Spin Dynamics of Density Wave and Frustrated Spin Systems Probed by Nuclear Magnetic Resonance Lloyd L

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2008 Spin Dynamics of Density Wave and Frustrated Spin Systems Probed by Nuclear Magnetic Resonance Lloyd L. (Lloyd Laporca) Lumata

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COLLEGE OF ARTS AND SCIENCES

SPIN DYNAMICS OF DENSITY WAVE AND FRUSTRATED SPIN SYSTEMS PROBED BY NUCLEAR MAGNETIC RESONANCE

LLOYD L. LUMATA

A Dissertation submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Degree Awarded: Fall Semester, 2008 The members of the Committee approve the Dissertation of Lloyd L. Lumata defended on October 31, 2008.

James S. Brooks Professor Directing Dissertation

Naresh Dalal Outside Committee Member

Arneil P. Reyes Committee Member

Pedro Schlottmann Committee Member

Christopher Wiebe Committee Member

Mark Riley Committee Member Approved:

Mark Riley, Chair Department of Physics

Joseph Travis, Dean, College of Arts and Sciences The Oﬃce of Graduate Studies has veriﬁed and approved the above named committee members.

ii To my family...

iii ACKNOWLEDGEMENTS

This is it, like an Oscar called Ph.D. after four and a half years... I would like to express my gratitude, first and foremost, to my advisor Prof. James S. Brooks for being a great mentor in person and in research. He is the type of advisor who can turn a novice, clumsy graduate student into an astute observer and skilled researcher. He is smart, open-minded, responsible, and very helpful to his students and I am honored to be his 23rd Ph.D. graduate. I am indebted to Dr. Arneil Reyes and Dr. Philip Kuhns, the two people who, together with my advisor, formed the triad which contributed much to my scientific education and training throughout my years of study at the National High Magnetic Field Laboratory. Thanks is also extended to Dr. Michael Hoch and Prof. William Moulton for their scientific guidance. It is my pleasure to collaborate and exchange ideas with Prof. Stuart Brown of UCLA Department of Physics. I would like to thank my colleagues Robert Smith, Tiglet Besara, Dr. David Graf, Dr. Takahisa Tokumoto, Dr. Eung Sang Choi, Ade Kismarahardja, Moaz Altarawneh, Eden Steven, and Zach Stegen for their company and assistance at NHMFL. Thanks to my predecessors Dr. Relja Vasic, Dr. Eric Jobiliong, and Dr. Andrew Harter for teaching me how to handle cryogenics and some instrumentation during my first time. Thanks to Dr. Kwang-Yong Choi for his brilliant ideas on hot condensed matter topics. Thanks is extended to Dr. Haidong Zhou and Prof. Chris Wiebe for introducing me to the physics of frustrated spin systems. Furthermore, I would like to thank: John Pucci and Dan Freeman for providing liquid Helium even in short notice for urgent experiments. The staff of DC field control room for giving extra minutes in the high magnetic field experiments. Bruce Brandt and Eric Palm for approving our magnet time proposals. Vaughan and the Machine shop staff for their fine work in making those little brass pieces for our probe and cryostats.

iv Alice Hobbs of NHMFL and Sherry Tointigh of the FSU Physics Department for all their help in the paperwoks and reminders. Laurel McKinney and Eva Crowdis for processing my tutorial timesheets. Connie Eudy for giving me an awesome opportunity to be an associate in the Program for Instructional Excellence (PIE). The Hinchliﬀe family: Pilar, Mark, and Bill for being my family here in Tally. The Filipino-American community for making me feel at home. My friends at Rogers Hall, especially Robin and Wolfgang, for the good ol’ times in Tennessee street. I’ll surely miss the Seminoles playing football at Doak Campbell stadium. Go Noles! The Ong family: Cromwell, Winston, Lionel, Madeleine, and Sir Poly Huang for their brilliant advice and generosity. Prof. Jose Perano and the WMSU Physics family for teaching me perseverance in physics. Thanks to the committee members for perusing this manuscript and for devoting a couple of their precious hours to my dissertation defense. This is also an opportune time to acknowledge the support for this work by the National Science Foundation Division of Materials Research through grants NSF DMR-0602859 and DMR-0654118, the U.S. Department of Energy, and the State of Florida. This dissertation is dedicated to my parents Jose and Evelyn Lumata and to my siblings Richard, Analyn, Edwin, and Jenica. I also dedicate this to Vivienne Anne Santos for her care and inspiration. Above all, I thank the Almighty God for all the blessings He has given me without which I could not have completed this long road to Ph.D.

v TABLE OF CONTENTS

List of Tables ...... viii

List of Figures ...... ix

Abstract ...... xvii

1. INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE ...... 1 1.1 Knight Shift: Probing the Internal Magnetism ...... 2 1.2 Hyperﬁne Coupling Terms of the Interaction Hamiltonian ...... 5 1.3 Measuring the Dynamics: Relaxation Rates ...... 6 1.4 Temperature-dependent Relaxation in Metals ...... 12 1.5 Hebel-Slichter Peak: Test of BCS Superconductivity ...... 15 1.6 NMR Instrumentation ...... 16

2. AN OVERVIEW OF LOW-DIMENSIONAL ORGANIC CONDUCTORS .. 23 2.1 Low Dimensional Instabilities ...... 24 2.2 The Bechgaard Salts ...... 29 2.3 Transport Properties ...... 33 2.4 Magnetic Properties ...... 35 2.5 Phase Diagram of (TMTSF)2X ...... 37

3. SIMULTANEOUS 77Se NMR AND TRANSPORT INVESTIGATION OF THE SPIN DENSITY WAVE SYSTEMS (TMTSF)2X, X=ClO4,PF6 ..... 41 3.1 Experimental Details ...... 42 3.2 Temperature Dependence of NMR Spectra ...... 46 3.3 The RF Enhancement Factor η ...... 47 3.4 RF Power dependence of NMR lineshapes in the FISDW State ...... 48 77 3.5 Field Dependence of 1/T1,Rzz, and Spectra at Constant Temperature . 51 77 3.6 Temperature Dependence of 1/T1 at Low Fields ...... 53 3.7 Angular Dependence of 77Se NMR and Transport in the Metallic State . 55 3.8 Angular Dependence of 77Se NMR and Transport in the FISDW State . 57 77 3.9 Temperature Dependence of 1/T1, Spectra, and Rzz at High Fields .. 63 77 3.10 A Comparative Study: Se NMR and Transport on (TMTSF)2PF6 ... 67 3.11 Summary and Conclusion ...... 72

4. NMR ON CHARGE DENSITY WAVE SYSTEMS ...... 75

vi 4.1 Coexisting CDW and Spin-Peierls States in (Per)2Pt[mnt]2 ...... 75 4.2 Crystal Structure and Electronic Properties ...... 77 4.3 Experimental Details ...... 78 4.4 Results and Discussion ...... 78 4.5 Conclusion ...... 82 4.6 CuxTiSe2: a new CDW-Superconductor ...... 82 4.7 Experimental methods ...... 84 77 63 4.8 Se and Cu NMR Studies of CuxTiSe2 ...... 85 4.9 Conclusion ...... 89

5. PROBING THE DYNAMICS OF FRUSTRATED SPIN SYSTEMS ..... 90 5.1 A survey of Frustrated Spin Systems ...... 90 5.2 The Rare-Earth Kagom´eR3Ga5SiO14 ...... 93 69,71 5.3 Ga NMR Probe of the Spin Dynamics of Pr3Ga5SiO14 ...... 93 5.4 Results and Discussion ...... 96 5.5 Conclusion ...... 102 93 5.6 Nb NMR Probe of Ba3NbFe3Si2O14 ...... 102 5.7 Conclusion ...... 110

6. CONCLUSION ...... 111 6.1 Future Work ...... 113

REFERENCES ...... 115

BIOGRAPHICAL SKETCH ...... 123

vii LIST OF TABLES

2.1 Broken symmetry ground states of metals: SS-singlet superconductivity, TS- triplet superconductivity, CDW-charge density wave, SDW-spin density wave (from Ref. [12])...... 24

2.2 The Bechgaard Family of Superconductors (from Ref. [9]) ...... 30

4.1 Korringa factor K(α) due to electron-electron interaction in CuxTiSe2 .... 88 6.1 NMR parameters relevant to the work done in this dissertation...... 111

viii LIST OF FIGURES

1.1 A simple mechanism of NMR: rf irradiation of the nuclear moment precessing at Larmor frequency. With energy equal to the Zeeman splitting, rf can flip the nuclear spin then it returns to equilibrium releasing a signal. The lower figure represents the energy levels when the field is off or on...... 2

1.2 The Knight shift: the resonant frequency of the sample shifts by γBint from the reference due to local internal magnetic ﬁeld...... 3

1.3 The spin-echo pulse sequence: the magnetization is tipped by a 900 pulse and the spins start to “fan out”on the XY plane forming a FID signal. A second pulse ﬂipped the spins 1800 and they regroup then fan out again forming a spin echo signal...... 4

1.4 Measuring the spin-lattice relaxation time T1: (a) a π/2 “saturation” pulse is followed by a variable delay time which allows the growth of longitudinal magnetization Mz as it increases. The π/2 π spin-echo sequence “inspects” the recovery of this magnetization which is reﬂected− in (b) where the constant of the exponential growth is T1...... 8

1.5 Measuring the spin-spin relaxation time T2: (a) the variable delay time τdelay after each pulse in the sequence π/2 π is increased. The constant of the − exponential decay of the transverse magnetization Mxy (b) is T2...... 10 1.6 The probability of occupied states f(E) and unoccupied states 1 f(E) where f(E) is the Fermi-Dirac function. The red area, which is the product− of the two probabilities, represents the electrons that participate in the relaxation process...... 13

1.7 Schematic diagram of the transmitter section: gated oscillating electrical signal from the rf synthesizer is ampliﬁed and transmitted to the probe. ... 16

1.8 The duplexer: two hybrid couplers direct RF into the probe (transmit mode) and divert the tiny NMR signal from the probe to the pre-ampliﬁer (receive mode)...... 17

ix 1.9 Resonant circuits for NMR probe: (a) Series-tuned parallel-match. (b) Parallel-tuned series-match. CT ,CM , and L are tuning, matching capacitors, and inductor, respectively...... 18

1.10 (a) Schematic diagram of the receiver section: the NMR signal from the probe and the reference frequency from the synthesizer are mixed in the quadrature receiver where real and imaginary NMR signals are generated. (b) Spin echo signal showing the real and imaginary components...... 18

1.11 NMR hardware: (a) NMR rack containing the spectrometer, PC, temperature controller, rf synthesizer, magnet power supply, and liquid Helium level sensor. (b) Photo of the goniometer and coil with sample mounted on a probe. (c) Top view of the magnet with probe...... 20 1.12 Making a microcoil...... 21

2.1 (a) The dispersion relation and the electron density in the metallic state (b) The modulation of the electron density in the CDW state causing an opening π of gap Δ with Fermi wave vector kF = 2a ...... 25 2.2 (a) An array of equidistant electrons with antiferromagnetic interaction J (Heisenberg spin chain). (b) Due to spin-lattice coupling, the electrons are dimerized and lattice distortion occurs at q =2kF (Spin-Peierls state) leading to stronger antiferromagnetic interaction J + δ among the dimerized electrons and weaker antiferromagnetic interaction J δ between the pairs...... 27 − 2.3 (a) The dispersion relation in spin density wave state showing the opening of the gap Δ at the Fermi wave vector kF . (b) The modulation of the two spin π subbands in SDW with wavelength λ0 = ...... 28 kF 2.4 Crystal structure of the TMTSF molecule...... 30

2.5 Crystal structure of (TMTSF)2ClO4 viewed along a-axis. The crystallographic axes are given by the arrows...... 31

2.6 Zero-ﬁeld cooldown of (TMTSF)2ClO4 showing a kink in resistance at 24 K which is due to anion ordering...... 32

2.7 Low-ﬁeld c*-axis magnetoresistance (MR) of (TMTSF)2ClO4 at diﬀerent temperatures. The kinks in MR correspond to the FISDW cascade phases. Inset: T-B phase diagram showing the cascade of FISDW phases...... 33

2.8 High ﬁeld c*-axis magnetoresistance (MR) of (TMTSF)2ClO4 at diﬀerent temperatures showing kinks in MR that correspond to the FISDW phases ∗ Bth,B1,B, and Bre. Above 15 T, periodic modulations in MR called “rapid oscillations” occur...... 34

x 2.9 (a) Electronic motion in momentum space: a pair of warped FS sheets within the ﬁrst Brillouin zone in a Q1D system. (b) Electronic motion in real space with amplitude A = 4tb and wavelength λ = h ...... 36 evF Bz ebBz

2.10 T-B phase diagram of (TMTSF)2ClO4 showing the metallic region, superconducting (SC) state, the cascade of FISDW phases, and the re-entrant FISDW region...... 38

2.11 T-B phase diagram of (TMTSF)2PF6 at 12 kbar hydrostatic pressure: this is similar to (TMTSF)2ClO4 except for the absence of the re-entrant FISDW region at high magnetic ﬁelds...... 39

3.1 The (TMTSF)2ClO4 FISDW phase diagram where the dots represent the FISDW transport features seen in this work. The arrows represent the regions in the phase diagram where NMR was measured...... 42 3.2 Simultaneous NMR and electrical transport setup in a goniometer...... 43

3.3 Portable He-4 and He-3 cryogenic systems...... 44

3.4 Simultaneous NMR and electrical transport setup at high magnetic ﬁelds: (a) Electrical transport and NMR racks. (b) Back view showing the gas pressure system and the 30 T resistive magnet at NHMFL Cell 7...... 45

77 3.5 Temperature dependence of Se NMR spectra in a relaxed (TMTSF)2ClO4 with B c∗. Part (a) shows frequency-swept spectra in the metallic state at constant magnetic field B =12.12 T while (b) shows field-swept spectra at constant NMR frequency 98.2 MHz. The broad, double-horned spectra indicate inhomogeneous local magnetic field in the FISDW state...... 47

3.6 (a) Field-swept 77Se NMR spectra at diﬀerent rf power level attenuation in the FISDW state of (TMTSF)2ClO4 at 1.8 K and constant NMR frequency 98.6 MHz with B c∗. The pulse sequence used is 450 ns - 900 ns. (b) Power- swept spin-echo intensity taken at ﬁelds indicated by dashed arrows (LP-left peak, CP-central peak, RP-right peak) in (a). The red arrow at the bottom indicates the direction of increasing rf power...... 49

77 3.7 Field dependence of 1/T1 and c-axis resistance Rzz (red curve) in (TMTSF)2ClO4 measured simultaneously at 2 K with the ﬁeld applied parallel to the c∗- axis. The peak in 1/T1 occurs at B1 FISDW transition. Solid circles denotes relaxation rate in the metallic state, the solid triangles indicates the enhanced relaxation in the FISDW region, and the open circles denote a coexistence region or depinned state of FISDW. Inset: the corresponding ﬁeld dependence of full-width half maximum (FWHM) of NMR spectra...... 52

xi 77 3.8 Representative temperature dependence of 1/T1 at low fields (B<20 T) where B c∗. The dashed lines above the peak are fits to the SCR theory for T itinerant antiferromagnets 1/T1 = A 1/2 and below the peaks are power (T −TN ) α law fits 1/T1 = AT ...... 54

77 3.9 (a) Angular dependence of Se NMR spectra along the a-axis of (TMTSF)2ClO4 in the metallic phase at 7.84 T and 4.2 K. (b) Corresponding angular- dependent magnetoresistance and frequency shift. Note that the peaks in Rzz and ν ν0 do not coincide. (c) Full-width half maximum (FWHM) and − 77 spin-lattice relaxation rate 1/T1 as a function of angle...... 56 3.10 Angular dependence of 77Se NMR lineshapes along the a-axis at constant NMR frequency 104.55 MHz (a) in the SDW state of “quenched” (TMTSF)2ClO4 at 2 K (b) in the metallic and FISDW states of “relaxed” (TMTSF)2ClO4 at 1.8 K...... 58 3.11 Angular-dependent NMR and electrical transport at 14 T and 1.5 K in 77 (TMTSF)2ClO4. (a) 1/T1 versus ﬁeld orientation θ. A dip in 1/T1 is marked X. (b) Corresponding magnetoresistance and rf enhancement η at 14 T and 1.5 K. (c) Schematic of sample rotation along a-axis and the orientation of the TMTSF molecule with respect to magnetic ﬁeld B at point X. (d) Angular- dependent 77Se NMR spectra...... 59

77 3.12 (a) The spin-lattice relaxation rate 1/T1 as the sample is rotated along 77 the a-axis at 12.86 T and 3.87 K. Note the dip in 1/T1 marked X. There is no distinct hysteresis in the result as sample is rotated clockwise and 77 counterclockwise. Inset: schematic of sample rotation. (b) 1/T1 versus θ at 12.86 T and 1.87 K. Inset: orientation of TMTSF at point X with respect to ﬁeld. (c) Corresponding enhancement factor η at 12.86 T and 1.87 K. .. 60

3.13 Angular-dependent NMR and electrical transport at B = 30 T and 1.47 K. (a) Metallic and FISDW transitions revealed in angular-dependent 1/T1. (b) The rf enhancement η vs. θ. Note that η = 1 above Bre. (c) Field-swept NMR spectra at ν0 = 243.9 MHz (30 T) taken at different angles and consequently 0 different phases: (i) FISDW phase at θ = 105 (above B1) (ii) Metallic phase 0 0 at θ =90 and (iii) Re-entrant FISDW phase at θ =0 (above Bre). The corresponding magnetoresistance data at different angles are shown in the lower right hand corner. For each trace, the NMR measurement was made at 30 T...... 62

77 3.14 1/T1 versus effective perpendicular magnetic field B⊥ = B0 cos θ. The red 77 arrows mark the peaks in 1/T1 which coincide with the different FISDW ∗ phase boundaries B1, B , and Bre seen in electrical transport measurements. 63

xii 77 3.15 Crossing the re-entrant FISDW phase: (a) Temperature dependence of 1/T1 and c*-axis resistance R of (TMTSF) ClO at 23 T with B c∗. 771/T zz 2 4 1 exhibits a peak at 5 K which is coincident with the upturn in Rzz. The re-entrant phase is denoted by another sharp increase in Rzz at around 3 K. (b) Normalized 77Se NMR spectra in the metallic (black) and FISDW phases (red). (c) Corresponding temperature-dependent NMR linewidth and Boltzmann-corrected NMR intensity...... 65

77 ∗ 3.16 (a) Temperature dependence of 1/T1 and c -axis resistance Rzz measured simultaneously at 29 T with B c∗. (b) Corresponding 77Se NMR spectra in the metallic (black) and FISDW (red) states. (c) Temperature dependence of full-width half maximum (FWHM) and NMR intensity at 29 T...... 66

77 3.17 (a) Temperature dependence of 1/T1 and Rzz in (TMTSF)2PF6 at 17 T where B c . Dashed lines are ﬁts to certain equations: self-consistent ∗ T renormalization (SCR) theory equation A 1/2 (blue dashed line), power (T −TN ) law AT α where α =3.2 (yellow dashed line), and AeT/Δ where Δ = 0.65 (green dashed line). (b) Temperature dependence of 77Se spectra in the metallic (narrow, multiple-peaked lineshapes) and SDW state (broad lineshape). Inset: Plot of the peak position versus temperature in the metallic state...... 69

77 3.18 (a) Angular-dependent Se NMR spectra in (TMTSF)2PF6 at 20 K. There are four inequivalent sites. (b) Plot of the peak position versus angle. The solid lines are ﬁtted according to dipolar interaction equation A (3 cos2 θ 1) 0 − where A0 is a constant. The deviation of the resonant peaks from this ﬁt is attributed to the triclinic structure of the crystal...... 70

77 3.19 (a) Angular-dependent Se spectra in (TMTSF)2PF6 at 17 T and 4.2 K. (b) The NMR lineshape at c-axis (0 deg) showing two sets of double-horned peaks (indicated by the model ﬁts). (c) Corresponding resistance at diﬀerent angles. 71

∗ 3.20 Phase diagram of (TMTSF)2ClO4 for B c derived from previous reports 77 (dashed lines) including a summary of the observed 1/T1 peaks (asterisks and open squares), and the corresponding features in the transport measurements (dark and gray circles) from this work. The ﬁeld labels are deﬁned in text. . 72

3.21 Fermi surface nesting models in (TMTSF)2ClO4 and the corresponding dispersion relations in the metallic (left panel), FISDW (middle panel), and re-entrant FISDW (right panel) regions of the phase diagram...... 73

4.1 Schematic of the approximate T-B phase diagram of polycrystalline (Per)2Pt[mnt]2 showing coexisting charge density wave (CDW) and spin-Peierls (SP) ground states below 20 T, and ﬁeld-induced CDW (FICDW) region above 20 T [from Graf et al.]. The dashed arrows show the regions in the phase diagram where NMR was measured...... 76

xiii 4.2 (a) Crystal structure of (Per)2Pt[mnt]2 viewed along a-axis. (b) Schematic of the perylene and dithiolate layers. Electronic conduction occurs in the perylene chains and is directed mainly on the b-axis...... 77

195 4.3 Temperature dependence of Pt NMR spectra in polycrystalline (Per)2Pt[mnt]2 at 14.8 T. Left Inset: Plot of the corresponding spectral intensity which deviates from the Boltzmann prediction. The NMR signal disappears at around 5 K which is coincident with the Spin-Peierls transition temperature. Right Inset: characteristic power lineshape pattern that resembles the 195Pt NMR spectra due to distribution of anisotropic Knight shifts...... 79

195 4.4 Pt NMR spectra of (Per)2Pt[mnt]2 at various ﬁelds at constant temperature 1.8 K. Note the loss of 195Pt NMR signal above 20 T...... 80

195 4.5 Field dependence of 1/T1 at 1.8 K (open circles) and 2.2 K (open squares). The spectral amplitude at 2.2 K shows the disappearance of 195Pt NMR signal above 20 T...... 81

4.6 T-x Phase diagram of newly-discovered CDW-superconductor CuxTiSe2 [from Morosan et al.]. Notice the similarity to the high-Tc phase diagram...... 83

77 4.7 (a) Temperature dependence of Se NMR spectra in Cu0.07TiSe2 at 7.193 T. (c) Plot of the peak position/shift from (a)...... 84

63 4.8 (a) Temperature dependence of Cu Knight shift for Cu0.05TiSe2 (red triangles) and Cu0.07TiSe2 (blue circles) at 8 T. (b) Temperature dependence of 63 1/T1 at8T...... 85

77 4.9 Temperature dependence of 1/T1 of parallel stacks of CuxTiSe2 (x=0.05, ∗ 77 0.07) platelets with B c , powder of TiSe2 (Dupree et. al.), and pure Se 77 metal. Inset: log-log plot of this graph. Note the non-linearity of 1/T1 vs T in TiSe2 due to CDW formation...... 86 4.10 Enhanced Korringa factor K(α) vs interaction parameter α from Moriya and a corrected version from Narath et. al. The blue and red dashed lines are the observed K(α) for 7% and 5% Cu dopings, respectively. The arrows indicate that the interaction parameter α is 0.8 for 7% and 0.93 for 5%...... 88

5.1 (a) The structurally perfect kagomélattice (b) Crystal structure of herbertsmithite (ZnCu3(OH)6Cl2), the S =1/2 structurally perfect kagomélattice network of Cu2+ spins (from Ref. [69])...... 91 5.2 (a) q = 0 and (b) q = √3 √3 (also called the weathervane mode) states in a structurally perfect kagomélattice.× ...... 92

5.3 Distorted kagom´esystem: crystal structure of Pr3Ga5SiO14 (from Ref. [89]) representative of other frustrated langasite systems...... 94

xiv 69 5.4 NMR field-scan spectrum of Pr3Ga5SiO14 showing quadrupolar split Ga (I =3/2) and 71Ga (I =3/2) components for the two non-equivalent Ga sites (the third site, embedded in a small peak, is not shown). The inset depicts the crystal structure showing three kagoméplanes. Two Ga sites lie between these planes and the third site is in plane...... 95 5.5 (a) 69Ga Knight shifts measured in an applied field of 9 T along the crystal c-axis plotted versus the magnetic susceptibility with temperature as the implicit parameter. For T>30 K the plot shows that but for lower T departures from a linear relationship are observed. (b) 69Ga NMR spectra as a function of T...... 98

69 5.6 Ga 1/T1 and 1/T2 for Pr3Ga5SiO14 as a function of T at 16.37 T, together with specific heat at 9 T whose peak is coincident with the broad maximum of 69 1/T2. The similarity in behavior of the two quantities is striking and points 69 to a common underlying mechanism. Inset: log-log plot of 1/T2 at different fields. Notice that the broad maximum sharpens as the field is increased and below the peak the behavior is close to T 2...... 99

5.7 Temperature dependence of the transverse spin correlation time τ2 at different 69 fields extracted from 1/T1 vs T plot (see lower inset). At high temperatures, the gap Δ 98 K obtained from the slope is field-independent. Below 10 K, ≈ the gap ΔNMR has field dependence (see upper inset) where the dashed line corresponds to the fit ΔNMR =Δ0 + αH with α = gμB =3.32μB and Δ0 =3.5 K. The field-dependence of the spin gap in the excitation spectrum as derived from 35 mK inelastic neutron scattering results (from Ref. [90]) is shown for comparison...... 101

5.8 Crystal structure of Ba3NbFe3Si2O14 viewed along (a) c-axis (b) b-axis. (c) Room temperature X-ray diﬀraction pattern. Inset: temperature dependence of the lattice parameters (from Ref. [92])...... 103

5.9 (a) Inverse DC susceptibility where the solid line fit is the Curie-Weiss law. (b) Temperature dependence of the specific heat of Ba3NbFe3Si2O14 (open circles) and Ba3Nb(Fe0.5Ga0.5)3Si2O14 (solid line) (c) The magnetic contribution to the specific heat and the calculated entropy (d) Temperature dependence of thermal conductivity (from Ref. [92]) ...... 104

93 5.10 Field-swept Nb spectra in the paramagnetic state of Ba3NbFe3Si2O14. Inset: Plot of the resonant ﬁeld versus temperature reﬂecting the spin susceptibility. 106

5.11 (a) Fieldswept 93Nb spectra at constant frequency 83.4 MHz versus field orientation (θ is the angle between B and a-b plane) at 4.2 K in the antiferromagnetic state. The small sharp peaks are 63,65Cu and 27Al NMR signals from the coil. (b) Plot of the left and right resonant peaks. The dashed lines are fits to the equation A cos θ where A is the hyperfine coupling constant...... 107

xv 93 5.12 (a) Temperature dependence of 1/T1T in Ba3NbFe3Si2O14 at 83.4 MHz in the paramagnetic state (PP-paramagnetic peak) and antiferromagnetic state (LP-left peak, MP-middle peak, and RP-right peak). The dashed in the paramagnetic region is ﬁt to SCR spin ﬂuctuation theory with TN 27.5 K. (b) Corresponding temperature-dependent spectra in the two states.≈ The location of the peaks are indicated and the middle sharp line is 27Al NMR signal from the probe...... 108

93 5.13 Temperature dependence of the spin-spin relaxation rate 1/T2 measured in the paramagnetic and antiferromagnetic regions at 83.4 MHz with B a b plane. The dashed line is a ﬁt similar to SCR spin ﬂuctuation behavior.⊥ − Inset: Dependence of the stretched exponential parameter β of the middle peak. The change in β at around 4 K corresponds to T ∗ seen in the relaxation rate measurements...... 109

xvi ABSTRACT

This dissertation encompasses my major experimental work using nuclear magnetic resonance (NMR) to probe the local magnetism and spin dynamics of two interesting systems in condensed matter: density wave and frustrated spin systems. Density waves are ordered ground states formed due to the instability in low-dimensions while frustrated spin systems inhibit long-range magnetic ordering due to their corner-shared triangular structure. The ﬁrst part of this dissertation entails a discussion of the broken symmetry ground states in low dimensional systems: spin density waves (SDW), charge density waves (CDW), and spin-Peierls (SP) states. Simultaneous 77Se NMR and electrical transport is employed to investigate the spin density wave (SDW) ground state in the quasi-one-dimensional

(Q1D) organic conductor (TMTSF)2PF6 and the field-induced spin density wave (FISDW) transitions in (TMTSF)2ClO4. Furthermore, angular-dependent measurements were taken at very high magnetic fields to probe the anisotropic properties of FISDW subphases, giving insight into the electronic structure in the quantum limit. The CDW and SP ground states 195 in another Q1D organic conductor (Per)2Pt[mnt]2 were studied using Pt NMR revealing the breaking of the SP state at high magnetic fields. The role of doping in the electronic 63 correlations of the newly discovered CDW-superconductor CuxTiSe2 is revealed by Cu and 77Se NMR. The later part of this dissertation focuses on the kagoméspin systems which show very interesting phenomena due to magnetic frustration. Using 69,71Ga NMR, the dynamical behavior of spins in the spin-liquid state in one of the first rare-earth kagomématerials

Pr3Ga5SiO14 is described and compared with other existing frustrated spin systems. On 93 the other hand, Nb NMR on structurally similar material Ba3NbFe3Si2O14 provides an opportunity to study multiferroicity in a geometrically frustrated lattice. This work shows how NMR contributes to the understanding of these two distinct classes of condensed matter systems.

xvii CHAPTER 1

INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE

Nuclear magnetic resonance (NMR) was developed in 1945 when E. M. Purcell detected radiofrequency (rf) signals from paraffin and independently, Felix Bloch observed rf signals from water. Since then a lot of technological applications have emerged in medical diagnostics, chemical, biological, materials characterization, and industrial applications [1]. NMR relies on the fact that the nuclei of different elements “resonate” at specific frequencies when subjected to applied magnetic field. These nuclear spins behave like gyrating tops in the presence of applied magnetic field B0, precessing with a frequency

ω = γnB0 called the Larmor frequency where γn is the gyromagnetic ratio. Each nucleus 1 has a specific gyromagnetic ratio: proton or H, for instance, has γn =42.5774 MHz/T and 77 Se has γn =8.13 MHz/T. A simple schematic in Fig. 1.1 shows how NMR works where rf irradiation, with energy equal to the Zeeman splitting, causes a precessing spin-1/2 nucleus to flip, then this nucleus returns to thermal equilibrium releasing a signal from which we can study [2, 3, 4]. NMR is a spectroscopic tool that looks at nuclear coupling with the environment in the MHz window range. Other magnetic resonance techniques are electron spin resonance (ESR) which utilizes microwave radiation to probe fluctuations in the GHz range, Mössbauer spectroscopy which relies on the recoilless emission and resonant absorption of gamma rays, and muon spin resonance which are ideal for studying small moment magnetism by observing the decay and dynamics of implanted muons on the material [5]. NMR is the most commonly used spectroscopic technique because nuclei are everywhere and rf is a non-ionizing radiation. This dissertation focuses on using NMR in investigating condensed matter systems as a local magnetic probe. NMR spectroscopy is one of the research tools that can give us

1 B0

ω=γB0

flipping rf signal

E=2µB0

ω=2µB0/h

B=B0 µB0

B=0 2µB0

−µB0

Figure 1.1: A simple mechanism of NMR: rf irradiation of the nuclear moment precessing at Larmor frequency. With energy equal to the Zeeman splitting, rf can flip the nuclear spin then it returns to equilibrium releasing a signal. The lower figure represents the energy levels when the field is off or on.

microscopic information about the spin dynamics and internal magnetism of materials via the coupling of the nuclei with the local environment. This chapter of the dissertation will attempt to describe the basic theory of NMR, the various parameters measured and what they tell us, and the generic instrumentation needed to perform these measurements.

1.1 Knight Shift: Probing the Internal Magnetism

When subjected to external magnetic ﬁeld, the nuclear moment μ = γnI where I is the nuclear spin, points parallel to the ﬁeld direction. In a solid, these moments add up and 2 2 Nγn I(I+1) collectively give the magnetization described by Curie law M0 = H for N number 3kB T of spin-1/2 nuclei. This magnetization is tipped to some angle by rf, then it spirals toward equilibrium inducing a decaying electrical signal in the time domain called free induction decay (FID) [1, 2, 3, 4]. The Fourier transformation of FID gives us the NMR spectrum which is a plot of the nuclear population resonating at a particular frequency. One can also utilize a spin-echo technique (see Fig. 1.3), which is a “back-to-back” FID to get the NMR

2 Shift

reference

sample

Frequency ω = γB ref 0 ωsample = γ(B0+Bint)

Figure 1.2: The Knight shift: the resonant frequency of the sample shifts by γBint from the reference due to local internal magnetic ﬁeld.

spectrum. The change in the linewidth and the shift of the resonant frequency of the NMR spectrum can give us information about the local internal magnetic field in the material. The lineshape broadens in the case of magnetic ordering like antiferromagnetism or spin density wave formation which will be discussed in great detail in the succeeding chapter. The resonant frequency for non-interacting nuclear spins like in ionic salts is the Larmor frequency ω = γnB0. However in most solids, the resonant frequency deviates from the expected frequency because of the internal magnetic field Bint in the material as depicted in Fig. 1.2. This shift is given by the effective nuclear-spin Zeeman Hamiltonian [2]:

H = γ H (I + K σ) I + H (1.1) − n 0 · s − · Q where H0 is the applied magnetic ﬁeld, γn is the gyromagnetic ratio, I is the nuclear spin vector, σ is the orbital or chemical shift, Ks is the Knight shift, and HQ is the interaction 1 of the nuclear quadrupole moment Q (for nuclei with spins I>2 ) with the local electric ﬁeld gradient (EFG). The chemical shift σ, also called orbital shift, emanates from orbital magnetism which includes diamagnetic shielding and Van Vleck paramagnetism. The Knight

3 o o 90 180

FID Spin Echo

Figure 1.3: The spin-echo pulse sequence: the magnetization is tipped by a 900 pulse and the spins start to “fan out”on the XY plane forming a FID signal. A second pulse ﬂipped the spins 1800 and they regroup then fan out again forming a spin echo signal.

shift, also called metallic shift, results from the polarization of the conduction electron spins by the applied magnetic ﬁeld (Pauli susceptibility) and the hyperﬁne coupling of the nuclear spin to this electron spin polarization [6, 7]. However in broad-line condensed matter NMR, the most dominant factor in the deviation from the expected resonant frequency is the Knight shift. This shift is a measure of the spin susceptibility χs of the material:

Ahf Ks(T )= χs(T ) (1.2) NμB where Ahf is the hyperﬁne coupling constant. To ﬁnd the Knight shift, the resonant frequency of the sample is subtracted from the resonant frequency of a certain reference. The reference sample is usually a salt since it has zero Knight shift. For instance, the reference salt for 13C is tetramethylsilane (TMS) Si(CH3)4. Ks is usually expressed in percent:

ωref ω Ks = − 100% (1.3) ωref × where ωref = γH0, the unshifted resonant frequency of a reference salt. For ﬁeld-swept spectra at constant frequency, the Knight shift is expressed as K = Href −H 100%. s H ×

4 1.2 Hyperﬁne Coupling Terms of the Interaction Hamiltonian

The Hamiltonian describing the interaction of nuclei with conduction electrons is given by [6]:

8π S 3r(S r) e r p H =2 μ γ I S(r)δ(r) 2μ γ I [ · ] γ [I × ] (1.4) · 3 B n · − B n · r3 − r5 − n mc · r3 where μB is the Bohr magneton, γn is the gyromagnetic ratio, I is the nuclear spin, S the electron spin, and r is the electron position vector with the nucleus as the origin. The ﬁrst term is the Fermi contact interaction, the second term is the spin dipolar interaction between nuclear and electron spins, and the third term is the interaction of the nuclear spin with the orbital motion of the electrons.

1.2.1 The Fermi Contact Term

The ﬁrst term in the interaction Hamiltonian H = 8π γ γ 2I Sδ(r) is the Fermi contact contact 3 e n · interaction between the resonating nucleus and the s-electrons where r is the electron position vector and the nucleus is taken to be at the origin. The resulting Knight shift for metals can be written as [6]:

8π K = ψ (0) 2 χ (1.5) s 3 | s | FS P where χ is the Pauli paramagnetic spin susceptibility per atom and ψ (0) 2 is the P | s | FS average over the Fermi surface of the squared magnitude of the Bloch wavefunctions evaluated at the site of the nucleus [6]. Non-s electrons will have no contact interaction because their probability at the site of the nucleus vanishes [7].

1.2.2 Dipolar coupling term

The magnetic dipolar interaction between two magnetic moments μj and μk is generally given by [2]:

1 N N μ μ 3(μ r )(μ r ) H = [ j · k j · jk k · jk ] (1.6) d 2 r3 − r5 j=1 jk jk k=1 5 where in this case of μj = γnI is the nuclear moment and μk = γeS is the electron moment.

In spherical polar coordinates, the position (rx,ry,rz)=(r sin θ cos φ, r sin θ sin φ, r cos θ) and 2 γnγe we can therefore write the dipolar Hamiltonian as Hd = r3 (A + B + C + D + E + F ) where, in terms of the raising operators I+ and lowering operators I− , A = I I (1 cos2 θ), 1,2 1,2 1z 2z − B = 1 (I+I− + I−I+)(1 cos2 θ), C = 3 (I+I + I I+) sin θ cos θe−iφ, D = 3 (I−I + 4 1 2 1 2 − − 2 1 2z 1z 2 − 2 1 2z I I−) sin θ cos θeiφ, E = 3 I+I+ sin2 θe−2iφ, and F = 3 I−I− sin2 θe2iφ. If both moments 1z 2 − 4 1 2 − 4 1 2 are aligned with magnetic ﬁeld, the splitting of NMR spectra is given by ΔH = A (3 cos2 θ 1). 2 − This equation is particularly useful in the analysis of angular-dependent NMR lineshapes in the spin density wave state which we will discuss later in Chapter 3.

1.2.3 Orbital Term

The orbital contribution to the total shift, which is important in transition metals, emanates from the orbital magnetic moments of conduction electrons induced by the applied magnetic field H. We can write this shift as K = b χ where b is the orbital hyperfine coupling orb orb constant. More specifically, the orbital shift is [6]:

2l·I 2μ i H l f f 3 i δ(kf ki) K = B | · | | r | − (1.7) orb IH E E i i f f − where i is the occupied Bloch state and f is the unoccupied Bloch state and the matrix | | elements are integrated over a Wigner-Seitz cell. A simpliﬁed version of the above equation n n 1 is K i f r3 where n is the number of occupied Bloch states and n is the number of orb ≈ Δ i f unoccupied Bloch states and Δ is the conduction electron bandwidth [6].

1.3 Measuring the Dynamics: Relaxation Rates

A gyroscopic motion dL = μ B, or similarly dμ = γμ B, is produced when a nuclear dt × dt × magnetic moment μ is subjected to an applied magnetic ﬁeld B where L is the angular momentum. For an ensemble of nuclei of the same isotope, the magnetization can be written as M = μ so that the dM = γM B.ForI =1/2 nuclei, the equilibrium value i i dt × μB of the magnetization is M0 = Nμtanh( ) where N is the number of nuclei. When the kBT magnetization is perturbed, the z-component of the magnetization returns to equilibrium at a rate proportional to the departure from the equilibrium magnetization value dMz = M0−Mz . dt T1 Thus, the z-component of the equation of motion is [2, 3]:

6 dM /dt = γ(M B) +(M M )/T (1.8) z × z 0 − z 1

T1 is the spin-lattice relaxation time. On the other hand, the transverse components Mx and

My will decay to zero as the magnetization returns to equilibrium. Thus,

dM /dt = γ(M B) M /T (1.9) x × x − x 2

dM /dt = γ(M B) M /T (1.10) y × y − y 2 where T2 is the spin-spin relaxation time. Eqs. 1.8, 1.9, and 1.10 are collectively called the Bloch equations which describe the equations of motion of the magnetization. The details of the relaxation processes are discussed below.

1.3.1 Spin-Lattice Relaxation Rate 1/T1

The spin-lattice relaxation rate 1/T1 measures how fast the longitudinal magnetization moves back to its thermal equilibrium value. It is called “spin-lattice” because the nuclear spins transfer energy to some repository which appear as translations, vibrations, and rotations in the electronic system collectively called the “lattice” [4]. The fluctuating electronic spin density, described by S+ and S− operators, gives rise to the fluctuating field detected by the nuclear spin as it makes a transition. We can therefore write the relaxation equation:

+∞ 1/T dτeiω0τ S+(τ)S−(0) (1.11) 1 ∝ q q −∞ which tells us that 1/T1 is a measure of the fluctuating magnetic field (spin-spin correlation function) perpendicular to applied magnetic field. In electronic systems, the fluctuation-dissipation theorem is used to relate the spin-spin correlation function to the spin susceptibility. In general, the nuclear spin-lattice relaxation rate for electronic systems is given by Moriya’s equation:

′′ 4kBT Aα(q) 2 χ (q,ω) 1/T1 = lim ( ) (1.12) ω→ωn∼0 γ ω q,α=xx,yy e ′′ where χ is the dynamic spin susceptibility which is the absorptive, imaginary part of the retarded electron spin susceptibility and Aα(q) is the hyperﬁne coupling.

7 (a) π/2 π/2 π τ recycle=5T1 τdelay τ0 τ0

(b)

400

300

200

100

0 Longitudinal Magnetization (arb. units)

3 4 5 6 10 10 10 10 delay time (µs)

Figure 1.4: Measuring the spin-lattice relaxation time T1: (a) a π/2 “saturation” pulse is followed by a variable delay time which allows the growth of longitudinal magnetization Mz as it increases. The π/2 π spin-echo sequence “inspects” the recovery of this magnetization − which is reﬂected in (b) where the constant of the exponential growth is T1.

One way to measure the spin-lattice relaxation time T1 is given in Fig. 1.4. There are two sets of pulses: the saturation and the “inspect” pulses. The saturation pulse is usually a π/2 pulse which knocks the spins down on the XY plane. After a certain delay time τdelay which allows some part of the longitudunal magnetization to grow, a π/2 π spin-echo pulse − sequence, “inspects” the magnitude of the NMR signal for a particular delay time. Thus a plot of the integrated spin-echo intensity versus delay time is generated in Fig. 1.4b, which reﬂects the growth of the z-component of the magnetization.

The rate at which the longitudinal component of the magnetization Mz goes back to thermal equilibrium can be expressed as dMz = M0−Mz . The growth of the longitudinal dt T1

8 M t magnetization M (t) can be solved by integrating z dMz = 1 dt from which z 0 M0−Mz T1 0 ln M0 = t . Finally, the recovery of the longitudinal magnetization is ﬁtted with a M0−Mz T1 single exponential equation1:

M = M (1 e−t/T1 ) (1.13) z 0 − where t is the delay time, M0 is the equilibrium value of the magnetization, and the constant

T1 is the spin-lattice relaxation time. A good single exponential recovery of the magnetization looks like an S curve in a semi-logarithmic plot shown in Fig. 1.4. In some cases, multiple exponential or stretched exponential ﬁtting is used if there is more than one relaxation mechanism. For S>1/2, the so-called master equations are used if there are quadrupolar contributions.

1.3.2 Spin-Spin Relaxation Rate 1/T2

The spin-spin relaxation rate 1/T2, also called the transverse relaxation rate, measures how fast the magnetization decays on the XY plane. T2 is the dephasing time in which the nuclear spin ensemble loses its coherence due to the diﬀerent local magnetic ﬁelds experienced by the spins. It should be noted that there is no energy loss in this relaxation process since there is no associated Zeeman level transitions. We can write this dephasing rate as:

+∞ 1/T dτeiω0τ S (τ)S (0) (1.14) 2 ∝ α α −∞ which tells us that 1/T2 is a measure of the fluctuating magnetic field parallel and perpendicular (since T1 process is also involved here; see Eq. 1.19) to the applied magnetic ′ field. In electronic systems, 1/T2 is proportional to χ (r,ω) which the non-dissipative, real part of the retarded spin susceptibility.

Figure 1.5 shows one way to measure the T2 where a spin-echo pulse sequence is generated with a delay time placed after each pulse. After the magnetization is knocked down on XY plane by the π/2 pulse, the spins lose their coherence or “fan out” as the delay time increases and the π pulse ﬂips the fanned-out spins where they regroup giving a spin echo signal. As

1 β Some systems have stretched exponential recovery Mz = M0([1 exp( (t/T1) )] where β ranges from 0 to 1. Others can have multi-exponential forms. − −

9 (a) π/2 π τrecycle=5T 1 variable τdelay variable τdelay

(b)

5 Transverse Magnetization (arb. units)

0 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 2 3 4 10 10 10 delay time (µs)

Figure 1.5: Measuring the spin-spin relaxation time T2: (a) the variable delay time τdelay after each pulse in the sequence π/2 π is increased. The constant of the exponential decay − of the transverse magnetization Mxy (b) is T2.

the delay time increases, the integrated spin echo decreases and for most systems a single exponential ﬁtting2 is appropriate. To obtain the time dependence of transverse magnetization decay, the Bloch equations

dMx Mx dMy Mxy dMz are written as = γB0My , = γB0Mx , and = 0. From these dt − T2 dt − − T2 dt −t/T2 −t/T2 we obtain Mx = m0e cos ωt and My = m0e sin ωt. To compute for the transverse 2 2 1/2 relaxation we used the equation Mxy =(Mx + My ) which yields:

2 − t 2 2T 2 In some solids, the form of transverse relaxation decay is Gaussian Mxy = M0e 2G or Lorentzian 1/T2 2 2 Mxy = M0 (1/T2) +t .ThusT2 is half of the full-width half maximum of these functions.

10 −t/T2 Mxy = M0e (1.15) where t is the delay time, the constant T2 is the spin-spin relaxation time, and M0 is the equilibrium value of the magnetization. A good single exponential ﬁt of the XY magnetization decay is a reversed S curve in a semi-logarithmic plot illustrated in Fig. 1.5.

1.3.3 Spectral Density and Correlation Time

A pair of spins brought together from inﬁnity to a certain distance will have a constant interaction energy for a particular static orientation. However the spins move randomly with respect to each other causing the interaction energy to be distributed in frequency and time [4]. The dependence of power on this frequency is called the spectral density denoted by J(ω). For a purely magnetic contribution, the spin-lattice relaxation rate 1/T1 is proportional to the value of this function at the Larmor frequency of the nuclear spin [1, 4].

By random ﬁeld approximation (RFA), the mechanism of 1/T1 is based on the following assumptions [1]:

The fluctuating fields have zero average H (t) = 0 where H (t) is the fluctuating • ⊥ ⊥ magnetic field perpendicular to the applied external field.

The magnitude, obtained by root-mean-square, of the ﬂuctuating ﬁelds is not zero: • H2 (t) =0. ⊥

The autocorrelation function given by G(τ)=H⊥(t)H⊥(t + τ) is not zero. The • − 2 τ autocorrelation function, also written as G(τ)= H e τc , tends to be large for small ⊥ values of τ and tends to be zero for large values of τ. τc is called the correlation time of the ﬁeld ﬂuctuations.

The spectral density is the Fourier transform of the autocorrelation function:

∞ J(ω)=2 G(t) exp( iωτ)dτ (1.16) − 0 For a transverse ﬂuctuating magnetic ﬁeld, the spectral density is:

2 τc J(ω)=2H⊥ 2 2 (1.17) 1+ω τc

11 Hence, the spin-lattice relaxation rate can now be expressed as:

2 2 τc 1/T1 = γn H⊥ 2 2 (1.18) 1+ω τc where H2 = H2 + H2 . If the transverse ﬁeld ﬂuctuates rapidly, the correlation time is ⊥ x y short and the spectral density is broad. Similarly, the spin-spin relaxation rate can also be written in terms of the correlation time [2]:

2 2 2 2 1 2 2 τc 1/T2 = γn H τc +1/2T1 = γn H τc + γn H⊥ 2 2 (1.19) 2 1+ω τc where H = Hz, the fluctuating magnetic field parallel to the applied external field. 1.4 Temperature-dependent Relaxation in Metals

In a metal, the relaxation process involves a transfer of energy to the conduction electrons. We may think of this as a scattering process where a simultaneous nuclear and electronic transition occur from initial state mks to the ﬁnal state nk′s′ where m,n are quantum | | numbers, k is the wavevector, and s is the spin orientation. The number of transitions per unit time is therefore given by [2]:

1 ′ ′ 2 W ′ ′ = mks V nk s δ(E + E E E ′ ′ ) (1.20) mks,nk s 2| | | | m ks − n − k s The parameter V is the contact interaction that drives the scattering. In the case of simple metals, V = 8π γ γ 2I Sδ(r) where the nucleus is chosen to be at the ori- 3 e n · gin. The total probability per transition is the sum of all the initial and ﬁnal electron states which is W = ′ ′ W ′ ′ where the state ks is occupied by an mn ks,k s mks,nk s | electron and the state k′s′ is unoccupied. Since this is an ensemble of electrons, we | ′ ′ employ the Fermi-Dirac statistics to write W = ′ ′ W ′ ′ f(k, s)[1 f(k ,s)] mn ks,k s mks,nk s − where f(k,s) is the Fermi function. The electronicwave function can be written as a product of the spin function and Bloch wave function: mks = m s u (r)eik·r. | | | k We may therefore write the matrix elements as mks V nk′s′ = 8π γ γ 2 m I n | | 3 e n | | · ′ ∗ s S s u (0)u ′ (0). Thus we can write the number of transitions per time as W ′ ′ = | | k k mks,nk s 2π 64π2 2 2 4 ′ ′ 2 2 γ γ ′ m I n n I ′ m s S s s S ′ s u (0) u ′ (0) δ(E +E E 9 e n α,α =x,y,z | α| | α | | α| | α | | k | | k | m ks− n− Ek′s′ ). Introducing the density of states g(Ek,A), we can write the sum of the previous expression [2]:

12 occupied states unoccupied states 1.0 f(E) 1-f(E) kBT

0.5

f(E)[1-f(E)]

0.0 0 1 2 EF

Figure 1.6: The probability of occupied states f(E) and unoccupied states 1 f(E) where f(E) is the Fermi-Dirac function. The red area, which is the product of the two− probabilities, represents the electrons that participate in the relaxation process.

2 2π 64π 2 2 4 ′ ′ W = γ γ m I n n I ′ m s S s s S ′ s (1.21) mn 9 e n | α| | α | | α| | α | ′ ′ α,α,s,s 2 2 ′ ′ ′ u (0) u ′ (0) f(k, s)[1 f(k ,s)]g(E ,A)g(E ′ ,A)δ(E E +E E ′ ′ )dE dAdE ′ dA | k | | k | − k k m− n ks− k s k k where the delta function ensures that E +E = E ′ ′ +E and E ′ = E +E E ′ +E E . ks m k s n k k s− s m− n Further, by introducing the density of states function ρ(Ek′ ) and the averaged wavefunction 2 ∞ 2 2 2 uk′ (0) allows us to write the integral simply as uk′ (0) ρ (E)f(E)[1 f(E)]dE | | 0 | | Ek − where f(E) is the Fermi function which denotes the probability of occupied states and [1 f(E)] is the probability of unoccupied states. The integrand, which is the product − f(E)[1 f(E)] is depicted in Fig. 1.6 where the only contributions come from the region near − ′ ′ the Fermi surface. The other matrix elements can be simpliﬁed: ′ s S s s S ′ s = s,s | α| | α | 1 1 s S S ′ s = TrS S ′ = δ ′ S(S + 1)(2S + 1) which for S = is equal to δ ′ /2. We s | α α | α α αα 3 2 αα can then write a more simpliﬁed form of the transition probability [2]:

13 64 W = π33γ2γ2 m I n n I m u (0) 2 ρ2(E)f(E)[1 f(E)]dE (1.22) mn 9 e n | α| | α| | k | Ek − α The integrand, in the limit T 0, can be written as a delta function f(E)[1 f(E)] = → − kTδ(E E ), then the above equation becomes [2]: − F

64 3 3 2 2 2 2 2 2 Wmn = π γ γ kT uk(0) m Iα n ρ (E) (1.23) 9 e n | | Ek | | α This is because not all of the electrons take part in relaxation process because some of them have no empty states to jump into; only the electrons near the tail of the distribution function or in the proximity of the Fermi energy, participate so that the integrand could be simpliﬁed to kT.ForN number of nuclei, the transition probability can assume the general form

Wmn = i,j aij α m Iiα n n Ijα m where for i = j the coeﬃcient is a00. Since the spin- | | | | 2 1 1 Pm,n Wmn(Em−En) lattice relaxation rate and the transition probability are related by = 2 , T1 2 Pn En then [2]:

2 1 1 m Iα n n Iα m (Em En) 1 m H, Iα n n H, Iα m = a m,n,α | | | | − = a m,n,α | | | | T 00 2 E2 00 2 E2 1 m m − m m (1.24) 2 1 1 Pα=x,y,z Tr[H,Iα] Now the above equation can be written as = a00 2 where H = γ H0Iz is T1 − 2 TrH − the Hamiltonian. We introduce the commutation relation [Ix,Iy]=iIz. The numerator can be simpliﬁed to Tr[H, I ]2 = γ22H2Tr[I2 + I2] and for the denominator TrH2 = α α − n 0 x y α P 2 22 2 2 2 2 2 α Tr[H,Iα] γn H0 TrIz . Using the property TrIx = TrIy = TrIz , the fraction 2 is equal to Pα TrH -2. Thus, remarkably, the spin-lattice relaxation rate is [2]:

1 64 33 2 2 2 2 2 = a00 = π γe γn uk(0) EF ρ (EF )kT (1.25) T1 9 | | K ΔH 8π u2 2 χs where the Knight shift is s = H = 3 k(0) EF e and for non-interacting spins 22 | | s γe χ0 = 2 ρ0(EF ). Using these expressions, the above equation reduces to the familiar form [2, 3]:

1 4πk γ2 = B n T (1.26) 2 2 T1 Ks γe This is the Korringa relation which is an expression of the linearity of spin-lattice relaxation rate with temperature, the slope of which varies with diﬀerent metals.

14 1.5 Hebel-Slichter Peak: Test of BCS Superconductivity

One of the most deﬁnitive tests of BCS superconductivity is the Hebel-Slichter peak [7, 8], named after C. P. Slichter and L. C. Hebel when they were investigating the nuclear relaxation rates in the superconducting state of aluminum. The spin-lattice relaxation rate in the superconducting state deviates from the Korringa behavior given by Eq. 1.26. By taking the ratio of the relaxation rate in the superconducting state RS =(1/T1)S to the relaxation rate in the normal state RN =(1/T1)N , Hebel and Slichter predicted and conﬁrmed that

RS there is an upturn or peak in in temperature just below Tc which is consistent with the RN BCS scenario. There are two causes of the Hebel-Slichter (HS) peak: ﬁrst, as the temperature is lowered below the transition temperature Tc a gap opens up and the states pile up at the edge of the gap, resulting in this form of density of states (DOS) [7]:

C E for E > Δ(T ) N = (E2−Δ2)1/2 (1.27) 0 for 0 Δ(T ) the normal state DOS is at the Fermi surface. The second source of the HS peak is the coherence factor which is related to the electron pairing mechanism in 1 2 the BCS model. Due to the coherence factor C+ = 2 [1 + Δ (T )/EiEf ], the eﬀective matrix element is then modiﬁed [7]:

i V f 2 C i V f 2 (1.28) | | e−n| | −→ +| | e−n| | The ratio RS can therefore be written as [7]: RN

R 2 ∞ Δ2 E E S = dE f(E )[1 f(E )](1 + )[ i ][ f ] (1.29) R k T i i − i E E (E2 Δ2)1/2 (E2 Δ2)1/2 N B Δ i f i − f − In addition, the spin-lattice relaxation rate below the HS peak follows the relation −∆ 1/T e kBT . This steep falloﬀ at low temperature is also known as the Yosida function 1 ∝ and Δ is the superconducting energy gap. This short discussion on the NMR in the metallic and superconducting states thus demonstrate the fair amount of information that can be extracted from the material using NMR.

15 Amplifier

Pulse Gate Pulse Programmer

Shifter RF Synthesizer

Figure 1.7: Schematic diagram of the transmitter section: gated oscillating electrical signal from the rf synthesizer is ampliﬁed and transmitted to the probe.

1.6 NMR Instrumentation

We brieﬂy discuss below how NMR instruments operate and some details about preparing for general NMR experiments. First, we discuss the transmitter and receiver sections of the NMR spectrometer.

1.6.1 Transmitter Section

The transmitter section (Fig. 1.7) is the part of the spectrometer that generates the radiofrequency needed for pulsed NMR. It consists of rf synthesizer, phase shifter and gating circuits, and rf amplifier. The rf synthesizer generates an oscillating electrical signal at a specific frequency which is the spectrometer reference frequency ωref . The output wave here is ssynth = cos(ωref t+φ(t)) where φ is the r.f. phase. This signal from the synthesizer is controlled by a pulse gate. The pulse gate is a fast switch which is opened at defined moments to allow the rf frequency reference wave to pass through. The gated rf pulse is then “scaled up” from a couple of watts to around 300 W output for transmission. The duplexer directs the strong rf pulse from the amplifier into cable leading to the probe, not into the sensitive signal detection circuitry. On the other hand, it diverts the weak signal

16 To Probe diode

50 Ω load

diode RF in To Pre-amp

Figure 1.8: The duplexer: two hybrid couplers direct RF into the probe (transmit mode) and divert the tiny NMR signal from the probe to the pre-ampliﬁer (receive mode).

coming from the probe to the receiver section (see Fig. 1.8). The probe is an important piece of apparatus because it contains the rf electronic circuit for “tuning” and “matching” of the resonant frequency, contains the coil and sample subjected in magnetic field, controls the temperature, and it may be equipped with goniometer for rotating the sample under magnetic field. Shown in Fig. 1.9 is the general probe circuit containing the tuning CT and matching CM capacitors. These are glass covered and have variable capacitance (ranging from 2 pF to 120 pF for Voltronics capacitors). The circuits in NMR probes are band pass filters where the properties of these filters are determined by the values of capacitance of the capacitors in the circuit and the inductance of the NMR coil. Turning the tuning capacitor will shift the band of this filter therefore allowing access on a certain frequency range. Changing the matching capacitance results to changing the efficiency of this band pass filter. These two capacitors must be properly adjusted to get the optimum resonance. The power transmitted to the coil depends on the quality factor Q which is a measure of the sharpness of the resonance. Q is the ratio Δf f0 where f is the frequency and Δf refers to a region around f where the resonance condition is satisfied.

17 CT coax (a)

L CM

CT coax (b)

L CM

Figure 1.9: Resonant circuits for NMR probe: (a) Series-tuned parallel-match. (b) Parallel- tuned series-match. CT ,CM , and L are tuning, matching capacitors, and inductor, respectively.

NMR signal from probe (a) (b) |s(t)| Re s(t) Duplexer Pre-amp Im s(t)

Receiver NMR Intensity (arb. units) RF synthesizer reference frequency 0 20 40 60 80 delay time (µs)

Figure 1.10: (a) Schematic diagram of the receiver section: the NMR signal from the probe and the reference frequency from the synthesizer are mixed in the quadrature receiver where real and imaginary NMR signals are generated. (b) Spin echo signal showing the real and imaginary components.

18 1.6.2 Receiver Section

The receiver section, illustrated in Fig. 1.10, is where the NMR signals are detected and processed. When the tiny NMR signal arrives at the duplexer, it is diverted to the signal preampliﬁer or commonly known as pre-amp, which amplify the tiny NMR signal to a convenient voltage level of about 30 dB gain. Strictly speaking, the NMR signal is not oscillating. The carrier, which is the oscillating part, is modulated by the NMR signal which is DC or near DC. The NMR signal is converted to digital form via specialized electronic circuits called ADCs (Analog-to-Digital Converters). However these NMR signals oscillate from a couple of MHz to several hundred MHz which are too fast for ADCs. A quadrature receiver is needed to down-convert the frequency that can be handled by the

ADCs. The quadrature receiver combines the NMR signal from the sample ω0 with the reference frequency ωref from the rf synthesizer, generating a new signal that oscillates at the relative Larmor frequency Ω = ω ω which is typically a MHz or less. This is similar 0 0 − ref to what happens in a radio receiver: the rf waves are down-converted to audible frequency range [1].

−t/T2 In the down-conversion, the free induction decay (FID) signal cos(ω0t)e is trans-

−t/T2 formed to cos(Ω0t)e . There is a problem here because this equation does not determine whether ω0 >ωref or ω0 <ωref . To solve this, the quadrature receiver generates two output signals: s (t) cos(Ω t)e−t/T2 and s sin(Ω t)e−t/T2 , or collectively s(t)=s (t)+is (t). A ≈ 0 B ≈ 0 A B These two signals may be interpreted as the real and imaginary components. The real and imaginary signals are then digitized via an ADC connected to each of the two outputs of the quadrature receiver [1].

1.6.3 NMR Hardware

Condensed matter NMR spectroscopy generally requires the use of homogeneous magnetic field, rf control (source, amplifier, synthesizer, and gating), cryostat, probe, duplexer, temperature controller, and a receiver section for data analysis (see Fig. 1.11). Depending on necessity and nature of experiments, accessories can include goniometer for sample rotation and pressure cells. In some or most occasions an NMR spectroscopist has to deal with very tiny single crystals where the spin count for getting an NMR signal is very low. The filling factor needs to be

19 (a) (b) rf synthesizer

liquid Helium level sensor goniometer

Magnet power supply

temperature controller coil with sample temperature sensor

NMR spectrometer

(c)

probe computer

oscilloscope

Magnet

Figure 1.11: NMR hardware: (a) NMR rack containing the spectrometer, PC, temperature controller, rf synthesizer, magnet power supply, and liquid Helium level sensor. (b) Photo of the goniometer and coil with sample mounted on a probe. (c) Top view of the magnet with probe.

maximized or near-perfect in order to get the most signal out of a single crystal, and as such coils of submillimeter size are needed. For instance, the organic conductor (Per)2Pt[mnt]2 single crystal has an average diameter of 35 μm and a coil of approximately 40 μm is needed. Below is a technique to make such a “microcoil” (refer to Fig. 1.12):

1. Mount a tungsten wire/rod (this is a good mandrel because of its stiﬀness and the diameter ranges from 4 mil – 10 mil and up) on a lathe chuck. Wind a No. 40 or 50 AWG copper wire around the tungsten rod by manually rotating the chuck under a

20 (1) (2)

(3) (4)

Figure 1.12: Making a microcoil.

microscope. Put a thin ﬁlm of vacuum grease around the tungsten rod before winding.

2. Secure the two ends of copper wire with a scotch tape on the bench once the desired number of windings and coil spacing is reached. Carefully mix a blob of ﬁve-minute epoxy and apply a thin ﬁlm of it to the coil. Avoid spilling the epoxy over the sides of the tungsten rod. You can use a thin copper wire in distributing the epoxy around the coil.

Before the epoxy cures completely, slide the coil back and forth; this is to make sure that the coil is not glued to the tungsten and that you can slide it out later. The thin ﬁlm of grease in step 1 was applied for this reason.

3. Put a small amount of mixed epoxy on a G-10 (copper or other material) platform and carefully attach the coil to it. Important: Do not pull out the tungsten mandrel yet

21 and be sure that the epoxy does not touch the tungsten rod. Wait for a few minutes until the epoxy cures completely.

4. Finally, you can pull the tungsten rod out of the coil. A caveat: make sure that the tungsten rod is cut clean at the edges – no extra sharp edge that will block the coil from sliding out. Then you can slowly slide in tiny single crystal samples in the coil. The coil shown in Fig. 1.12 is 40 μm in diameter.

Additional experimental details regarding the use of goniometer and portable cryogenic systems at high magnetic ﬁelds are discussed in Chapter 3.

22 CHAPTER 2

AN OVERVIEW OF LOW-DIMENSIONAL ORGANIC CONDUCTORS

Organic materials are made up of Carbon and usually combined with Hydrogen, Oxygen, Nitrogen, and a plethora of other elements. They were generally regarded as electrical insulators like plastics, nylon, and other polymers. However a couple of organic crystals were synthesized with improved conductivity like TCNQ (7,7,8,8-tetracyano-p-quinodimethane) in 1962 and TTF (tetrathiafulvalene) in 1970. It was not until 1973 when these two materials when reacted to form TTF-TCNQ which exhibits very high electrical conductivity. TTF- TCNQ, an organic charge transfer salt where TTF is the electron donor and TCNQ is the electron acceptor, is the ﬁrst true synthetic metal because it exhibits metallic-like electrical conductivity although it does not have metal atoms in its conducting network [9, 10, 11]. The search from organic superconductivity was sparked as early as 1964 when W. A. Little suggested the possibility of high temperature polymeric superconductor via phonon-mediated or excitonic mechanism. Little’s prediction materialized in 1979 when the group of Denis Jerome found that the organic system (TMTSF)2PF6 superconduct at 0.9 K under 12 kbar of hydrostatic pressure. 1 (TMTSF)2PF6 is a member of Bechgaard salts which has the general formula (TMTSF)2X, where X = ClO4, ReO4, AsF6, and other anions [9]. In 1981 (TMTSF)2ClO4 was found to exhibit superconductivity at Tc = 1.2 K at ambient pressure. Although most Bechgaard salts have low Tc at about 1 K, they exhibit another interesting low-dimensional phenomenon: spin density waves (SDW). This part of the dissertation is devoted to the discussion of the properties of the Bechgaard salts. We shall ﬁrst discuss the instabilities in low-dimensional

1These materials are often referred to as a laboratory of solid state physics because they are rich in magnetic and electronic properties.

23 Table 2.1: Broken symmetry ground states of metals: SS-singlet superconductivity, TS- triplet superconductivity, CDW-charge density wave, SDW-spin density wave (from Ref. [12]).

Ground State Pairing Spin Momentum Broken Symmetry SS electron-electron S=0 q=0 gauge TS electron-electron S=1 q=0 gauge CDW electron-hole S=0 q=2kF translational SDW electron-hole S=1 q=2kF translational

materials. 2.1 Low Dimensional Instabilities

Low dimensional metals are materials with highly anisotropic crystal and electronic structures. Here we discuss the effect on electronic, structural, and magnetic properties when electronic motion or conduction is confined to one dimension (1D). In the 1D electron gas 2 e q+2kF model, the wavevector q-dependent Lindhard response function χ(q)= ln − π vF | q−2kF | where vF is the Fermi velocity, diverges at 2kF [12]. This divergence in the response function is caused by a particular topology of the Fermi surface (FS) called perfect nesting. In quasi- one-dimensional (Q1D) system discussed in this dissertation, application of magnetic field makes the FS becomes increasingly more 1D. As a result, low-dimensional instabilities (see Table 2.1) such as charge density wave or Peierls distortion, spin-Peierls effect, and spin density wave ground states are formed.

2.1.1 Charge Density Waves

The charge density wave (CDW) ground state is a broken symmetry in low dimensional metals driven by electron-phonon, and in some cases Coulomb interactions. CDW is a periodic modulation of the electronic charge density, both periods are related to the Fermi wavevector kF . A strongly renormalized phonon spectrum known as Kohn anomaly is formed as a consequence of this electron-phonon interaction and the divergent electronic response at q =2kF . This renormalization is strongly temperature-dependent by virtue of χ(q, T) and within the mean ﬁeld theory framework, the renormalized frequency ωren at q =2kF

24 (a) Metal (b) CDW ε (k) ε (k)

0 k π/a π/a 0 π/a kF F kF kF π/a

ρ(r) ρ(r)

Figure 2.1: (a) The dispersion relation and the electron density in the metallic state (b) The modulation of the electron density in the CDW state causing an opening of gap Δ with π Fermi wave vector kF = 2a .

approaches zero at a phase transition where a periodic static lattice distortion and charge density variation are formed [12, 13]. There are two types of CDWs: a commensurate CDW occurs when the periodicity of the electronic charge modulation is a integral multiple of the lattice constant, otherwise it is an incommensurate CDW. As illustrated in Fig. 2.1, the lattice distortion opens up a single particle gap at the Fermi level, turning a metal into an insulator. This is generally known as the Peierls transition, after R. Peierls who predicted this phenomenon in low-dimensional metals in 1955. Next we brieﬂy discuss the mean-ﬁeld treatment of the CDW transition. The full details of the derivations can be found in Refs. [12, 13]. The 1D electron-phonon Hamiltonian2 for CDW can be expressed as:

+ 0 + + + H = ǫkckσckσ + ωq bq bq + g(k)ck+q,σck,σ(bq + b−q) (2.1) k,σ k,σ k,q,σ + + where ck (ck), bq (bq) are the electron and phonon creation (annihilation) operators with 2This also known as the Fr¨ohlich Hamiltonian.

25 0 momenta k, q, and spin σ, ǫk and ωq the electron and phonon dispersions, and g(k) the electron-phonon coupling constant [12]. The renormalized phonon frequency goes to zero close to the transition temperature MF TCDW where a “frozen-in” distortion occurs. This is apparent in the mean ﬁeld prediction:

MF T TCDW 1/2 ωren,2kF = ω2kF ( −MF ) (2.2) TCDW iφ + Introducing a complex order parameter Δe = g(2kF ) b2k + b , the displacement F −2kF of the ions is:

2Δ b + b+ e2ikF x + const. = cos(2k x + φ) (2.3) 2kF −2kF F g(2kF ) Invoking the above equation, the periodic, spatially-dependent electron density at T =0 in CDW can now be written as:

Δρ0 ρx = ρ0 + cos(2kF x + φ)=ρ0 + ρ1 cos(2kF x + φ) (2.4) vF kF λ where ρ0 is the unperturbed electronic density in the metallic state, Δ is the BCS-like CDW gap, λ is a dimensionless electron-phonon coupling constant, and vF is the Fermi velocity. This charge density is associated with a collective mode wherein an applied electric ﬁeld above a certain threshold ET can de-pin or slide the CDW condensate. Electronic potentials due to impurities, grain boundaries, etc. break the translational symmetry and lead to the pinning of the CDW condensate. Pinning gives rise to non-linear electrical conductivity and frequency-dependent transport properties in CDW [12].

2.1.2 Spin-Peierls State

The spin-Peierls system consists of a quasi-one dimensional spin-1/2 antiferromagnetic spin chain with spin-lattice coupling. The antiferromagnetic exchange coupling J, which is dependent on distance, is uniform between equidistant atoms. Due to the softening of the lattice along the chain direction, a lattice distortion called dimerization occurs at certain temperature TSP where electronic spins are paired in a periodic manner. Due to the change in the distance between the atoms, the dimerized electrons have a stronger exchange interaction J +δ while the antiferromagnetic coupling between the pairs is reduced to J δ (see Fig. 2.2). − A well-studied spin-Peierls system is the inorganic compound CuGeO3 in which spin-1/2

26 (a) J J J J J J J

(b)

J+δ J-δ J+δ J-δ J+δ J-δ J+δ

Figure 2.2: (a) An array of equidistant electrons with antiferromagnetic interaction J (Heisenberg spin chain). (b) Due to spin-lattice coupling, the electrons are dimerized and lattice distortion occurs at q =2kF (Spin-Peierls state) leading to stronger antiferromagnetic interaction J + δ among the dimerized electrons and weaker antiferromagnetic interaction J δ between the pairs. −

Cu2+ atoms form a linear chain that dimerize at 14 K [14]. In this dissertation, the focus will be on the organic conductor (Per)2Pt[mnt] which exhibits CDW-spin Peierls at 8 K at zero ﬁeld and suppressed to 0 K around 20 T. The details will be discussed in Chapter 4.

2.1.3 Spin Density Waves

The spin-density wave ground state is a periodic modulation of the electronic spin density π in low-dimensional metals with period λ0 = caused by electron-electron interactions. As kF illustrated in Fig. 2.3, the development of SDW results in opening of a gap at the Fermi level and a metal-to-insulator transition occurs if there is a complete removal of the Fermi surface. The interaction in SDW can be described in the Hubbard Hamiltonian [12]:

+ U + + H = ǫkak,σak,σ + ak,σak+q,σak′,−σak′−q,−σ (2.5) N ′ k,σ k,k,q where the ﬁrst term is the kinetic energy and the second is the electron-electron interaction + term, U is the on-site Coulomb interaction, N is the number of electrons per unit length, ak and ak are the creation and annihilation operators with momenta k, q and spin σ.

Mean-ﬁeld approximation, with the assumption that U<<ǫF similar to weak coupling in

27 ε (k) (a)

k k π/a F 0 kF π/a

(b) ρup(r)

ρdown(r)

λ0 = π/kF

Figure 2.3: (a) The dispersion relation in spin density wave state showing the opening of the gap Δ at the Fermi wave vector kF . (b) The modulation of the two spin subbands in SDW π with wavelength λ0 = . kF

superconductors, is employed to describe in detail the SDW ground state and its parameters. A complete review of SDW is found in Refs. [12, 15]. We begin with an equation for an applied external ﬁeld that varies along the chain direction given by:

iqx H(x)= Hqe (2.6) q ′ An additional term to the above ﬁeld is H = q MqH−q where Mq is the qth component of the magnetization. The expectation value of the magnetization is M = μ ( n q B q,↑− eff eff U(nq,↑−nq,↓) nq,↓ )=χ0(q)Hq where Hq = Hq + (the arrows denote the spin directions) 2μB and χ0(q) is the susceptibility that is dependent on wavevectors in the absence of Coulomb

28 interactions. Introducing Δn = n n , the self-consistent equation is μ Δn = q q,↑− q,↓ B q UΔnq χ0(q)(Hq + ). The average magnetization is therefore [12]: 2μB

χ0(q) Mq = Hq = χ (q)Hq (2.7) Uχ0(q) 1 2μ2 − B χ0(q) 2 where χ (q)= Uχ (q) which is an enhanced susceptibility. For q =0,χ0(0)=2μ n(ǫF ) 1− 0 B 2μ2 B 2μ2 n(ǫ ) and the static susceptibility is χ(0) = B F which is enhanced by the Stoner factor. 1−Un(ǫF )

At q =2kF , χ0(q) exhibits a peak which is strongly temperature dependent so that

Uχ0(2kF ,T ) 1.14ǫ0 2 = Un(ǫF )ln = 1. This allows us to write the mean-ﬁeld prediction: 2μB kBT

MF 1 kBTSDW =1.14ǫ0 exp( ) (2.8) −λe where λe = Un(ǫF ), the dimensionless electron-electron coupling constant. Another result MF of the SDW mean-ﬁeld treatment is the weak coupling BCS relation 2Δ = 3.52kBTSDW . Finally, due to a spatially-dependent magnetic moment μ(x) = μ cos(2k x + φ), the 0 F spatially-dependent spin modulation can be written as:

ΔS(x)=ΔS0 cos(2kF x + φ) (2.9) where ΔS0 is the unperturbed spatially dependent spin density modulation along the linear chain direction x. As illustrated in Fig. 2.3, SDW can be visualized as split charge density waves with spin- Δ Δ up ρ↑(x)=ρ0[1+ cos(2kF x+φ)] and spin-down ρ↓(x)=ρ0[1+ cos(2kF x+φ+π)] vF kF λe vF kF λe subbands. Just like CDW, SDW also manifests collective mode, pinning, and non-linear transport properties. Thus, a certain threshold electric ﬁeld ET can slide or de-pin the SDW condensate. The main diﬀerence is that SDW is a magnetic transition while CDW is a structural phase transition. 2.2 The Bechgaard Salts

First synthesized by Klaus Bechgaard in 1979, (TMTSF)2X are 2:1 charge transfer salts which are formed by transferring one electron from two TMTSF3 (tetramethyltetraselenafulvalene or (CH3)4C6Se4; see the crystal structure in Fig. 2.4) molecules to one monovalent anion

′ 3The IUPAC name of TMTSF is ∆2,2 -bi-4,5-dimethyl-1,3-diselenolyldine.

29 Table 2.2: The Bechgaard Family of Superconductors (from Ref. [9])

Bechgaard Salt Anion Symmetry Tc(K) Pressure(kbar) TSDW (K) (TMTSF)2PF6 Octahedral 0.9 12 12 (TMTSF)2AsF6 Octahedral 1.1 12 12 (TMTSF)2SbF6 Octahedral 0.4 11 17 (TMTSF)2TaF6 Octahedral 1.4 12 11 (TMTSF)2ReO4 Tetrahedral 1.3 9.5 180 (TMTSF)2FSO3 Tetrahedral-like 3 5 86 (TMTSF)2ClO4 Tetrahedral 1.4 ambient -

Figure 2.4: Crystal structure of the TMTSF molecule.

X. To date, Bechgaard salts have been synthesized from these monovalent anions: PF6,

AsF6,TaF6, NbF6, SbF6, ClO4, ReO4,BF4, BrO4,IO4,NO3, FSO3,CF3SO3, and TeF5 [10]. Among these, only the Bechgaard salts listed in Table 2.2 are superconducting. However, due to their low dimensionality, other interesting properties emerged.

2.2.1 Crystal Structure

The Bechgaard salt crystals have a triclinic structure with room temperature lattice parameters a =7.297 A,˚ b =7.711 A,˚ c =13.522 A,˚ α =83.390, β =86.270, and γ =71.010 [9, 10]. The brick-like TMTSF molecules are stacked in columns; these columns are formed in the a-axis which is the most conducting direction due to the overlap of the π orbitals of the Se atoms. Sandwiched between the columns are the anions which act as barriers

30 TMTSF ClO4

Figure 2.5: Crystal structure of (TMTSF)2ClO4 viewed along a-axis. The crystallographic axes are given by the arrows.

to the electron transfer between TMTSF columns; this forms the c-axis which is the least conducting direction. The b crystallographic axis is along the direction where the TMTSF molecules are in contact side by side; this has the intermediate electrical conductivity among the three crystallographic directions (see Fig. 2.5). The symmetry of the monovalent anions also contributes to the physical properties of the

Bechgaard salts. The Bechgaard salts with centrosymmetric anions like the octahedral PF6 and AsF are superconducting (T 1 K) only when pressure is applied. On the other hand, 6 c ≈ the Bechgaard salts with non-centrosymmetric anions like ClO4 and ReO4 exhibit anion ordering at 24 K and 180 K, respectively. This is because the tetrahedral anions can take two orientations with respect to the crystalline system. The anion ordering in (TMTSF)2ClO4 was conﬁrmed by diﬀuse x-ray scattering where it was revealed that a superlattice with wave number Q =(0, 1/2, 0) formed at TAO = 24 K. The case of (TMTSF)2ClO4 is interesting because to achieve anion ordering (see Fig. 2.6), slow-cooling at around 10 mK/min is needed; superconductivity at 1.2 K ensues in this “relaxed state”. However, when (TMTSF)2ClO4

31 10 Anion Ordering (AO) TAO=24 K 1 Ω) ( zz

R First Cooldown 0.1 Second Cooldown

0.01 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 10 100 T (K)

Figure 2.6: Zero-ﬁeld cooldown of (TMTSF)2ClO4 showing a kink in resistance at 24 K which is due to anion ordering.

is cooled rapidly or quenched from a high temperature to around 15 K, the anions become frozen at random orientations. SDW appears at around 6.5 K and superconductivity is absent.

2.2.2 Electronic Properties

The Bechgaard salts are quasi-one-dimensional molecular conductors. They are quasi-one- dimensional because electrical conduction is directed mainly on one direction. This is due to the fact that the transfer energies ti of the delocalized π electrons are anisotropic: ta >> tb >> tc. From the tight-binding model, the energy dispersion is given by [9, 11]:

ǫ(k)=2tacos(amk)+2tbcos(bmk)+2tccos(cmk) (2.10) where am, bm, and cm are the intermolecular distances in the respective crystallographic directions a, b, and c, k is the electronic wave vector, and ti is the electron transfer energy along the i direction [11]. The transfer energies for Bechgaard salts were determined from optical studies: ta =0.28 eV, tb =0.022 eV, and tc =0.001 eV. The a direction, which is the stacking direction, is the most conducting axis because the π electrons from the four

32 10

4 6 0.347 K 5 4 3

3 T (K) 0.478 K 2 2 1.692 K 0.978 K 1 1.981 K 1 1.372 K Ω)

( 6 4 6 8 10 12

zz 5 Field (T) 1.591 K

R 4 3 2.910 K 2 3.930 K 5.940 K 0.1

6 5 4 3 4 5 6 7 8 9 10 B (T)

Figure 2.7: Low-ﬁeld c*-axis magnetoresistance (MR) of (TMTSF)2ClO4 at diﬀerent temperatures. The kinks in MR correspond to the FISDW cascade phases. Inset: T-B phase diagram showing the cascade of FISDW phases.

Selenium atoms are overlapping allowing large electron transfers along the stacks. The b direction is the next most conducting axis with two Selenium atoms in side by side contact while the c axis is the least conducting axis due to the anions which act as barrier to the electron transfers between the TMTSF molecules. 2.3 Transport Properties

2.3.1 Rapid Oscillations

The rapid oscillations are seen in magnetoresistance and magnetization measurements in several Bechgaard salts which are periodic in 1/B with a frequency of a few hundred inverse Teslas (see Fig. 2.8). They are called “rapid” because these oscillations have higher frequency

33 100 B//c*, (TMTSF)2ClO4

1.57 K B 80 B* re

2 K

40 Resistance (Ohms)

20 4.5 K B1

Bth 6 K 7.5 K 0 0 5 10 15 20 25 30 Field (T)

Figure 2.8: High ﬁeld c*-axis magnetoresistance (MR) of (TMTSF)2ClO4 at diﬀerent ∗ temperatures showing kinks in MR that correspond to the FISDW phases Bth,B1,B, and Bre. Above 15 T, periodic modulations in MR called “rapid oscillations” occur.

than the structures coming from the FISDW transitions seen in Figs. 2.7 and 2.8. They are generally temperature-independent and occur both in the metallic and FISDW regions in the 4 case of (TMTSF)2ClO4. This is distinct from de Haas-van Alphen (dHvA) or Shubnikov- de Haas (SdH)5 oscillations because they are not aﬀected by FISDW transitions and Hall eﬀect measurements do not give any evidence of a closed Fermi surface in the a b plane − [11]. Instead, the rapid oscillations in the metallic phase is attributed to the Stark quantum

4dHvA is the oscillation of the magnetic moment of a metal as a function of applied external magnetic field. The periodicity in dHvA effect measures the extremal cross sectional area in k space of the Fermi surface. 5SdH is the oscillation in the resistivity of a material at low temperatures and high magnetic fields. It is similar to dHvA where the frequency of the magnetoresistance oscillations indicate the extremal section of the Fermi surface.

34 interference eﬀect but the mechanism of the rapid oscillation in the spin density wave state is still an open question [16].

2.3.2 Lebed Magic Angles

Dips or weak structures in the angular-dependent magnetoresistance appear at some angle θ if the field is rotated in the b c plane (perpendicular to the a-axis) in the case of − (TMTSF)2ClO4 [17] or (TMTSF)2PF6 [18]. They are called the Lebed “magic angles” (MA), after A. G. Lebed who first predicted such features in the angular magnetoresistance of quasi-one-dimensional organic conductors due to weak electron tunnelling between the layers [19]. In general, the condition for the appearance of MA effects is given by:

p b sin γ tan θ = cot α (2.11) q c sin β sin α − where p and q are integers and b, c, α, β, and γ are the lattice parameters of the crystal. A recent proposal, which is a field dependent renormalization of the coherent c-axis hopping tc to zero, attempt to explain the magic angle effects. Single particle hopping between spin- charge separated Luttinger liquids is incoherent for weak interliquid hopping. Away from the magic angle directions and at strong magnetic fields, the Q1D material becomes 2D because tc is vanishing. At the magic angle directions, higher order hops are generated because of the renormalization of tb and tc which gives rise to the dips in the magnetoresistance [11]. 2.4 Magnetic Properties

Experimentally SDW in the Bechgaard salts is seen as a rise in the resistivity as the temperature is lowered through the transition which is well described by the Arrhenius activation relation ρ = ρ0 exp(Δ/kBT ). The amplitude of the SDW modulation is given by μ = 4|Δ| where U is the on-site Coulomb interaction and Δ is the single particle μB U gap. The magnetic behavior of the SDW state is close to an antiferromagnet with the localized moments separated at a distance λ0/2, reduced magnetic moment μ, and an eﬀective exchange constant Jeff . This leads to the Hamiltonian [9]:

μ μ μ H =( )2J S S ( )2D SxSx +( )2E (SzSz SySy ) gμH S μ eff i · i+1 − μ i i+1 μ i i+1 − i i+1 − i B i B i B i (2.12)

35 k (a) b (b) π/b Bz X a

kc QN b −π/a π/a ka λ

A −π/b -kF kF

Figure 2.9: (a) Electronic motion in momentum space: a pair of warped FS sheets within the ﬁrst Brillouin zone in a Q1D system. (b) Electronic motion in real space with amplitude A = 4tb and wavelength λ = h . evF Bz ebBz

where Jeff is the interaction along the chains, and D and E represent the hard and intermediate anisotropy energies respectively.

At zero temperature, this Hamiltonian leads to anisotropic susceptibilities: χ 0 and 2 2 2 2 → Neg μ0 Neg μB χ⊥ = μ 2 = . The parallel component refers to the easy axis direction where 2( ) Jeff 2Jeff μ0 the electronic spins align even without external magnetic field [12, 15]. The anisotropic susceptibilities were measured in (TMTSF)2AsF6 which is representative of the other Bechgaard salts [20]. The effective interaction energy is found to be 740 K. The Hamiltonian ∗ 1/2 π(2E Jeff ) also gives us the spin flop field Hsf = γ where γ is the gyromagnetic ratio.

Experimentally the spin ﬂop ﬁeld is around Hsf =0.6 T and the anisotropy energy is E =3 10−5 K[12, 20]. × 2.4.1 Fermi surfaces

The Fermi surface (FS) of a strictly 1D conductor is composed of a pair of planar sheets, one located at +k and the other at k . In the case of a Q1D conductor, where there is a F − F considerable transverse electron transfer energy, these sheets are warped [9]. For low warping,

36 one FS sheet can be nested with the other sheet given by the wave vector Q =( 2k , π/b, 0). ± F Nesting is necessary to induce broken symmetries like CDW and SDW. If the warping is increased, the nesting condition is deteriorated and the low dimensional instabilities are suppressed. Thus, as illustrated in Fig. 2.9, Q1D organic conductors have weakly warped FS sheets. Another characteristic of the FS of Q1D organic conductors is its openness. This is in contrast to most metals where the FS sheets are mostly closed within the Brillouin zone. The FS of a Q1D conductor exists only in one direction, therefore the electrons are conﬁned to move in a certain direction in the momentum space. When magnetic ﬁeld H is applied perpendicular to the a-b plane, the dynamics of the electron is given by dka = eH v and dkb = eH v = eH 2ata sin(ak ) where we used the dt − c b dt c a c a 1 ∂ǫ(k) equation for velocity v = ∂k . The frequency of the electron traversing the Brillouin zones in the b-direction is ω = eH k b. We note that in the case of closed FS in metals, the c mec F eH electrons move in a circular motion with cyclotron frequency ω = mc .

With open FS the electrons are strongly oscillating in the kb direction. In real space, the electrons make a sinusoidal motion with the amplitude 4tbb . ωc It is important to note that increasing the magnetic field narrows the width of the electron motion both in the momentum space and the real space. That is, when ωc >> 4tb, the electrons are confined to the 1D column. The one-dimensionalization of the FS by the applied magnetic field is the basis of the peculiar behavior of the electrons in the quantum limit.

2.5 Phase Diagram of (TMTSF)2X

At first, the oscillation of the magnetoresistance of (TMTSF)2PF6 under pressure was 2πe thought to be Shubnikov-de Haas (SdH) type with periodicity Δ(1/H)= cA where A is the area of the extremum in momentum space [9]. This suggests the presence of cylindrical pockets parallel to the c-axis and presumably there are compensated electron and hole pockets. However this idea does not fit in the band model with strongly anisotropic transfer integrals and the fact that the oscillation only appears above a certain threshold field. Successive experiments such as specific heat, magnetization, and NMR later proved that the oscillation is not SdH but is magnetic in nature. The broadened double-horned spectra from NMR confirmed that these magnetic transitions are spin density waves. Similar behavior

37 7 c* B 6 a b' 5 FISDW 4 Metallic Phase

2 Re-entrant FISDW 1 SC Cascade 0 0 5 10 15 20 25 30

Figure 2.10: T-B phase diagram of (TMTSF)2ClO4 showing the metallic region, superconducting (SC) state, the cascade of FISDW phases, and the re-entrant FISDW region.

was found in (TMTSF)2ClO4.

It is interesting to note the peculiar similarities of these two Bechgaard salts. (TMTSF)2PF6 has a single pair of warped Fermi surfaces and exhibits FISDW only under pressure of

12 kbar. On the other hand, (TMTSF)2ClO4 has two pairs of warped Fermi surfaces due to ordering of the tetrahedral ClO4 anions at 24 K and exhibits FISDW at ambient pressure. Both materials exhibits superconductivity at around 1 K under these conditions. The generalized phase diagram of these two materials are given in Figs. 2.10 and 2.11. Gorkov and Lebed were the first to provide explanation to the occurrence of FISDW transitions in the Bechgaard salts. They showed that the spin susceptibility χ(Q) diverge in response to spatially periodic magnetic field with wave vector Q, giving rise to the step-like kinks in magnetoresistance which correspond to the series of FISDW transitions. Starting from the simple band model ε = v ( k k )+ε (k ) where the second term is k F | x|− F ⊥ ⊥ ε⊥(k⊥)=2tb cos(bky)+2tc cos(ckz)+2tb cos(2bky)+2tc cos(2ckz), they obtained the eigenvalue and eigenfunction from the Schrödinger equation with Landau gauge A =(0,Hx,0). The

38 7 c* B P=12 kbar 6 a b' 5

4 Metallic Phase

3 FISDW 2

1 SC Cascade 0 0 5 10 15 20 25 30

Figure 2.11: T-B phase diagram of (TMTSF)2PF6 at 12 kbar hydrostatic pressure: this is similar to (TMTSF)2ClO4 except for the absence of the re-entrant FISDW region at high magnetic ﬁelds.

spin susceptibility χ(Q), after employing Green’s functions, can be written as [9]:

+∞ 4t 4t x dx χ(Q)=N(0) J [( b ) sin(κx)]J ( c ) (2.13) 0 κv 0 v x sinh(x/x ) d F F T T Nas vF eHb where J0 is the Bessel function, N(0) = is the state density, xT = , κ = , 2πkBT 2πkBT c χ0(Q) and k is the Boltzmann constant. From the divergence in χQ = , the cutoﬀ of the B 1−Uχ0(Q) lower limit (d) of the integral is obtained from N(0)U ln vF = 1. Using the relation · πkBT0d 1 π 2 1 π 0 J0(z sin φ)dφ = J0 ( 2 z), one can reduce the above equation to [9]:

2 2tb vF χ0(Q0) N(0)J0 ( )ln( ) (2.14) ≃ κvF πkBTd π π where Q0 =(2kF , b , c ) is the optimal nesting wavevector. Chaikin proposed that the transition temperature T = T exp( HA ) where T and H are constants. SDW 0 − H 0 A Later reﬁnements in the theory of FISDW have been made and is now called the standard theory for FISDW [9]. This theory works well for (TMTSF)2PF6, giving the essential features

39 of the phase diagram. However in the case of (TMTSF)2ClO4, some deviations from the calculations have been found especially the high ﬁeld re-entrant FISDW phases. This may 1 be attributed to the presence of a superlattice with wave vector Q =(0, 2 , 0) due to the anion ordering of ClO4 anions.

In the case of relaxed (TMTSF)2ClO4 and pressurized (TMTSF)2PF6, a superconducting pocket in the phase diagram is found with T 1 K and H 0.5 T. At low temperature, c ≈ c2 ≈ say 500 mK, the so-called “cascade” of FISDW transitions appear at 4 T to 9 T (see Fig. 2.7). These are series of first-order FISDW transitions, seen as kinks in magnetoresistance and plateaus in Hall effect measurements, that correspond to the quantization of the nesting vector Q =(2k N2π/λ,π/b) where λ = h . F ± ebB The main difference between the two Bechgaard salts is that additional FISDW features are present in (TMTSF)2ClO4: one in 15 T and the so-called re-entrant FISDW at around 26 T. These features were not predicted by the standard model. One of the main points of this dissertation is the investigation of the nature of these high field FISDW transitions via NMR which is discussed in the next chapter.

40 CHAPTER 3

SIMULTANEOUS 77Se NMR AND TRANSPORT INVESTIGATION OF THE SPIN DENSITY WAVE SYSTEMS (TMTSF)2X, X=ClO4,PF6

The effects of high magnetic fields on the quasi-one- and two-dimensional electronic structure of organic conductors is a rich area of investigation [9]. In Bechgaard and related salts [21, 22] which remain metallic at low temperatures, a magnetic field applied parallel to the least conducting direction (perpendicular to the conducting chains) produces a field- induced spin density wave (FISDW) ground state [23]. A simple description of this effect is that the magnetic field decreases the amplitude of the lateral motion of the carriers as they move along the conducting chains, thereby making the electronic structure increasingly more one-dimensional. Hence eventually 1D instabilities become favorable. In reference to (TMTSF)2ClO4 (at ambient pressure) and (TMTSF)2ClO4 (under pressure), a nested quasi-1D Fermi surface is induced at a second-order phase boundary where a FISDW gap opens above a threshold field Bth [24]. Due to quantization of the nesting vector Q=(2k N2π/λ,π/b) (where λ=h/ebB), increasing magnetic field produces a first-order F ± “cascade” of FISDW subphases. In the quantum limit, the optimum nesting vector (where

N=0) yields the ﬁnal FISDW state in the case of (TMTSF)2PF6 which has a single quasi- one-dimensional Fermi surface (Q1D FS) [25].

However, (TMTSF)2ClO4 experiments show additional phase boundaries in the quantum limit [26, 27, 28, 29, 30]. Since the ordering of the tetrahedral ClO4 anions below 24 K doubles the unit cell along the inter-chain direction b′, zone folding produces two Q1D FS sheets, leading to complex high ﬁeld behavior [29, 30]. Recently a model [31] has been proposed where both FS sheets are gapped at the Fermi level EF in the FISDW region, but above the

“re-entrant” phase boundary Bre [27], only one of the sheets is gapped at EF . This leads to

41 7

4 B 3 th

Temperature (K) 2

1 B 1 B* Bre 0 0 5 10 15 20 25 30 Magnetic Field (T)

Figure 3.1: The (TMTSF)2ClO4 FISDW phase diagram where the dots represent the FISDW transport features seen in this work. The arrows represent the regions in the phase diagram where NMR was measured.

an explanation for the oscillatory sign reversal of the Hall eﬀect [32] above Bre. In this chapter, the experimental results of simultaneous 77Se nuclear magnetic resonance and electrical transport measurements at diﬀerent regions of the FISDW phase diagram are presented (see Fig. 3.1) and discussed in light of the present nesting and Fermi surface models for these materials. 3.1 Experimental Details

The idea behind the simultaneous NMR and electrical transport measurements is to avoid ambiguity in the locations of the FISDW transitions seen by both measurements. NMR, being a microscopic magnetic probe, will provide additional insights into the nature of these transitions. Due to its location (see Figs. 2.4 and 2.5), Se is an appropriate nucleus to do NMR in order to study the electron physics in the metallic and SDW states. 77Se is a spin-1/2 nucleus with a gyromagnetic ratio γn =8.13 MHz/T, and 7.5 % natural abundance.

42 NMR Coil transport leads

Sample

Figure 3.2: Simultaneous NMR and electrical transport setup in a goniometer.

3.1.1 Sample and Probe Preparation

The (TMTSF)2ClO4 single crystals were grown by electrochemical methods the details of which are described elsewhere [10]. Single crystals of (TMTSF)2ClO4 with typical dimensions 5 0.6 0.5mm3 were inserted into miniature coils made from # 40 AWG copper wire with × × filling factors of order 70 to 90 % by volume. Four 12 μm Au wires were attached to the sample ends with carbon paste or DuPont silver paste for concurrent ac four-terminal electrical transport measurements. The coil is glued to a G-10 platform using five-minute epoxy. The G-10 sample/coil holder is custom-built to a miniature gear-type goniometer which can be attached to the NMR probe (see Fig. 3.2). The NMR probe has extra twisted pairs of copper wires connected to the 19-pin connector of the probe head for electrical transport measurement. Different NMR coils are used in the low and high magnetic field measurements because of the limited tuning range for a series-tuned parallel-matched NMR circuit. The capacitor space of the NMR probe is pumped to prevent or minimize arcing. Silicon bathroom sealant are used to cover the soldered joint between the high-power rf feedtrough and the coil. The NMR probe is cooled down slowly to prevent thermal shock that causes cracks in the glass-covered Voltronics capacitors.

43 Figure 3.3: Portable He-4 and He-3 cryogenic systems.

3.1.2 Cryogenic System and Magnets

Home-built doubled-walled cryostats with needle valve for liquid Helium flow control are used. The cryostat has superinsulation in between its walls and is evacuated which allows us to measure at high temperatures down to 4.2 K with ease. To access the temperature range 4.2 K – 1.5 K, the sample space of the probe is pumped. For measurements below 1.5 K, a He-3 cryogenic system is used. Low field measurements are done using superconducting magnets (SCM) with fields up to 17.5 T1. Here SCMs are the reservoir of the liquid Helium for cooling down the sample. For high field measurements up to 30 T in the resistive magnets (RM), portable Janis dewar with tail size compatible with the RM bore size is used. Portable He-3 system is also available for measurements down to 300 mK (Fig. 3.3).

3.1.3 Cooldown

It should be noted that slowcooling the (TMTSF)2ClO4 sample at a rate 10 mK/min from 30 K to 18 K is necessary to achieve a well-relaxed state, a prerequisite to obtaining cleaner features of FISDW transitions. The zero-ﬁeld cooldown curve of the resistivity is monitored.

Anion ordering at TAO = 24 K characterized by a small kink in resistance at that temperature

1The deﬁnition of low ﬁeld here is, of course, relative.

44 Figure 3.4: Simultaneous NMR and electrical transport setup at high magnetic ﬁelds: (a) Electrical transport and NMR racks. (b) Back view showing the gas pressure system and the 30 T resistive magnet at NHMFL Cell 7.

45 should be observed. In a well-relaxed state, the sample should be metallic down to 1.2 K below which it is superconducting.

On the other hand, if the sample is cooled quickly down to temperatures below TAO, the

ClO4 anions will be in a frozen state with random orientations. This leads to the appearance of SDW transition at around 6.5 K and remains in this state down to the lowest temperature. Thus, in this “quenched” state the SDW is not ﬁeld-induced.

3.1.4 Sample Rotation

All angular-dependent NMR-transport results mentioned in this chapter were done along the a-axis of the crystal, which is the longer side of the needle-like sample and its most conducting axis. The goniometer shown in Fig. 3.2 is controlled manually at the probe head where one complete turn corresponds to 20-degree rotation and each small line in the knob is 0.2 degree. Typically rotations are done in one direction, either clockwise or counterclockwise, to avoid error in angles because of backlash between the gears in the goniometer. The extent of rotation is limited by the length of the transport and NMR leads.

3.2 Temperature Dependence of NMR Spectra

The broadening of the 77Se NMR spectra at around 3.7 K shown in Fig. 3.5 is an indication of inhomogeneous local magnetic field coming from the SDW (field-induced) state. The double-peak feature shows that it is an incommensurate SDW as predicted by the spatial variation of the internal magnetic field seen by 77Se nuclei given by [12]:

μ δH(r)= a0 H0 cos(2Q r + φ) (3.1) μB · μ where H0 is the applied magnetic field, a0 is the hyperfine coupling constant, and μB is the ratio of the effective magnetic moment in the SDW and the Bohr magneton. At high temperatures, multiple peaks are observed in the spectra indicating the four chemically 77 inequivalent sites of Se nuclei. Near the anion ordering temperature TAO = 24 K, the multiple peaks merged into one indicating that the movement of the ClO4 anions slightly smeared out the local field detected by the chemically inequivalent 77Se nuclei. The NMR spectra remain single-peak until 3.7 K where (FI)SDW transition occurs.

46 77 Figure 3.5: Temperature dependence of Se NMR spectra in a relaxed (TMTSF)2ClO4 with B c∗. Part (a) shows frequency-swept spectra in the metallic state at constant magnetic field B =12.12 T while (b) shows field-swept spectra at constant NMR frequency 98.2 MHz. The broad, double-horned spectra indicate inhomogeneous local magnetic field in the FISDW state.

3.3 The RF Enhancement Factor η

The spin-echo pulse sequence used for typical NMR measurements in the metallic state is about 2 μs-4μs with about 20 dB rf power attenuation. However, one has to reduce the rf power and/or shorten the pulse widths when measuring in the SDW or FISDW state otherwise there will be no NMR signal. This is ﬁrst observed by Takigawa and Saito [33]in

(TMTSF)2ClO4 in which they attribute this effect to the rf enhancement commonly observed in ferromagnets. In ferromagnets, the rf magnetic field H1 causes the small oscillation of the electronic moment or domain wall motion that produces a far greater oscillating magnetic field at the nuclear sites [33, 34]. Normally in the metallic state the tipping angle of the

47 π magnetization is given by θtip = 2 = γnH1τm where γn is the nuclear gyromagnetic ratio,

H1 is the rf magnetic ﬁeld, and τM is the pulse width in the metallic state. However in the

SDW or FISDW state θtip = ηγnH1τsdw where η is the rf enhancement factor. Assuming the rf power level used is the same in the metallic and SDW/FISDW states, thereby H1 is the same in both phases, then we deﬁne the enhancement factor to be:

τ η = m (3.2) τsdw This is only an approximation since η is the ratio of the integrated pulse-power of the metallic state and the ordered state, an important consideration especially if the pulse shapes are not rectangular.

Near the SDW transition TSDW , the optimum pulse width can be as short as 50 ns or 100 ns giving an enhancement factor of about 40. This is small compared to ferromagnets where domain wall motion gives an η that ranges from 102 to 103 [34].

In the case of (TMTSF)2ClO4 and (TMTSF)2PF6, the rf enhancement is attributed to oscillatory motion of the SDW condensate driven by an electric field induced in the coil. The rf magnetic field H1 causes small oscillations of the SDW phases, consequently oscillating the spin polarization at the nuclear sites giving an enhanced rf magnetic field perpendicular to the applied magnetic field [33]. The electric fields associated with de-pinning are typically of order 5 mV/cm or less [35, 36], and in the work, the estimated ac electric field in the NMR coil is of similar magnitude ( 4 mV/cm in the b′-c∗ plane) (see also Ref. [33]). ≈ 3.4 RF Power dependence of NMR lineshapes in the FISDW State

The NMR lineshapes of incommensurate systems are predicted as doubled-horned spectra with the assumption that the modulation wave is static and the condensate is pinned to the lattice or impurity. However, incommensurate systems (IC) such as IC-CDW in NbSe3 can be depinned by applying an electric ﬁeld greater than a threshold value E0. Thus the NMR lineshapes will dramatically change due to the sliding of the modulation wave.

Here we consider sliding of incommensurate SDW in (TMTSF)2ClO4 by electric ﬁeld induced by current in the NMR coil with increasing the rf power. We apply the model developed by Kogoj et al.[37] of NMR lineshape calculation by assuming uniform siding of

48 Figure 3.6: (a) Field-swept 77Se NMR spectra at diﬀerent rf power level attenuation in the FISDW state of (TMTSF)2ClO4 at 1.8 K and constant NMR frequency 98.6 MHz with B c∗. The pulse sequence used is 450 ns - 900 ns. (b) Power-swept spin-echo intensity taken at ﬁelds indicated by dashed arrows (LP-left peak, CP-central peak, RP-right peak) in (a). The red arrow at the bottom indicates the direction of increasing rf power.

the modulation wave u(r, t)=A0 cos Φ(x, t) where Φ(x, t) is the phase of the displacement.

In uniform motion, Φ(x, t)=φ(x, t)+φ0 where φ(x, t)=φ(x+vt). The atomic displacement can now be written as:

u(x, t)=A0 cos[φ(x + vt)+φ0] (3.3)

In the in-plane limit, Φ(x, t)=k1(x + vt)+φ0 = k1x +Ωt + φ0 where k1 is the diﬀerence between the incommensurate and commensurate critical wavevector and Ω describes the harmonic motion of a given atom due the sliding motion. The NMR frequency can be

49 written as ω(x, t)=ωL + a1u(x, t)=ωL + ω1 cos(k1x +Ωt + φ0) where ω1 = a1A. The NMR lineshape I(ω) is the Fourier transform of the autocorrelation function:

1 +∞ I(Δω)= G(τ)eiΔωτ dτ (3.4) 2π −∞ where the autocorrelation function G(τ)= exp( i t+τ ω cos[(k x+Ωt′)+φ)]dt′ is periodic. − t 1 1 The variable w = Ω = vk1 is introduced as a measurement of the velocity of the sliding ω1 ω1 condensate. The autocorrelation function can be modiﬁed as

1 2 G(τ)= exp[ i sin(πτ) cos(πτ + φ +2πu)]du (3.5) 0 − w w ′ w w where τ = ω1τ 2π , y = ω1t 2π , and u = ω1t 2π are dimensionless numbers. The above 2 equation can be simpliﬁed to G(τ)=J0( w sin(πτ)) where J0 is the zeroth order cylindrical Bessel function. The NMR spectrum can be calculated as the Fourier transformation of this correlation function depending on the sliding velocity w:

In the limit of small sliding velocities (w 0), the NMR lineshape is given by: • →

1 +∞ 1 2 2π −∞ J0(z) exp(iωz)dz = π(1−ω2)1/2 for ω 1 Iw→0(ω)= ≤ (3.6) 0 for ω2 > 1 which gives us the double-horned NMR lineshape as expected for a static or nearly static IC system.

In the limit of large sliding velocities (w>>1), the NMR lineshape is: •

I(ω)=δ(ω) (3.7)

which is expected for a motionally-averaged spectrum: a single narrow line. This was 1 observed by Clark et al. in the H NMR spectra of sliding SDW in (TMTSF)2PF6

caused by an applied electric ﬁeld above the threshold E0.

In the intermediate limit (w>1), the other higher order terms of the Bessel function • series expansion have to be included resulting in a three-peak spectrum:

1 1 I(ω)=δ(ω)[1 ]+ [δ(ω w)+δ(ω + w)] (3.8) − 2w2 4w2 − 50 where the central peak is at ω = ω and the side peaks are at ω = ω Ω=ω vk . L L ± L ± 1 This three-peak spectrum prediction is observed experimentally in this work as depicted in Fig. 3.6 (bold black NMR spectrum). The intensity of the side peaks decreases as the sliding velocity increases w = vk1 . ω1 →∞ The periodic correlation function can also be expressed in terms of Fourier series:

2 +∞ 1 G(τ)=J ( sin(πτ)) = J 2( ) cos(2πnτ) (3.9) 0 w n w n=−∞ and the lineshape can now be written as I(ω)= +∞ J 2( 1 )δ(ω nw). This equation n=−∞ n w − gives sharp lines appearing at ω = ω nΩ because of the periodic nature of the correlation L ± function. Assuming the individual lines have Gaussian form g(ω ω′, σ), we modify the − NMR lineshape equation as F (ω)= +∞ J 2( 1 ) +∞ δ(ω′ nw)g(ω ω′, σ)dω′ where n=−∞ n w −∞ − − σ = σ . Finally, the continuous NMR lineshape equation is: ω1

1 +∞ 1 (ω nw)2 F (ω)= J 2( ) exp( − ) (3.10) √ n w − 2σ2 2πσ n=−∞ A much more detailed calculation of NMR spectra of sliding IC systems involving (i) phase soliton limit where the phase φ is not a linear function of x and (ii) when there is a distribution of sliding velocities is given in Ref. [37] which may explain the additional asymmetric peaks in FISDW spectra displayed in Fig. 3.6.

77 3.5 Field Dependence of 1/T1,Rzz, and Spectra at Constant Temperature

The scientific motivation of performing simultaneous field-dependent NMR and electrical 77 transport is to determine where exactly does 1/T1 exhibits a maximum and the development of order parameter in NMR linewidths as we measure across several FISDW transitions seen by electrical transport. The main results of this experiment are given in Fig. 3.7 where the field was swept from 0 to 15 T with B c∗ at a constant temperature 2 K. The first kink in the magnetoresistance (MR) at 6.3 T corresponds to the Bth FISDW transition and the second MR feature occurs at 7.6 T which is the B1 FISDW phase boundary. 77 Note that the peak in 1/T1 occurs in B1 FISDW phase boundary. Above B1 =7.6 T, two 77 sets of relaxation values are obtained: the enhanced 1/T1 is due to the optimized NMR

51 0 10 0.6 9 8 2 7 0.5 6 0.4 5 0.3 4

0.2 3

0.1

2 B 1 0.0 1 4 6 8 10 12 14 Bth

-1 10 B//c*, 2 K 9 8 7 6 B1 5 0 0 5 Bth 10 15

77 Figure 3.7: Field dependence of 1/T1 and c-axis resistance Rzz (red curve) in ∗ (TMTSF)2ClO4 measured simultaneously at 2 K with the ﬁeld applied parallel to the c -axis. The peak in 1/T1 occurs at B1 FISDW transition. Solid circles denotes relaxation rate in the metallic state, the solid triangles indicates the enhanced relaxation in the FISDW region, and the open circles denote a coexistence region or depinned state of FISDW. Inset: the corresponding ﬁeld dependence of full-width half maximum (FWHM) of NMR spectra.

pulses due to the rf enhancement eﬀect while the slower relaxation rates, accompanied by slightly narrower NMR linewidths, were obtained using metallic pulses. The NMR signal obtained by the metallic pulses, however, diminishes at higher ﬁelds around 10 T. This suggests that (i) there is a coexistence region of metallic and FISDW states even above

B1 (ii) the metallic pulses cause a sliding or back-and-forth sloshing motion of FISDW condensate and (iii) the RF power using metallic pulses is heating up the sample. The third scenario is ruled out because the same results were obtained when the recycle time after the pulses was increased from 200 ms to about 5 s. The second scenario which suggests moving FISDW condensate is more favored.

52 It is important to note that the full-width half maximum (FWHM) of the NMR spectra has a linear increase from B1 phase boundary to Bth, reﬂecting the increase in gap opening.

There is no rf enhancement in the ﬁeld range Bth B1.

77 3.6 Temperature Dependence of 1/T1 at Low Fields

77 The spin-lattice relaxation rate 1/T1 directly probes the electronic correlations of the title material by measuring the magnetic fluctuations detected by the 77Se nuclei above and below the FISDW phase boundaries. In this section we discuss the temperature dependence 77 of 1/T1 illustrated in Fig. 3.8, cutting through the low-field (B<15 T) FISDW phase 77 boundaries . The behavior of 1/T1 in (i) metallic region (ii) spin-fluctuation range and (iii) FISDW state is discussed in light of current theories. At high temperatures in the metallic region, the spin dynamics of the system obeys Korringa-like behavior where 771/T T and, as mentioned earlier in Chapter 1, is due to 1 ∝ the scattering of the conduction electrons near the Fermi surface. Note that since TSDW depends on the applied magnetic field along the c∗-axis, the temperature regions where Korringa behavior is observed vary: (a) T>20 K for B =12.86 T (b) T>3.5 K for B =8 T and (c) T>0.76 K for B =6T.

The spin fluctuation region is the temperature range between TSDW and below the temperature in which Korringa behavior breaks down (denoted hereafter by TBD). It is characterized by the upturn in the relaxation rate as the temperature approaches TSDW .As manifested in the fits to the experimental data in Fig. 3.8, this behavior can be explained by the self-consistent renormalization (SCR) theory for weakly itinerant antiferromagnets T developed by T. Moriya [38] which predicts that 1/T1 = A 1/2 where A is a constant (T −TN ) and TN is the Neel temperature. This precursor effect to SDW is also evident in the gradual broadening of the NMR linewidths (not shown) as the temperature is decreased near the SDW/FISDW transition. Another important observation is the increase of the spin fluctuation region with applied magnetic field in which we quantify by employing T T : BD− SDW (a) 0.11 K for 6 T (b) 1.7 K for 8 T and (c) 16 K for 12.86 T. These results point to ≈ ≈ ≈ larger FISDW gap opening at higher fields. 77 The peak in 1/T1 at TSDW is attributed to the enhancement of the relaxation rate due

53 3.88 K 2 B//c* 1 9 8 7 1.81 K 6 5 B=12.86 T 4

2 0.65 K B=8 T 0.1 9 8 7 6 5 B=6 T 4

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 1 10

77 Figure 3.8: Representative temperature dependence of 1/T1 at low fields (B<20 T) where B c∗. The dashed lines above the peak are fits to the SCR theory for itinerant T α antiferromagnets 1/T1 = A 1/2 and below the peaks are power law fits 1/T1 = AT . (T −TN )

to critical slowing down of magnetic ﬂuctuations around a magnetic transition, which, in this case, the FISDW phase boundary.

Below TSDW , the spin-lattice relaxation rate exhibits a power-law behavior given by 77 α 1/T1 = AT where A is a constant and α is the exponent. Note that the relaxation rates at different fields apparently obey almost the same power law behavior: α 1.2 for ≈ 6 T, 1.25 for 8 T, and 1.29 for 12.86 T. This point to the same underlying mechanism of relaxation in the low-field region of the FISDW phase diagram. Because of the proximity of this exponent to 1 (linear behavior), another plausible explanation can be attributed to the phason2 fluctuation model in SDW developed by Clark et al.[39]:

2The phasons here are ﬂuctuations of the SDW phase.

54 γ k T δH2 β′′(ω) 1/T = n B ⊥ (3.11) 1 2λ2ωǫ where λ is the SDW wavelength, δH2 is average fluctuating magnetic field perpendicular ⊥ to the applied external field, ω is the NMR frequency, ǫ is the dielectric constant, and β′′(ω) is the imaginary part of the polarizability (related to dielectric constant by ǫ′′ =4πNβ′′). Equation 3.11 tells us that the magnetic fluctuations seen by 77Se nuclei are dominantly due to the phason fluctuation rate.

3.7 Angular Dependence of 77Se NMR and Transport in the Metallic State

Due to the anisotropic nature of (TMTSF)2ClO4, the NMR and transport features shown in Fig. 3.9 are strongly dependent on sample orientation. As the sample is rotated along the a-axis under a constant magnetic ﬁeld B b′ c∗ plane, the resonant frequency of the NMR − spectra shifts. This can be explained by the presence of the anisotropic Knight shift given by:

K(θ)=K + K (3 cos2 θ 1) (3.12) iso ax −

where K is the total Knight shift, Kiso is the isotropic Knight shift, Kax is the axial Knight shift, and θ is the angle between the applied magnetic field B and c∗-axis. From the data in Fig. 3.9b, the estimated axial Knight shift is around 40 Gauss. It is interesting to note that the simultaneous NMR and electrical transport measurement do not yield the same extrema in the angular magnetoresistance Rzz and resonant frequency ν ν data. The c∗-axis, which is defined as the minimum in R versus θ, is offset by − 0 zz around 250 from the minimum seen in NMR resonant frequency shift. Here we found that this intrinsic offset is an important consideration when aligning (TMTSF)2ClO4 samples without electrical transport reference. 77 The NMR linewidth (FWHM) and 1/T1 are also angular-dependent as shown in

Fig. 3.9c. The angular dependence of FWHM is ﬁt empirically with the equation A0 + A cos θ 3/2; deviation from this empirical ﬁt is seen at around 250 to the right of the b′-axis 1| | (maximum in the NMR shift). It shows slight narrowing of NMR spectra at this particular

55 (b) (a) 7.84 T, 4.2 K 5 10

o 120 4 5 o 100 6 3 o 0 80

o 2 60 -5

o 40 1 -10 o 20 4 o -120 -90 -60 -30 0 30 60 90 120 0

o -20 45 (c) o 65 -40 40

o -60 60 2 35 o -80 55 o -100 30 50 o -120 25

o 45 -140 0 -120 -90 -60 -30 0 30 60 90 120 63.40 63.44 63.48 63.52

77 Figure 3.9: (a) Angular dependence of Se NMR spectra along the a-axis of (TMTSF)2ClO4 in the metallic phase at 7.84 T and 4.2 K. (b) Corresponding angular-dependent magnetoresistance and frequency shift. Note that the peaks in Rzz and ν ν0 do not coincide. (c) − 77 Full-width half maximum (FWHM) and spin-lattice relaxation rate 1/T1 as a function of angle.

angle. In the ﬁrst place, the fact that the NMR linewidth changes with angle hints that more than one chemically inequivalent 77Se sites are embedded in the apparently single-peak spectra. This is because the spectral FWHM of single-site nuclei in the metallic state is expected to be isotropic, that is, constant at all angles. It was shown earlier in Fig. 3.5 that anion ordering in (TMTSF)2ClO4 causes the four-peaked spectra to merge into one below 40 K. It is therefore plausible to say that this anisotropy of FWHM is attributed to the diﬀerent

56 anisotropic Knight shifts of each 77Se site. 77 The 1/T1 data in Fig. 3.5 show slight anisotropy despite the large error bars. The extra dips and peaks in the angular-dependent relaxation may be due to the Lebed magic angle eﬀect (MAE) discussed in Chapter 2, but there were no corresponding MAE signatures in FWHM data as one might expect. This absence of any MAE signatures in NMR relaxation is consistent with similar studies in (TMTSF)2PF6 [40]. However, the current data was done on relatively high temperature (4.2 K) where MAE eﬀects are not visible even in electrical transport. Thus angular-dependent relaxation measurement at lower temperature below 1 K is suggested to resolve this question.

3.8 Angular Dependence of 77Se NMR and Transport in the FISDW State

Orbital effects are primarily responsible for driving the FISDW transitions in Fig. 3.1. Hence by fixing the field and frequency at, for instance B0 = 30 T, the entire (TMTSF)2ClO4 phase diagram can be probed by rotating the sample since the NMR frequency depends only on

B0, and the FISDW transitions depend on the eﬀective perpendicular ﬁeld B⊥ = B0 cos θ. In all cases reported herein, the rotation was in the b′ c∗ plane. − 3.8.1 77Se NMR Spectra in the Partially Quenched and Relaxed Samples

The angular-dependent NMR spectra of a partially quenched sample (the cooling rate was of order 1 K/s or greater through the TAO range), shown in Fig. 3.10a, were measured just below the FISDW phase boundary. The double-peaked spectrum due to the SDW order is visible below the threshold ﬁeld region for the FISDW (in the range 60 to 120 degrees), but then grows dramatically when the FISDW state is entered. We believe the additional pinning of the coexisting SDW and FISDW phases leads to an enhanced double-peaked structure, based on comparisons with well-ordered samples to be discussed below. As mentioned before, a zero-ﬁeld SDW ground state at about 5 K is developed for a rapidly cooled (TMTSF)2ClO4 sample. The metallic and FISDW regions were accessed by rotating the relaxed sample along the a-axis at 12.86 T and 1.8 K shown in Fig. 3.10b. A clear transition from narrow, single peak spectrum (metal) to a broad, double-peak lineshape (FISDW) is seen. This transition takes

57 Partially Quenched Relaxed 1.8 K 2 K 0 (a) (b) θ=65

0 c*-axis, θ=0

0 θ=55

0 X b'-axis, θ=90 0 c*-axis,θ=0

0 0 c*-axis, θ=180 θ=−55 0 θ=−65 12.75 12.80 12.85 12.90 12.95 12.75 12.80 12.85 12.90 12.95

Figure 3.10: Angular dependence of 77Se NMR lineshapes along the a-axis at constant NMR frequency 104.55 MHz (a) in the SDW state of “quenched” (TMTSF)2ClO4 at 2 K (b) in the metallic and FISDW states of “relaxed” (TMTSF)2ClO4 at 1.8 K.

place at θ = 600, so that B =6.4 T which is the expected FISDW threshold ﬁeld when ± ⊥ B c∗ at 1.8 K. This agrees with the orbital nature of the FISDW phases. In both cases dipolar interaction between the spins govern the angular dependence of the resonant frequencies given by A0 (3 cos2 θ 1) where A is the dipolar coupling constant. A 2 − 0 0 here is estimated to be 0.2 T. Based on the previous NMR lineshape model based on sliding IC systems, we infer that SDW condensate in the quenched state is more “rigid” than the FISDW condensate in the relaxed state. The presence of a third peak or spectral structure in the middle indicates (FI)SDW collective sloshing motion in the intermediate velocity range. On the other hand the NMR lineshapes in the quenched state retains the double peak structure expected for a static incommensurate SDW.

58 B (a) 14 T, 1.5 K (c) 0.6 c* B B1 θ B 0.4 1 b' a 0.2 B Bth x th c* (d) 8 b' c* o (b) 20 90 6 o 15 60 4 X Metal 10 o B1 B1 30 B B 5 o 2 th th 0 FISDW FISDW 0 -0.4 0.0 0.4 0 30 60 90 120 150 180 ν−ν0 (MHz)

Figure 3.11: Angular-dependent NMR and electrical transport at 14 T and 1.5 K in 77 (TMTSF)2ClO4. (a) 1/T1 versus ﬁeld orientation θ. A dip in 1/T1 is marked X. (b) Corresponding magnetoresistance and rf enhancement η at 14 T and 1.5 K. (c) Schematic of sample rotation along a-axis and the orientation of the TMTSF molecule with respect to magnetic ﬁeld B at point X. (d) Angular-dependent 77Se NMR spectra.

3.8.2 Low-Field Angular Dependence of 77 NMR and Transport in the FISDW State

For angular dependent NMR data, the angle was set, and the field swept up from the metallic state to the field of interest (B0) and the MR was recorded. In Fig. 3.11 an example of the correspondence between the resistance (Rzz) and 1/T1 are shown, starting well within the FISDW phase at 1.5 K and B =14 T for θ = 0 or 180 degrees (c∗ b′ c∗-axes rotation). 0 − − It is evident how the features in the magnetoresistance, which define the FISDW phases, correlate with the NMR data. Specifically, for B b′, the system is in the metallic state, but as B⊥ = B0 cos θ increases, the threshold field is approached at 1/T1 is observed to increase. However, the peak in 1/T1 occurs not at the second-order threshold field Bth, but at a FISDW subphase transition B1. It is interesting to note that the magnetoresistance

59 1.0 (a) 0.8 12.86 T, 3.87 K

0.6 X

clockwise 0.4 counterclockwise

0.2 b' c* b'

-120 -90 -60 -30 0 30 60 90 120

0.5 (b) 12.86 T, 1.88 K

0.4

0.3 X

0.2

0.1 b' c* b' -120 -90 -60 -30 0 30 60 90 120 25 (c) 12.86 T, 1.88 K 20

15 X

5 b' c* b' 0 -120 -90 -60 -30 0 30 60 90 120

77 Figure 3.12: (a) The spin-lattice relaxation rate 1/T1 as the sample is rotated along the 77 a-axis at 12.86 T and 3.87 K. Note the dip in 1/T1 marked X. There is no distinct hysteresis in the result as sample is rotated clockwise and counterclockwise. Inset: schematic of sample 77 rotation. (b) 1/T1 versus θ at 12.86 T and 1.87 K. Inset: orientation of TMTSF at point X with respect to ﬁeld. (c) Corresponding enhancement factor η at 12.86 T and 1.87 K.

60 at 14 T, at the critical angle where Bth = B0 cos θ, exhibits a sharp dip that indicates the 77 position of the second-order phase boundary. Hence the peak in 1/T1 clearly occurs within the FISDW phase in a subphase transition (i.e. B1; see Fig. 3.1). Angular dependent 1/T1 results are shown in Fig. 3.10 for two different temperatures (b′ c∗ b′-axes rotation). The − − enhancement factor in Fig. 3.12c also shows a significant change with angle, exhibiting a metallic value η = 1 for B B. ⊥ 1 1 ≈ ⊥ 1 77 0 Also prominent is a dip in 1/T1 for θ =25 which occurs when the field is parallel to the long axis of the donor molecule, where a narrowing of the NMR spectral line also occurs (see “X” in Fig. 3.10b). This feature does not show up in the electrical transport.

3.8.3 High-Field Angular Dependence of 77Se NMR and Transport in the FISDW State

To explore the sub-phase transitions in the FISDW phase, high field (B0 = 30 T) experiments were carried out in a resistive magnet, and the main results are shown in Fig. 3.13. The trends are similar to Fig. 3.11 for rotation away from B b′ where there is a slow increase 77 77 in 1/T1 as the threshold field is approached, followed by a peak in 1/T1 at the first-order ∗ B1 FISDW phase boundary. At higher effective B⊥ fields, a significant peak, termed B , appears in the range 15 to 17 T, and another feature appears near 26 T which corresponds to the so-called re-entrant FISDW phase boundary Bre (see Fig. 3.1). The rf enhancement parameter η also has a significant change as Bre is crossed, going from η =5toη =1.

However, the corresponding spectra above Bre are still representative of the presence of antiferromagnetic order [41]. The data upon crossing Bre are consistent with a model [31] (see also Fig. 3.1) where both Fermi surfaces are nested at lower ﬁelds, (e.g. in the “dome” ∗ region between B and Bre) but above Bre only one of the Fermi surfaces is nested. The un-nested Fermi surface could, for instance “short out” the enhancement mechanism, but the nested Fermi surface would still provide antiferromagnetic order evident in the broad, double-peak NMR lineshape shown in Fig. 3.13c. 77 A summary of the angular-dependent 1/T1 plotted versus the eﬀective perpendicular

field B⊥ = B0 cos θ is shown in Fig. 3.14. There is a spin fluctuation behavior as B approaches 77 B1 FISDW transition evident in the gradual upturn in 1/T1. Peaks in the spin-lattice ∗ relaxation rate are observed when B⊥ =(B1,B,Bre). These peaks indicate critical slowing down of magnetic fluctuations as the nesting conditions in the FISDW subphases change.

61 B* 0.5 (a) 30 T, 1.47 K

0.4 B*

B Bre 0.3 1 B1 0.2

0.1

0.0 0 30 60 90 120 150 180

20 B1 (b) B1

η 10 B* B* Bre Bre

0 0 30 60 90 120 150 180

o (c) 300 105 FISDW (d) X1.5 η>1

o 0 X1 Metal η=1 B* o B1 90 Re-entrant X50 η=1 0 0 10 20 30 -0.1 0.0 0.1 Bre Field (T)

Figure 3.13: Angular-dependent NMR and electrical transport at B = 30 T and 1.47 K. (a) Metallic and FISDW transitions revealed in angular-dependent 1/T1. (b) The rf enhancement η vs. θ. Note that η = 1 above Bre. (c) Field-swept NMR spectra at ν0 = 243.9 MHz (30 T) taken at different angles and consequently different phases: (i) FISDW phase at θ = 1050 0 0 (above B1) (ii) Metallic phase at θ =90 and (iii) Re-entrant FISDW phase at θ =0 (above Bre). The corresponding magnetoresistance data at different angles are shown in the lower right hand corner. For each trace, the NMR measurement was made at 30 T.

62 B1

1.0

B0=12.9 T, 1.88 K B0=12.9 T, 3.87 K B0=30 T, 1.47 K 0.8 )

-1 B* 0.6 (ms

1 B1 1/T 77 0.4 Bre B1

0.2

0 5 10 15 20 25 30

B0cosθ (T)

77 Figure 3.14: 1/T1 versus effective perpendicular magnetic field B⊥ = B0 cos θ. The red 77 arrows mark the peaks in 1/T1 which coincide with the different FISDW phase boundaries ∗ B1, B , and Bre seen in electrical transport measurements.

77 3.9 Temperature Dependence of 1/T1, Spectra, and Rzz at High Fields

The sample resistance was measured concurrently with spectra and the spin lattice relaxation time 1/T1 to correlate previously known features of the FISDW phase diagram with the new 77Se NMR behavior. An example is shown first for the metallic state in Fig. 3.9a where the resistance and relative Knight shift of a partially quenched sample are plotted versus angle, and in Fig. 3.9b where the corresponding full width - half maximum (FWHM) of the spectra and 1/T1 are shown. (The temperature is too high to see the “magic angle” effects in Rzz, but in general find no evidence for magic angle effects in the NMR signal in accord with previous investigations [40].) The relative changes in 1/T1 with angle in the metallic state are comparatively smaller than the changes observed in the FISDW phases. Many of the details of Fig. 3.9b, including the dips in 1/T1 and the systematic departure of the FWHM from the B cos(θ) behavior remain unexplained at present. | |

63 3.9.1 Spin Dynamics in the Re-entrant Phase

On cooling down the sample from the metallic state at 23 T where B c∗, two phase transitions at 5.5 K and 3.5 K are observed in electrical transport and magnetization measurements [28]. The FISDW phase diagram in Fig. 3.1 shows that the 5.5 K transition

(part of the second-order phase boundary Bth) stays field-independent above 15 T, while a dome-like region with a plateau at 3.5 K is the so-called re-entrant FISDW phase. These transport features were observed in the temperature dependence of Rzz at 23 T shown in Fig. 3.15a where an upturn the resistance is observed around 5.5 K and yet another sharp increase occurs at 3.2 K. 77 We measured 1/T1 across this temperature range and found that it exhibits a peak, due to critical slowing down at the FISDW phase boundary, at around 5 K. Above 5 K, there is a spin fluctuation region reminiscent of the low field data in Fig. 3.8. Below the peak at 5 K, the relaxation rate gradually decrease until 3 K where there is a discontinuous drop in 77 1/T1. This sharp drop in the relaxation rate is coincident with crossing the re-entrant phase boundary. Also, there are two sets of relaxation values in the temperature range 2.8 K-4 K which maybe attributed to a coexistence region. Below the re-entrant phase, the values of 77 1/T1 are almost the same as that of the metallic phase, indicating the semi-metallic nature of the FISDW region. Analysis of the NMR linewidth in Fig. 3.15b shows the gradual broadening of spectra as it approaches the 5 K transition and a BCS-type of the development of order parameter (FWHM in this case) is evident. There is no distinct difference in the local internal magnetic field below the re-entrant phase and the FISDW region above it as suggested by their equal linewidths which is around 400 kHz from base to base. Another important point to mention in this experiment is the reduction of the Boltzmann- corrected integrated NMR intensity from the metallic state to FISDW region shown in Fig. 3.15c. As expected, the NMR intensity in the metallic state obeys 1/T Boltzmann behavior and so the corrected NMR intensity is 1. However, as the temperature approaches

TFISDW the Boltzmann-corrected NMR intensity drops and below the transition its value is only 0.15. This indicates that we are only seeing 15 % of the spins and that we are not getting part of the spectrum (although the pulses used in this region is 500 ns which should cover a 2 MHz range of frequency). To resolve this issue, ﬁeld-swept spectra were also taken and

64 T 2.0 (a) FISDW

23 T, B//c* 40 1.5

SCR fit 30 1.0

20 0.5 Re-entrant FISDW

10 0.0 0 2 4 6 8 10 12

5 0.14 TFISDW (b) (c) 1.0 12 K 4 9 K 0.12 0.8

7 K 3 6 K 0.6 0.10 FISDW Metal 5.7 K 2 0.4 4.9 K 4 K 0.08 1 0.2 3.3 K

1.9 K 0.06 0 0.0 -200 0 200 0 2 4 6 8 10 12

77 Figure 3.15: Crossing the re-entrant FISDW phase: (a) Temperature dependence of 1/T1 and c*-axis resistance R of (TMTSF) ClO at 23 T with B c∗. 771/T exhibits a peak at zz 2 4 1 5 K which is coincident with the upturn in Rzz. The re-entrant phase is denoted by another 77 sharp increase in Rzz at around 3 K. (b) Normalized Se NMR spectra in the metallic (black) and FISDW phases (red). (c) Corresponding temperature-dependent NMR linewidth and Boltzmann-corrected NMR intensity.

65 TFISDW 80 (a) 1.5 29 T, B//c* 70

1.0 50

0.5 30

0.0 10 0 2 4 6 8 10 12

TFISDW 1.0 (b) 160 (c) 8.5 K 0.8 140 7 K 0.6 6 K 120 FISDW Metal 5 K 0.4 3.75 K 100

2.69 K 80 0.2

2.18 K 60 0.0 -200 -100 0 100 200 0 2 4 6 8 10 12 Frequency (kHz)

77 ∗ Figure 3.16: (a) Temperature dependence of 1/T1 and c -axis resistance Rzz measured simultaneously at 29 T with B c∗. (b) Corresponding 77Se NMR spectra in the metallic (black) and FISDW (red) states. (c) Temperature dependence of full-width half maximum (FWHM) and NMR intensity at 29 T.

66 we obtain the same results. It is interesting to note the temperature dependence of FWHM mirrors that of the Boltzmann-corrected NMR intensity (see Fig. 3.15c). In addition, their intersection point corresponds to the FISDW transition temperature.

3.9.2 Spin Dynamics Above the Re-entrant Phase

Electrical transport and magnetization measurements above 27 T with B c∗ indicate that the 5.5 K transition is the only FISDW phase boundary observed [27, 28]. This is corroborated by our electrical transport data in Fig. 3.16a where the upturn in c∗-axis resistance is only at 5.6 K and there is no hint of the re-entrant phase at 3.5 K. The main ﬁnding of this experiment is the monotonic decrease or continuity of the temperature-dependent relaxation rate below TFISDW shown in Fig. 3.16a, reminiscent of the relaxation behavior at low FISDW phase boundaries in Fig. 3.8. This is consistent with the absence of the re-entrant phase boundary at this ﬁeld. In addition, a power-law behavior 77 with exponent close to 1.35 is observed. The location of the 1/T1 peak is slightly lower than TFISDW seen in electrical transport. The NMR linewidths and NMR intensity point to the same type of behavior seen in the 23 T data on the re-entrant phase.

3.10 A Comparative Study: 77Se NMR and Transport on (TMTSF)2PF6

77Se NMR and electrical transport measurements in the metallic and SDW states of

(TMTSF)2PF6 were done to compare with the spin dynamics and anisotropy in the FISDW states of (TMTSF)2ClO4.

3.10.1 Temperature-dependent Relaxation Rates and Lineshapes

There are three regions of interest in the temperature-dependent spin lattice relaxation rate 77 1/T1 shown in Fig. 3.17a. The ﬁrst is in the metallic or paramagnetic phase (T>12.6 77 K) where at high temperatures 1/T1 obey Korringa-like behavior and it enters the spin ﬂuctuation below 20 K as indicated by the gradual increase in the spin-lattice relaxation rate.

Similar to (TMTSF)2ClO4, the magnetization recovery Mz(t) in the metallic state is ﬁtted well with a single exponential equation while in the SDW state, Mz(t) deviates from single

67 exponential ﬁtting by around 10 to 15 percent due to broadening of the NMR lineshapes. The divergence toward T 12.6 K follows the spin ﬂuctuation behavior given by Moriya’s N ≈ self-consistent renormalization (SCR) theory [38] for weak itinerant antiferromagnets: 1 = T1 AT 1/2 . The second region of interest is below the Neel temperature where a power-law (T −TN ) dependence, 1 = AT α is observed where A is a constant and α =3.2. A thermally activated T1 region, 1 = AeT/Δ is observed below 4 K where Δ = 0.65 K, consistent with previous NMR T1 measurements [39] where an anomaly at 3.5 K is observed.

Radio frequency (rf) enhancement is also observed, similar to the behavior in (TMTSF)2ClO4, where the rf power has to be attenuated and the pulse length shortened to see and optimize the NMR signal. Typical pulses are 100 ns to 500 ns in the SDW state and 2 μs-4μs in the metallic state.

It is important to point out that the peak or divergence in 1/T1 versus T for ∗ (TMTSF)2PF6 occurs on the shoulder of the upturn in the c -axis resistance Rzz where

TN or TSDW is deﬁned. In the paramagnetic phase, Rzz appears to trace out the 1/T1 curve especially in the spin ﬂuctuation region. This indicates similarity of the development of the order parameter between these two sets of measurements. On the other hand, the divergence of 1/T1 in (TMTSF)2ClO4 occurs slightly below TSDW seen by electrical transport.

The spin-spin relaxation rate 1/T2 (not shown) has slight or no dependence on temperature in the metallic and SDW states. The spin-spin relaxation time is ﬁtted with a Gaussian 2 −t /2T2 form of the transverse magnetization decay Mxy(t)=M0e ; T2 is half the full-width half maximum (FWHM) of the Gaussian curve. The NMR lineshapes in the metallic phase show four distinct peaks which correspond to four chemically inequivalent sites of Se. The base-to-base linewidth is around 80 kHz for the multiple-peak lineshape and the individual peak is around 20-30 KHz wide. Unlike

(TMTSF)2ClO4 in which the multiple-peaked lineshapes turned into single-peak lineshapes below 40 K (near the anion ordering temperature TAO = 24 K), the spectra in (TMTSF)2PF6 show the distinct multiple peaks in the whole metallic region and even in the SDW phase. The peaks in the paramagnetic phase show positive shift as the temperature is lowered reﬂecting the spin susceptibility behavior. At T

68 TSDW=12.66 K (a) 2 17 T, B//c* (TMTSF)2PF6

1 10 9 8 7 T* 6 5

0.1 4

2 0.01

1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 10 100 (b) 20 3 80 K 10

0 2 60 K 40 K -10

22 K -20 1 14 K 20 40 60 80 13 K 0

11 K -1 -600 -400 -200 0 200 400 600

77 Figure 3.17: (a) Temperature dependence of 1/T1 and Rzz in (TMTSF)2PF6 at 17 T where B c . Dashed lines are ﬁts to certain equations: self-consistent renormalization ∗ T α (SCR) theory equation A 1/2 (blue dashed line), power law AT where α =3.2 (yellow (T −TN ) dashed line), and AeT/Δ where Δ = 0.65 (green dashed line). (b) Temperature dependence of 77Se spectra in the metallic (narrow, multiple-peaked lineshapes) and SDW state (broad lineshape). Inset: Plot of the peak position versus temperature in the metallic state.

69 (a) 17 T, 20 K 20 (b)

o 0.5 200

o 180 0

o 0.4 160

o 140

o -20 0.3 120

o 100

o 0.2 80 -40 o 60

o 0.1 40

o 20 -60

o 0.0 0 -60 -40 -20 0 20 40 0 50 100 150 200

77 Figure 3.18: (a) Angular-dependent Se NMR spectra in (TMTSF)2PF6 at 20 K. There are four inequivalent sites. (b) Plot of the peak position versus angle. The solid lines are 2 ﬁtted according to dipolar interaction equation A0(3 cos θ 1) where A0 is a constant. The deviation of the resonant peaks from this ﬁt is attributed− to the triclinic structure of the crystal.

3.10.2 Angular-dependent 77Se Spectra in the Metallic Phase

Due to its electronically anisotropic nature, the NMR lineshapes in (TMTSF)2PF6 are strongly angular-dependent. Fig. 3.18a shows the NMR spectra when the sample is rotated along the a-axis at a constant magnetic field 17 T and constant temperature 20 K . As previously mentioned the multiple peaks in the spectra correspond to multiple Se sites. The separation between the peaks depends on the angle due to dipolar nature of the interaction. Fig. 3.18b tracks the angular dependence of each peak and confirms the dipolar interaction because these peaks are fitted with f = A (3 cos2 θ 1) where f is the peak position, A 0 − 0 is a constant, and θ is the angle between the applied magnetic field and the c∗-axis. The deviation from the fit is attributed to the triclinic crystal structure of this material.

70 (a) 0 4.2 K, 17 T (b) B//c*, θ=0 o -80 8 o -70

o site 1 -60

o -50 6 o -40 site 2 -400 0 400 800 o -30 (c) 4 o -20 120 o -10 100 B//b'

o c*-axis 0 80

2 o 10 60

o 20 40 B//c*

o 30 20 0 -600 -400 -200 0 200 400 600 800 0 -50 0 50 100

77 Figure 3.19: (a) Angular-dependent Se spectra in (TMTSF)2PF6 at 17 T and 4.2 K. (b) The NMR lineshape at c-axis (0 deg) showing two sets of double-horned peaks (indicated by the model ﬁts). (c) Corresponding resistance at diﬀerent angles.

3.10.3 Angular-dependent 77Se Spectra in the SDW Phase

77 Fig. 3.19 shows the Se NMR lineshapes as (TMTSF)2PF6 is rotated along the a-axis at constant magnetic field 17 T and constant temperature 4.2 K in the SDW state. The linewidth and shape of the spectra change as the sample is rotated. The linewidth changes from a maximum of 800 kHz to a minimum of 200 kHz at some angles where, as established by previous measurements [42], dipolar interaction is the driving mechanism. This gives us an estimate that the dipolar hyperfine coupling Ahf is around 0.1 T. In the c-axis, two sets of double-horned peaks were found indicating two sites have slightly different dipolar hyperfine coupling. This is quite different from (TMTSF)2ClO4 where only one set of double-horned peak is found in the FISDW spectra. The NMR lineshapes show the spatial distribution of the internal magnetic field seen at the nuclear sites. For an incommensurate SDW, which is the case for (TMTSF)2PF6 and

71 7 1/T1 Peaks:

B||c* B⊥= Bcos(θ) 6

4 B 3 th

1 B 1 B* Bre 0 0 5 10 15 20 25 30

∗ Figure 3.20: Phase diagram of (TMTSF)2ClO4 for B c derived from previous reports 77 (dashed lines) including a summary of the observed 1/T1 peaks (asterisks and open squares), and the corresponding features in the transport measurements (dark and gray circles) from this work. The ﬁeld labels are deﬁned in text.

(TMTSF)2ClO4, the spectrum can be plotted as:

A 1 ,forω <ω0 π ω2−ω2 f(ω)= √ 0 | | (3.13) 0, otherwise where ω0 represents the distribution centroid. The NMR lineshape shown in Fig. 3.19bis plotted as the sum of two functions f1(ω)+f2(ω) with diﬀerent distribution widths and amplitude.

3.11 Summary and Conclusion

An important ﬁnding in this investigation for the FISDW behavior in (TMTSF)2ClO4 is that, when approaching the FISDW phase from the metallic phase, either with increasing

ﬁeld or decreasing temperature, the peak in 1/T1 does not occur at the second-order phase

72 2k F1 Q 1 Q1

2kF2

Metal FISDW Re-entrant

E Ek Ek k kF1 kF1

EF EF k k EF F1 kF2 x kx k kx F2 kF2

1,2 = (2kF1,2 ± n2π/λ, π/b) 1,2 = (2kF1,2,π/b)

Figure 3.21: Fermi surface nesting models in (TMTSF)2ClO4 and the corresponding dispersion relations in the metallic (left panel), FISDW (middle panel), and re-entrant FISDW (right panel) regions of the phase diagram.

boundary, but within it as illustrated in Fig. 3.20. In comparison, for a SDW transition (i.e. not ﬁeld induced as in (TMTSF)2PF6), the peak in 1/T1 is nearly coincident with the onset of the semimetallic transition.

To understand the origin of the peak in 1/T1 and its relationship to the opening of the FISDW gap, we have considered a simple model based on the Hebel-Slichter eﬀect for a BCS superconductor [8]. Invoking Eq. 1.29, the ratio of the spin-lattice relaxation rate in the superconducting state to the normal metal is given by [8]:

∞ T /T =2 ρ2(x, T )[1 + η2(T )/x2] f(x, t)[1 f(x, T )]dx (3.14) n s s 0 ∗ − 0 where x =(E E )/kT , η (T )=ǫ (T )/kT , and f(E,T) is the Fermi distribution function. − F 0 0 The density of states ρs is weighted by the BCS density of states, taking into account the width of the energy levels, according to [8]:

73 x+δ −1 ρs(x, T )=(2δ) ρBCS(y, T)dy (3.15) −δ where δ =Δ/kT and Δ is the BCS gap. This description leads to the “Hebel-Slichter” peak in 1/T1 which appears below Tc due to the relative contributions of the temperature dependence terms in Eq. 3.14. To compute the temperature-dependent 1 , we can employ a T1 Ginzburg-Landau gap model Δ = A 1 T where A is constant. Although the computed − Tc peak in 1/T1 appears below Tc, we expect that in the case of the FISDW in (TMTSF)2ClO4, due to imperfect nesting and the presence of two Fermi surface sheets, only part of the Fermi surface becomes gapped at the FISDW threshold, and we speculate that only below the “dome” region do both FS sheets become nested (see Fig. 3.21). Hence the parameters and functions in Eqs. 3.14 and 3.15 must be modiﬁed to account for the more complex

FISDW behavior.The peaks in (TMTSF)2ClO4 that appear at the ﬁrst order subphase FISDW transitions can also be modelled, in principle, by including the changes in Δ at ∗ B1, B , and Bre.

We note that “dynamic” eﬀects have been used [43] to describe some features in 1/T1 such as the 3 K anomaly that appears in (TMTSF)2PF6, as shown in Fig. 3.17. The mechanism is essentially that which arises when the Larmor frequency coincides with the inverse of the characteristic relaxation rate of the system, which in the case of (TMTSF)2PF6) may involve phason ﬂuctuations. The 3 K anomaly has also been attributed to an improved nesting of the Fermi surface below main SDW phase transition [44].

74 CHAPTER 4

NMR ON CHARGE DENSITY WAVE SYSTEMS

As discussed in Section 2.1.1, the charge density wave (CDW) ground state is a low-energy ordered state in low-dimensional solids characterized by periodic modulation of the electronic charge density. The modulation of the charge density is given by ρ(r)=ρ +ρ cos(2k r+φ) 0 1 F · where ρ0 represents the electron density before CDW formation, ρ1 the CDW amplitude, and φ is the phase of the CDW condensate. The electron-phonon interactions in materials with highly anisotropic band structures lead to a gap in the single-particle excitation spectrum and a collective mode with periodicity q =2kF where kF is the Fermi wavevector. Electrical transport evidence of collective motion or moving CDWs by applied electric ﬁelds have been demonstrated on these materials [12]. NMR also provide direct evidence of moving CDW condensate because electric quadrupolar eﬀects are sensitive to the inhomogeneous broadening of CDW modulation [12]. In this chapter, we discuss two distinct CDW systems: the quasi-one-dimensional organic conductor (Per)2Pt[mnt]2 which exhibits coexisting spin-Peierls and CDW, and the newly- discovered CDW-superconductor CuxTiSe2. In these studies, however, we investigate the electronic correlations of these materials through the use of I =1/2 nuclei which is sensitive to magnetic couplings but not structural distortions.

4.1 Coexisting CDW and Spin-Peierls States in (Per)2Pt[mnt]2

The quasi-one-dimensional organic conductor (Per)2Pt[mnt]2 exhibits a charge density wave (CDW) ground state at around8Kinzero ﬁeld. Application of magnetic ﬁeld suppresses the CDW ground state and even becoming more metallic under some conditions at around 20 T and that above 20 T a new kind of CDW ground state is formed [45, 46, 47] (see Fig. 4.1).

75 10

(Per)2Pt(mnt)2 8

6 BSP

4 CDW+SP

2 FICDW

0 0 10 20 30 40

Figure 4.1: Schematic of the approximate T-B phase diagram of polycrystalline (Per)2Pt[mnt]2 showing coexisting charge density wave (CDW) and spin-Peierls (SP) ground states below 20 T, and ﬁeld-induced CDW (FICDW) region above 20 T [from Graf et al.]. The dashed arrows show the regions in the phase diagram where NMR was measured.

The mechanism has to do with relative Zeeman splitting of the multiple Fermi surface sheets [48] where the CDW nesting condition changes with increasing magnetic field from favorable to unfavorable and then terminating above 45 T, a result that has been recently put on firm theoretical ground by Lebed and Wu [49]. Although this general phenomenon can be described by the standard CDW behavior of the perylene chains, there are very strange quantum steps that appear in the transport properties above 20 T. Based on theoretical conditions [50] and the coincidence of the Peierls and spin-Peierls (SP) transition temperature at 8 K, the conducting perylene chains and the Pt[mnt]2 chains (see Fig. 4.2) are strongly interacting. It is therefore possible that these steps arise from the breaking of the spin-Peierls state in the Pt[mnt]2 chains and since these chains are insulating, a local magnetic probe such as NMR is appropriate. NMR has been proven to be a valuable probe to the SP ground state as demonstrated in 13C NMR studies in organic SP material [51] and the high field

76 Pt (a) Pt (b)

Dithiolate Chain a b Perylene Chain Perylene Chain

Figure 4.2: (a) Crystal structure of (Per)2Pt[mnt]2 viewed along a-axis. (b) Schematic of the perylene and dithiolate layers. Electronic conduction occurs in the perylene chains and is directed mainly on the b-axis.

63 Cu NMR investigation in the inorganic SP compound CuGeO3 [52]. Since Pt is located in the mnt chain where SP is believed to occur, field-dependent 195Pt NMR will probe the local magnetic field in the title material and will determine the field at which the SP state breaks down. 4.2 Crystal Structure and Electronic Properties

The perylene family of organic conductors is composed of a conducting chain which is the perylene stack and a magnetic or non-magnetic chain given by mnt 1 (bismaleonitriledithio- late) with a metal M (which can be one of the following: Ni, Pt, Au, Co, Fe, Cu, and Pd). There are two kinds of structural phases in perylene compounds: the α phases which are metallic at high temperatures but exhibit metal-insulator transitions at low temperatures and the β phases which are semiconductors with a lattice modulation along the chain direction. The perylene chain has a three-quarter-ﬁlled conduction band while the mnt chains which are

1The IUPAC name of mnt is cis-(2,3-dimercapto-2-butene-dinitrile).

77 insulating are either Mott-Hubbard type or have closed shells. Due to this crystal structure b∗ and band-ﬁlling, the perylene chains are tetramerized at 4 which correspond to a wavevector P b∗ 2kF . On the other hand, dimerization of the dithiolate chain at 2 corresponds to a wave P D vector 4kF =2kF . The superscripts P and D refer to the perylene and dithiolate chains, respectively. In the case of Pt-perylene (Per)2Pt[mnt]2, the dithiolate layer has localized spin 1 because of Pt (S = 2 ) that interacts with the conduction electrons in the perylene layers. At high temperatures they interact via fast spin exchange interaction. The total susceptibility per mnt is χs(T )=χs (T )+χs (T ) where the superscripts refer to perylene (per ) and dithiolate

(mnt) chains. (Per)2Pt[mnt]2 is a Q1D conductor, with electronic conduction occurring mainly in the b-direction with estimated transfer integrals t 150 meV, t 2 meV, and b ≈ a ≈ t 0 meV [53]. The typical sample size is around 1 mm 0.050 mm 0.025 mm. c ≈ × × 4.3 Experimental Details

195 Pt is a spin-1/2 nucleus with a gyromagnetic ratio γn =9.094 MHz/T and a natural abundance of 33.8%. This magnetic ion sits on the mnt chain where SP state is thought to occur at the same time with CDW on the perylene chain. For single crystal measurements (results not shown here), a microcoil is used to have better filling factor. The details of making NMR microcoil for single crystal of (Per)2Pt[mnt]2 is given in Chapter 1. For higher field experiments, a sub-millimeter coil was used with stacks of around 50 single crystals with their longer axes more or less parallel to the coil axis. Temperature and field-dependent 195Pt NMR spectra and relaxation rates were taken.

4.4 Results and Discussion

We will ﬁrst discuss the temperature dependence of 195Pt NMR spectra at constant ﬁeld 14.8 T displayed in Fig. 4.3. The NMR spectrum at 1.74 K resembles the powder pattern lineshape due to the distribution of anisotropic Knight shifts. We invoke the total Knight shift equation for a single crystal [6]:

K(θ, φ)=K + K (3 cos2 θ 1) + K sin2 θ cos2φ (4.1) iso ax − asym where the ﬁrst, second, and third terms refer to the isotropic, axial, and asymmetric contributions to the Knight shift, respectively. The angles θ and φ represents the orientation

78 1.0 1.0

0.8 0.8 K || Kperp 0.6

0.6 Boltzmann prediction 0.4 Spectral Intensity (arb. units) 0.4 0.2 14.8 T T SP 1.74 K NMR Intensity (arb. units) 2.8 K 0.0 Pt 4.2 K 0.2 0 5 10 15 20 195 Temperature (K) 6 K 12 K

0.0 135.0 135.2 135.4 135.6 135.8 Frequency (MHz)

195 Figure 4.3: Temperature dependence of Pt NMR spectra in polycrystalline (Per)2Pt[mnt]2 at 14.8 T. Left Inset: Plot of the corresponding spectral intensity which deviates from the Boltzmann prediction. The NMR signal disappears at around 5 K which is coincident with the Spin-Peierls transition temperature. Right Inset: characteristic power lineshape pattern that resembles the 195Pt NMR spectra due to distribution of anisotropic Knight shifts.

of the principal axes of the shift tensor with respect to the magnetic ﬁeld. There are only two components of the axial Knight shift: K⊥ = Kx = Ky and K = Kz. However for powder samples, the total Knight shift will be averaged over all values of (θ, φ) and so the following simpliﬁcations have to be made [6]:

1 1 K = (K + K + K )= (2K + K ) (4.2) iso 3 x y z 3 ⊥

1 1 1 1 K = (K K K )= (K K ) (4.3) ax 3 z − 2 y − 2 x 3 − ⊥

1 K = (K K ) 0 (4.4) asym 2 y − x →

79 3.0 500 400

300 1.8 K BSP 1.8 K 200 2.5 100 Intensity (arb. units) 18 20 22 24 Field (T) 17.4 T 18.2 T 2.0

17 T 19 T 1.5

16.5 T 20 T 1.0 NMR Intensity (arb. units) NMR Intensity (arb. units) 21 T 12.84 T 0.5 22 T

12.31 T 23 T 0.0 -0.6 -0.4 -0.2 0.0 0.2 0.4 -0.6 -0.4 -0.2 0.0 0.2 0.4 Frequency (MHz) Frequency (MHz)

195 Figure 4.4: Pt NMR spectra of (Per)2Pt[mnt]2 at various ﬁelds at constant temperature 1.8 K. Note the loss of 195Pt NMR signal above 20 T.

This gives the NMR lineshape depicted in the inset of Fig. 4.3. On warming up the sample at 14.8T, the NMR intensity drops faster than the Boltzmann 1/T prediction as it crosses the spin-Peierls transition temperature T 5 K and that above 5 K, we completely lose SP ≈ the NMR signal. Field-swept scans were done on the possibility of the 195Pt NMR spectrum having a large Knight shift at T>TSP , but so far no such evidence is found. Temperature- dependent spin-echo measurements of spectra at different fields in the range 10 17 T yield − 195 the same result where Pt NMR signal disappears at T>TSP and it tracks down the SP phase boundary. Next we comment on the results in Fig. 4.4 where the magnetic field was swept at constant

80 1.2

2.5 195 1/T1 at 2.2 K 1.0 Spectral amplitude at 2.2 K 195 1/T1 at 1.8 K 2.0

0.8

1.5 0.6

1.0 0.4 BSP

0.5 0.2

0.0 0.0 10 15 20 25 30

195 Figure 4.5: Field dependence of 1/T1 at 1.8 K (open circles) and 2.2 K (open squares). The spectral amplitude at 2.2 K shows the disappearance of 195Pt NMR signal above 20 T.

temperature. The main finding of the experiment is the disappearance of the 195Pt NMR signal above 20 T, confirming the theoretical prediction of the breakdown of the SP state at 20 T where electrical transport data [45] show suppression of the CDW state. To complement the NMR spectra, the field dependence of the spin-lattice relaxation rate 195 195 1/T1 was measured (see Fig. 4.5). 1/T1 increases by a factor of 8 when the field is increased from 12 T to 19 T; the large error bars reflect the weak 195Pt NMR signal-to-noise ratio in this experiment. Above 20 T, the relaxation rate can not be measured because of the disappearance of the NMR signal.

81 4.5 Conclusion

195Pt NMR results indicate that the SP state breaks down at 20 T evident in the disappearance of the NMR signal beyond this ﬁeld, even in the FICDW region. The loss of NMR signal may be attributed to very fast relaxation rates outside the SP phase boundary. Preliminary results on the polycrystalline sample indicate no discernible changes in the 195Pt NMR linewidth; single crystal measurements are desired to avoid the problem of mixed phases in multiple crystal setup. 1H NMR at high magnetic ﬁeld is proposed to track down changes in the internal magnetism of this material in regions where 195Pt NMR signal disappears.

4.6 CuxTiSe2: a new CDW-Superconductor

CuxTiSe2, synthesized in 2006 by Morosan et al. [54], is the first system that provides an opportunity to investigate the nature of the CDW to superconducting transition by controlled chemical doping. Here the CDW transition in TiSe2 can be suppressed by continuous Cu doping of around x=0.04, and superconductivity emerges with a maximum Tc =4.15 K at x =0.08 Cu doping (see Fig. 4.6). The parent compound TiSe2 is one of the first CDW compounds known, where around 200 K a commensurate CDW with wave vector Q =(2a, 2a, 2c) is formed. However, photoemission measurements [55] show that the CDW in this material is not driven by the nesting of the Fermi surface. Apparently it is due to a transition from a small indirect gap like in a normal semiconductor state into a state with a larger indirect gap at a slightly different location in the Brillouin zone [54, 55]. TiSe2 is a layered compound with trigonal symmetry where the Ti atoms are in octahedral coordination with Se. The Cu atoms occupy the positions between the TiSe2 layers which leads to an expansion of the lattice parameters in CuxTiSe2 (see inset of Fig. 4.6). Various measurements were made to further investigate the role of Cu doping in the suppression of CDW and the emergence of superconductivity. Thermal conductivity suggests that this system exhibits conventional s-wave superconductivity due to the absence of a

κ0 residual linear term T at very low temperatures [56]. The weak magnetic ﬁeld dependence of thermal conductivity indicates a single gap that is uniform across the Fermi surface. Further, it is suggested that the 4p Se band is below the Fermi level and superconductivity is induced because of the Cu doping into the 3d Ti bands [56]. An optical spectroscopy investigation on Cu0.07TiSe2 reveal that the compound has a low carrier density and has an anomalous

82 Figure 4.6: T-x Phase diagram of newly-discovered CDW-superconductor CuxTiSe2 [from Morosan et al.]. Notice the similarity to the high-Tc phase diagram.

metallic state because of the substantial shift of the screened plasma frequency [57]. This is corroborated by the temperature-independent Hall coeﬃcient RH [58] found in heavily Cu- doped samples such as this mentioned concentration. Raman scattering measurements of

CuxTiSe2 at diﬀerent doping suggest that the x-dependent mode softening is associated with the reduction of the electron-phonon couplings and the presence of a quantum critical point within the superconducting region [59]. The momentum-space distribution of the electronic states has been mapped out by angle-resolved photoemission spectroscopy (ARPES). The parent compound TiSe2 is a small gap semiconductor or semimetal with a trigonal structure and hexagonal Brillouin zone. The main ﬁnding in this study is the CDW order parameter competing microscopically with superconductivity in the same band [60]. Another ARPES

83 3 (a) (b) 7.193T 2

2.1K 4.19K 10K 20K 1 30K 40K 50K 70K 100K 140K 0 180K

-1

-40 -20 0 20 40 0 50 100 150 200 250

77 Figure 4.7: (a) Temperature dependence of Se NMR spectra in Cu0.07TiSe2 at 7.193 T. (c) Plot of the peak position/shift from (a).

study [61] provided evidence that the parent compound TiSe2 is a correlated semiconductor. Cu doping enhances the density of states and raises the chemical potential which weakens the CDW and favors superconductivity. Here we present 77Se and 63Cu nuclear magnetic resonance studies to characterize the electronic correlations in the title material. 4.7 Experimental methods

The plate-like samples with average thickness of 30 μm along the c-axis were cut in rectangular shapes with approximate dimensions 4 mm 2 mm. Parallel stacks of rectangular × samples were placed in a rectangular-shaped coil for better ﬁlling factor. Magnetic ﬁeld is applied along the c-axis for NMR measurements. Two doping levels (x =0.05, 0.07) of 77 63 CuxTiSe2 were studied under the same conditions. Se and Cu spectra, Knight shift, and spin-lattice relaxation rate data were taken. 63Cu is a spin-3/2 nucleus with γ =11.285 MHz/T and 69.1 % natural abundance.

84 -0.140 100 8 (a) 8 T 6 (b) 8 T 4 Cu0.07TiSe2

-0.145 10 Cu0.07TiSe2 8 6

-0.150 Cu TiSe 1 0.05 2 8 6

-0.155 0.1 2 4 6 8 2 4 6 8 2 2 4 6 8 2 4 6 8 2 1 10 100 1 10 100

63 Figure 4.8: (a) Temperature dependence of Cu Knight shift for Cu0.05TiSe2 (red triangles) 63 and Cu0.07TiSe2 (blue circles) at 8 T. (b) Temperature dependence of 1/T1 at 8 T.

77 63 4.8 Se and Cu NMR Studies of CuxTiSe2

Fig. 4.7 shows the temperature-dependent 77Se NMR spectra and the relative shift at 7.193 77 T for Cu0.07TiSe2. The behavior of relative shift δν as a function of temperature more or less reﬂect the susceptibility curve (see Ref. [54]) for this particular concentration. 63 As mentioned earlier, the Cu atoms are located in between the TiSe2 layers. The Cu Knight shift data in Fig. 4.8a reveal weak temperature dependence, with K = 0.152 % − for Cu TiSe and K = 0.143 % for Cu TiSe . The spin-lattice relaxation rate data in 0.05 2 − 0.07 2 Fig. 4.8b show monotonic increase with increasing temperature, obeying Korringa behavior. On the other hand, the Se atoms are located on the layers and may reveal more information about the electronic correlation in this material than Cu. Fig. 4.9 shows 77 77 comparative temperature-dependent behavior of 1/T1 in a pure Se metal, powdered TiSe2 data from Dupree et al.[62], and in parallel stacks of CuxTiSe2 (x =0.05, 0.07) single crystals from this work. It is apparent that an increase of Cu doping level (from 0.05 to 0.07) enhances the relaxation rate values particularly the slope of the Korringa-like behavior increases with Cu doping. We henceforth invoke a model involving a Korringa factor K(α) due to the eﬀect

85 7 Cu0.05TiSe2, 11.1 T (B//c*) 4 2 6 1

4 2 5 0.1

4 2

4 2 4 6 8 2 4 6 8 2 1 10 100

Cu TiSe , 11.1 T (B//c*) Se metal 0.05 2 2

powder TiSe2 (Dupree et. al.) 0 0 50 100 150 200

77 Figure 4.9: Temperature dependence of 1/T1 of parallel stacks of CuxTiSe2 (x=0.05, 0.07) ∗ 77 platelets with B c , powder of TiSe2 (Dupree et. al.), and pure Se metal. Inset: log-log 77 plot of this graph. Note the non-linearity of 1/T1 vs T in TiSe2 due to CDW formation.

of electron-electron interaction in metals and alloy compounds developed by T. Moriya [63] and later refined by Narath et al. [64]. As discussed earlier in Section 1.4, the relaxation mechanism of nuclear spins in a metal are largely due to their interaction with the conduction electron spins. This is given by the Korringa relation 1/T T wherein only Fermi contact interaction is taken into account for 1 ∝ noninteracting electrons. This approximation works for most metals but not for alloys. This is the case where electron-electron interaction cannot be neglected. The Coulomb interaction has an effect on the magnetic susceptibility which is proportional to the Knight shift. The internal magnetic field in a nucleus is given by H = H + δH where H loc loc loc is average internal magnetic field and δH is the fluctuating magnetic field. The nuclear spin lattice relaxation rate due to fluctuating magnetic field is written as 1/T1 = 2 ( γ ) +∞ dt cos ωt δH (t)δH (0) . 2 −∞ + − 86 In terms of the wavelength and frequency-dependent magnetic susceptibilities, the spin lattice relaxation rate is [38, 63]:

′′ γ2 +∞ 2γ2k T χ (q, ω ) 1/T =( ) A A dt cos ω t s+(t)s−(0) =( B ) A A ⊥ 0 (4.5) 1 2 q −q 0 q q g2μ2 q −q ω q −∞ B q 0 1/2 where sq is the Fourier-transformed electronic spin density, Aq (from the relation V Ak+q,k = ′′ Aq) is a constant related to the Fermi contact interaction, and χ⊥(q, ω0) is the imaginary part of the susceptibility. The susceptibility can be calculated by using random phase approximation (RPA) the end result of which is [63]:

2 πγ kBT k(nk nk+q)δ( ω0 εk+q + εk) 1/T =( ) A A − ′′ − ′′ (4.6) 1 V q −q ω [1 wχ (q, ω )]2 +[wχ (q, ω )]2 q 0 0 0 0 0 − ′ This equation can be simpliﬁed by several approximations: limω0→0 χ0(q, ω)=χ0(q = ′′ 2 0, 0)G(q) and lim χ (q, ω) = 0. Here G(q)=(1 )[1 + 1−x log 1+x ] from the free electron ω0→0 0 2 2x | 1−x | approximation with x = q . The simpliﬁed equation is [63]: 2kF

A A 1/T = πγ2k T a −q [n(ε )]2 [1 αG(q)]−2 (4.7) 1 B V F − 3vn where n(εF ) is the DOS of electrons at the Fermi level and α = . Since the Knight shift 2εF is:

V −1/2μ A n(ǫ ) K = B q F (4.8) s 1 α − and introducing

1 (1 α)2xdx K(α)=2 − (4.9) [1 αG(x)]2 0 − the modiﬁed Korringa relation (in SI units) can therefore be written as:

4πk T γ2 1/T =( B ) n K2K(α) (4.10) 1 2 s γe where the Korringa factor K(α) is a function of the interaction parameter α, a measure of electron-electron interaction the value of which ranges from 0 to 1. For α =0,K(α)=1 which gives us the Korringa relation [63, 64].

87 1.0

0.8

0.6 (α) K

0.4 α0.07 α0.05

Cu0.07TiSe2 0.2 Cu0.05TiSe2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 Interaction Parameter α

Figure 4.10: Enhanced Korringa factor K(α) vs interaction parameter α from Moriya and a corrected version from Narath et. al. The blue and red dashed lines are the observed K(α) for 7% and 5% Cu dopings, respectively. The arrows indicate that the interaction parameter α is 0.8 for 7% and 0.93 for 5%.

Table 4.1: Korringa factor K(α) due to electron-electron interaction in CuxTiSe2

Metal 771/T T (s K)−1 K(α) 1 · Se metal 0.15625 1 Cu0.07TiSe2 0.036353 0.23266 Cu0.05TiSe2 0.0259313 0.1659615

88 4.9 Conclusion

77 63 In conclusion, the electronic correlation of CuxTiSe2 was investigated via Se and Cu 77 NMR. The temperature-dependent 1/T1 data show enhanced-Korringa behavior where the slope changes with Cu doping. Based on the modiﬁed Korringa model, the interaction 63 parameter α is around 0.8 for Cu0.07TiSe2 and 0.93 for Cu0.05TiSe2. 1/T1 reveal the same temperature-dependent behavior. A summary of the Korringa factors obtained for the two Cu dopings is given in Table 4.1.

89 CHAPTER 5

PROBING THE DYNAMICS OF FRUSTRATED SPIN SYSTEMS

The hunt for model materials showing spin liquid states has been one of the hottest topics in condensed matter today. Due to their corner-shared triangular lattice (see Fig. 5.1), these “frustrated” spin systems remain paramagnetic down to the lowest temperature when normally one should expect a Néel-ordered state or some other long-range magnetic ordering. In a triangular lattice, two adjacent spins are oriented antiparallel to each other and the third spin faces a problem because whichever choice is made is not energetically favorable, thus an ordered state cannot be achieved. Instead this spin system possesses a multiplicity of equally unsatisfied states due to the geometry of the lattice [5]. This effect is usually observed in Heisenberg spins in two dimensions on triangular and corner-sharing kagomé (named after a Japanese basket showing the same interweaving pattern) lattices. Materials with pyrochlore structure (corner-sharing tetrahedra occupied by magnetic ions) have also been studied as possible model materials for spin-liquid state in three dimensions. A ground state called cooperative paramagnetism is expected for these materials in which only short range correlations between spins occur and ideally this leads to the persistence of spin fluctuations down to the lowest temperature [5]. This chapter of the dissertation introduces a new 69,71 candidate of spin liquid model material, the langasite Pr3Ga5SiO14 studied via Ga NMR. 5.1 A survey of Frustrated Spin Systems

Over the last few decades a number of geometrically frustrated materials have been studied in search of a true spin liquid ground state. Among them are SCGO, the jarosites, volvorthites, herbertsmithite, and recently the langasites.

SCGO, short for the family of magnetoplumbite compounds SrCr12−xGaxO19,have

90 Cu (a) O (b) H

Figure 5.1: (a) The structurally perfect kagom´elattice (b) Crystal structure of herbert- 2+ smithite (ZnCu3(OH)6Cl2), the S =1/2 structurally perfect kagom´elattice network of Cu spins (from Ref. [69]).

3 3+ attracted scientific attention because of the triangular arrangement of the S = 2 Cr spins. Ga NMR studies have been done in (SrCr8Ga4O19)[65] which reveal the existence of paramagnetic “clusters” of spins. From the Ga Knight shift data, it appears that the intrinsic kagomésusceptibility χ displays a broad maximum at T J and that the macroscopic frus ∼ 2 susceptibility at low temperatures is dominated by defects coupled to the frustrated network [65]. This is in conjunction with neutron scattering measurements [66] which indicates that the short-range antiferromagnetic order has only a correlation length of ξ 7 A.˚ ≈ 3+ In the case of the jarosites RFe3(OH)6(SO4)2 (where R=K, Na, etc.), the Fe spins with 5 S = 2 form a triangular network and are antiferromagnetically coupled to each other. Most of the jarosites order and of particular interest is the potassium variant KFe3(OH)6(SO4)2,a good example of nearly two-dimensional kagomélattice antiferromagnet. 1H NMR confirmed aNéel-type magnetic ordering below 65 K and in this ordered state, the spin structure is a q = 0 120-degree type with positive chirality as shown in Fig. 5.2a. In the ordered state, the spin-lattice relaxation rate 1/T1 decreases sharply with decreasing temperature; this is well-described by a two-magnon process with an energy gap of about 15 K [67]. 51V NMR studies on kagome-like S = 1 volvorthite Cu V O (OH) 2H O[68] reveal a 2 3 2 7 2· 2 mixture of different spin configurations of Cu2+. Estimates point to 20% are in the short-

91 (a) (b)

Figure 5.2: (a) q = 0 and (b) q = √3 √3 (also called the weathervane mode) states in a structurally perfect kagom´elattice. ×

range order of q = √3 √3 type (see Fig. 5.2b), and only 40 % of the Cu moments are × frozen. 2+ In the case of herbertsmithite ZnCu3(OH)6Cl2,Cu form a network of corner-sharing 1 triangles where the Cu S = 2 spins interact antiferromagnetically with J = 170 K. Thermodynamic and magnetic susceptibility measurements, neutron scattering, and later local probes such as μSR and NMR studies have established that this material remains in paramagnetic state down to 50 mK [69]. 63Cu, 35Cl, and 1H NMR [69] show that the spin- lattice relaxation rate at low temperatures obey a power law 1/T T α (α 0.45 for 4-8 T 1 ∝ ≈ and α 0.2 for fields lower than 2.4 T) dependence at high magnetic fields. This apparently ≈ agrees with the power law calculations based on the Dirac Fermion approach to spin liquid 1 systems. This S = 2 Heisenberg kagoméantiferromagnet is believed to be a good model material for low spin, pure spin liquid ground state. 3+ On the other hand, the recently discovered langasite Nd3Ga5SiO14 where the Nd mag- 9 netic moments with S = 2 provides a unique opportunity for investigating competing single- ion anisotropy and spin-liquid state. Magnetic susceptibility [70] and neutron scattering [71] studies show no evidence of long range magnetic ordering down to 50 mK. This material apparently has higher frustration index f |θ| 1300 than any other kagomésystems [71]. ∼ Tc ≥ Moreover, complementary Ga NMR and μSR measurements [72] show a relaxation plateau

92 below 10 K down to 60 mK, indicative of a fluctuating collective paramagnetism. However a partial magnetic ordering is apparently induced by application of magnetic field greater than 0.5 T [71]. Nd3Ga5SiO14 is believed to be the first realization of a high spin, rare-earth based model material for spin liquid state arising from competing single-ion anisotropy and exchange interaction.

The sister langasite compound Pr3Ga5SiO14 is the main subject of the NMR studies in this chapter. NMR will show the eﬀect of substituting the Kramers ion Nd by non-Kramers ion Pr in the spin dynamics of the kagom´earrangement of magnetic moments.

5.2 The Rare-Earth Kagom´eR3Ga5SiO14

The langasite compounds (from the prototype material La3Ga5SiO14) have a structure similar to Ca3Ga2Ge4O14. They exhibit piezoelectric properties with better electromechanical coupling than quartz and as such they are now used as wave ﬁlters in telecommunication devices and high temperature sensors. Analysis of magnetic susceptibility suggests that Pr and Nd magnetic moments can be modelled as coplanar elliptic rotators perpendicular to the threefold axis of the crystal structure that interact antiferromagnetically. The Langasite- type compounds belong to the trigonal space group P 321. The Ga3+ ions are sitting in three crystallographically inequivalent sites.

5.3 69,71Ga NMR Probe of the Spin Dynamics of Pr3Ga5SiO14

Various two-dimensional (2D) triangular lattices, including kagomésystems, exhibit in- triguing low-temperature spin dynamics induced by a geometrical frustration [73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]. The prototypical kagoméexamples 2+ 3+ are ZnCu3(OH)6Cl12 [Cu (S=1/2)], SrCr8−xGa4+xO19 [Cr (S=3/2)], and the jarosite 3+ (D3O)Fe3(SO4)2(OD)6 [Fe (S=5/2)] [75, 76, 77]. These are all 3d transition metal based compounds in which the exchange interactions and geometrical frustration are of dominant importance while Dzyaloshinsky-Moriya interactions, single-ion anisotropies, and off-stoichiometry serve as small perturbations. Nonetheless, investigation of the predicted spin-liquid state is complicated by the presence of these perturbations. For rare-earth (RE) compounds, however, the situation is less complicated: the single-

93 Pr Ga O Si

c a

Figure 5.3: Distorted kagom´esystem: crystal structure of Pr3Ga5SiO14 (from Ref. [89]) representative of other frustrated langasite systems.

ion anisotropies govern the magnetic behavior, while frustration-induced spin dynamics is expected to emerge at energies well below the crystal ﬁeld splitting. In this light, the recently discovered RE based kagom´ecompounds R3Ga5SiO14 (R=Nd or Pr) provide an unprecedented opportunity to study cooperative magnetic correlations in a strong crystal-

ﬁeld background. In R3Ga5SiO14, the rare-earth ionsR=NdorProccupy sites in corner sharing triangles as illustrated in Fig. 5.3. The geometric frustration results in a low-T disordered state [89, 90, 70, 72].

Pr3Ga5SiO14, which belongs to the langasite family, has a trigonal crystal structure (P 321) with lattice parameters a = 8.0661(2)A˚ and c = 5.0620(2)A.˚ The RE Pr3+ ions (5f 2 : J = 4) are networked by corner sharing triangles in well-separated planes to form a distorted lattice (see the inset of Fig. 5.4) topologically equivalent to the kagom´elattice. Plots of 1/χ versus T for H parallel and perpendicular to the c-axis show a crossover at 127 K with

χ slightly larger than χ⊥ below this temperature [89]. The high T susceptibility data has been interpreted using an expression which allows for antiferromagnetic (AFM) correlations

94 pnsse a emdle selpi lnrrttr epniua otecrysta the to perpendicular rotators [ planar elliptic axis as modelled be can system spin o antcodrn ont down ordering magnetic for ewe h Pr the between iue54 M edsa pcrmo Pr of spectrum field-scan NMR 5.4: Figure he ao´ lns w astslebtenteepae n h hr iei in sho is structure site crystal third the the and depicts planes inset these The between lie shown). sites not Ga is Two kagomé peak, planes. three small a in embedded n cssetblt ontrva ogrnemgei reignrasi ls transit [ glass spin mK a 35 nor to ordering down magnetic temperatures long-range reveal at not do susceptibility ac and ftresnlt ihgp Δ gaps with singlets three of ( antcseicha ugssta h rsa edeeg ee tutr o structure level energy field crystal the that suggests heat specific magnetic rudsaecrepnigt ihfutainidxf= f index frustration high a to corresponding state ground I =3 89 / )and 2) .TeetmtdCreWistmeaueis temperature Curie-Weiss estimated The ]. NMR Intensity (arb. units) 5 71 3+ a( Ga pn n rsa edeet [ effects field crystal and spins I 6 71 =3 C(1) / )cmoet o h w o-qiaetG ie tetidsite, third (the sites Ga non-equivalent two the for components 2) o2Kin 71 1 C(2) 7 5KadΔ and K 25 = 90 pt ftecmaaieylreAMitrcin.The interactions. AFM large comparatively the of spite .Ti hw htPr that shows This ]. Field (T) 8 69 95 C(1) 89 3 2 Ga .Tessetblt aaso oevidence no show data susceptibility The ]. 8K n Δ and K, 68 = 5 69 SiO θ C(2) 9 CW 14 = hwn uduoa split quadrupolar showing 3 Ga − θ CW 2 5 . u eto diffraction neutron but K 3 SiO 10 / 4.2 Å 3 T c 14 8 .Seicheat Specific K. 780 = ≥ a spin-liquid-like a has 6 nlsso the of Analysis 66. 100 K 11 2.9 K Pr f 3+ threefold l consists plane. wing 69 ion Ga measurements below 1 K are again consistent with the 2D Pr3+ ions forming a spin liquid ground state in zero field showing no evidence of a phase transition at temperatures down to 35 mK [90]. The distorted kagomélattice results in XY anisotropy and neutron scattering experiments show that short-range magnetic order is induced by an applied magnetic field H directed along the crystal c-axis [90].

5.4 Results and Discussion

μSR and NMR measurements on the sister RE kagomécompound Nd3Ga5SiO14 reveal that a disordered state persists down to low T in zero field [70, 72]. Spin fluctuations are suppressed by an applied field and for H = 0.5 T a field-induced transition is found at 60 mK. Although 3+ 3+ the two isostructural compounds R3Ga5SiO14 (R = Pr or Nd ) share, to some extent, common physics, detailed cooperative spin dynamics will vary with the RE ions. This is because Pr3+ and Nd3+ have different crystal field splittings, single-ion anisotropies, and exchange interactions leading to a substantial difference in a low-energy spin dynamics both at zero field and/or in external field.

It is clearly of interest to compare the behavior of the two kagomésystems R3Ga5SiO14 69 (R = Nd, Pr). Ga NMR measurements of Pr3Ga5SiO14 were taken as a function of temperature in applied fields in the range 2 17 T. The objective is to follow the spin − dynamical behavior with temperature in the spin liquid regime and to determine how the dynamical behavior depends on large applied magnetic fields. Apparently the spin-spin relaxation rate is sensitive to field-induced short-range magnetic ordering. This enables separation of frustration-induced cooperative phenomena from single-ion physics.

Single crystal samples of Pr3Ga5SiO14 were grown using the traveling floating-zone technique and characterized by X-ray diffraction as described previously [90]. The NMR spectra and relaxation rate measurements were made a using a pulsed spectrometer with spin echo techniques. Fig. 5.4 shows the NMR field sweep spectrum obtained by integrating spin echo signals. Multiple peaks are found corresponding to three non-equivalent sites for the I =3/2 69Ga and 71Ga isotopes each having quadrupolar splittings which give rise to a central line and two satellites due to non-cubic site symmetry. The resulting spectrum consisting of eighteen overlapping lines is similar to that of Nd3Ga5SiO14 [72]. The location of the three Ga sites between the kagoméplanes is shown in the inset in Fig. 5.4. Note that

96 site 3 is randomly occupied by Ga3+ and Si4+ ions. Figure 2(a) plots the measured spectral 69 Knight shift K for H = 9 T along c versus χ with T as the implicit parameter while Fig. 2(b) shows the spectra in a stacked plot. The linewidth increases significantly as T is 69 lowered. For T>30 K a linear relationship between K and χ is found, but departures from this relationship become important below 30 K pointing to the growing importance of spin correlation effects with decreasing T. 3+ In order to study the Ga ion spin dynamics spin-lattice (1/T1) and spin-spin (1/T2) relaxation rates were measured as a function of temperature for the central 69Ga spectral component corresponding to site 1. Measurements were made in several different applied magnetic fields directed parallel to the c-axis and the results are given in Figs. 5.6 and 5.7. Similar results were found for the other spectral components. In contrast to the behavior found in Nd3Ga5SiO14 [72] and in other transition metal-based frustrated 2D antiferromagnets such as NiGa2S4 [86], in which wipe-out of the NMR signal occurs at low temperatures due to low frequency magnetic fluctuations, the NMR signal in Pr3Ga5SiO14 could be observed in all applied fields over the temperature range 300 mK to 290 K. Fig. 5.5 shows that while the NMR linewidth in Pr3Ga5SiO14 does increase significantly below 100 K it reaches a plateau value of 0.22 T below 10 K. The relaxation rates could be measured over the entire temperature range showing well defined maxima below 30 K, as seen in Figs. 5.6 and 5.7, which permit spin correlation times to be extracted from the data. As noted above magnetic contributions to the specific heat are accounted for by assuming low-lying crystal field split states for the Pr3+ ions [90]. Following this model one may expect the low temperature dependence of the correlation time for transitions between the ground

Δ1/T state and the ﬁrst excited state to be given at low T by τ = τ0e with Δ1 the lowest energy gap and the pre-exponential factor τ 10−11 s. This predicts that for T<Δ the 0 ∼ 1 correlation time for spin ﬂuctuations will increase rapidly with decreasing T. Behavior of this kind has been found in μSR and Ga NQR relaxation rate measurements in the quasi-2D

AF NiGa2S4 [86]. However, the present relaxation rate behavior cannot be accounted for using the Arrhenius expression with a ﬁeld-independent energy gap. The slope of the low

T region of the log-log plot of 1/T1 versus T in Fig. 5.7 (inset) gives a ﬁeld-dependent gap

ΔNMR which is discussed below. Spin-lattice relaxation is attributed to fluctuating hyperfine fields, produced by the electron moments on nearby Pr3+ ions, which induce nuclear spin state transitions. The

97 0.0 (a) (b)

69 5 C(1) 230 K -0.5 110 K 4 70 K

40 K -1.0 3 30 K

2 20 K -1.5 69 10 K C(2) 1 7 K 69 69 C(1) C(2) 4 K -2.0 20 40 60 80 9.2 9.3 9.4 9.5 9.6

Figure 5.5: (a) 69Ga Knight shifts measured in an applied ﬁeld of 9 T along the crystal c-axis plotted versus the magnetic susceptibility with temperature as the implicit parameter. For T>30 K the plot shows that but for lower T departures from a linear relationship are observed. (b) 69Ga NMR spectra as a function of T.

fluctuating local field HL at a Ga nuclear site is mainly due to dipolar interactions with neighboring electron spins and any transferred hyperfine interaction plays a subsidiary role.

The dipolar interaction induces nuclear transitions at the Larmor frequency ωI while an isotropic hyperﬁne interaction will induce mutual electron-nucleus transitions at the much higher frequency ω ω where ω is the electron frequency. For long correlation times I ± S S the dipolar process is dominant and we can neglect any contributions due to the hyperﬁne

τ2 coupling. The relaxation rate may be written as 1/T1 = C⊥( 2 2 ) where τ2 is the transverse 1+ω1τ2 correlation time for electron spins and where C = γ2 H2 is the transverse component of ⊥ I ⊥ τ2 the local ﬁeld [2, 3]. Similarly we have 1/T2 = C⊥( 2 2 )+C τ1 with τ1 the longitudinal 1+ω1τ2 correlation time, C = γ2 H2 and H the z-component of the local ﬁeld. The introduction of I

98 4 0.4

1/T 2 10 1 1/T2 10 C 6 mag 4 0.3 8 2 7.09 T 8.90 T 1 12.06 T 6 16.37 T 6 2 4 T behavior 0.2 2 4 6 2 4 6 2 1 10 100 4

0.1 2

0 0.0 2 3 4 5 6 7 2 3 4 5 6 7 2 1 10 100

69 Figure 5.6: Ga 1/T1 and 1/T2 for Pr3Ga5SiO14 as a function of T at 16.37 T, together 69 with specific heat at 9 T whose peak is coincident with the broad maximum of 1/T2. The similarity in behavior of the two quantities is striking and points to a common underlying 69 mechanism. Inset: log-log plot of 1/T2 at different fields. Notice that the broad maximum sharpens as the field is increased and below the peak the behavior is close to T 2.

both transverse and longitudinal electron spin correlation times allows for the possibility of diﬀerent mechanisms being of dominant importance for τ1 and τ2 respectively. A maximum in 1/T1 occurs for ωI τ2 = 1 while for higher T in the short correlation time case (ω1τ2 < 1) the dipolar mechanism leads to 1/T C τ and 1/T C τ + C τ . Inspection of Fig. 5.6 1 ∼ ⊥ 2 2 ∼ ⊥ 2 1 shows that for T>30 K, the 1/T1 and 1/T2 curves lie close together suggesting that the transverse ﬂuctuations are of dominant importance, corresponding to C⊥ >C , for both relaxation rates in this interval. It is likely that τ1 = τ2 at high T. The condition for

99 the maximum in 1/T1 permits τ2 values to be obtained as a function of T as shown in Fig. 5.7. Assuming the Arrhenius relation holds in the high - T region the fitted curve gives τ 10−11s and the energy gap for spin excitations Δ = 98 K. The energy gap obtained 0 ∼ from the slopes of the curves in the low T region of Fig. 5.7, denoted ΔNMR, is clearly field-dependent with values plotted versus H in the upper inset. The fitted curve in this plot has the form Δ =Δ + αH where the slope α gμ with μ the Bohr magneton and NMR 0 ≈ B B 3+ g =3.32 close to the g value for Pr ion. The magnitude of the zero-field gap Δ0 =3.5K (see Fig. 5.7 caption) obtained from the low temperature (T<10 K) NMR relaxation data is much smaller than the high T NMR value or that from the specific heat results. The finding of the linear field dependence of ΔNMR is new. It is likely that field-suppressed magnetic

ﬂuctuations, similar to those that have been observed in Nd3Ga5SiO14 [72], are responsible for the observed ﬁeld-dependence but the mechanism is not clear.

While 1/T1 decreases at temperatures below the maximum shown in Fig. 5.6 1/T2 continues to increase as T is lowered before passing through a maximum and then decreasing dramatically below 7 K. The behavior is strongly field-dependent and this again points to field suppression of magnetic fluctuations. Fig. 5.7 compares the behavior of 1/T2 at 16.37 T with that of the specific heat with the scales adjusted so that the maxima roughly coincide. The magnetic specific heat Cmag(T ), which is obtained from the measured CP (T ) by subtracting 2 the lattice contribution using La3Ga5SiO14 as a reference, shows T behavior for H = 0.15 T. The entropy saturates at R ln 3 as expected for a system with three low-lying crystal field states. The temperature dependence of 1/T2 is strikingly similar to that of Cmag(T ) as seen in Fig. 5.6 for H = 9 T and this similarity in form points to a common underlying process involved in both observed behaviors. Similar behavior is observed in NiGa2S4 where the spin

“freezing” temperature Tf is coincident with the peak in Cmag but there is a NMR wipeout region in which the relaxation rates could not be measured [86]. Pr3Ga5SiO14 however provides a complete range of temperature which we can study the spin dynamics. The peaks in 1/T2 and Cmag occur near7Kbelowwhich occupation of the singlet ground state by the electron spins rapidly increases and both 1/T2 and Cmag show a sharp decrease. When local ordering of spins becomes important, a reduction in H may occur. This eﬀect will be particularly marked if spin ordering occurs in the a-b plane. The NMR spectra for T<1K show little change in behavior of the linewidth and Knight shift in this range.

The quadratic T -dependence of Cmag for H = 0 T is interpreted as evidence for gapless

100 0 40 16.37 T 12.06 T 30 Δ

20 (K)

10 -3 NMR Neutron 0 7.09 T 0 5 10 15 20 Field (T)

(s) 4.47 T τ 2.42 T

log -6

1 10 100 2 4 6 8 2 4 6 8 2

0 10 ) -1 -2 10

(ms 3

-9 1 ~T -4 1/T 10 69

-6 10 0.0 0.2 0.4 0.6 0.8

Figure 5.7: Temperature dependence of the transverse spin correlation time τ2 at different 69 fields extracted from 1/T1 vs T plot (see lower inset). At high temperatures, the gap Δ 98 K obtained from the slope is field-independent. Below 10 K, the gap Δ has field ≈ NMR dependence (see upper inset) where the dashed line corresponds to the fit ΔNMR =Δ0 + αH with α = gμB =3.32μB and Δ0 =3.5 K. The field-dependence of the spin gap in the excitation spectrum as derived from 35 mK inelastic neutron scattering results (from Ref. [90]) is shown for comparison.

101 Goldstone modes. For H>0T,Cmag(T ) has a minimum at 80 mK with the low-T upturn ascribed to nuclear contributions to the specific heat [90]. Elastic neutron scattering measurements for T<1 K show that nanoscale ordering occurs in the presence of an applied field consistent with 2D short-range order and give a correlation length of 29 A(˚ 6to7 ∼ in-plane lattice spacings) for H =9T[90]. The present relaxation rate results suggest that at low temperatures the spin correlation time τ becomes long as shown in Fig. 5.7 and a possible reduction in the c-axis component of the dipolar field at Ga sites results from in-plane short range ordering leading to the anomalous behavior of 1/T2.

5.5 Conclusion

In conclusion, evidence supporting a short-range spin ordered state at low T in Pr3Ga5SiO14 has been obtained from 69Ga NMR measurements. The correlation time τ for spin fluctuations extracted from the spin-lattice relaxation rate values exhibits novel features; τ increases with decreasing T and below 10 K the results are consistent with a field-dependent energy gap in the excitation spectrum. The spin-spin relaxation rate shows a maximum close to that in the specific heat and the form of the temperature dependence of these two quantities is very similar below 10 K. The decrease in 1/T2 is attributed to a decrease in the local field at Ga sites linked to field-induced nanoscale ordering of the electron spins in the a-b plane.

93 5.6 Nb NMR Probe of Ba3NbFe3Si2O14

Previously magnetic frustration in langasite compounds were discussed in light of single- ion anisotropy and exchange interaction. Now another material with similar network of triangular spins is investigated due to a probable multiferroicity in a geometrically frustrated lattice. Multiferroic materials exhibit two coexisting phenomena: magnetism (ferromagnetism or antiferromagnetism) and ferroelectricity. These materials are of great technological importance and intense amount of research are presently devoted to create more compact, high density memory devices [91]. A notable example of multiferroic with a triangular motif are the hexagonal manganites RMnO (R = Ho Lu, Y ). YMnO for instance exhibits ferroelectricity at very high 3 − 3 T 700 K and at the same time exhibit antiferromagnetic transition at a much lower c ≈

102 Figure 5.8: Crystal structure of Ba3NbFe3Si2O14 viewed along (a) c-axis (b) b-axis. (c) Room temperature X-ray diﬀraction pattern. Inset: temperature dependence of the lattice parameters (from Ref. [92]).

temperature T 70 K. Near the the N´eel temperature this material shows multiferroic N ≈ properties.

Ba3NbFe3Si2O14, also known for its piezoelectric properties, is a member of the langasite family of materials discovered in the USSR in the early 1980s. It has subnet magnetic Fe3+ cations (S =5/2, top spin, L = 0) consisting of a triangular arrangement of triangles isolated, making it particularly useful for studying the magnetic frustration. Previous studies have shown that there is a magnetic transition at 26 K. The Fe3+ ions form a network of triangular units in the a b plane. The Nb5+ cations are located above or beneath these triangles along − the c-axis separating the Fe layers, which is similar to the structural arrangements of the hexagonal RMnO3.

The single crystals of Ba3NbFe3Si2O14 were grown by travelling-solvent ﬂoating-zone technique. The crystals appear as black rods with average diameter of 3 mm and length 1 cm.

X-ray diﬀraction measurements show that Ba3NbFe3Si2O14 has a single phase with hexagonal

103 Figure 5.9: (a) Inverse DC susceptibility where the solid line fit is the Curie-Weiss law. (b) Temperature dependence of the specific heat of Ba3NbFe3Si2O14 (open circles) and Ba3Nb(Fe0.5Ga0.5)3Si2O14 (solid line) (c) The magnetic contribution to the specific heat and the calculated entropy (d) Temperature dependence of thermal conductivity (from Ref. [92])

104 non-centrosymmetric P 321 structure (see Fig. 5.8). The DC susceptibility measurement illustrated in Fig. 5.9a shows a sharp drop at 26 K indicating Neel magnetic ordering . The Curie-Weiss ﬁt of the high temperature susceptibility gives a Weiss constant θ = 190 K, − thus the frustration index of this material is f = |θ| =7.3, a much lower value than the Nd- Tc and Pr-langasites. From the susceptibility analysis, the eﬀective moment is 5.58 μB which 3+ 5 is closed to the bare Fe value (S = 2 , 5.9 μB).

The λ-shaped peak in the specific heat at TN = 26 K shown in Fig. 5.9b indicates second- order transition. To isolate the magnetic contribution to the specific heat, Ga was substituted to 50% of the Fe sites resulting in the compound Ba3Nb(Fe0.5Ga0.5)3Si2O14 which shows no magnetic ordering down to 2 K. The Ga substitution weakened the interaction among the Fe3+ spins, making the resulting compound a good reference to isolate the magnetic contribution to the specific heat of Ba3NbFe3Si2O14. Moreover there is about 40% loss in the total magnetic entropy above TN . This large release of entropy is attributed to the formation of strong spin fluctuations above TN . An abrupt jump in thermal conductivity is seen at 26 K (see Fig. 5.9d), corroborating the antiferromagnetic nature of this sample. At high temperatures, the thermal conductivity has a relatively weak temperature dependence and a broad maximum is seen at 50 K which could be due to the spin fluctuations above TN . There is a low-temperature peak below

TN which may be attributed to the phonon contribution κph(T) restored by the long-range magnetic ordering. On the other hand, the dielectric constant exhibits a broad transition around 30 K which is concomitant with the sharp increase in the polarization around 24 K, which is a little bit lower than TN =26K. The diﬀerence in the electric and magnetic transitions occurs in other multiferroic systems. We report on the use of 93Nb nuclear magnetic resonance to probe the internal magnetism in the geometrically frustrated multiferroic system Ba3NbFe3Si2O14. The spin-lattice relaxation rate 93Nb 1/T shows a magnetic transition at T 26 K and the spectra changes 1 c ≈ when crossing Tc. These results point to the magnetoelastic motion of the Nb atoms in the lattice. 93Nb is a spin-9/2 nucleus with 100% natural abundance and has Q = 0.22. − The spin dynamics of this multiferroic candidate is shown in the temperature dependence 93 of the spin-lattice relaxation rate 1/T1 (see Fig. 5.12a) and the spin-spin lattice relaxation

105 40 K 35 K 7.80 1.5

50 K 7.75

60 K 7.70 1.0

7.65 80 K 50 100 150 200 250

100 K 0.5

125 K 150 K 175 K 200 K225 K 250 K

0.0 7.60 7.65 7.70 7.75 7.80 7.85

93 Figure 5.10: Field-swept Nb spectra in the paramagnetic state of Ba3NbFe3Si2O14. Inset: Plot of the resonant ﬁeld versus temperature reﬂecting the spin susceptibility.

93 rate 1/T2 (see Fig. 5.13). The antiferromagnetic spin fluctuation is evident in the huge upturn of both relaxation rates near the Néel temperature T 27.5K. This type of N ≈ relaxation behavior can be explained by the similar mechanism of the fluctuation near TSDW in organic conductors described in the previous chapter of this dissertation. The fluctuation region fits very well with the equation for the self-consistent renormalization (SCR) theory for weak itinerant antiferromagnets [38]:

B A + 1/2 forT >TN 1 (T −TN ) = C forT >T (5.1) T1T (1− T )α N TN Note that the Korringa behavior describes the high temperature relaxation as expected for paramagnetic region, but it starts to deviate below 50 K – the same temperature where the broad maximum in thermal conductivity was observed. This conﬁrms the fact that spin

106 o (a)40 (b) 4.2 K 8.5 Left Peak o 8.0 60 o 4.2 K 90 65 27 7.5 Cu Al 63 Cu 7.0 Right Peak

6.5

6 7 8 9 -100 -50 0 50 100 150

Figure 5.11: (a) Fieldswept 93Nb spectra at constant frequency 83.4 MHz versus field orientation (θ is the angle between B and a-b plane) at 4.2 K in the antiferromagnetic state. The small sharp peaks are 63,65Cu and 27Al NMR signals from the coil. (b) Plot of the left and right resonant peaks. The dashed lines are fits to the equation A cos θ where A is the hyperfine coupling constant.

fluctuations drive this particular feature in thermal conductivity as mentioned earlier. There is a small region in temperature near the Néel temperature where the relaxation rates are too fast to measure, thus no spectra or NMR in general can be gathered. This is commonly referred to as the NMR wipeout effect; this was seen in a wider temperature range in the Nd-langasite relaxation at lower fields. At lower temperatures, there is a small bump in 93Nb 1/T and 93Nb 1/T around T ∗ 4K which could be due to spin reorientation effect 1 2 ≈ but further studies like neutron scattering are needed to confirm this. This low temperature feature in the relaxation rate is also seen in thermal conductivity (see Fig. 5.9d). In Fig. 5.10, the temperature dependence of the 93Nb spectra is shown and a strong 93 temperature-dependent Knight shift is evident. The Knight shift Ks reflects the local spin susceptibility measured at the nuclear site and the corresponding Clogston-Jaccarino plot 93 ( Ks vs χs) reveals a large hyperfine coupling constant. Notice that the NMR spectra become broader as the temperature is lowered down to 35 K in the paramagnetic region.

107 TN=27 K PP (a) PP (b) LP 2.5 1.5 RP MP

93 Figure 5.12: (a) Temperature dependence of 1/T1T in Ba3NbFe3Si2O14 at 83.4 MHz in the paramagnetic state (PP-paramagnetic peak) and antiferromagnetic state (LP-left peak, MP- middle peak, and RP-right peak). The dashed in the paramagnetic region is ﬁt to SCR spin ﬂuctuation theory with TN 27.5 K. (b) Corresponding temperature-dependent spectra in the two states. The location≈ of the peaks are indicated and the middle sharp line is 27Al NMR signal from the probe.

9 This is attributed to quadrupolar broadening (since Nb has spin I = 2 ) and spin fluctuation effects as it gets close to the antiferromagnetic transition. Also, there is another 93Nb peak which becomes more pronounced as the the temperature is lowered. This peak barely shifts with temperature and this could be another chemically inequivalent site of Nb. It seems that the shifting 93Nb spectrum gets closer to the other peak as it gets near the Néel temperature.

Near TN , there is a small NMR wipeout eﬀect region where the relaxation rates are too fast to measure. Below TN , a broad NMR lineshape with peak distinct peaks is observed. This is characteristic of the inhomogeneous local magnetic ﬁeld due to antiferromagnetic ordering. As mentioned earlier, this is associated with the onset of an increase of polarization and a concomitant decrease in the dielectric constant. Clearly, there is a coexisting electric and magnetic ordering in this material which makes it a multiferroic.

108 T 0.5 N

MP 80 0.4 T* 0.3 60 0.2 PP

40 0 5 10 15 20 25 T* LP 20 RP MP

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 10 100

93 Figure 5.13: Temperature dependence of the spin-spin relaxation rate 1/T2 measured in the paramagnetic and antiferromagnetic regions at 83.4 MHz with B a b plane. The dashed ⊥ − line is a ﬁt similar to SCR spin ﬂuctuation behavior. Inset: Dependence of the stretched exponential parameter β of the middle peak. The change in β at around 4 K corresponds to T ∗ seen in the relaxation rate measurements.

With regards to the relaxation in the ordered phase, the two side peaks have more or less the same relaxation values while the spins in the middle peak relax at a much slower rate. The relaxation rate curves of the two side peaks appear to continue from the relaxation rate of the shifting 93Nb lineshapes in the paramagnetic phase, thus it can be inferred that they refer to the same Nb site. A broad double-horned NMR lineshape is a typical feature of incommensurate SDW or antiferromagnetic ordering. The origin of the middle peak however is not clear at present. Inhomogeneous local magnetic ﬁeld is clearly the cause of the broadening of the middle peak but it is probably on another site. The movement of the Nb atoms suggested by polarization and dielectric measurements [92] may explain the inhomogeneous local magnetic ﬁeld seen by the Nb atoms. 93Nb, being a quadrupolar nucleus, should be able to detect structural distortions via the coupling of the quadrupolar

109 moment 93Q with the electric field gradient in the ordered phase but the atomic displacements for typical multiferroic is as low as 0.05 A.˚ Another clue to the spin structure in the ordered phase of this material comes from recent powder neutron diffraction data which suggest the presence of a long-ranged incommensurate spiral ordering of spins along the c-axis with the moments sitting on the a b plane [93]. − In the ordered phase, the angular dependence of the 93Nb field-swept NMR spectra at constant NMR frequency and temperature reveal the antiferromagnetic ordering does not follow the conventional dipolar coupling behavior given by ΔH = A0 (3 cos2 θ 1). Instead ± 2 − the angular dependence of the peak splitting is best described by the fitting A cos(θ). ± 0 This deviation reveals that the coupling among the spins is not simply dipolar in nature.

5.7 Conclusion

93 Nb NMR spectroscopy in Ba3NbFe3Si2O14 confirmed the existence of antiferromagnetic order below T 27 K evident in the divergence of both 931/T and 931/T and the N ≈ 1 2 broadening of 93Nb NMR spectra. A small NMR wipeout region is seen in the vicinity of the Néel ordering temperature. The spectra below TN show three-peak structure where the middle peak has a different relaxation behavior than the side peaks, suggesting two chemically inequivalent sites of 93Nb. Further analysis is needed to ascertain whether the existence of the middle peak in the spectrum is due to the movement of Nb atoms which 93 93 could lead to a different magnetic environment. The peak in 1/T1 and 1/T2 at around 4 K may be attributed to spin reorientation effect but further magnetic measurements are needed to confirm this.

110 CHAPTER 6

CONCLUSION

Nuclear magnetic resonance provided information about the internal magnetism and spin dynamics of the different condensed matter systems discussed in this dissertation. Through analysis of the linewidth, resonant frequency shift, and intensity of NMR spectra, information on the local magnetic field of the material can be extracted. On the other hand, the behavior of the NMR relaxation rates and correlation times can be one of the definitive tests to establish the magnetic character of a material. As such, I listed the various NMR parameters in Table 6.1 used in this work. The major findings in this work are listed below: The simultaneous NMR and electrical transport measurements in the quasi-one-dimensional 77 organic conductor (TMTSF)2ClO4 reveal that the peaks in 1/T1 are not coincident with the second-order FISDW phase boundary, pointing to a mechanism similar to the Hebel-Slichter peak in conventional superconductors where the peak in 1/T1 is slightly below Tc. The existence of the re-entrant FISDW region is confirmed by NMR due to the distinct relaxation

Table 6.1: NMR parameters relevant to the work done in this dissertation.

parameter what they measure Ks spin susceptibility spectra magnetic structure FWHM internal magnetic field 1/T1 fluctuating local magnetic field 1/T2 fluctuating local magnetic field τ magnetic fluctuations η enhanced susceptibility Intensity number of spins

111 behavior above and below this phase boundary. The angular-dependent measurements facilitate the crossing of FISDW phase boundaries at constant temperature and magnetic 77 field, where peaks in 1/T1 are also observed, indicating changes in the nesting configurations at these transitions. At high fields, the drop in the rf-enhancement factor η upon crossing the re-entrant phase boundary is consistent with the expectation that only one FS sheet is nested above Bre. The magnetic character of FISDW in (TMTSF)2ClO4 and SDW in

(TMTSF)2PF6 is close to a weakly itinerant antiferromagnet due to the agreement of the data with theory. The breaking of the spin-Peierls state at 20 T at low temperatures (500 mK to 3 K in this case) in the quasi-one-dimensional organic conductor (Per)2Pt[mnt]2 was conﬁrmed by the loss of 195Pt NMR signal beyond this ﬁeld. Charge density wave (CDW) in perylene chains and spin-Peierls (SP) in the Pt chain are thought to coexist until B>20TSPis broken and CDW is suppressed to zero. The loss of 195Pt NMR signal is most likely due to

NMR wipeout eﬀect where the relaxation rates, either or both 1/T1 and 1/T2, are too fast to measure.

The second CDW system discussed is the newly discovered superconductor CuxTiSe2. 63Cu and 77Se NMR measurements show that the temperature-dependent spin-lattice relaxation rate can be explained by a modified Korringa relation. As the Cu content increases near the optimum superconducting doping, the electron-electron interaction parameter increases thereby enhancing the spin-lattice relaxation rate. 69,71Ga nuclear magnetic resonance probed the spin dynamics in the rare-earth kagomé 69 system Pr3Ga5SiO14. The spin-lattice relaxation rate 1/T1 exhibits a maximum around 30 K, below which the Pr3+ spin correlation time τ shows novel field-dependent behavior consistent with a field-dependent gap in the excitation spectrum. The spin-spin relaxation 69 rate 1/T2 exhibits a peak at a lower temperature (10 K) below which field-dependent 2 69 power-law behavior close to T is observed. In addition, the peak in 1/T2 is coincident with the broad maximum in specific heat pointing to a common underlying mechanism. These results point to the interplay of single-ion anisotropy and field-induced formation of nanoscale magnetic clusters consistent with recent neutron scattering measurements. On the other hand, 93Nb NMR studies on the structurally similar langasite system

Ba3NbFe3Si2O14 show antiferromagnetic ordering below TN = 27 K. Hints of multiferroic behavior due to movement of Nb atoms maybe evident in the 93Nb NMR spectra, but

112 further analysis is needed to conﬁrm this assumption. 135,137Ba NMR spectroscopy on this material may provide additional information. 6.1 Future Work

It would be interesting to do a comparative NMR study of the spin dynamics and the development of order parameter in the FISDW transitions in (TMTSF)2PF6 (under 12 kbar hydrostatic pressure) which has a single pair of nested Fermi surfaces and in (TMTSF)2ClO4 which has two pairs of nested Fermi surfaces. The present thought is that the ordering of the tetrahedral ClO4 anions has an important consequence in the development of extra high

field FISDW features seen in (TMTSF)2ClO4, but not in (TMTSF)2PF6. 195 In addition to Pt NMR lineshapes, a follow-up work on (Per)2Pt[mnt]2 by measuring 195 field dependence of 1/T1 cutting through boundary where Spin-Peierls break (around 20 T) would give additional insight into its electronic structure. A single crystal NMR measurement is highly desired though some improvement has to be made in the filtering of noise in the spectrometer at high fields. This will eliminate the problem of mixed phases encountered in multiple crystal measurements. It is of considerable interest to investigate the effect of very high magnetic fields up to 69,71 69,71 35 T on the temperature dependence of the relaxation rates 1/T1 and 1/T2 of Nd- and Pr-langasites. The partial field-induced magnetic order in Nd-langasite revealed by neutron measurements [71] is particularly interesting from an NMR point of view because this will tell us how spin-liquid behavior is disrupted at high magnetic fields. The other variants of the Pr-langasite system by substitution of Si with group IVB elements (Sn, Ge, Pb) provide an opportunity to study the effects of chemical pressure on the competing single-ion anisotropy and spin-liquid behavior in these materials. I propose to do Ga-NMR measurements on these materials and probably apply for high magnetic field time in the

DC magnet facilities. Recently, a spinel-related oxide Na4Ir5O8 is claimed to be the first realization of three-dimensional (3D) kagomélattice (corner-shared triangular arrangement of S =1/2Ir4+ spins) showing spin-liquid state [94, 95]. I would like to investigate spin dynamics on this “hyperkagomé” material via 23Na relaxation rates and compare with the spin dynamical behavior in 2D kagomématerials like the langasites. In the near future, I plan to get involved in the physics of iron-based superconductors which is one of the hottest topics in condensed matter today [96]. Recently the synthesis

113 and characterization of an iron-based superconductor LiFeAs (Tc =18K) was reported [97].

This is quite interesting because of its very high Hc2 (around 80 T) and the absence of SDW in contrast with the other iron-based superconductors. I propose to do 7Li and 75As NMR on this material with particular measurements of temperature-dependent Knight shift and spin-lattice relaxation rate at diﬀerent ﬁelds. This would give a nice comparative NMR study with the other iron-based superconductors investigated earlier.

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122 BIOGRAPHICAL SKETCH

Lloyd L. Lumata

Lloyd L. Lumata was born on the 17th of November 1981 in Zamboanga City, Philippines. He attended Recodo Elementary School (1988-1994) and Ayala National High School (1994- 1998) where he graduated class valedictorian. With a scholarship from the Philippine Department of Science and Technology, he completed his Bachelor’s degree in physics at the Western Mindanao State University (1998-2002) where he graduated Magna Cum Laude and gave the valedictory address to the class of 2002. While at the university, he had a stint as Editor-in-Chief of the University Digest, the oﬃcial English-based student magazine of the university. He taught fundamental physics courses in his Alma Mater for two years. He attended the Department of Physics graduate school of the Florida State University in the Fall of 2004 and served as a teaching assistant for two years. He got his master’s degree in the Fall of 2006. He worked as a research assistant (RA) at the National High Magnetic Field Laboratory (NHMFL) under the supervision of Prof. James S. Brooks. His Ph.D. work focused on nuclear magnetic resonance (NMR) investigation of density waves in organic conductors and the spin liquid behavior in frustrated spin systems. His NMR training was done under the auspices of Dr. Arneil Reyes and Dr. Philip Kuhns. While at FSU, he served as an associate for physics in the Program for Instructional Excellence (PIE) for two years (2006-2008) where he organized teaching seminars to train the new teaching assistants of the FSU Department of Physics. Lloyd has four siblings: Richard, Analyn, Edwin, and Jenica. He is the eldest son of Jose and Evelyn Lumata. He is planning to continue working on NMR in strongly-correlated electron systems and the application of NMR in medical research.

123 Publications:

1. Low electrical conductivity threshold and crystalline morphology of single-walled carbon nanotubes-high density polyethylene nanocomposites characterized by SEM, Raman spectroscopy and AFM, Keesu Jeon, Lloyd Lumata, Takahisa Tokumoto, Eden Steven, James Brooks, and Ruﬁna G. Alamo, Polymer 48 Issue 16, 4751-4764 (2007).

2. Magnetic-polaron-driven magnetoresistance in the pyrochlore Lu2V2O7, H.D. Zhou, E.S. Choi, J.A. Souza, J. Lu, Y. Xin, L.L. Lumata, B.S. Conner, L. Balicas, J.S. Brooks, J.J. Neumeier, and C.R. Wiebe, Phys. Rev. B 77, 020411(R) (2008).

77 3. Se NMR probe of the ﬁeld-induced spin density wave transitions in (TMTSF)2ClO4, L.L. Lumata, J.S. Brooks, P.L. Kuhns, A.P. Reyes, S.E. Brown, H.B. Cui, and R.C. Haddon, Phys. Rev. B 78, 020407(R) (2008); also arXiv:0807.3119.

3+ 4. Ba3NbFe3Si2O14: a new multiferroic with a 2D triangular Fe motif, H.D. Zhou, L.L. Lumata, P.L. Kuhns, A.P. Reyes, E.S. Choi, J. Lu, Y.J. Jo, L. Balicas, J.S. Brooks, and C.R. Wiebe (accepted for publication in Chemistry of Materials, 2008).

5. Angular and temperature-dependent 77Se NMR in the metallic and ﬁeld-induced spin

density wave phases of (TMTSF)2ClO4, L.L. Lumata, J. S. Brooks, P.L. Kuhns, A.P. Reyes, H.B. Cui, S.E. Brown, R.C. Haddon, and J.-I. Yamada, J. Phys.: Conf. Ser. 132, 012014 (2008).

6. Chemical pressure-induced spin freezing phase transition in kagom´ePr-Langasites H. D. Zhou, C. R. Wiebe, Y.-J. Jo, L. Balicas, L. L. Lumata, J. S. Brooks, P. L. Kuhns, A. P. Reyes, Y. Qiu, R. D. Copley, and J. S. Gardner (submitted to PRL, 2008).

7. Dynamical behavior of spins in the spin-liquid Kagome Pr3Ga5SiO14, L.L. Lumata, K.- Y. Choi, H.D. Zhou, M.J.R. Hoch, J.S. Brooks, P.L. Kuhns, A.P. Reyes, T. Besara, N.S. Dalal, and C.R. Wiebe (submitted to PRL, 2008).

Manuscripts in preparation:

93 8. Nb local magnetic probe of the geometrically-frustrated multiferroic Ba3NbFe3Si2O14, L.L. Lumata, H.D. Zhou, J.S. Brooks, P.L. Kuhns, A.P. Reyes, and C.R. Wiebe.

124 9. NMR probe of the Electronic Correlations in CuxTiSe2 L.L. Lumata, K.-Y. Choi, J.S. Brooks, P.L. Kuhns, A.P. Reyes, T. Wu, and X.H. Chen.

10. 195Pt NMR Investigation of the Breaking of the Spin-Peierls State in the Organic

Conductor (Per)2Pt(mnt)2, L.L. Lumata, D. Graf, J.S. Brooks, S.E. Brown, P.L. Kuhns, A.P. Reyes, M. Almeida

11. 51V NMR study of the S =1/2 quasi one-dimensional Ising-like antiferromagnet

BaCo2V2O8

135,137 12. Ba NMR Study of Ba3Mn2O8 S. Suh, S. E. Brown, L. L. Lumata, et. al.

Posters, Talks, and Presentations

1. Angular and temperature-dependent 77Se NMR in the metallic and spin-density wave

phases in (TMTSF)2ClO4, American Physical Society Conference (Denver, CO March 2007).

2. Spin dynamics in the ﬁeld-induced spin-density wave phases of (TMTSF)2ClO4 Amer- ican Physical Society Conference (New Orleans, LA March 2008).

3. High-Current Hunt for Bardeen-Stephen ﬂux motion in A15 Superconductor V3Si at High Fields, R. Khadka, A.A. Gapud, A.P. Reyes, P. Kuhns, L.L. Lumata, D.K. Christen, and J.R. Thompson, 74th Annual Meeting of the Southeastern Section of the American Physical Society, Nashville, TN, November 8-10 (2007).

4. Towards ordered flux flow in A15 superconductor V3Si at high fields R. Khadka, A.A. Gapud, A.P. Reyes, L. Lumata , P.L. Kuhns , and D.K. Christen New Orleans, LA APS March Meeting 2008.

5. Electrical Conductivity and Crystalline Morphology of Single-Walled Carbon Nan- otube/Linear Polyethylene Nanocomposites, K. Jeon, R.G. Alamo, L.L. Lumata, T. Tokumoto, and J. Brooks, ACS National Meeting. Division of Polymer Chemistry, Chicago, IL, March 25-29 (2007).

135,137 6. Ba NMR study of Ba3Mn2O8 Steve Suh, W.G. Clark, Guoqing Wu, S.E. Brown, E.C. Samulon, I.R. Fisher, C.D. Batista, A.P. Reyes, P.L. Kuhns, and L.L. Lumata, New Orleans, LA APS March Meeting 2008.

125 7. Unusual transport properties in the orbitally-ordered system Lu2V2O7 H.D. Zhou, B. Conner, B.W. Vogt, C.R. Wiebe, L.L. Lumata, J.S. Brooks, E.S. Choi, and Y. Xin Denver, CO APS March Meeting 2007.

8. Orbital ordering transitions in the vanadium oxides Sr2VO4 and Lu2V2O7, C. R. Wiebe, H. D. Zhou, B. S. Conner, Y.-J. Jo, L. Balicas, Y. Xin, J. Lu, J. J. Neumeier, L. L. Lumata, and J. S. Brooks, International Workshop on Synthesis of Functional Oxide Materials, Santa Barbara, CA August 2007.

9. An Introduction to Student Teaching in the Department of Physics–a talk on teaching responsibities, tips on lecture delivery, grading, and teaching resources available at FSU. This was given at the FSU Physics TA Training for the new graduate students (August 2006).

10. Student Teaching in the Department of Physics: An Introduction FSU Physics TA Workshop, Tallahassee, FL August 2007.

11. Student Teaching in the Department of Physics: An Introduction FSU Physics TA Workshop, Tallahassee, FL August 2008.

12. Spin Dynamics of Density Wave and Frustrated Spin Systems Probed by NMR Special Seminar, 241 Compton, Department of Physics, Washington University in St. Louis, St. Louis, MO, November 6, 2008.

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