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Electronic Theses, Treatises and Dissertations The Graduate School
2008
Spin Dynamics of Density Wave andFrustrated Spin Systems Probed by NuclearMagnetic ResonanceLloyd L. (Lloyd Laporca) Lumata
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FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCES
SPIN DYNAMICS OF DENSITY WAVE AND FRUSTRATED SPIN
SYSTEMS PROBED BY NUCLEAR MAGNETIC RESONANCE
By
LLOYD L. LUMATA
A Dissertation submitted to theDepartment of Physics
in partial fulfillment of therequirements 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. BrooksProfessor Directing Dissertation
Naresh DalalOutside Committee Member
Arneil P. ReyesCommittee Member
Pedro SchlottmannCommittee Member
Christopher WiebeCommittee Member
Mark RileyCommittee Member
Approved:
Mark Riley, ChairDepartment of Physics
Joseph Travis, Dean, College of Arts and Sciences
The Office of Graduate Studies has verified and approved the above named committee members.
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To my family...
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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.
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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.
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TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1. INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE . . . . . . . . 11.1 Knight Shift: Probing the Internal Magnetism . . . . . . . . . . . . . . . 21.2 Hyperfine Coupling Terms of the Interaction Hamiltonian . . . . . . . . 51.3 Measuring the Dynamics: Relaxation Rates . . . . . . . . . . . . . . . . 61.4 Temperature-dependent Relaxation in Metals . . . . . . . . . . . . . . . 121.5 Hebel-Slichter Peak: Test of BCS Superconductivity . . . . . . . . . . . 151.6 NMR Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2. AN OVERVIEW OF LOW-DIMENSIONAL ORGANIC CONDUCTORS . . 232.1 Low Dimensional Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 242.2 The Bechgaard Salts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Phase Diagram of (TMTSF)2X . . . . . . . . . . . . . . . . . . . . . . . 37
3. SIMULTANEOUS 77Se NMR AND TRANSPORT INVESTIGATION OFTHE SPIN DENSITY WAVE SYSTEMS (TMTSF)2X, X=ClO4, PF6 . . . . . 413.1 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Temperature Dependence of NMR Spectra . . . . . . . . . . . . . . . . . 463.3 The RF Enhancement Factor η . . . . . . . . . . . . . . . . . . . . . . . 473.4 RF Power dependence of NMR lineshapes in the FISDW State . . . . . . 483.5 Field Dependence of 771/T1, Rzz, and Spectra at Constant Temperature . 513.6 Temperature Dependence of 771/T1 at Low Fields . . . . . . . . . . . . . 533.7 Angular Dependence of 77Se NMR and Transport in the Metallic State . 553.8 Angular Dependence of 77Se NMR and Transport in the FISDW State . 573.9 Temperature Dependence of 771/T1, Spectra, and Rzz at High Fields . . 633.10 A Comparative Study: 77Se NMR and Transport on (TMTSF)2PF6 . . . 673.11 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4. NMR ON CHARGE DENSITY WAVE SYSTEMS . . . . . . . . . . . . . . . 75
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4.1 Coexisting CDW and Spin-Peierls States in (Per)2Pt[mnt]2 . . . . . . . . 754.2 Crystal Structure and Electronic Properties . . . . . . . . . . . . . . . . 774.3 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.6 CuxTiSe2: a new CDW-Superconductor . . . . . . . . . . . . . . . . . . 824.7 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.8 77Se and 63Cu NMR Studies of CuxTiSe2 . . . . . . . . . . . . . . . . . . 854.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5. PROBING THE DYNAMICS OF FRUSTRATED SPIN SYSTEMS . . . . . 905.1 A survey of Frustrated Spin Systems . . . . . . . . . . . . . . . . . . . . 905.2 The Rare-Earth Kagome R3Ga5SiO14 . . . . . . . . . . . . . . . . . . . . 935.3 69,71Ga NMR Probe of the Spin Dynamics of Pr3Ga5SiO14 . . . . . . . . 935.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.6 93Nb NMR Probe of Ba3NbFe3Si2O14 . . . . . . . . . . . . . . . . . . . . 1025.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
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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
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LIST OF FIGURES
1.1 A simple mechanism of NMR: rf irradiation of the nuclear moment precessingat Larmor frequency. With energy equal to the Zeeman splitting, rf can flipthe nuclear spin then it returns to equilibrium releasing a signal. The lowerfigure 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 fromthe reference due to local internal magnetic field. . . . . . . . . . . . . . . . 3
1.3 The spin-echo pulse sequence: the magnetization is tipped by a 900 pulse andthe spins start to “fan out”on the XY plane forming a FID signal. A secondpulse flipped the spins 1800 and they regroup then fan out again forming aspin echo signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Measuring the spin-lattice relaxation time T1: (a) a π/2 “saturation” pulseis followed by a variable delay time which allows the growth of longitudinalmagnetization Mz as it increases. The π/2− π spin-echo sequence “inspects”the recovery of this magnetization which is reflected in (b) where the constantof 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 theexponential 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) wheref(E) is the Fermi-Dirac function. The red area, which is the product of thetwo probabilities, represents the electrons that participate in the relaxationprocess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Schematic diagram of the transmitter section: gated oscillating electricalsignal 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 (receivemode). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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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 probeand the reference frequency from the synthesizer are mixed in the quadraturereceiver where real and imaginary NMR signals are generated. (b) Spin echosignal showing the real and imaginary components. . . . . . . . . . . . . . . 18
1.11 NMR hardware: (a) NMR rack containing the spectrometer, PC, temperaturecontroller, 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 openingof gap Δ with Fermi wave vector kF = π
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 aredimerized and lattice distortion occurs at q = 2kF (Spin-Peierls state) leadingto stronger antiferromagnetic interaction J + δ among the dimerized electronsand weaker antiferromagnetic interaction J − δ between the pairs. . . . . . . 27
2.3 (a) The dispersion relation in spin density wave state showing the opening ofthe gap Δ at the Fermi wave vector kF . (b) The modulation of the two spinsubbands in SDW with wavelength λ0 = π
kF. . . . . . . . . . . . . . . . . . . 28
2.4 Crystal structure of the TMTSF molecule. . . . . . . . . . . . . . . . . . . . 30
2.5 Crystal structure of (TMTSF)2ClO4 viewed along a-axis. The crystallographicaxes are given by the arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Zero-field cooldown of (TMTSF)2ClO4 showing a kink in resistance at 24 Kwhich is due to anion ordering. . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7 Low-field c*-axis magnetoresistance (MR) of (TMTSF)2ClO4 at differenttemperatures. 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 differenttemperatures showing kinks in MR that correspond to the FISDW phasesBth, B1, B∗, and Bre. Above 15 T, periodic modulations in MR called “rapidoscillations” occur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
x
2.9 (a) Electronic motion in momentum space: a pair of warped FS sheets withinthe first Brillouin zone in a Q1D system. (b) Electronic motion in real spacewith amplitude A = 4tb
evF Bzand wavelength λ = h
ebBz. . . . . . . . . . . . . . . 36
2.10 T-B phase diagram of (TMTSF)2ClO4 showing the metallic region, supercon-ducting (SC) state, the cascade of FISDW phases, and the re-entrant FISDWregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.11 T-B phase diagram of (TMTSF)2PF6 at 12 kbar hydrostatic pressure: thisis similar to (TMTSF)2ClO4 except for the absence of the re-entrant FISDWregion at high magnetic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 The (TMTSF)2ClO4 FISDW phase diagram where the dots represent theFISDW transport features seen in this work. The arrows represent the regionsin 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 pressuresystem and the 30 T resistive magnet at NHMFL Cell 7. . . . . . . . . . . . 45
3.5 Temperature dependence of 77Se NMR spectra in a relaxed (TMTSF)2ClO4
with B ‖ c∗. Part (a) shows frequency-swept spectra in the metallic stateat constant magnetic field B = 12.12 T while (b) shows field-swept spectraat constant NMR frequency 98.2 MHz. The broad, double-horned spectraindicate inhomogeneous local magnetic field in the FISDW state. . . . . . . . 47
3.6 (a) Field-swept 77Se NMR spectra at different rf power level attenuation inthe FISDW state of (TMTSF)2ClO4 at 1.8 K and constant NMR frequency98.6 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-leftpeak, CP-central peak, RP-right peak) in (a). The red arrow at the bottomindicates the direction of increasing rf power. . . . . . . . . . . . . . . . . . . 49
3.7 Field dependence of 771/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 denotesrelaxation rate in the metallic state, the solid triangles indicates the enhancedrelaxation in the FISDW region, and the open circles denote a coexistenceregion or depinned state of FISDW. Inset: the corresponding field dependenceof full-width half maximum (FWHM) of NMR spectra. . . . . . . . . . . . . 52
xi
3.8 Representative temperature dependence of 771/T1 at low fields (B < 20 T)where B ‖ c∗. The dashed lines above the peak are fits to the SCR theory foritinerant antiferromagnets 1/T1 = A T
(T−TN )1/2 and below the peaks are power
law fits 1/T1 = ATα. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.9 (a) Angular dependence of 77Se 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 inRzz and ν − ν0 do not coincide. (c) Full-width half maximum (FWHM) andspin-lattice relaxation rate 771/T1 as a function of angle. . . . . . . . . . . . 56
3.10 Angular dependence of 77Se NMR lineshapes along the a-axis at con-stant 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(TMTSF)2ClO4. (a) 771/T1 versus field orientation θ. A dip in 1/T1 is markedX. (b) Corresponding magnetoresistance and rf enhancement η at 14 T and1.5 K. (c) Schematic of sample rotation along a-axis and the orientation of theTMTSF molecule with respect to magnetic field B at point X. (d) Angular-dependent 77Se NMR spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.12 (a) The spin-lattice relaxation rate 771/T1 as the sample is rotated alongthe a-axis at 12.86 T and 3.87 K. Note the dip in 771/T1 marked X. Thereis no distinct hysteresis in the result as sample is rotated clockwise andcounterclockwise. Inset: schematic of sample rotation. (b) 771/T1 versus θat 12.86 T and 1.87 K. Inset: orientation of TMTSF at point X with respectto 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 NMRspectra at ν0 = 243.9 MHz (30 T) taken at different angles and consequentlydifferent phases: (i) FISDW phase at θ = 1050 (above B1) (ii) Metallic phaseat θ = 900 and (iii) Re-entrant FISDW phase at θ = 00 (above Bre). Thecorresponding magnetoresistance data at different angles are shown in thelower right hand corner. For each trace, the NMR measurement was made at30 T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.14 771/T1 versus effective perpendicular magnetic field B⊥ = B0 cos θ. The redarrows mark the peaks in 771/T1 which coincide with the different FISDWphase boundaries B1, B∗, and Bre seen in electrical transport measurements. 63
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3.15 Crossing the re-entrant FISDW phase: (a) Temperature dependence of 771/T1
and c*-axis resistance Rzz of (TMTSF)2ClO4 at 23 T with B ‖ c∗. 771/T1
exhibits a peak at 5 K which is coincident with the upturn in Rzz. There-entrant phase is denoted by another sharp increase in Rzz at around 3K. (b) Normalized 77Se NMR spectra in the metallic (black) and FISDWphases (red). (c) Corresponding temperature-dependent NMR linewidth andBoltzmann-corrected NMR intensity. . . . . . . . . . . . . . . . . . . . . . . 65
3.16 (a) Temperature dependence of 771/T1 and c∗-axis resistance Rzz measuredsimultaneously at 29 T with B ‖ c∗. (b) Corresponding 77Se NMR spectra inthe metallic (black) and FISDW (red) states. (c) Temperature dependence offull-width half maximum (FWHM) and NMR intensity at 29 T. . . . . . . . 66
3.17 (a) Temperature dependence of 771/T1 and Rzz in (TMTSF)2PF6 at 17 Twhere B ‖ c∗. Dashed lines are fits to certain equations: self-consistentrenormalization (SCR) theory equation A T
(T−TN )1/2 (blue dashed line), power
law ATα where α = 3.2 (yellow dashed line), and AeT/Δ where Δ = 0.65 (greendashed 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
3.18 (a) Angular-dependent 77Se NMR spectra in (TMTSF)2PF6 at 20 K. Thereare four inequivalent sites. (b) Plot of the peak position versus angle. Thesolid lines are fitted according to dipolar interaction equation A0(3 cos2 θ− 1)where A0 is a constant. The deviation of the resonant peaks from this fit isattributed to the triclinic structure of the crystal. . . . . . . . . . . . . . . . 70
3.19 (a) Angular-dependent 77Se 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(dashed lines) including a summary of the observed 771/T1 peaks (asterisks andopen 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 dis-persion relations in the metallic (left panel), FISDW (middle panel), andre-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]2showing coexisting charge density wave (CDW) and spin-Peierls (SP) groundstates below 20 T, and field-induced CDW (FICDW) region above 20 T [fromGraf et al.]. The dashed arrows show the regions in the phase diagram whereNMR was measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
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4.2 (a) Crystal structure of (Per)2Pt[mnt]2 viewed along a-axis. (b) Schematicof the perylene and dithiolate layers. Electronic conduction occurs in theperylene chains and is directed mainly on the b-axis. . . . . . . . . . . . . . . 77
4.3 Temperature dependence of 195Pt NMR spectra in polycrystalline (Per)2Pt[mnt]2at 14.8 T. Left Inset: Plot of the corresponding spectral intensity whichdeviates from the Boltzmann prediction. The NMR signal disappears ataround 5 K which is coincident with the Spin-Peierls transition temperature.Right Inset: characteristic power lineshape pattern that resembles the 195PtNMR spectra due to distribution of anisotropic Knight shifts. . . . . . . . . 79
4.4 195Pt NMR spectra of (Per)2Pt[mnt]2 at various fields at constant temperature1.8 K. Note the loss of 195Pt NMR signal above 20 T. . . . . . . . . . . . . . 80
4.5 Field dependence of 1951/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 signalabove 20 T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6 T-x Phase diagram of newly-discovered CDW-superconductor CuxTiSe2 [fromMorosan et al.]. Notice the similarity to the high-Tc phase diagram. . . . . . 83
4.7 (a) Temperature dependence of 77Se NMR spectra in Cu0.07TiSe2 at 7.193 T.(c) Plot of the peak position/shift from (a). . . . . . . . . . . . . . . . . . . 84
4.8 (a) Temperature dependence of 63Cu Knight shift for Cu0.05TiSe2 (red trian-gles) and Cu0.07TiSe2 (blue circles) at 8 T. (b) Temperature dependence of631/T1 at 8 T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.9 Temperature dependence of 771/T1 of parallel stacks of CuxTiSe2 (x=0.05,0.07) platelets with B ‖ c∗, powder of TiSe2 (Dupree et. al.), and pure 77Semetal. Inset: log-log plot of this graph. Note the non-linearity of 771/T1 vs Tin TiSe2 due to CDW formation. . . . . . . . . . . . . . . . . . . . . . . . . 86
4.10 Enhanced Korringa factor K(α) vs interaction parameter α from Moriya anda corrected version from Narath et. al. The blue and red dashed lines are theobserved K(α) for 7% and 5% Cu dopings, respectively. The arrows indicatethat the interaction parameter α is 0.8 for 7% and 0.93 for 5%. . . . . . . . . 88
5.1 (a) The structurally perfect kagome lattice (b) Crystal structure of herbert-smithite (ZnCu3(OH)6Cl2), the S = 1/2 structurally perfect kagome latticenetwork of Cu2+ spins (from Ref. [69]). . . . . . . . . . . . . . . . . . . . . . 91
5.2 (a) q = 0 and (b) q =√
3 ×√
3 (also called the weathervane mode) states ina structurally perfect kagome lattice. . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Distorted kagome system: crystal structure of Pr3Ga5SiO14 (from Ref. [89])representative of other frustrated langasite systems. . . . . . . . . . . . . . . 94
xiv
5.4 NMR field-scan spectrum of Pr3Ga5SiO14 showing quadrupolar split 69Ga(I = 3/2) and 71Ga (I = 3/2) components for the two non-equivalent Gasites (the third site, embedded in a small peak, is not shown). The insetdepicts the crystal structure showing three kagome planes. Two Ga sites liebetween 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 crystalc-axis plotted versus the magnetic susceptibility with temperature as theimplicit parameter. For T > 30 K the plot shows that but for lower Tdepartures from a linear relationship are observed. (b) 69Ga NMR spectraas a function of T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.6 69Ga 1/T1 and 1/T2 for Pr3Ga5SiO14 as a function of T at 16.37 T, togetherwith specific heat at 9 T whose peak is coincident with the broad maximum of691/T2. The similarity in behavior of the two quantities is striking and pointsto a common underlying mechanism. Inset: log-log plot of 691/T2 at differentfields. Notice that the broad maximum sharpens as the field is increased andbelow the peak the behavior is close to T 2. . . . . . . . . . . . . . . . . . . . 99
5.7 Temperature dependence of the transverse spin correlation time τ2 at differentfields extracted from 691/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 linecorresponds to the fit ΔNMR = Δ0 + αH with α = gμB = 3.32μB and Δ0 = 3.5K. The field-dependence of the spin gap in the excitation spectrum as derivedfrom 35 mK inelastic neutron scattering results (from Ref. [90]) is shown forcomparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.8 Crystal structure of Ba3NbFe3Si2O14 viewed along (a) c-axis (b) b-axis. (c)Room temperature X-ray diffraction pattern. Inset: temperature dependenceof 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 tothe specific heat and the calculated entropy (d) Temperature dependence ofthermal conductivity (from Ref. [92]) . . . . . . . . . . . . . . . . . . . . . . 104
5.10 Field-swept 93Nb 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 fieldorientation (θ is the angle between B and a-b plane) at 4.2 K in theantiferromagnetic state. The small sharp peaks are 63,65Cu and 27Al NMRsignals from the coil. (b) Plot of the left and right resonant peaks. Thedashed lines are fits to the equation A cos θ where A is the hyperfine couplingconstant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
xv
5.12 (a) Temperature dependence of 931/T1T in Ba3NbFe3Si2O14 at 83.4 MHz inthe paramagnetic state (PP-paramagnetic peak) and antiferromagnetic state(LP-left peak, MP-middle peak, and RP-right peak). The dashed in theparamagnetic region is fit to SCR spin fluctuation theory with TN ≈ 27.5K. (b) Corresponding temperature-dependent spectra in the two states. Thelocation of the peaks are indicated and the middle sharp line is 27Al NMRsignal from the probe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.13 Temperature dependence of the spin-spin relaxation rate 931/T2 measured inthe paramagnetic and antiferromagnetic regions at 83.4 MHz with B ⊥ a − bplane. The dashed line is a fit similar to SCR spin fluctuation behavior.Inset: Dependence of the stretched exponential parameter β of the middlepeak. The change in β at around 4 K corresponds to T ∗ seen in the relaxationrate 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
in another Q1D organic conductor (Per)2Pt[mnt]2 were studied using 195Pt NMR revealing
the breaking of the SP state at high magnetic fields. The role of doping in the electronic
correlations of the newly discovered CDW-superconductor CuxTiSe2 is revealed by 63Cu and
77Se NMR. The later part of this dissertation focuses on the kagome spin 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 kagome materials
Pr3Ga5SiO14 is described and compared with other existing frustrated spin systems. On
the other hand, 93Nb 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
has a specific gyromagnetic ratio: proton or 1H, for instance, has γn = 42.5774 MHz/T and
77Se 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, Mossbauer
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
B=B0 µB0
−µB0
2µB0
ω=γB0
B0
E=2µB0
ω=2µB0/h
flipping
rf signal
B=0
Figure 1.1: A simple mechanism of NMR: rf irradiation of the nuclear moment precessing atLarmor frequency. With energy equal to the Zeeman splitting, rf can flip the nuclear spinthen it returns to equilibrium releasing a signal. The lower figure represents the energy levelswhen 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
collectively give the magnetization described by Curie law M0 = Nγ2n
2I(I+1)3kBT
H for N number
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
reference
sample
ωref = γB0 ωsample = γ(B0+Bint)
Shift
Frequency
Figure 1.2: The Knight shift: the resonant frequency of the sample shifts by γBint from thereference 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 = −γnH0 · (I + Ks − σ) · I + HQ (1.1)
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
of the nuclear quadrupole moment Q (for nuclei with spins I > 12) 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
180o90
o
FID Spin Echo
Figure 1.3: The spin-echo pulse sequence: the magnetization is tipped by a 900 pulse andthe spins start to “fan out”on the XY plane forming a FID signal. A second pulse flippedthe 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:
Ks(T ) =Ahf
NμB
χs(T ) (1.2)
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:
Ks =ωref − ω
ωref
× 100% (1.3)
where ωref = γH0, the unshifted resonant frequency of a reference salt. For field-swept
spectra at constant frequency, the Knight shift is expressed as Ks =Href−H
H× 100%.
4
1.2 Hyperfine Coupling Terms of the InteractionHamiltonian
The Hamiltonian describing the interaction of nuclei with conduction electrons is given by
[6]:
H = 2 · 8π
3μBγnI · S(r)δ(r) − 2μBγnI · [ S
r3− 3r(S · r)
r5] − γn
e
mc[I · r × p
r3] (1.4)
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 Hcontact = 8π3
γeγn2I·Sδ(r) is the Fermi contact
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]:
Ks =8π
3〈|ψs(0)|2〉FSχP (1.5)
where χP is the Pauli paramagnetic spin susceptibility per atom and 〈|ψs(0)|2〉FS is the
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]:
Hd =1
2
N∑
j=1
N∑
k=1
[μj · μk
r3jk
− 3(μj · rjk)(μk · rjk)
r5jk
] (1.6)
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
we can therefore write the dipolar Hamiltonian as Hd = γnγe2
r3 (A + B + C + D + E + F )
where, in terms of the raising operators I+1,2 and lowering operators I−
1,2, A = I1zI2z(1−cos2 θ),
B = 14(I+
1 I−2 + I−
1 I+2 )(1 − cos2 θ), C = −3
2(I+
1 I2z + I1zI+2 ) sin θ cos θe−iφ, D = −3
2(I−
1 I2z +
I1zI−2 ) sin θ cos θeiφ, E = −3
4I+1 I+
2 sin2 θe−2iφ, and F = −34I−1 I−
2 sin2 θe2iφ. If both moments
are aligned with magnetic field, the splitting of NMR spectra is given by ΔH = A2(3 cos2 θ−1).
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 Korb = 〈b〉χorb where 〈b〉 is the orbital hyperfine coupling
constant. More specifically, the orbital shift is [6]:
Korb =2μB
IH
∑
i
∑
f
〈i|H · l|f〉〈f |2l·Ir3 |i〉δ(kf − ki)
Ei − Ef
(1.7)
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
is Korb ≈ninf 〈
1r3 〉
Δwhere ni is the number of occupied Bloch states and nf is the number of
unoccupied Bloch states and Δ is the conduction electron bandwidth [6].
1.3 Measuring the Dynamics: Relaxation Rates
A gyroscopic motion dLdt
= μ × B, or similarly dμdt
= γμ × B, is produced when a nuclear
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 =∑
i μi so that the dMdt
= γM × B. For I = 1/2 nuclei, the equilibrium value
of the magnetization is M0 = Nμ tanh( μBkBT
) where N is the number of nuclei. When the
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
dt= M0−Mz
T1.
Thus, the z -component of the equation of motion is [2, 3]:
6
dMz/dt = γ(M × B)z + (M0 − Mz)/T1 (1.8)
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,
dMx/dt = γ(M × B)x − Mx/T2 (1.9)
dMy/dt = γ(M × B)y − My/T2 (1.10)
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/T1 ∝∫ +∞
−∞
dτeiω0τ 〈S+q (τ)S−
q (0)〉 (1.11)
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:
1/T1 =4kBT
lim
ω→ωn∼0
∑
q,α=xx,yy
(Aα(q)
γe)2χ
′′
(q, ω)
ω(1.12)
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
400
300
200
100
0
Long
itudi
nal M
agne
tizat
ion
(arb
. uni
ts)
103
104
105
106
delay time (µs)
π/2 π/2 π
τdelay τ0τrecycle=5T1
(a)
(b)
τ0
Figure 1.4: Measuring the spin-lattice relaxation time T1: (a) a π/2 “saturation” pulse isfollowed 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 magnetizationwhich 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
dt= M0−Mz
T1. The growth of the longitudinal
8
magnetization Mz(t) can be solved by integrating∫ Mz
0dMz
M0−Mz= 1
T1
∫ t
0dt from which
ln M0
M0−Mz= t
T1. Finally, the recovery of the longitudinal magnetization is fitted with a
single exponential equation1:
Mz = M0(1 − e−t/T1) (1.13)
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/T2 ∝∫ +∞
−∞
dτeiω0τ 〈Sα(τ)Sα(0)〉 (1.14)
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
1Some systems have stretched exponential recovery Mz = M0([1 − exp(−(t/T1)β)] where β ranges from
0 to 1. Others can have multi-exponential forms.
9
20
15
10
5
0
Tra
nsve
rse
Mag
netiz
atio
n (a
rb. u
nits
)
4 5 6 7
102
2 3 4 5 6 7
103
2 3 4 5 6 7
104
2
delay time (µs)
π/2 πvariable τdelay variable τdelay
τrecycle=5T1
(a)
(b)
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 decayof 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
are written as dMx
dt= γB0My − Mx
T2, dMy
dt= −γB0Mx − Mxy
T2, and dMz
dt= 0. From these
we obtain Mx = m0e−t/T2 cos ωt and My = m0e
−t/T2 sin ωt. To compute for the transverse
relaxation we used the equation Mxy = (M2x + M2
y )1/2 which yields:
2In some solids, the form of transverse relaxation decay is Gaussian Mxy = M0e−
t2
2T2
2G or Lorentzian
Mxy = M01/T2
(1/T2)2+t2 . Thus T2 is half of the full-width half maximum of these functions.
10
Mxy = M0e−t/T2 (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
autocorrelation function, also written as G(τ) = 〈H2⊥〉e
−ττc , 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
∫ ∞
0
G(t) exp(−iωτ)dτ (1.16)
For a transverse fluctuating magnetic field, the spectral density is:
J(ω) = 2〈H2⊥〉
τc
1 + ω2τ 2c
(1.17)
11
Hence, the spin-lattice relaxation rate can now be expressed as:
1/T1 = γ2n〈H2
⊥〉τc
1 + ω2τ 2c
(1.18)
where 〈H2⊥〉 = 〈H2
x〉+ 〈H2y 〉. If the transverse field fluctuates rapidly, the correlation time is
short and the spectral density is broad. Similarly, the spin-spin relaxation rate can also be
written in terms of the correlation time [2]:
1/T2 = γ2n〈H2
‖ 〉τc + 1/2T1 = γ2n〈H2
‖ 〉τc +1
2γ2
n〈H2⊥〉
τc
1 + ω2τ 2c
(1.19)
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]:
Wmks,nk′s′ =1
2|〈mks|V |nk′s′〉|2δ(Em + Eks − En − Ek′s′) (1.20)
The parameter V is the contact interaction that drives the scattering. In the case of
simple metals, V = 8π3
γeγn2I · Sδ(r) where the nucleus is chosen to be at the ori-
gin. The total probability per transition is the sum of all the initial and final elec-
tron states which is Wmn =∑
ks,k′s′ Wmks,nk′s′ where the state |ks〉 is occupied by an
electron and the state |k′s′〉 is unoccupied. Since this is an ensemble of electrons, we
employ the Fermi-Dirac statistics to write Wmn =∑
ks,k′s′ Wmks,nk′s′f(k, s)[1 − f(k′, s′)]
where f(k,s) is the Fermi function. The electronic wave function can be written as
a product of the spin function and Bloch wave function: |mks〉 = |m〉|s〉uk(r)eik·r.
We may therefore write the matrix elements as 〈mks|V |nk′s′〉 = 8π3
γeγn2〈m|I|n〉 ·
〈s|S|s′〉u∗k(0)uk′(0). Thus we can write the number of transitions per time as Wmks,nk′s′ =
2π
64π2
9γ2
eγ2n
4∑
α,α′=x,y,z〈m|Iα|n〉〈n|Iα′ |m〉〈s|Sα|s′〉〈s′|Sα′ |s〉|uk(0)|2|uk′(0)|2δ(Em+Eks−En−Ek′s′). Introducing the density of states g(Ek, A), we can write the sum of the previous
expression [2]:
12
1.0
0.5
0.0210
EF
f(E) 1-f(E)
f(E)[1-f(E)]
kBT
occupied states unoccupied states
Figure 1.6: The probability of occupied states f(E) and unoccupied states 1 − f(E) wheref(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.
Wmn =2π
64π2
9γ2
eγ2n
4∑
α,α′ ,s,s′
〈m|Iα|n〉〈n|Iα′ |m〉〈s|Sα|s′〉〈s′|Sα
′ |s〉 (1.21)
∫
|uk(0)|2|uk′(0)|2f(k, s)[1−f(k′, s′)]g(Ek, A)g(Ek′ , A)δ(Em−En+Eks−Ek′s′)dEkdAdEk′dA′
where the delta function ensures that Eks+Em = Ek′s′+En and Ek′ = Ek+Es−Es′+Em−En.
Further, by introducing the density of states function ρ(Ek′) and the averaged wavefunction
〈|uk′(0)|2〉 allows us to write the integral simply as∫ ∞
0〈|uk′(0)|2〉2Ek
ρ2(E)f(E)[1 − f(E)]dE
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|Sα|s′〉〈s′|Sα′ |s〉 =∑
s〈s|SαSα′ |s〉 = TrSαSα′ = δαα′13S(S + 1)(2S + 1) which for S = 1
2is equal to δαα′/2. We
can then write a more simplified form of the transition probability [2]:
13
Wmn =64
9π3
3γ2
eγ2n
∑
α
〈m|Iα|n〉〈n|Iα|m〉∫
〈|uk(0)|2〉Ekρ2(E)f(E)[1 − f(E)]dE (1.22)
The integrand, in the limit T → 0, can be written as a delta function f(E)[1 − f(E)] =
kTδ(E − EF ), then the above equation becomes [2]:
Wmn =64
9π3
3γ2
eγ2nkT 〈|uk(0)|2〉2Ek
∑
α
〈m|Iα|n〉2ρ2(E) (1.23)
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. For N 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-
lattice relaxation rate and the transition probability are related by 1T1
= 12
P
m,n Wmn(Em−En)2P
n E2n
,
then [2]:
1
T1
= a001
2
∑
m,n,α〈m|Iα|n〉〈n|Iα|m〉(Em − En)2
∑
m E2m
= −a001
2
∑
m,n,α〈m|H, Iα|n〉〈n|H, Iα|m〉∑
m E2m
(1.24)
Now the above equation can be written as 1T1
= −a0012
P
α=x,y,z Tr[H,Iα]2
TrH2 where H = −γH0Iz is
the Hamiltonian. We introduce the commutation relation [Ix, Iy] = iIz. The numerator can
be simplified to∑
α Tr[H, Iα]2 = −γ2n
2H20Tr[I2
x + I2y ] and for the denominator
∑
α TrH2 =
γ2n
2H20TrI2
z . Using the property TrI2x = TrI2
y = TrI2z , the fraction
P
α Tr[H,Iα]2P
α TrH2 is equal to
-2. Thus, remarkably, the spin-lattice relaxation rate is [2]:
1
T1
= a00 =64
9π3
3γ2
eγ2n〈|u2
k(0)|〉2EFρ2(EF )kT (1.25)
where the Knight shift is Ks = ΔHH
= 8π3〈|u2
k(0)|〉2EFχs
e and for non-interacting spins
χs0 = γ2
e 2
2ρ0(EF ). Using these expressions, the above equation reduces to the familiar form
[2, 3]:
1
T1
=4πkB
K2s
γ2n
γ2e
T (1.26)
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 BCSSuperconductivity
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
there is an upturn or peak in RS
RNin temperature just below Tc which is consistent with the
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]:
N =
C E(E2−Δ2)1/2 for E > Δ(T )
0 for 0 < E < Δ(T )(1.27)
where for E > 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
the BCS model. Due to the coherence factor C+ = 12[1 + Δ2(T )/EiEf ], the effective matrix
element is then modified [7]:
|〈i|Ve−n|f〉|2 −→ C+|〈i|Ve−n|f〉|2 (1.28)
The ratio RS
RNcan therefore be written as [7]:
RS
RN
=2
kBT
∫ ∞
Δ
dEif(Ei)[1 − f(Ei)](1 +Δ2
EiEf
)[Ei
(E2i − Δ2)1/2
][Ef
(E2f − Δ2)1/2
] (1.29)
In addition, the spin-lattice relaxation rate below the HS peak follows the relation
1/T1 ∝ e−∆kBT . This steep falloff at low temperature is also known as the Yosida function
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
Pulse Programmer
RF SynthesizerShifter
Amplifier
Pulse Gate
Figure 1.7: Schematic diagram of the transmitter section: gated oscillating electrical signalfrom 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
RF in To Pre-amp
50 Ω load
diode
diode
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 Δff0
where f is the frequency and Δf refers to a region around f where the resonance condition
is satisfied.
17
coaxCT
CML
coaxCT
CML
(a)
(b)
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.
Duplexer Pre-amp
RF synthesizer
Receiver
NM
R In
tens
ity (
arb.
uni
ts)
806040200delay time (µs)
|s(t)| Re s(t) Im s(t)
(a) (b)NMR signal from probe
reference frequency
Figure 1.10: (a) Schematic diagram of the receiver section: the NMR signal from the probeand the reference frequency from the synthesizer are mixed in the quadrature receiver wherereal and imaginary NMR signals are generated. (b) Spin echo signal showing the real andimaginary 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 Ω0 = ω0−ωref which is typically a MHz or less. This is similar
to what happens in a radio receiver: the rf waves are down-converted to audible frequency
range [1].
In the down-conversion, the free induction decay (FID) signal cos(ω0t)e−t/T2 is trans-
formed to cos(Ω0t)e−t/T2 . 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: sA(t) ≈ cos(Ω0t)e−t/T2 and sB ≈ sin(Ω0t)e
−t/T2 , or collectively s(t) = sA(t) + isB(t).
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
(b)
(c)
Magnet power supply
temperature controller
NMR spectrometer
rf synthesizer
oscilloscope
computer
liquid Helium level sensor
Magnet
probe
goniometer
temperature sensorcoil with sample
(a)
Figure 1.11: NMR hardware: (a) NMR rack containing the spectrometer, PC, temperaturecontroller, rf synthesizer, magnet power supply, and liquid Helium level sensor. (b) Photo ofthe goniometer and coil with sample mounted on a probe. (c) Top view of the magnet withprobe.
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)
(4)(3)
(2)
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.
(TMTSF)2PF6 is a member of Bechgaard salts1 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 inmagnetic 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 SymmetrySS electron-electron S=0 q=0 gaugeTS electron-electron S=1 q=0 gauge
CDW electron-hole S=0 q=2kF translationalSDW electron-hole S=1 q=2kF translational
materials.
2.1 Low Dimensional Instabilities
Low dimensional metals are materials with highly anisotropic crystal and electronic struc-
tures. 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
model, the wavevector q-dependent Lindhard response function χ(q) = − e2
πvFln | q+2kF
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) (b)Metal CDW
ε (k)ε (k)
π/a π/a π/akFkF kF kF
Δ
π/a0 0
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 withFermi wave vector kF = π
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:
H =∑
k,σ
ǫkc+kσckσ +
∑
k,σ
ω0qb
+q bq +
∑
k,q,σ
g(k)c+k+q,σck,σ(bq + b+
−q) (2.1)
where c+k (ck), b+
q (bq) are the electron and phonon creation (annihilation) operators with
2This also known as the Frohlich Hamiltonian.
25
momenta k, q, and spin σ, ǫk and ω0q 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
TMFCDW where a “frozen-in” distortion occurs. This is apparent in the mean field prediction:
ωren,2kF= ω2kF
(T − TMF
CDW
TMFCDW
)1/2 (2.2)
Introducing a complex order parameter Δeiφ = g(2kF )〈b2kF+ b+
−2kF〉, the displacement
of the ions is:
〈b2kF+ b+
−2kF〉e2ikF x + const. =
2Δ
g(2kF )cos(2kF x + φ) (2.3)
Invoking the above equation, the periodic, spatially-dependent electron density at T = 0
in CDW can now be written as:
ρx = ρ0 +Δρ0
vF kF λcos(2kF x + φ) = ρ0 + ρ1 cos(2kF x + φ) (2.4)
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)
(b)
J JJJJJJ
J+δ J+δJ-δJ+δJ-δJ+δJ-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 andlattice distortion occurs at q = 2kF (Spin-Peierls state) leading to stronger antiferromagneticinteraction J + δ among the dimerized electrons and weaker antiferromagnetic interactionJ − 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
in low-dimensional metals with period λ0 = πkF
caused by electron-electron interactions. As
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]:
H =∑
k,σ
ǫka+k,σak,σ +
U
N
∑
k,k′,q
a+k,σak+q,σa
+k′,−σak′−q,−σ (2.5)
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, a+k
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)
Δ
0 kFkF π/aπ/a
k
(a)
(b)
λ0 = π/kF
ρdown(r)
ρup(r)
Figure 2.3: (a) The dispersion relation in spin density wave state showing the opening of thegap Δ at the Fermi wave vector kF . (b) The modulation of the two spin subbands in SDWwith wavelength λ0 = π
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:
H(x) =∑
q
Hqeiqx (2.6)
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 〈Mq〉 = μB(〈nq,↑〉 −〈nq,↓〉) = χ0(q)H
effq where Heff
q = Hq +U(〈nq,↑〉−〈nq,↓〉)
2μB(the arrows denote the spin directions)
and χ0(q) is the susceptibility that is dependent on wavevectors in the absence of Coulomb
28
interactions. Introducing Δnq = 〈nq,↑〉 − 〈nq,↓〉, the self-consistent equation is μBΔnq =
χ0(q)(Hq + UΔnq
2μB). The average magnetization is therefore [12]:
〈Mq〉 =χ0(q)
1 − Uχ0(q)
2μ2B
Hq = χ‖(q)Hq (2.7)
where χ‖(q) = χ0(q)
1−Uχ0(q)
2μ2B
which is an enhanced susceptibility. For q = 0, χ0(0) = 2μ2Bn(ǫF )
and the static susceptibility is χ(0) =2μ2
Bn(ǫF )
1−Un(ǫF )which is enhanced by the Stoner factor.
At q = 2kF , χ0(q) exhibits a peak which is strongly temperature dependent so thatUχ0(2kF ,T )
2μ2B
= Un(ǫF ) ln 1.14ǫ0kBT
= 1. This allows us to write the mean-field prediction:
kBTMFSDW = 1.14ǫ0 exp(− 1
λe
) (2.8)
where λe = Un(ǫF ), the dimensionless electron-electron coupling constant. Another result
of the SDW mean-field treatment is the weak coupling BCS relation 2Δ = 3.52kBTMFSDW .
Finally, due to a spatially-dependent magnetic moment 〈μ(x)〉 = μ0 cos(2kF x + φ), the
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-
up ρ↑(x) = ρ0[1+ ΔvF kF λe
cos(2kF x+φ)] and spin-down ρ↓(x) = ρ0[1+ ΔvF kF λe
cos(2kF x+φ+π)]
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
ClO4TMTSF
Figure 2.5: Crystal structure of (TMTSF)2ClO4 viewed along a-axis. The crystallographicaxes 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 AsF6 are superconducting (Tc ≈ 1 K) only when pressure is applied. On the other hand,
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
0.01
0.1
1
10
Rzz
(Ω)
2 3 4 5 6 7 8 910
2 3 4 5 6 7 8 9100
2
T (K)
First Cooldown Second Cooldown
Anion Ordering (AO)TAO=24 K
Figure 2.6: Zero-field cooldown of (TMTSF)2ClO4 showing a kink in resistance at 24 Kwhich 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
3
456
0.1
2
3
456
1
2
3
456
10
Rzz
(Ω
)
10987654B (T)
0.347 K
0.478 K
0.978 K
1.372 K
1.591 K
1.692 K
1.981 K
2.910 K
3.930 K5.940 K
4
3
2
1
T (K
)
1210864Field (T)
Figure 2.7: Low-field c*-axis magnetoresistance (MR) of (TMTSF)2ClO4 at different tem-peratures. The kinks in MR correspond to the FISDW cascade phases. Inset: T-B phasediagram 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
80
60
40
20
0
Resis
tan
ce (
Oh
ms)
302520151050
Field (T)
1.57 K
2 K
4.5 K
6 K7.5 K
BreB*
Bth
B1
B//c*, (TMTSF)2ClO4
Figure 2.8: High field c*-axis magnetoresistance (MR) of (TMTSF)2ClO4 at differenttemperatures 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
case of (TMTSF)2ClO4. This is distinct from de Haas-van Alphen (dHvA)4 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 magneticfield. The periodicity in dHvA effect measures the extremal cross sectional area in k space of the Fermisurface.
5SdH is the oscillation in the resistivity of a material at low temperatures and high magnetic fields. It issimilar to dHvA where the frequency of the magnetoresistance oscillations indicate the extremal section ofthe 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:
tan θ =p
q
b sin γ
c sin β sin α− cot α (2.11)
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 μμB
= 4|Δ|U
where U is the on-site Coulomb interaction and Δ is the single particle
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]:
H = (μ
μB
)2Jeff
∑
i
Si ·Si+1−(μ
μB
)2D∑
i
Sxi Sx
i+1+(μ
μB
)2E∑
i
(Szi S
zi+1−Sy
i Syi+1)−gμH
∑
Si
(2.12)
35
kb
ka
kc
−π/b
π/a
π/b
−π/a
kF-kF
QN
X
a
b
A
λ
(a) (b)
Bz
Figure 2.9: (a) Electronic motion in momentum space: a pair of warped FS sheets withinthe first Brillouin zone in a Q1D system. (b) Electronic motion in real space with amplitudeA = 4tb
evF Bzand wavelength λ = h
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
χ⊥ =Neg2μ2
0
2( μμ0
)2Jeff=
Neg2μ2B
2Jeff. The parallel component refers to the easy axis direction where
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
also gives us the spin flop field Hsf =π(2E∗Jeff )1/2
γwhere 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 +kF and the other at −kF . In the case of a Q1D conductor, where there is a
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 = (±2kF , π/b, 0).
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
dt= − eH
cvb and dkb
dt= eH
cva = eH
c2ata
sin(aka) where we used the
equation for velocity v = 1
∂ǫ(k)∂k
. The frequency of the electron traversing the Brillouin zones
in the b-direction is ωc = eHmec
kF b. We note that in the case of closed FS in metals, the
electrons move in a circular motion with cyclotron frequency ω = eHmc
.
With open FS the electrons are strongly oscillating in the kb direction. In real space, the
electrons make a sinusoidal motion with the amplitude 4tbbωc
.
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
thought to be Shubnikov-de Haas (SdH) type with periodicity Δ(1/H) = 2πecA
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
6
5
4
3
2
1
0302520151050
Cascade
Re-entrant FISDW
SC
FISDW
Metallic Phase
c*
b'a
B
Figure 2.10: T-B phase diagram of (TMTSF)2ClO4 showing the metallic region, supercon-ducting (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 εk = vF (|kx| − kF )+ ε⊥(k⊥) where the second term is
ε⊥(k⊥) = 2tb cos(bky)+2tc cos(ckz)+2tb cos(2bky)+2tc cos(2ckz), they obtained the eigenvalue
and eigenfunction from the Schrodinger equation with Landau gauge A = (0, Hx, 0). The
38
7
6
5
4
3
2
1
0302520151050
CascadeSC
FISDW
Metallic Phase
c*
b'a
B P=12 kbar
Figure 2.11: T-B phase diagram of (TMTSF)2PF6 at 12 kbar hydrostatic pressure: this issimilar to (TMTSF)2ClO4 except for the absence of the re-entrant FISDW region at highmagnetic fields.
spin susceptibility χ(Q), after employing Green’s functions, can be written as [9]:
χ(Q) = N(0)
∫ +∞
d
J0[(4tb
κvF
) sin(κx)]J0(4tcx
vF
)dx
xT sinh(x/xT )(2.13)
where J0 is the Bessel function, N(0) = Nas
2πkBTis the state density, xT = vF
2πkBT, κ = eHb
c,
and kB is the Boltzmann constant. From the divergence in χQ = χ0(Q)1−Uχ0(Q)
, the cutoff of the
lower limit (d) of the integral is obtained from N(0)U · ln vF
πkBT0d= 1. Using the relation
1π
∫ π
0J0(z sin φ)dφ = J2
0 (12z), one can reduce the above equation to [9]:
χ0(Q0) ≃ N(0)J20 (
2tbκvF
)ln(vF
πkBTd) (2.14)
where Q0 = (2kF , πb, π
c) is the optimal nesting wavevector. Chaikin proposed that the
transition temperature TSDW = T0 exp(−HA
H) where T0 and HA are constants.
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
be attributed to the presence of a superlattice with wave vector Q = (0, 12, 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 Tc ≈ 1 K and Hc2 ≈ 0.5 T. At low temperature,
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 = (2kF ± N2π/λ, π/b) where λ = hebB
.
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=(2kF±N2π/λ,π/b) (where λ=h/ebB), increasing magnetic field produces a first-order
“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
6
5
4
3
2
1
0
Tem
pera
ture
(K
)
302520151050Magnetic Field (T)
Bth
B* BreB1
Figure 3.1: The (TMTSF)2ClO4 FISDW phase diagram where the dots represent the FISDWtransport features seen in this work. The arrows represent the regions in the phase diagramwhere 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
Sample
transport leads
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 andthe 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]:
δH(r) = 〈a0〉H0μ
μB
cos(2Q · r + φ) (3.1)
where H0 is the applied magnetic field, 〈a0〉 is the hyperfine coupling constant, and μμB
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
inequivalent sites of 77Se 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
Figure 3.5: Temperature dependence of 77Se NMR spectra in a relaxed (TMTSF)2ClO4 withB ‖ c∗. Part (a) shows frequency-swept spectra in the metallic state at constant magneticfield B = 12.12 T while (b) shows field-swept spectra at constant NMR frequency 98.2MHz. The broad, double-horned spectra indicate inhomogeneous local magnetic field in theFISDW 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 - 4 μs 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
magnetization is given by θtip = π2
= 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:
η =τm
τsdw
(3.2)
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 theFISDW 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 theFISDW state of (TMTSF)2ClO4 at 1.8 K and constant NMR frequency 98.6 MHz withB ‖ c∗. The pulse sequence used is 450 ns - 900 ns. (b) Power-swept spin-echo intensitytaken 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:
I(Δω) =1
2π
∫ +∞
−∞
G(τ)eiΔωτdτ (3.4)
where the autocorrelation function G(τ) = 〈exp(−i∫ t+τ
tω1 cos[(k1x+Ωt′)+φ)]dt′〉 is periodic.
The variable w = Ωω1
= vk1
ω1is introduced as a measurement of the velocity of the sliding
condensate. The autocorrelation function can be modified as
G(τ) =
∫ 1
0
exp[−i2
wsin(πτ) cos(πτ + φ + 2πu)]du (3.5)
where τ = ω1τw2π
, y = ω1t′ w2π
, and u = ω1tw2π
are dimensionless numbers. The above
equation can be simplified to G(τ) = J0(2w
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:
Iw→0(ω) =
12π
∫ +∞
−∞J0(z) exp(iωz)dz = 1
π(1−ω2)1/2 for ω2 ≤ 1
0 for ω2 > 1(3.6)
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
observed by Clark et al. in the 1H 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:
I(ω) = δ(ω)[1 − 1
2w2] +
1
4w2[δ(ω − w) + δ(ω + w)] (3.8)
50
where the central peak is at ω = ωL and the side peaks are at ω = ωL ±Ω = ωL ± vk1.
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:
G(τ) = J0(2
wsin(πτ)) =
+∞∑
n=−∞
J2n(
1
w) cos(2πnτ) (3.9)
and the lineshape can now be written as I(ω) =∑+∞
n=−∞ J2n( 1
w)δ(ω − nw). This equation
gives sharp lines appearing at ω = ωL ±nΩ because of the periodic nature of the correlation
function. Assuming the individual lines have Gaussian form g(ω − ω′, σ), we modify the
NMR lineshape equation as F (ω) =∑+∞
n=−∞ J2n( 1
w)∫ +∞
−∞δ(ω′ − nw)g(ω − ω′, σ)dω′ where
σ = σω1
. Finally, the continuous NMR lineshape equation is:
F (ω) =1√2πσ
+∞∑
n=−∞
J2n(
1
w) exp(−(ω − nw)2
2σ2 ) (3.10)
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.
3.5 Field Dependence of 771/T1, Rzz, and Spectra at
Constant Temperature
The scientific motivation of performing simultaneous field-dependent NMR and electrical
transport is to determine where exactly does 771/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.
Note that the peak in 771/T1 occurs in B1 FISDW phase boundary. Above B1 = 7.6 T, two
sets of relaxation values are obtained: the enhanced 771/T1 is due to the optimized NMR
51
5
6
789
10-1
2
3
4
5
6
789
100
151050
2
1
0Bth
B1
0.6
0.5
0.4
0.3
0.2
0.1
0.0141210864
Bth
B1
B//c*, 2 K
Figure 3.7: Field dependence of 771/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 themetallic 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: thecorresponding 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 < B < B1 which supports the current idea
that this region is semi-metallic [28]. rf enhancement only occurs for B > B1.
3.6 Temperature Dependence of 771/T1 at Low Fields
The spin-lattice relaxation rate 771/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
of 771/T1 illustrated in Fig. 3.8, cutting through the low-field (B < 15 T) FISDW phase
boundaries . The behavior of 771/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/T1 ∝ T and, as mentioned earlier in Chapter 1, is due to
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 = 6 T.
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
developed by T. Moriya [38] which predicts that 1/T1 = A T(T−TN )1/2 where A is a constant
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 TBD−TSDW :
(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.
The peak in 771/T1 at TSDW is attributed to the enhancement of the relaxation rate due
53
2
3
4
5
6789
0.1
2
3
4
5
6789
1
2
3 4 5 6 7 8 91
2 3 4 5 6 7 8 910
2 3
B//c*
B=12.86 T
B=8 T
B=6 T
0.65 K
1.81 K
3.88 K
Figure 3.8: Representative temperature dependence of 771/T1 at low fields (B < 20 T)where B ‖ c∗. The dashed lines above the peak are fits to the SCR theory for itinerantantiferromagnets 1/T1 = A T
(T−TN )1/2 and below the peaks are power law fits 1/T1 = ATα.
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
771/T1 = ATα 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
1/T1 =γnkBT 〈δH2
⊥〉β′′(ω)
2λ2ωǫ(3.11)
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 Transportin 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) = Kiso + Kax(3 cos2 θ − 1) (3.12)
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 − ν0 data. The c∗-axis, which is defined as the minimum in Rzz versus θ, is offset by
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.
The NMR linewidth (FWHM) and 771/T1 are also angular-dependent as shown in
Fig. 3.9c. The angular dependence of FWHM is fit empirically with the equation A0 +
A1| cos θ|3/2; deviation from this empirical fit is seen at around 250 to the right of the b′-axis
(maximum in the NMR shift). It shows slight narrowing of NMR spectra at this particular
55
5
4
3
2
1
-120 -90 -60 -30 0 30 60 90 120
10
5
0
-5
-10
45
40
35
30
25
-120 -90 -60 -30 0 30 60 90 120
65
60
55
50
45
(b)
(c)
6
4
2
063.5263.4863.4463.40
7.84 T, 4.2 K
-120o
-100o
-80o
-60o
-40o
-20o
0o
20o
40o
60o
80o
100o
120o
-140o
(a)
Figure 3.9: (a) Angular dependence of 77Se 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 magnetore-sistance and frequency shift. Note that the peaks in Rzz and ν − ν0 do not coincide. (c)Full-width half maximum (FWHM) and spin-lattice relaxation rate 771/T1 as a function ofangle.
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.
The 771/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 Transportin 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 RelaxedSamples
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
12.9512.9012.8512.8012.75
c*-axis, θ=1800
b'-axis, θ=900
(a) (b)
12.9512.9012.8512.8012.75
Partially Quenched Relaxed
c*-axis, θ=00
c*-axis,θ=00
θ=−550
θ=650
X
θ=550
X
2 K 1.8 K
θ=−650
Figure 3.10: Angular dependence of 77Se NMR lineshapes along the a-axis at constant NMRfrequency 104.55 MHz (a) in the SDW state of “quenched” (TMTSF)2ClO4 at 2 K (b) inthe 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
2(3 cos2 θ − 1) where A0 is the dipolar coupling constant. A0
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
0.6
0.4
0.2
x
14 T, 1.5 K
8
6
4
2
1801501209060300
20
15
10
5
0
Metal
FISDW FISDW
a
b'
c*θ B
b'c* c*
BthBth
B1
B1
B1B1Bth Bth
B
-0.4 0.0 0.4ν−ν0 (MHz)
(a)
(b)
(c)
(d)
X
0o
30o
60o
90o
Figure 3.11: Angular-dependent NMR and electrical transport at 14 T and 1.5 K in(TMTSF)2ClO4. (a) 771/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) Schematicof sample rotation along a-axis and the orientation of the TMTSF molecule with respect tomagnetic field B at point X. (d) Angular-dependent 77Se NMR spectra.
3.8.2 Low-Field Angular Dependence of 77 NMR and Transportin 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 B0 = 14 T for θ = 0 or 180 degrees (c∗ − b′ − c∗-axes rotation).
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
25
20
15
10
5
0-120 -90 -60 -30 0 30 60 90 120
0.5
0.4
0.3
0.2
0.1
-120 -90 -60 -30 0 30 60 90 120
12.86 T, 1.88 K
(a)
(b)
1.0
0.8
0.6
0.4
0.2
-120 -90 -60 -30 0 30 60 90 120
clockwise counterclockwise
12.86 T, 3.87 K
(c) 12.86 T, 1.88 K
X
X
X
c*
c*
c*
b'
b'
b'
b'
b'
b'
Figure 3.12: (a) The spin-lattice relaxation rate 771/T1 as the sample is rotated along thea-axis at 12.86 T and 3.87 K. Note the dip in 771/T1 marked X. There is no distinct hysteresisin the result as sample is rotated clockwise and counterclockwise. Inset: schematic of samplerotation. (b) 771/T1 versus θ at 12.86 T and 1.87 K. Inset: orientation of TMTSF at pointX 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
position of the second-order phase boundary. Hence the peak in 771/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⊥ < B1, a peak at B1, and then a decrease ( η ≈ 5) for B⊥ > B1.
Also prominent is a dip in 771/T1 for θ = 250 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 Transportin 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
in 771/T1 as the threshold field is approached, followed by a peak in 771/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 = 5 to η = 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.
A summary of the angular-dependent 771/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
B1 FISDW transition evident in the gradual upturn in 771/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
20
10
0
η
1801501209060300
0.5
0.4
0.3
0.2
0.1
0.01801501209060300
30 T, 1.47 K(a)
300
03020100
-0.1 0.0 0.1Field (T)
X1.5
X1
X50
FISDW
Metal
Re-entrantη=1
η=1
η>1
90o
0o
105o
(b)
(c)(d)
B1
BreB*
B1
Bre
B*
BreB1
B*
B1
B*
Bre
B1B*
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 enhancementη vs. θ. Note that η = 1 above Bre. (c) Field-swept NMR spectra at ν0 = 243.9 MHz (30T) taken at different angles and consequently different phases: (i) FISDW phase at θ = 1050
(above B1) (ii) Metallic phase at θ = 900 and (iii) Re-entrant FISDW phase at θ = 00 (aboveBre). The corresponding magnetoresistance data at different angles are shown in the lowerright hand corner. For each trace, the NMR measurement was made at 30 T.
62
1.0
0.8
0.6
0.4
0.2
771/
T1
(ms-1
)
302520151050
B0cosθ (T)
B0=12.9 T, 1.88 K B0=12.9 T, 3.87 K B0=30 T, 1.47 K
Bre
B*
B1
B1
B1
Figure 3.14: 771/T1 versus effective perpendicular magnetic field B⊥ = B0 cos θ. The redarrows mark the peaks in 771/T1 which coincide with the different FISDW phase boundariesB1, B∗, and Bre seen in electrical transport measurements.
3.9 Temperature Dependence of 771/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.
We measured 771/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
771/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
771/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
2.0
1.5
1.0
0.5
0.0121086420
40
30
20
10
0.14
0.12
0.10
0.08
0.06
121086420
1.0
0.8
0.6
0.4
0.2
0.0
(a)
(c)(b)5
4
3
2
1
02000-200
12 K
9 K
7 K
6 K
5.7 K
4.9 K
4 K
3.3 K
1.9 K
Re-entrant FISDW
23 T, B//c*
SCR fit
TFISDW
TFISDW
MetalFISDW
Figure 3.15: Crossing the re-entrant FISDW phase: (a) Temperature dependence of 771/T1
and c*-axis resistance Rzz of (TMTSF)2ClO4 at 23 T with B ‖ c∗. 771/T1 exhibits a peak at5 K which is coincident with the upturn in Rzz. The re-entrant phase is denoted by anothersharp 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 andBoltzmann-corrected NMR intensity.
65
1.5
1.0
0.5
0.0121086420
80
70
60
50
40
30
20
10
-200 -100 0 100 200Frequency (kHz)
160
140
120
100
80
60121086420
1.0
0.8
0.6
0.4
0.2
0.0
(a)
(b)
29 T, B//c*
8.5 K
7 K
6 K
5 K
3.75 K
2.69 K
2.18 K
(c)
TFISDW
TFISDW
MetalFISDW
Figure 3.16: (a) Temperature dependence of 771/T1 and c∗-axis resistance Rzz measuredsimultaneously 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
with exponent close to 1.35 is observed. The location of the 771/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 Transporton (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
771/T1 shown in Fig. 3.17a. The first is in the metallic or paramagnetic phase (T > 12.6
K) where at high temperatures 771/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 TN ≈ 12.6 K follows the spin fluctuation behavior given by Moriya’s
self-consistent renormalization (SCR) theory [38] for weak itinerant antiferromagnets: 1T1
=
AT(T−TN )1/2 . The second region of interest is below the Neel temperature where a power-law
dependence, 1T1
= ATα is observed where A is a constant and α = 3.2. A thermally activated
region, 1T1
= AeT/Δ is observed below 4 K where Δ = 0.65 K, consistent with previous NMR
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 -4 μs 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 tempera-
ture in the metallic and SDW states. The spin-spin relaxation time is fitted with a Gaussian
form of the transverse magnetization decay Mxy(t) = M0e−t2/2T2 ; 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 < TN , the NMR lineshape is broadened
indicating the presence of inhomogeneous local magnetic ordering.
68
0.01
0.1
1
12 3 4 5 6 7 8 9
102 3 4 5 6 7 8 9
100
1
2
3
4
5
6
78910
2
3
2
1
0
-16004002000-200-400-600
(a)
(b)
11 K
13 K
14 K
22 K
40 K
60 K
80 K
TSDW=12.66 K
17 T, B//c* (TMTSF)2PF6
20
10
0
-10
-20
80604020
T*
Figure 3.17: (a) Temperature dependence of 771/T1 and Rzz in (TMTSF)2PF6 at 17 Twhere B ‖ c∗. Dashed lines are fits to certain equations: self-consistent renormalization(SCR) theory equation A T
(T−TN )1/2 (blue dashed line), power law ATα where α = 3.2 (yellow
dashed line), and AeT/Δ where Δ = 0.65 (green dashed line). (b) Temperature dependenceof 77Se spectra in the metallic (narrow, multiple-peaked lineshapes) and SDW state (broadlineshape). Inset: Plot of the peak position versus temperature in the metallic state.
69
0.5
0.4
0.3
0.2
0.1
0.0-60 -40 -20 0 20 40
-60
-40
-20
0
20
200150100500
0o
20o
40o
120o
140o
160o
180o
200o
80o
100o
60o
(a) (b)17 T, 20 K
Figure 3.18: (a) Angular-dependent 77Se NMR spectra in (TMTSF)2PF6 at 20 K. Thereare four inequivalent sites. (b) Plot of the peak position versus angle. The solid lines arefitted according to dipolar interaction equation A0(3 cos2 θ− 1) where A0 is a constant. Thedeviation of the resonant peaks from this fit is attributed to the triclinic structure of thecrystal.
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 = A0(3 cos2 θ − 1) where f is the peak position, A0
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
8
6
4
2
08006004002000-200-400-600
120
100
80
60
40
20
0100500-50
30o
-70o
-60o
-50o
-40o
-30o
-20o
-10o
0o
10o
20o
-80o
(a) (b)
(c)
4.2 K, 17 T
8004000-400
site 1
site 2
c*-axis
B//c*, θ=00
B//c*
B//b'
Figure 3.19: (a) Angular-dependent 77Se 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 bythe model fits). (c) Corresponding resistance at different angles.
3.10.3 Angular-dependent 77Se Spectra in the SDW Phase
Fig. 3.19 shows the 77Se 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
6
5
4
3
2
1
0302520151050
1/T1 Peaks:B||c* B⊥= Bcos(θ)
Bth
B* BreB1
Figure 3.20: Phase diagram of (TMTSF)2ClO4 for B ‖ c∗ derived from previous reports(dashed lines) including a summary of the observed 771/T1 peaks (asterisks and opensquares), and the corresponding features in the transport measurements (dark and graycircles) from this work. The field labels are defined in text.
(TMTSF)2ClO4, the spectrum can be plotted as:
f(ω) =
Aπ
1√ω2
0−ω2, for |ω| < ω0
0, otherwise(3.13)
where ω0 represents the distribution centroid. The NMR lineshape shown in Fig. 3.19b is
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
Q1
Q2
2kF1
2kF2
kF1
kF2
Ek
EF kxEF
Ek
kF1
kF2
kx
EF
Ek
kF1 kF2
kx
Metal FISDW Re-entrant
1,2 = (2kF1,2 ± n2π/λ, π/b) 1,2 = (2kF1,2,π/b)
Q1
Figure 3.21: Fermi surface nesting models in (TMTSF)2ClO4 and the correspondingdispersion relations in the metallic (left panel), FISDW (middle panel), and re-entrantFISDW (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]:
Tn/Ts = 2
∫ ∞
0
ρ2s(x, T )[1 + η2
0(T )/x2] ∗ f(x, t)[1 − f(x, T )]dx (3.14)
where x = (E −EF )/kT , η0(T ) = ǫ0(T )/kT , and f(E, T ) is the Fermi distribution function.
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
ρs(x, T ) = (2δ)−1
∫ x+δ
−δ
ρBCS(y, T )dy (3.15)
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 1T1
, we can employ a
Ginzburg-Landau gap model Δ = A√
1 − TTc
where A is constant. Although the computed
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) = ρ0+ρ1 cos(2kF ·r+φ)
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 around 8 K in zero 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
8
6
4
2
0403020100
(Per)2Pt(mnt)2
BSP
CDW+SP
FICDW
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) groundstates 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
Perylene Chain Dithiolate Chain Perylene Chain
Pt(a) (b)
a b
c
Pt
Figure 4.2: (a) Crystal structure of (Per)2Pt[mnt]2 viewed along a-axis. (b) Schematic ofthe perylene and dithiolate layers. Electronic conduction occurs in the perylene chains andis directed mainly on the b-axis.
63Cu 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 mnt1 (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
and band-filling, the perylene chains are tetramerized at b∗4
which correspond to a wavevector
2kPF . On the other hand, dimerization of the dithiolate chain at b∗
2corresponds to a wave
vector 4kPF = 2kD
F . 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
because of Pt (S = 12) that interacts with the conduction electrons in the perylene layers. At
high temperatures they interact via fast spin exchange interaction. The total susceptibility
is χs(T ) = χpers (T ) + χmnt
s (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 tb ≈ 150 meV, ta ≈ 2 meV, and
tc ≈ 0 meV [53]. The typical sample size is around 1 mm × 0.050 mm × 0.025 mm.
4.3 Experimental Details
195Pt 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) = Kiso + Kax(3 cos2 θ − 1) + Kasym sin2 θ cos 2φ (4.1)
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
0.8
0.6
0.4
0.2
0.0
195 P
tNM
R In
tens
ity (
arb.
uni
ts)
135.8135.6135.4135.2135.0
Frequency (MHz)
1.74 K 2.8 K 4.2 K 6 K 12 K
14.8 T
1.0
0.8
0.6
0.4
0.2
0.0
Spe
ctra
l Int
ensi
ty (
arb.
uni
ts)
20151050Temperature (K)
Boltzmann prediction
TSP
K|| Kperp
Figure 4.3: Temperature dependence of 195Pt NMR spectra in polycrystalline (Per)2Pt[mnt]2at 14.8 T. Left Inset: Plot of the corresponding spectral intensity which deviates from theBoltzmann prediction. The NMR signal disappears at around 5 K which is coincident withthe Spin-Peierls transition temperature. Right Inset: characteristic power lineshape patternthat 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]:
Kiso =1
3(Kx + Ky + Kz) =
1
3(2K⊥ + K‖) (4.2)
Kax =1
3(Kz −
1
2Ky −
1
2Kx) =
1
3(K‖ − K⊥) (4.3)
Kasym =1
2(Ky − Kx) → 0 (4.4)
79
3.0
2.5
2.0
1.5
1.0
0.5
0.0
NM
R In
tens
ity (
arb.
uni
ts)
-0.6 -0.4 -0.2 0.0 0.2 0.4Frequency (MHz)
12.31 T
12.84 T
16.5 T
17 T
17.4 T
1.8 K
NM
R In
tens
ity (
arb.
uni
ts)
-0.6 -0.4 -0.2 0.0 0.2 0.4
Frequency (MHz)
1.8 K
18.2 T
19 T
20 T
21 T
22 T
23 T
500
400
300
200
100
Inte
nsity
(ar
b. u
nits
)
24222018Field (T)
BSP
Figure 4.4: 195Pt NMR spectra of (Per)2Pt[mnt]2 at various fields at constant temperature1.8 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 TSP ≈ 5 K and that above 5 K, we completely lose
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
the same result where 195Pt 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
1.0
0.8
0.6
0.4
0.2
0.03025201510
2.5
2.0
1.5
1.0
0.5
0.0
1951/T1 at 2.2 K
Spectral amplitude at 2.2 K195
1/T1 at 1.8 K
BSP
Figure 4.5: Field dependence of 1951/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
1951/T1 was measured (see Fig. 4.5). 1951/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 disappear-
ance 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
residual linear term κ0
Tat 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 [fromMorosan 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
2
1
0
-1
250200150100500-40 -20 0 20 40
2.1K4.19K10K20K30K40K50K70K100K140K180K
7.193T
(a) (b)
Figure 4.7: (a) Temperature dependence of 77Se 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 rect-
angular 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
CuxTiSe2 were studied under the same conditions. 77Se and 63Cu 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
-0.155
-0.150
-0.145
-0.140
12 4 6 8
102 4 6 8
1002
0.1
2
4
68
1
2
4
68
10
2
4
68
100
12 4 6 8
102 4 6 8
1002
(a) (b)Cu0.07TiSe2
Cu0.05TiSe2
Cu0.07TiSe2
8 T 8 T
Figure 4.8: (a) Temperature dependence of 63Cu Knight shift for Cu0.05TiSe2 (red triangles)and Cu0.07TiSe2 (blue circles) at 8 T. (b) Temperature dependence of 631/T1 at 8 T.
4.8 77Se and 63Cu NMR Studies of CuxTiSe2
Fig. 4.7 shows the temperature-dependent 77Se NMR spectra and the relative shift at 7.193
T for Cu0.07TiSe2. The behavior of relative shift 77δν as a function of temperature more or
less reflect the susceptibility curve (see Ref. [54]) for this particular concentration.
As mentioned earlier, the Cu atoms are located in between the TiSe2 layers. The 63Cu
Knight shift data in Fig. 4.8a reveal weak temperature dependence, with K = −0.152 %
for Cu0.05TiSe2 and K = −0.143 % for Cu0.07TiSe2. The spin-lattice relaxation rate data in
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
comparative temperature-dependent behavior of 771/T1 in a pure 77Se 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
6
5
4
3
2
1
0200150100500
Cu0.05TiSe2, 11.1 T (B//c*)
Cu0.05TiSe2, 11.1 T (B//c*)Se metal
powder TiSe2 (Dupree et. al.)
2
4
0.1
2
4
1
2
4
12 4 6 8
102 4 6 8
1002
Figure 4.9: Temperature dependence of 771/T1 of parallel stacks of CuxTiSe2 (x=0.05, 0.07)platelets with B ‖ c∗, powder of TiSe2 (Dupree et. al.), and pure 77Se metal. Inset: log-logplot of this graph. Note the non-linearity of 771/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/T1 ∝ T wherein only Fermi contact interaction is taken into account for
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 Hloc = 〈Hloc〉 + δH where 〈Hloc〉is average internal magnetic field and δH is the fluctuating magnetic field. The nu-
clear spin lattice relaxation rate due to fluctuating magnetic field is written as 1/T1 =
(γ2
2)∫ +∞
−∞dt cos ωt〈δH+(t)δH−(0)〉.
86
In terms of the wavelength and frequency-dependent magnetic susceptibilities, the spin
lattice relaxation rate is [38, 63]:
1/T1 = (γ2
2)∑
q
AqA−q
∫ +∞
−∞
dt cos ω0t〈s+q (t)s−q (0)〉 = (
2γ2kBT
g2μ2B
)∑
q
AqA−qχ
′′
⊥(q, ω0)
ω0
(4.5)
where sq is the Fourier-transformed electronic spin density, Aq (from the relation V 1/2Ak+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]:
1/T1 = (πγ2kBT
V)∑
q
AqA−q
∑
k(nk − nk+q)δ(ω0 − εk+q + εk)
ω0[1 − wχ′′
0(q, ω0)]2 + [wχ′′
0(q, ω0)]2(4.6)
This equation can be simplified by several approximations: limω0→0 χ′
0(q, ω) = χ0(q =
0, 0)G(q) and limω0→0 χ′′
0(q, ω) = 0. Here G(q) = (12)[1+ 1−x2
2xlog |1+x
1−x|] from the free electron
approximation with x = q2kF
. The simplified equation is [63]:
1/T1 = πγ2kBT〈AaA−q
V〉[n(εF )]2〈[1 − αG(q)]−2〉 (4.7)
where n(εF ) is the DOS of electrons at the Fermi level and α = 3vn2εF
. Since the Knight shift
is:
Ks =V −1/2μB〈Aq〉n(ǫF )
1 − α(4.8)
and introducing
K(α) = 2
∫ 1
0
(1 − α)2xdx
[1 − αG(x)]2(4.9)
the modified Korringa relation (in SI units) can therefore be written as:
1/T1 = (4πkBT
)γ2
n
γ2e
K2s K(α) (4.10)
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
0.4
0.2
0.0
K(α
)
1.00.80.60.40.20.0Interaction Parameter α
Cu0.07TiSe2
Cu0.05TiSe2
α0.07α0.05
Figure 4.10: Enhanced Korringa factor K(α) vs interaction parameter α from Moriya and acorrected 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 parameterα 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/T1T (s · K)−1 K(α)Se metal 0.15625 1
Cu0.07TiSe2 0.036353 0.23266Cu0.05TiSe2 0.0259313 0.1659615
88
4.9 Conclusion
In conclusion, the electronic correlation of CuxTiSe2 was investigated via 77Se and 63Cu
NMR. The temperature-dependent 771/T1 data show enhanced-Korringa behavior where the
slope changes with Cu doping. Based on the modified Korringa model, the interaction
parameter α is around 0.8 for Cu0.07TiSe2 and 0.93 for Cu0.05TiSe2.631/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 Neel-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 kagome (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
candidate of spin liquid model material, the langasite Pr3Ga5SiO14 studied via 69,71Ga 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
CuOH
(a) (b)
Figure 5.1: (a) The structurally perfect kagome lattice (b) Crystal structure of herbert-smithite (ZnCu3(OH)6Cl2), the S = 1/2 structurally perfect kagome lattice network of Cu2+
spins (from Ref. [69]).
attracted scientific attention because of the triangular arrangement of the S = 32
Cr3+
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
kagome susceptibility χfrus displays a broad maximum at T ∼ J2
and that the macroscopic
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.
In the case of the jarosites RFe3(OH)6(SO4)2 (where R=K, Na, etc.), the Fe3+ spins with
S = 52
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 kagome lattice antiferromagnet. 1H NMR confirmed
a Neel-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 = 12
volvorthite Cu3V2O7(OH)2·2H2O [68] reveal a
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 astructurally perfect kagome lattice.
range order of q =√
3 ×√
3 type (see Fig. 5.2b), and only 40 % of the Cu moments are
frozen.
In the case of herbertsmithite ZnCu3(OH)6Cl2, Cu2+ form a network of corner-sharing
triangles where the Cu S = 12
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/T1 ∝ Tα (α ≈ 0.45 for 4-8 T
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
systems. This S = 12
Heisenberg kagome antiferromagnet is believed to be a good model
material for low spin, pure spin liquid ground state.
On the other hand, the recently discovered langasite Nd3Ga5SiO14 where the Nd3+ mag-
netic moments with S = 92
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|Tc
≥ 1300 than any other kagome systems [71].
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 kagome arrangement of magnetic moments.
5.2 The Rare-Earth Kagome R3Ga5SiO14
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 P321. The Ga3+ ions are sitting in three
crystallographically inequivalent sites.
5.3 69,71Ga NMR Probe of the Spin Dynamics ofPr3Ga5SiO14
Various two-dimensional (2D) triangular lattices, including kagome systems, 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 kagome examples
are ZnCu3(OH)6Cl12 [Cu2+ (S=1/2)], SrCr8−xGa4+xO19 [Cr3+ (S=3/2)], and the jarosite
(D3O)Fe3(SO4)2(OD)6 [Fe3+ (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
PrGaOSi
b
ac
Figure 5.3: Distorted kagome system: 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 kagome compounds 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 ions R = Nd or Pr occupy 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
(P321) with lattice parameters a = 8.0661(2)A and c = 5.0620(2)A. The RE Pr3+ ions
(5f2 : 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 kagome lattice. 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
NM
R I
nte
nsity (
arb
. u
nits)
111098765Field (T)
2.9 K
100 K
71C(1)
69C(1)71
C(2)
69C(2)
4.2 Å
Figure 5.4: NMR field-scan spectrum of Pr3Ga5SiO14 showing quadrupolar split 69Ga(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 showingthree kagome planes. Two Ga sites lie between these planes and the third site is in plane.
between the Pr3+ spins and crystal field effects [89]. The susceptibility data show no evidence
for magnetic ordering down to 2 K in spite of the comparatively large AFM interactions. The
spin system can be modelled as elliptic planar rotators perpendicular to the crystal threefold
axis [89]. The estimated Curie-Weiss temperature is θCW = −2.3 K but neutron diffraction
and ac susceptibility do not reveal long-range magnetic ordering nor a spin glass transition
at temperatures down to 35 mK [90]. This shows that Pr3Ga5SiO14 has a spin-liquid-like
ground state corresponding to a high frustration index f = θCW/Tc ≥ 66. Analysis of the
magnetic specific heat suggests that the crystal field energy level structure of Pr3+ consists
of three singlets with gaps Δ1 = 25 K and Δ2 = 68 K, and Δ3 = 780 K. Specific heat
95
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 kagome lattice 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 kagome compound 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
the two isostructural compounds R3Ga5SiO14 (R = Pr3+ or Nd3+) 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 kagome systems R3Ga5SiO14
(R = Nd, Pr). 69Ga 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 kagome planes 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
Knight shift 69K 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
lowered. For T > 30 K a linear relationship between 69K 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.
In order to study the Ga3+ 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
state and the first excited state to be given at low T by τ = τ0eΔ1/T with Δ1 the lowest
energy gap and the pre-exponential factor τ0 ∼ 10−11 s. This predicts that for T < Δ1 the
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
-2.0
-1.5
-1.0
-0.5
0.0
80604020
(a)5
4
3
2
1
9.69.59.49.39.2
4 K
7 K
10 K
20 K
30 K
40 K
70 K
110 K
230 K
(b)
69C(2)
69C(1)
69C(2)
69C(1)
Figure 5.5: (a) 69Ga Knight shifts measured in an applied field of 9 T along the crystal c-axisplotted versus the magnetic susceptibility with temperature as the implicit parameter. ForT > 30 K the plot shows that but for lower T departures from a linear relationship areobserved. (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 ωI ± ωS where ωS is the electron frequency. For long correlation times
the dipolar process is dominant and we can neglect any contributions due to the hyperfine
coupling. The relaxation rate may be written as 1/T1 = C⊥( τ21+ω2
1τ22) where τ2 is the transverse
correlation time for electron spins and where C⊥ = γ2I 〈H2
⊥〉 is the transverse component of
the local field [2, 3]. Similarly we have 1/T2 = C⊥( τ21+ω2
1τ22) + C‖τ1 with τ1 the longitudinal
correlation time, C‖ = γ2I 〈H2
‖ 〉 and H‖ the z-component of the local field. The introduction of
98
10
8
6
4
2
0
12 3 4 5 6 7
102 3 4 5 6 7
1002
0.4
0.3
0.2
0.1
0.0
1/T1
1/T2
Cmag
4
6
1
2
4
6
10
2
4
12 4 6
102 4 6
1002
7.09 T 8.90 T 12.06 T 16.37 T
T2
behavior
Figure 5.6: 69Ga 1/T1 and 1/T2 for Pr3Ga5SiO14 as a function of T at 16.37 T, togetherwith specific heat at 9 T whose peak is coincident with the broad maximum of 691/T2. Thesimilarity in behavior of the two quantities is striking and points to a common underlyingmechanism. Inset: log-log plot of 691/T2 at different fields. Notice that the broad maximumsharpens 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 ωIτ2 = 1 while for higher T in the short correlation time case (ω1τ2 < 1)
the dipolar mechanism leads to 1/T1 ∼ C⊥τ2 and 1/T2 ∼ C⊥τ2 +C‖τ1. Inspection of Fig. 5.6
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
τ0 ∼ 10−11s and the energy gap for spin excitations Δ = 98 K. The energy gap obtained
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 ΔNMR = Δ0 + αH where the slope α ≈ gμB with μB the Bohr magneton and
g = 3.32 close to the g value for Pr3+ ion. The magnitude of the zero-field gap Δ0 = 3.5 K
(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
the lattice contribution using La3Ga5SiO14 as a reference, shows T 2 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 near 7 K below which 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 < 1 K
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
-9
-6
-3
0
log
τ (s
)
0.80.60.40.20.0
16.37 T12.06 T
7.09 T
4.47 T
2.42 T
10-6
10-4
10-2
100
691/
T1
(m
s-1)
12 4 6 8
102 4 6 8
1002
40
30
20
10
0
Δ (K
)
20151050Field (T)
NMR Neutron
~T3
Figure 5.7: Temperature dependence of the transverse spin correlation time τ2 at differentfields extracted from 691/T1 vs T plot (see lower inset). At high temperatures, the gapΔ ≈ 98 K obtained from the slope is field-independent. Below 10 K, the gap ΔNMR has fielddependence (see upper inset) where the dashed line corresponds to the fit ΔNMR = Δ0 + αHwith α = gμB = 3.32μB and Δ0 = 3.5 K. The field-dependence of the spin gap in theexcitation spectrum as derived from 35 mK inelastic neutron scattering results (from Ref.[90]) is shown for comparison.
101
Goldstone modes. For H > 0 T, 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 (∼ 6 to 7
in-plane lattice spacings) for H = 9 T [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.
5.6 93Nb 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
RMnO3 (R = Ho − Lu, Y ). YMnO3 for instance exhibits ferroelectricity at very high
Tc ≈ 700 K and at the same time exhibit antiferromagnetic transition at a much lower
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 latticeparameters (from Ref. [92]).
temperature TN ≈ 70 K. Near the the Neel temperature this material shows multiferroic
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) andBa3Nb(Fe0.5Ga0.5)3Si2O14 (solid line) (c) The magnetic contribution to the specific heat andthe calculated entropy (d) Temperature dependence of thermal conductivity (from Ref. [92])
104
non-centrosymmetric P321 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|Tc
= 7.3, a much lower value than the Nd-
and Pr-langasites. From the susceptibility analysis, the effective moment is 5.58 μB which
is closed to the bare Fe3+ value (S = 52, 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 mag-
netism in the geometrically frustrated multiferroic system Ba3NbFe3Si2O14. The spin-lattice
relaxation rate 93Nb 1/T1 shows a magnetic transition at Tc ≈ 26 K and the spectra changes
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
of the spin-lattice relaxation rate 931/T1 (see Fig. 5.12a) and the spin-spin lattice relaxation
105
1.5
1.0
0.5
0.07.857.807.757.707.657.60
7.80
7.75
7.70
7.65
25020015010050
250 K225 K200 K
175 K150 K
125 K
100 K
80 K
60 K
50 K
35 K40 K
Figure 5.10: Field-swept 93Nb spectra in the paramagnetic state of Ba3NbFe3Si2O14. Inset:Plot of the resonant field versus temperature reflecting the spin susceptibility.
rate 931/T2 (see Fig. 5.13). The antiferromagnetic spin fluctuation is evident in the huge
upturn of both relaxation rates near the Neel temperature TN ≈ 27.5K. This type of
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]:
1
T1T=
A + B(T−TN )1/2 for T > TN
C(1− T
TN)α for T > TN
(5.1)
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
8.5
8.0
7.5
7.0
6.5
150100500-50-1009876
(a) (b)
90o
60o
40o
65Cu
63Cu
27Al
Left Peak
Right Peak
4.2 K
4.2 K
Figure 5.11: (a) Fieldswept 93Nb spectra at constant frequency 83.4 MHz versus fieldorientation (θ is the angle between B and a-b plane) at 4.2 K in the antiferromagneticstate. The small sharp peaks are 63,65Cu and 27Al NMR signals from the coil. (b) Plot ofthe left and right resonant peaks. The dashed lines are fits to the equation A cos θ where Ais the hyperfine coupling constant.
fluctuations drive this particular feature in thermal conductivity as mentioned earlier. There
is a small region in temperature near the Neel 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/T1 and 93Nb 1/T2 around T ∗ ≈ 4K which could be due to spin reorientation effect
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
temperature-dependent Knight shift is evident. The Knight shift 93Ks reflects the local spin
susceptibility measured at the nuclear site and the corresponding Clogston-Jaccarino plot
(93Ks 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
1.5
1.0
0.5
0.0
12 3 4 5 6 7
102 3 4 5 6 7
1002
PP LP RP MPT<TN SCR fit
2.5
2.0
1.5
1.0
0.5
0.09876
30 K
27 K
15 K
4.2 KLPMP
RP
PPTN=27 K
T*=4 K
27Al
(a) (b)
Figure 5.12: (a) Temperature dependence of 931/T1T in Ba3NbFe3Si2O14 at 83.4 MHz in theparamagnetic 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 spinfluctuation theory with TN ≈ 27.5 K. (b) Corresponding temperature-dependent spectra inthe two states. The location of the peaks are indicated and the middle sharp line is 27AlNMR signal from the probe.
This is attributed to quadrupolar broadening (since Nb has spin I = 92) 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 Neel 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
80
60
40
20
12 3 4 5 6 7 8 9
102 3 4 5 6 7 8 9
100
0.5
0.4
0.3
0.2
2520151050
T*
MP
PP
LP
RP
MP
T*
TN
Figure 5.13: Temperature dependence of the spin-spin relaxation rate 931/T2 measured in theparamagnetic and antiferromagnetic regions at 83.4 MHz with B ⊥ a− b plane. The dashedline is a fit similar to SCR spin fluctuation behavior. Inset: Dependence of the stretchedexponential parameter β of the middle peak. The change in β at around 4 K corresponds toT ∗ 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
2(3 cos2 θ− 1). Instead
the angular dependence of the peak splitting is best described by the fitting ±A0 cos(θ).
This deviation reveals that the coupling among the spins is not simply dipolar in nature.
5.7 Conclusion
93Nb NMR spectroscopy in Ba3NbFe3Si2O14 confirmed the existence of antiferromagnetic
order below TN ≈ 27 K evident in the divergence of both 931/T1 and 931/T2 and the
broadening of 93Nb NMR spectra. A small NMR wipeout region is seen in the vicinity
of the Neel 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
could lead to a different magnetic environment. The peak in 931/T1 and 931/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
organic conductor (TMTSF)2ClO4 reveal that the peaks in 771/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 measureKs spin susceptibility
spectra magnetic structureFWHM internal magnetic field1/T1 fluctuating local magnetic field1/T2 fluctuating local magnetic field
τ magnetic fluctuationsη 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
field, where peaks in 771/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 > 20 T SP is
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 relax-
ation 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 kagome
system Pr3Ga5SiO14. The spin-lattice relaxation rate 691/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
rate 691/T2 exhibits a peak at a lower temperature (10 K) below which field-dependent
power-law behavior close to T 2 is observed. In addition, the peak in 691/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.
In addition to 195Pt NMR lineshapes, a follow-up work on (Per)2Pt[mnt]2 by measuring
field dependence of 1951/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
35 T on the temperature dependence of the relaxation rates 69,711/T1 and 69,711/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) kagome lattice (corner-shared triangular arrangement
of S = 1/2 Ir4+ spins) showing spin-liquid state [94, 95]. I would like to investigate spin
dynamics on this “hyperkagome” material via 23Na relaxation rates and compare with the
spin dynamical behavior in 2D kagome materials 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.
114
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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).
3. 77Se 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.
4. Ba3NbFe3Si2O14: a new multiferroic with a 2D triangular Fe3+ 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 kagome Pr-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:
8. 93Nb 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
12. 135,137Ba 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).
6. 135,137Ba 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.
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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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