 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 IPA-CuCl3 H_|_C planeT (K)H (T) Figure 64. Magnetocaloric effect curves at various temperatures. (a) (left panel) H B plane and (b) (right panel) H C plane. Red traces are for upward field sweeps and blue traces for downward field sweeps. In the left panel, temperature variation has been amplified by a factor of 2 for clar ity.
110 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 12 14 16 18 20 Energy gap 3D long-range order Spin liquidH (T)T (K) 0.0 0.5 1.0 9.0 9.5 10.0 10.5 11.0 11.5 3D long-range order Spin liquidH (T)T (K) F igure 65. Magnetic phase diagram of IPA CuCl3. Red squares are for H B plane and purple circles for H C plane. Above the critical field, at which the energy gap of triplet magnons becomes zero, 3D long range order appears. Solid line for H B plane and dash line for H C plane are power law fits for BEC.
111 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 IPA-CuCl3 H B planeExponentTmax (K) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 IPA-CuCl3 H C planeExponentTmax (K) Figure 66. Exponent of the power law fit as a function of the highest transition temperature, for H B plane and H C plane.
112 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 12 14 16 18 20 22 H C plane H B planegH/2 (T)T (K) Figure 67. Transition field scaled with the g factor for the two applied field directions. Red squares and purple circles are for H B plane and H C plane, respectively. The scaled phase diagrams are almost the same for the two field directi ons.
113 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4 0 2 4 6 8 10 IPA-CuCl3H B plane 86 mK 430 mK 395 mK 323 mK 280 mK 230 mK 184 mK 23.8 mK 124 mK 27.6 mK 121 mK 38.5 mK 97.5 mK 79.7 mK 47.8 mK 48.3 mK 53.8 mK 62.2 mK 72.6 mK /H (arb. unit)H (T) Figure 68. Magnetic torque divided by magnetic field at various temperatures, for field applied perpendicular to the B plane. Blue lines are data below 100 mK, red lines at and above 124 mK, and magenta for 121 mK. For clarity, curves at different temperatures have been shifted by arbitrary amounts.
114 8.6 8.8 9.0 9.2 9.4 9.6 9.8 10.0 0 5 10 15 20 25 30 8.6 8.8 9.0 9.2 9.4 9.6 9.8 10.0 IPA-CuCl3H C plane 373 mK 216 mK 108 mK 97.8 mK 59.3 mK 28.9 mK /H (arb.unit)H (T)d2( /H ) /dH2 (arb. unit)H (T) Figure 69. Magnetic torque for H C plane. For comparison, all curves are forc ed to be zero at 8.6 T. Inset: the second derivative of torque divided by H with respect to H
115 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4 -1 0 1 2 3 4 5 6 7 (a) IPA-CuCl3H B plane 23.8 mK 27.6 mK 47.8 mK 48.3 mK 53.8 mK 62.2 mK 72.6 mK 86 mK 97.5 mK 121 mK 79.7 mK 38.5 mK 124 mK 184 mK 230 mK 280 mK 323 mK 395 mK 430 mKd( /H)/dH (arb. unit)H (T) 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4 -5 0 5 10 15 20 25 30 (b) IPA-CuCl3H B plane 23.8 mK 27.6 mK 38.5 mK 47.8 mK 48.3 mK 53.8 mK 62.2 mK 72.6 mK 79.7 mK 86 mK 97.5 mK 121 mK 124 mK 184 mK 230 mK 280 mK 323 mK 395 mK 430 mKd2( /H)/dH2 (arb. unit)H (T) Figure 610 Derivatives of magnetic torque divided by magnetic field for H B plane at various temperatures. (a) the first derivati ve and (b) the second derivative.
116 9.0 9.5 10.0 10.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 9.0 9.5 10.0 10.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 (b) d2( /H)/dH2IPA-CuCl3H B planeT (K)H (T)(a) d( /H)/dHT (K)H (T) Figure 611 Comparison of the magnetic torque data shown in Figure 6 10 with the phase boundary determined from specific heat (solid squares) and the magnetocaloric effect (open squares) shown in Figure 6 5. Blue lines are the power law fit. (a) T he first deriv ativ e and (b) the second derivative of the torque data. Each curve has been drawn so that the value at 9.1 T indicates the temperature at which the data were taken.
117 8.5 9.0 9.5 10.0 0.0 0.1 0.2 0.3 0.4 0.5 8.5 9.0 9.5 10.0 0.0 0.1 0.2 0.3 0.4 0.5 (b) d2( /H)/dH2 T (K)H (T)(a) IPA-CuCl3H C plane d( /H)/dHT (K)H (T) Figure 612. Comparison of the magnetic torque data for H C plane, partly shown in the inset to Figure 6 9, with the phase boundary determined from the magnetocaloric effect (open ci rcles) shown in Figure 6 5. Dash lines are the power law fit. (a) T he first deriv ativ e and (b) th e second derivative of the torque data. Each curve has been drawn so that the value at 8.6 T indicates the temperature at which the data were taken.
118 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 0 5 10 15 20 25 30 35 (a) 12.5 T, 15.5 T 6 T, 9 TIPA-CuCl3Tb= 180 mK T (mK)H (T) 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 0 10 20 30 40 50 (b)IPA-CuCl3Tb= 99.7 mK 11 T T (mK)H (T) 9 T Figure 613 T between the sample and the thermal reservoir as a function of the magnetic field during field sweeps at 0.1 T/min. The peaks to th e right were observed during downward field sweeps and those to the left during upward sweeps. (a ) The ther mal reservoir was held at 180 mK. T emperature Tq from which the sample has been quenched ranged from 253 mK to 1.42 K. The quenching fields were 6 T (green) 9 T (blue) 12.5 T (red) and 15.5 T (magenta) (b ) The thermal reservoir was held at 99.7 mK. Tq ranges from 222 m K to 796 mK. Quenching fields we re 9 T (blue) and 11 T (red) The QCP is at Hc = 9.995 0.002 T.
119 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 IPA-CuCl3 Q ( J)Tq (K) 6 T 9 T 12.5 T 15.5 T 6 T 9 T 12.5 T 15.5 TTb=180 mK(a) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) Q ( J)Tq (K) 9 T 11 T 9 T 11 TIPA-CuCl3Tb=99.7 mK Figure 61 4 Amount of heat released at (a) 180 mK and (b) 99.7 mK as a function of Tq, the temperature from which the sample was quenched. Quenching fields Hq are given in the legends. The lines represent Eq. 4 5 with nH = 7 for (a), and nH = 5 ( 9 T) and nH = 4 ( 11 T ) for (b).
120 Figure 615. C rystal structure of IPA CuCl3. Brown spheres are Cu, green Cl, gray C, blue N, and white H. The triclinic unit cell, indicated by thin lines, contains two chemical formula units. Drawn with Balls & Sticks [ 71]
121 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 9.0 9.5 10.0 10.5 11.0 11.5 AF P Quantum critical region H (T)T (K) SpecificHeat MCE UpSweepPeak DownSweepPeak Figure 616 T curves (triangles and circles), marking a quantum critical region of IPA CuCl3 delimited by two lin es through the data points. Squares represent the phase boundary detected by the specific heat and the magnetocaloric effect (MCE) between the antiferromagnetic (AF) and quantum mechanically disordered (P) phases The line through the squares is a power law fit of data points up to 0.6 K.
122 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 Above QCP Below QCP 1/ (s-1)H-Hc (T) Figure 617. Relaxation rate 1/ vs the field at which field sweep was stopped. Blue and red squares are for Tq = 779 mK at Hq = 9 T and Tq = 827 mK at Hq = 12.5 T, respectively. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Q (J)|H-Hc| (T) Above Hc Below Hc Figure 618. Amount of heat released at T0 = 180 mK as a function of the field at which field sweep ended. Horizontal line s indicate the amounts of heat released in complete field sweeps through Hc. Blue and red squares are for Tq = 779 mK at Hq = 9 T and Tq = 827 mK at Hq = 12.5 T, respectively.
123 CHAPTER 7 SUM M A RY We have studied phase transitions, thermodynamics, and magnetic properties of t w o quantum magnets, Cr(di e n ) and IPA CuCl3. T hese compounds are excellent laboratory model s of quantum magnets to investi gate quantum critical phenomena. The first material Cr(dien) is a quasi two dimensional squarelattice antiferromagnet with a saturation field of 12.392 T 0.003 T, at which the quantum phase transition occurs between the antiferromagnetic ally ordered phase and the fully pol arized state The critical exponent of the phase boundary is = 2.01 0.02 indicating a 3D Ising criticality inst ead of a BEC of magnons Our most significant finding in this material is heat release from proton spins caused by div ergent quantum fluctuation s of Cr4+ near the QCP. The results present clear evidence that temperature quenching of Cr(dien) l eaves the proton spins frozen in a highenergy nonequilibrium state and, as the magnetic field is then brought near the QCP, quantum fluctuations of Cr4+ spins quickly anneal them. W e have also used NMR to probe the dynamics of Cr4+ spins very near the QCP The results show that the spinlattice relaxation time, T1, of the proton spins becomes significantly shorter near the QCP with increasing ly stretch exponential behavior. These are caused by the divergent quantum fluctuations of the Cr4+ spins. How the quantum fluctuations of ionic spins near a QCP lead to stretched exponential relaxation of nuclear spins is a question whose answer will require theoret ical investigation. The second compound, IPACuCl3, is a quasi onedimensi onal antiferromagnet Although the spin quantum number of Cu2+ is 1/2, low lying excitations in this material have a Haldane gap, usually associated with integer spin quasi onedimensional
124 antiferromagnets, because the spins form twoleg ladders with antiferromagnetic leg interactions. The QCP of IPA CuCl3 is at 9.995 0.002 T for H B plane and 9.399 0.004 T for H C plane [ 69 ] These critical fields translate to a Haldane gap of 1 3.831 0.003 K < < 14.166 0.003 K for H B plane and = 14.268 0.006 K for H C plane, in good agreement with the value di rectly obtained by zerofield inelastic neutron scattering, = 13.589 K [ 65] Above the critical fields, the spins order antiferro magnet ic ally. The critical exponent of the boundary between the ordered phase above and the quantum mechanically disordered spin liquid phase below is = 1.47 0.06 for H B plane and = 1. 50 0.06 for H C plane, in excellent agreement with = 3/2 expected for 3D BEC of magnons As in Cr(d ien ), temperature quenching of IPA CuCl3 leaves the proton spins frozen in a highenergy nonequilibrium state and, as the magnetic field is then brought close to the QCP, quantum fluctuations of the Cu2+ spins quickly anneal them to reach thermal equilibr ium with the lattice. Our discovery of heat release from proton spins i n Cr(dien) and IPA CuCl3, involving two different kinds of QCPs with different dimensionality and spin quantum number s, strongly suggests that it is a generic phenomenon near a QCP. Thi s phenomenon promises to be a useful probe for the dynamics of quantum fluctuations that underlie quantum criticality in a variety of systems.
125 LIST OF REFERENCES  S. Sachdev, Nat. Phys. 4 185 (2008).  S. Sachdev, Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 1999).  S. Sachdev, Science 288, 475 (2000).  C. M. Ramsey et al. Chem. Mater. 15, 92 (2003).  Kaur N. (2010). Magnetic and Thermodynamics Studies on Spin 1 Compounds (Doctoral dissertation) Florida State University  Y. H. Kim et al. Ph ys. Rev. Lett. 103, 247201 (2009).  J. A. Hertz, Phys. Rev. B 14, 1165 (1976).  H. v. Lhneysen, A. Rosch, M. Vojta, and P. Wlfle, Rev. Mod. Phys. 79, 1015 (2007).  P. Gegenwart, Q. Si, and F. Steglich, Nat. Phys. 4 186 (2008).  N. D. Mathur et al. Nature 394, 39 (1998).  H. Q. Yuan et al. Science 302, 2104 (2003).  M. P. Lilly, et al. Phys. Rev. Lett. 82, 394 (1999).  E. Fradkin and S. A. Kivelson, Phys. Rev. B 59, 8065 (1999).  R. A. Borzi et al. Science 315, 214 (2007).  Y. S. Oh et al. Phys. Rev. Lett. 98, 016401 (2007).  F. Bloch, Z. Physik 61, 206 (1930).  T. Holstein and H. Primakoff, Phys. Rev. 58 1098 (1940).  J. Van Kranendonk and J. H. Van Vleck, Rev. Mod. Phys. 30, 1 (1958).  C. Kittel, Introduction to Solid State Physics 7th edition (Wiley New York, 1996).  H. A. Bethe, Z. Phys. 71 205 (1931).  J. des Cloizeaux and J. J. Pearson, Phys. Rev. 128, 2131 (1962).  F. D. M. Haldane, Phy s. Rev. Lett. 50, 1153 (1983).
126  M. P. Nightingale and H. W. J. Blte Phys. Rev. B 33, 659 (1986).  J. P. Renard et al. Europhys. Lett. 3 945 ( 1987).  T. Giamarchi and A. M. Tsvelik, Phys. Rev. B 59, 11398 (1999).  I. Affleck, J. Phys.: Condens. Matter 1 3047 (1989).  V. F. Sears and E. C. Svensson Phys. Rev. Lett. 43, 2009 (1979).  E. C. Svensson V. F. Sears A. D. B. Woo ds and P. Martel Phys. Rev. B 21, 3638 (1980).  F. London, Phys. Rev. 54, 947 (1938).  O. Penrose and L. Onsager Phys. Rev. 104, 576 (1956).  K. B. Davis et al. Phys. Rev. Lett. 75, 3969 (1995).  T. Nikuni, et al. Phys. Rev. Lett. 84, 5868 (2000).  Y. Shindo and H. Tanaka, J. Phys. Soc. Jpn. 73, 2642 (2004).  S. E. Sebastian et al. Nature 441 617 (2006).  T. Matsubara and H. Matsuda, Prog. Theor. Phys. 16, 569 (1956).  H. Tsujii et al. Physica B 329 333 1638 (2003).  G. W. Hull, Jr., as cited in R. E. Schwall, R. E. Howard, and G. R. Stewart, Rev. Sci. Instrum. 46, 1054 (1975).  R. Bachmann et al. Rev. Sci. Instrum. 43, 205 (1972).  R. E. Schwall, R. E. Howard, and G. R. Stewart, Rev. Sci. Instrum. 46, 1054 (1975).  G. R. Stewart, Rev. Sci. Instrum. 54, 1 (1983).  H. Tsujii, B. Andraka, K. A. Muttalib, and Y. Takano, Physica B 329 333, 1552 (2003).  A. M. Tishin and Y. I. Spichkin, The Magnet ocaloric Effect and Its Applications (Institute of Physics, Bristol and Philadelphia, 2003).  V. K. Pecharsky and K. A. Gsc hneidner, Jr., Phys. Rev. Lett. 78, 4494 (1997).  J. S. Brooks et al. Rev. Sci. Instrum. 58, 117 (1987).
127  M. J. Naughton et al. Rev. Sci. Instrum. 68, 4061 (1997).  I. Dzyaloshinskii, Sov. Phys. JETP 5 1259 (1957).  T. Moriya, Phys. Rev. 120, 91 (1960).  B Cowan, Nuclear Magnetic Resonance and Relaxation (Cambridge Univ. Press, Cambridge, 1997).  S. Blundell, Magnetism in Condensed Matter (Oxford Univ. Press, New York, 2001).  Sherline, T. E. (2006). Antiferromagnetism in Cesium Tetrabromocuprate(II) and Bod yCenteredCubic Solid Helium Three (Doctoral dissertation). University of Florida.  C. Domb, The Critical Point: a Historical Introduction to the Modern Theory of Critical Phenomena (Taylor & Francis, London, 1996).  S. L. Sondhi, S. M. Girvin, J. P. Carin i, and D. Shahar, Rev. Mod. Phys. 69, 315 (1997).  Q. Si, Adv. Solid State Phys. 44, 253 (2004).  J. Brooke, D. Bitko, T. F. Rosenbaum, and G. Aeppli, Science 284 779 (1999).  295, 2427 (2002).  D. A. House and C. S. Garner, Nature 208, 776 (1965).  B. Andraka and Y. Takano, Rev. Sci. Instrum. 67, 4256 (1996).  D. Patan et al. Phys. Rev. Lett. 101, 175701 (2008).  D. Patan, A. Silva, F. Sols, and L. Amico, Phys. Rev. Lett. 102, 245701 (2009).  Ramsey C. M. (2004). Thermomagnetic and EPR Probing of Magnetism in Low Dimensional Lattices and SingleMolecule Magnets (Doctoral dissertation). Florida State University.  H. Takano, H. Nakanishi, and S. Miyashita, Phys. Rev. B 37, 3716 (1988).  S. A. Roberts, D. R. Bloomquist, R. D. Willett, and H. W. Dodgen, J. Am. Chem. Soc. 103, 2603 (1981).  H. Manaka, I. Yamada, and K. Yamaguchi, J. Phys. Soc. Jpn. 66, 564 (1997).  H. Manaka et al. J. Phys. Soc. Jpn. 76, 023002 (2007).
128  T. Masuda et al. Phys. Rev. Lett. 96, 047210 (2006).  T. Fischer, S. Duffe, an d G. S. Uhrig, arXiv 1009.3375.  V. O. Garlea et al. Phys. Rev. Lett. 98, 167202 (2007).  A. Zheludev et al. Phys. Rev. B 76, 054450 (2007).  H. Tsujii et al. J. Phys.: Conf. Ser. 150, 042217 (2009).  H. Manaka et al. J. Phys. Soc. Jpn. 67, 3913 (1998).  T. C. Ozawa and S. J. Kang, J. Appl. Cryst 37, 679 (2004).
129 BIOGRAPHICAL SKETCH Younghak Kim was born in Daegu, Korea. He graduated from Kyungpook National University. While an undergraduate student at the universi ty majoring in physics, he decided to continue his study of the subject abroad as a graduate student He came to the University of Florida in the fall of 2005 to pursue Ph. D. in physics with interest in experimental condensedmatter research. He joined P rof. Yasumasa Takanos group in 2006 and graduated in the spring of 2011 with Ph. D. in physics.