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DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University
By Victor A. Stenger, B. S., M. S.
The Ohio State University
1996
Dissertation Committee: , Approved by
Dr. Charles H. Pennington %•
Dr. Thomas R. Lemberger ^
Dr. Daniel L. Cox Adviser
Department of Physics UMI Number: 9620075
UMI Microform 9620075 Copyright 1996, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103 NMR Studies of Alkali Fulleride Superconductors
by Victor A. Stenger. Ph. D. The Ohio State University, 1996 Professor Charles H. Pennington, Adviser
There has been much debate over the mechanism of superconductivity in the alkali fulleride superconductors A 3C60 (A - alkali metal). Some researchers think an exotic pairing mechanism, possibly involving an electronic interaction, is manifest in A 3 C60 where others believe that the more prosaic phonon-mediated interaction proposed in the theory of
Bardeen, Cooper, and Schrielfer (the BCS theory) is appropriate. Also, if the framework of the BCS theory is indeed proper, there is the issue of identifying the relevant modes of the pairing phonons: either the intermolecular or intramolecular modes. This thesis explores these questions with the use of nuclear magnetic resonance (NMR). We find that the body of NMR data on A 3C60 are consistent within the weak coupling limit of BCS theory with a BCS gap parameter less than or equal to l.OkgTc, suggesting that the intramolecular phonons are responsible for electron pairing iti the superconducting state of
A3C60* In addition, we use NMR to probe the normal state of A 3 C60 and find that the electronic properties of A 3Q 0 are that of a normal metal with electron-electron interactions having minor importance. Acknowledgments
I would first like to thank my lab colleagues (and friends) for their support and guidance. Thanks to Charles Recchia, Joseph Vance, D.-R. Buffinger, S.-M. Lee,
Krzysztof Gomy, Christopher Hahm, Valerie Nandor, and Dr. Joseph Martindale. Thanks also to Armen Ezekeilian, James Reynolds and Dave Robertson. It really makes a difference having a bunch of fun and intelligent people to inspire me on a daily basis. With all of you I made lasting friendships which I value forever.
Thank you to Dr. Charles Pennington, it was an honor being your first graduate student. Thanks for teaching me about NMR and about superconductivity, even more so for teaching me how to be enthusiastic and excited about science and have confidence in my own intellect. My ties with you will surely entwine into a deep appreciation as life goes on.
Thanks also to Dr. Robin Ziebarth for always taking time out of your day to help by providing me with not only outstanding samples but also outstanding insight. There was absolutely no way my work could have been so prolific without you.
Thanks to the faculty in the Physics Department at The Ohio State University.
Especially, thanks to Dr. Daniel Cox, Dr. Thomas Lemberger, Dr. Ratnasingham
Sooryakumar, Dr. David Stroud, and Dr. John Wilkins. Every member with whom I interacted with at OSU physics was nothing but superb. Thanks to the faculty in the
Physics Department at the University of Hawaii: Dr. James Gaines, Dr. Peter Crooker, Dr.
Xerxes Tata, Dr. Chester Vauss, and Dr. Frederick Harris. My association with both departments was indeed a privilege. I would finally like to thank my family. Thank you to my lovely wife Helenna, you
are truly my inspiration and sunshine. Thank you for sacrificing your time and love for my
pursuits. Thank you Mom, your infinite support and wisdom is something that I can
always rely on to guide me through life's changes. I would especially like to thank Dad.
You are the one who not only taught me about the laws of physics also the laws of life.
The path you cleared in front of me has afforded me so many advantages that I humbly
hope to make you proud. From one Dr. Stenger to the next I dedicate this thesis to you. Vita
Sept. 28, 1966 ...... Bom, Honolulu, Hawaii
1984 ...... Graduated from Iolani High School, Honolulu, Hawaii
1989 ...... Bachelor of Science, University of Hawaii at Manoa
1989-1992 ...... Teaching Assistant, Department of Physics, The Ohio State University
1993-1995 ...... Research Assistant, Department of Physics, The Ohio State University
1993 ...... Master of Science, The Ohio State University, Columbus, Ohio
Publications
1. V. A. Stenger, C. H. Pennington, D. R. Buffinger, and R. P. Ziebarth, Physical Review Letters 74, 1649 (1995).
2. V. A. Stenger, C. H. Recchia, J. E. Vance C. H. Pennington, D. R. Buffinger, and R. P. Ziebarth, Physical Review B Rapid Communications 48, 9942 (1993).
3. V. A. Stenger, C. H. Recchia, C. H. Pennington, D. R. Buffinger, and R. P. Ziebarth, Journal o f Superconductivity 7,931 (1994). 4. V. A. Stenger, C. H. Pennington, D. R. Buffinger, and R. P. Ziebarth, to be submitted to Physical Review B.
5. C. H. Pennington and V. A. Stenger, submitted to Reviews o f Modem Physics.
6. C. H. Pennington, V. A. Stenger, C. H. Recchia, C. D. Hahm, K. R. Gomy, V. A. Nandor, D. R. Buffinger, S.-M. Lee, and R. P. Ziebarth, to be published in Physical Review B Rapid Communications.
7. D. R. Buffinger, R. P. Ziebarth, V. A. Stenger, C. H. Recchia, and C. H. Pennington, Journal o f the American Chemical Society 115, 9267 (1993).
8. D. R. Buffinger, S.-M. Lee, R. P. Ziebarth, V. A. Stenger, and C. H. Pennington, Recent Advances in the Physics and Chemistry, Proceedings of the 185th Meeting of the Electrochemical Society, May 22-27,1994.
Field of Study
Major Field: Physics
Branches of Research: Condensed Matter Physics Superconductivity Nuclear Magnetic Resonance
Research Adviser Charles H. Pennington, Assistant Professor of Physics Research Co-Adviser Thomas R. Lemberger, Professor of Physics Table of Contents
Page
A bstract ...... ii
Acknowledgments ...... iii
V ita...... v
Table of Contents ...... vii
List of Tables ...... x
List of Figures ...... xi
Chapter Page
I. Introduction ...... 1
II. Alkali Fulleride Superconductors ...... 4 HA. Chapter Overview IIB. Superconductivity HBl. Phenomenology of Superconductivity HB2. The BCS Theory EC. Alkali Fulleride Superconductors IIC1. Fullerenes and Solid C60 HC2. A3C60 HC3. Superconductivity in A 3C60 BC4. Phonons in A 3C60 HC5. The Eneigy Gap in A 3C60
vii III. Nuclear Magnetic Resonance ...... 23 IDA Chapter Overview DIB. NMR of Metals HIBl. The Nuclear Spin Hamiltonian mB2. The Knight Shift 1HB3. The Korringa Relation m e. NMR of Superconductors mCl. Electron Spin Susceptibility mC2. The Hebel-Slichter Peak mC3. NMR of Type II Superconductors HID. NMR Results of Other Groups on A 3C60 IDD1. Knight Shift IHD2. Spin Lattice Relaxation Rate
IV. Experimental Methods ...... 40 IVA. Chapter Overview IVB. Experimental Apparatus IVB1. NMR Spectrometer IVB2. NMR Probe IVC. Experimental Techniques IVC1. The Bloch Equations IVC2. The Spin Echo IVC3. Pulse Sequences and the Config.con File IVC4. FID and Spin Echo Sequences IVC5. Phase Cycling IVC6. Saturation /Inversion Recovery
V. Sample Characterization ...... 57 VA Chapter Overview VB. Sample Preparation VB1. Synthesis of A 3C60 VB2. Samples Used VC. Sample Characterization VC1. SQUID Magnetization Curves VC2. NMR Lineshapes
VI. The Knight Shift ...... 71 VIA. Chapter Overview VIB. Electron Spin Susceptibility VIB1. Subtraction Method VIB2. Rb 2CsC60 Lineshapes VIB3. Rb 2CsC6o Line Shifts VIB4. KdT) of Nuclei in Octahedral Sites VIB5. X s (T) in Superconducting State of Rb 2CsC6o VIC. Normal State Knight Shift VIC1. A Priori Estimate of K u s VIC2. A Priori Estimates of Alkali Knight Shifts VIC3. Measurement of Demagnetization Factor
viii VII. Spin-Lattice Relaxation Rates ...... 93 VHA. Chapter Overview VIIB. Relocation in the Normal State VTTB1. Recovery Curves VIIB2. Normal State Relaxation Rates VIIB3. Stoner Enhanced Density of States VIIB4. (l/TiT)lJ2 vs. AT for 87Rb and 133Cs vnc. Relaxation in the Superconducting State V nC l. The Hebel-Slichter Coherence Peak VHC2. Arhenius Plots of the Relaxation Data VHC3. BCS Fits to Rs/Rn Data VHC4. Field-Induced Mechanisms of Peak Suppression
VIII. Discussion of Results ...... 126 VmA. Chapter Overview VmB. Superconducting State VUIBl. Summary of Superconducting State Data V111B2. Comparison with other Superconductors Vine. Normal State Vm Cl. Summary of Normal State Data VHIC2. Strong Correlations in A 3C60
Appendix Page
A. Nuclear Quadrupole Effects in A 3 C 6 0 ...... 142
B. The Normal Modes Problem ...... 148
Page
R eferences ...... 155
ix List of Tables
Table Page
2.1 Table of superconductivity parameters for Al, Pb, Yba 2Cu3 0 7 , K3C60, and Rb3C 6o ...... 6
2.2 Table of phonon modes and strong coupling theory parameters 22
5.1 Table of A 3C 60 compounds and the related sample identification code ...... 60
5.2 Table of NMR shifts of aqueous alkali ions and that of the same ion in free space ...... 60
5.3 NMR shifts for A 3 C6O compounds at room temperature ...... 70
x List of Figures
Figure Page
2.1 Schematic of a Ceo molecule ...... 12
2.2 The three forms of solid carbon: diamond, graphite, and C 60...... 13
2.3 Motionally narrowing l3C of Cqq lineshape ...... 15
2.4 Unit cell of A 3 C 6 0 ...... 17
2.5 39k spectrum of K 3 C6O at room temperature ...... 18
2.6 Phonon spectrum of A 3 C 6 0 ...... 20
3.1 Spin susceptibility data for aluminum in the superconducting state...... 30
3.2 Plot of superconducting density of states ...... 31
3.3 Hebel-Slichter coherence peak in the superconducting state of alum inum ...... 33
3.4 Rs/Rn data for 51V in the mixed state of V 3 Sn ...... 35
3.5 13^ shift data of Tycko et a l in RI 33C6 0 ...... 37
3.6 13C l/T! T data of Tycko et a l in Rb 3C6o at « 9 T ...... 38
4.1 Schematic of NMR spectrometer ...... 42
4.2 Schematic of NMR probe and the equivalent circuit ...... 44
4.3 Diagram showing the formation of a spin echo ...... 48
4.4 Pulse sequence for the formation of a free induction decay ...... 49
4.5 Configuration file for pulse sequences in the MacNMR software application...... 49
xi 4.6 Pulse sequence for the formation of a spin echo ...... 51
4.7 Schematic of how add/subtract phase cycling works ...... 53
4.8 Pulse sequence for a saturation recovery experiment ...... 56
4.9 Pulse sequence for an inversion recoveiy experiment ...... 56
5.1 Superconducting fraction vs. temperature for A 3C60 samples prepared by different methods ...... 62
5.2 spectra for 13C, 87Rb, 39K, and 133Cs at room temperature for A 3C60 com pounds ...... 63
5.3 87Rb an(j 85Rb spectra of Rb 3 C6 o ...... 67
5.4 39r spectra in K 3 C6O as a function of temperature ...... 69
6.1 Lineshapes for 13C, 87Rb, and 133Cs in Rt>2CsC60 as a function of temperature ...... 74
6.2 Total measured isotropic shifts of 13C, 87Rb, and 133Cs isotopes in RbjCsCeo as a function of temperature ...... 76
6.3 Total measured isotropic shifts of 13C, 87Rb, and 133Cs isotopes in Rl^CsCgo as a function of temperature and applied magnetic field...... 78
6.4 Total measured isotropic shifts of 87Rb in RbCs 2C6o and 133Cs in Rb2CsC6o as a function of temperature ...... 79
6.5 Superconducting state spin susceptibility vs. temperature of Rb 2CsC6o at 8.8 Tesla ...... 81
6.6 Superconducting state spin susceptibility vs. temperature of Rb2CsC6o for different applied field strengths ...... 83
6.7 Magnetic moment vs. temperature for Rb 2CsC6o at 3 and 5 T esla...... 90
6.8 Magnetic moment vs. time in a field-cooled sample of Rb2CsC6o at 1,3, and 5 Tesla ...... 92
7.1 Saturation recovery curves for 13C in Rb2CsC6o at 8.8 Tesla ...... 95
7.2 inversion recoveiy curves for 87Rb and 85Rb in Rb 3C6o at 8.8 Tesla and 80 Kelvin ...... 97
xii 7.3 87Rb 1/TiTvs. temperature forthe “O “T,” and “T ” lines in R b 3C 60 ...... 99
7.4 Inversion recovery curves for 87Rb in Rb 2CsC6o at 8.8 Tesla 100
7-5 Saturation recoveiy curves for 133Cs in Rb2CsC60 at 8.8 T esla ...... 101
7.6 Temperature dependence of 1/TiTfor 13C, 87Rb, and 133Cs isotopes in R b 2 C sC 60 at 8.8 Tesla ...... 103
7.7 13c \IT\ Tasa function of lattice parameter a for AsCeo with dopant and temperature as implicit parameters ...... 104
7-8 ( 1/ r i 7) 1/2 vs tota] isotropic shift K for 87Rb in Rb 2 CsC60...... 108
7.9 (l/T \T )lf2 vs. total isotropic shift K for 133Cs in Rb2CsC60 ...... 109
7.10 Saturation recovery curves for 13C at 3 Tesla...... 112
7.11 13C Rs!Rn vs. temperature in Rb 2CsC«) as a function of applied magnetic field strength ...... 113
7.12 T\ vs. 1/Tfor l3C in Rb 2CsC«) as function of applied magnetic field ...... 114
7.13 T\ vs. 1/7 for the 13C, 87Rb, and 133Cs isotopes in Rb 2CsCgo at 8.8 T esla...... 115
7.14 13c RsIRn vs. temperature in Rb2CsC6o at 3 Tesla with theoretical fits...... 118
7.15 13c Rs/Rn vs. TITC for Rb 2CsC60 at 3 Tesla with Eliashberg theoretical fits ...... I I 9
7.16 13c R s/R n vs. T!TC for Rb2CsC<>o and K 3C60 at 3 Tesla...... 123
7.17 13c Rs/Rn vs. temperature for Rb 2CsC6o at 3 Tesla and a fit assuming a weighted average of superconducting and normal state relaxation rates...... 125
8.1 Weak coupling BCS predictions for the temperature dependence of the electron spin susceptibility with and without Stoner enhancement ...... 128
xiii 8.2 Normalized superconducting state spin susceptibility vs. TITC as measured by the Knight shift for Al, Rb 2CsC60, and Y B a 2C u 30 7...... 130
8.3 RsIRn vs. T/Tc for Al, Rb2CsC6o, and YBa2Cu306.9l (NQR) 132
8.4 Arhenius plots for Al, Rb2CsCgo, and YBa2Cu3C >7 (NQR)...... 133
8.5 17q and 63Cu spin-lattice relaxation rates in the normal state of Yba2Cu306.63 ...... ^ 6
8.6 Lineshape and l/T\ as function of frequency in Rt^CsCeo at 8.8 T and 80 K ...... 138
8.7 Theoretical fits to 13C lineshape in Rt^CsCeo ...... 1 ^
A.1 Nutation curves for 87Rb in RbCl and Rb 2CsC60 ...... 1^3
A.2 Nutation curves for 133Cs in Cs2C03 and Rt^CsCeo ...... ^
xiv Chapter I
Introduction
The technique of nuclear magnetic resonance (NMR) has proven to be an extremely powerful tool which can be used to elucidate the properties of superconducting materials
[MacLaughlin, 1976]. Groundbreaking NMR experiments by Hebei and Slichter [Hebei,
1959] and Masuda and Redfield [Masuda, 1962] in the late 1950's on conventional superconductors such as aluminum helped confirm the standard theory of superconductivity: the Bardeen-Cooper-Schrieffer (BCS) theory, 1957 [Bardeen, 1957].
These same pioneering works also provided stringent tests and guidelines which are still followed today in the investigations of new superconducting materials and the new theories required to describe them. Among these materials are the copper-oxide or “high-Tc” superconducting materials [Pennington, 1990; Slichter, 1994], the heavy fermion superconductors [Asayama, 1988], and superfluid 3He [Volhardt, 1989]. In particular, the results of NMR experiments on the high-Tc superconducting materials (discovered in 1986
[Bednorz, 1986]) have revealed the possible existence of an “exotic” form of superconductivity where electrons are paired via antiferromagnetic interactions into a d-wave state [Monthoux, 1991]. The reconciliation of these NMR results in the high-Tc materials with what is already understood about conventional superconductors could require a modification to the original BCS model or even more general theory of superconductivity. It is this kind of important role that NMR measurements play that make it a vital and necessary probe into the properties of superconducting materials.
Recently, much attention has been paid to the alkali fulleride superconductors
A3C60 (A = alkali metal) [Hebard, 1992]. Transition temperatures (TV) as high as 33 K
(RbCs2C6o) are obtainable in these materials. A 3 C60 has the highest Tc of any three dimensional superconductor and would have had the world record highest Tc prior to the discovery of the copper-oxide superconductors. Indeed, a significant amount of the efforts to study the alkali fullerides involves the raising of this maximum Tc of 33 K by synthesizing new fiillerene based compounds. Efforts are also underway to understand the mechanisms of superconductivity in these materials. Can superconductivity in the A 3C6O materials be understood within the BCS framework, where electrons are paired through a phonon-mediated interaction, or do the A 3 C60 materials exhibit an exotic form of superconductivity similar to the high-TV materials? If the BCS framework is indeed appropriate, what are the relevant phonon modes? Are the low frequency “/nrermolecular” phonon modes important or are the high frequency “/n/ramolecular” modes dominant in the formation of electron pairs? The normal state properties of A 3C60 are also of interest. Can the normal state be understood with conventional band theory calculations or are there “strong correlations” as in the high-!Tc materials? The technique of NMR will certainly give important insight into these issues.
It is the focus of this thesis to explore the superconducting and normal state electronic properties of A 3 C60 with NMR techniques (with the former receiving the majority of our attention).
Chapters n and in introduce the reader to some of the general features of superconductivity, the properties of A 3 C60, and how NMR can probe these features.
Chapters IV and V present the experimental methods and sample characterization respectively. 3
The experimental determinations of the Knight shifts in A 3C6O are discussed in
Chapter VI. We will show that the behavior of the Knight shift in the superconducting
state is veiy conventional: using multinuclear shift measurements we deduce the
temperature dependence of the electron spin susceptibility in the superconducting state and
show that it can be well described by the BCS theoiy. From these measurements we find
that the wtframolecular phonon modes are responsible for electron pairing. We also present
estimates and measurements which attempt to determine the sign and magnitude of the
Knight shifts of the nuclei in the normal state of A 3C60.
In Chapter VII we present the results of spin-lattice relaxation time T\ measurements in A3C60- We will show that the suppression of the Hebel-Slichter coherence peak in \IT\Tis a result of the high applied field strengths and is not a “smoking gun” for exotic superconductivity. We will show that as the field is reduced the Hebel-
Slichter peak is recovered to a maximum value of approximately 1.2 times the normal state value. Using a Migdal-Eliashberg extension of the BCS theoiy we find that the T\ data, unlike the Knight shift, suggest that some electron coupling to the low frequency mfermolecular modes may be necessary to understand A 3 C6 O superconductivity.
However, given other sources of peak suppression and the possibility of a more complicated phonon spectrum we will see that it is reasonable to conclude that the relaxation data are consistent with intramolecular phonon mediated pairing.
In Chapter Vm we will summarize the results presented. We will compare the superconducting state NMR data with that of other superconductors, namely the conventional superconductor aluminum and the high-Tc superconductor YBa 2Cu3 0 7 -5 .
We will also discuss what the NMR data tell us about the normal state of A 3C60 and the possibility of strong electron correlations. Chapter n
Alkali Fulleride Superconductors
IIA. Chapter Overview
This chapter provides a discussion of the basic principles of the superconducting phenomenon and an introduction to A 3 C60 and its superconducting properties. Further reading on the principles of superconductivity may be done in the texts by deGennes and
Tinkham [deGennes, 1989; Tinkham, 1980]. Section IIB presents some fundamental results regarding superconductivity in A 3 C60 - Section HC discusses the structural, electronic, and superconducting properties of solid C 60 and A 3C60. The review articles by
Dresselhaus et al., Ramirez, and Gelfand [Dresselhaus, 1994; Ramirez, 1994; Gelfand,
1994] can be consulted for more detail. Although we will present some NMR data in this chapter, any specific discussion of magnetic resonance in superconductors (including
A3C60) will be deferred until Chapter IE.
4 5
IIB. Superconductivity
HB1. Phenomenology of Superconductivity: Superconductivity was first
discovered in 1911 when H. Kamerlingh Onnes noticed that the electrical resistance in
samples of mercury dropped steeply to zero as the samples were cooled below 4.19 K
[Onnes, 1911]. This property of zero resistance or perfect conductivity is the first of two
fundamental properties that characterize superconductors and the temperature at which this occurs is called the critical temperature Tc. The second property is p erfect diamagnetism , first seen 1933 by Meissner and Ochsenfeld [Meissner, 1933]. Meissner and Ochsenfeld found that magnetic field was expelled from superconductors cooled through Tc in zero applied field and those cooled in the field (perfect conductivity would explain the field expulsion in the first case only). Perfect diamagnetism in superconductors is often called the “Meissner effect” after one of its discoverers. The existence of the Meissner effect in superconductors implies that there will be a critical field Hc above which superconductivity becomes destroyed. Table 2.1 lists some values of Tc for an assortment of metals including copper oxide and alkali fulleride materials.
In 1935 F. London and H. London [London, 1935], in an attempt to understand the Meissner effect, found that a magnetic field parallel to the surface of a superconductor,
B ||, penetrates into the material but is exponentially screened to zero: jBm= 5,|[0)exp(-^±/ / L) (d± is a measure of distance perpendicular to the surface of the superconductor). The characteristic length scale over which the field is screened, A/,, is called the London penetration depth.
# _ jzeL_ ^ ~ A m / 6
Table 2.1. Listing of superconducting parameters for conventional superconductors Al and Pb; the high-Tc superconductor YBa 2Cus0 7 ; the heavy fermion superconductor UBei 3; and the alkali fulleride superconductors K 3C60 and Rb3C60. For the high-Tc materials ab and c mean along the a£-plane or along the c-axis respectively (for the critical field values, the applied field is along this direction ).______
A l1 Pb1 YBa2Cu3<)72 U B ei3 3 K 3C 6O Rb3CfiO
Tc {K) 1.20 7.20 93 0.85 19.7a 30.0*
2A(0)lkBTc 3.4 4.3 - - 5.2b,4.0d, 5.3°,3.ld, 3.6e 3.0eJ.6f iBums [1992]. 2Martindale [1993]. 3Hess [1993]. aAverages of values given by Dresselhaus [1994] and the references therein. bSTM measurements of Zhang [1991]. CSTM measurements of Zhang [1991] dNMR measurements of Tycko [1992]. eFar IR measurements of Digioigi [1992]. fjiSR measurements of Kiefl [1992]. which is related to the superfluid density ns. For elemental superconductors calculated values for Al (assuming that ns a n where n is the total number density of electrons) range from 100to 1,000 A. Table 2.1 gives values of Al for A3C60- A second length scale was introduced in 1950 in the theoiy of V. Ginzburg and L. Landau [Ginzberg, 1950]. The Ginzburg-Landau (GL) theory related the superfluid density ns to the square of the modulus of a complex order parameter ip: ns = \y)\2. The GL theory expands the free energy in powers of ip and uses a variational calculation to obtain differential equations for ip and the supercurrent Js (these equations are called the Ginzburg-Landau equations). From the GL equations one can parameterize the spatial variation of the superconducting order parameter with a length scale £ known as the GL coherence length. The GL coherence length is the average length scale over which the superfluid density can change without a significant change in the free energy. It is convenient to characterize superconductors by the GL parameter k. Table 1 lists some values of £ and xfor superconductors including AjC^q. One of the most important predictions of the GL theoiy was realized in 1957 by A. Abrikosov [Abrikosov, 1957]. Abrikosov examined the case when k is large (for elemental superconductors k is less than approximately unity) and discovered that in such a situation the GL equations predicted a mixed or vortex state. He found that below an applied field strength Hc\ (the lower critical field) the large k material behaved like an elemental superconductor and above a larger applied field strength HC2 (the upper critical field) superconductivity is destroyed. For Hc\ < B a p p lie d< Hc2 Abrikosov discovered that a state existed where instead of the applied field being completely excluded from the 8 bulk of a superconductor the field punches through in quantized units of flux the mixed state behavior are known as Type n superconductors and those that possess the more conventional Meissner effect are Type I. The exact cross over from Type I to Type II superconductivity occurs for *= m . We see from Table 2.1 that A 3 C60 is a Type II superconductor. IIB2. The BCS Theory: Much of the above phenomenology of superconductors was well known prior to any understanding of the actual microscopic interaction responsible for superconductivity. It was not until 1957 when J. Bardeen, R. Schrieffer, and L. Cooper published their Nobel prize winning theory of superconductivity (the BCS theory) that the phenomenological properties of superconductors could be calculated from first principles [Bardeen, 1957]. The BCS theory is arguably the most successful theory in condensed matter physics because of the wide variety materials it encompasses and because of the experimental verifications of the subtle macroscopic manifestations of quantum mechanics it predicts. To this day the BCS theory is the benchmark by which all new superconducting materials must be tested. In 1956 Cooper [Cooper, 1956] realized that the Fermi sea of conduction electrons was unstable to the formation of pairs (“Cooper pairs”) bound through some attractive interaction regardless of how small the strength of the interaction. The attractive interaction in conventional superconductors results from the exchange of a lattice vibration or phonon between the two electrons in the pair. As the first electron travels through the lattice it polarizes the positive ionic background by distorting the lattice (it creates a phonon). The second electron will be attracted to this distorted region and its energy will be subsequently reduced. Earlier experiments such as those studying Tc as a function of isotope [Maxwell, 1950] had already suggested such a mechanism. (This dependence of Tc on isotope is 9 - 1/2 referred to as the isotope effect where if M is the mass of the isotope then Tc <*M ). Electrons bound in a Cooper pair will have equal and opposite momenta and opposite spin (k t ,-k i ). The BCS theory extended the Cooper pair idea to an N particle system using a many body theoretical approach. The details of the theoiy are out of the scope of this thesis so we will present only the important results and let the reader consult the references for more details. Assuming that the interaction potential V between electrons is weak and isotropic (essentially ignoring the presence of the phonons) the BCS theory predicts that the transition temperature can be expressed as tc - u 4 ^ V l w m>v . Here N(0) is the density of electron states at the Fermi energy Ep and wc is a phonon cut off frequency characteristic of the Debye frequency (Od . One can define the coupling strength KeP as being equal to iV(0)V. The BCS theoiy also predicts that a gap A(7) opens in the spectrum of allowed states about Ep. The zero temperature value for the gap is given by 4(0) * 4ft tx>ce~l ,N(0)V. (H.4) Combining equations (H3) and (H.4) we obtain the BCS value for the gap parameter in the weak coupling limit: 4(0) =1.76kBTc. If the phonon mediated interaction V is not “weak” then modifications to the BCS theoiy are required. In the explicit presence of phonons, the strong coupling limit, Migdal in 1958 showed that the electron-phonon interaction could be perturbatively incorporated into the BCS theoiy in a series of terms that go like Jm /M (m = electron mass and M = ion mass) [Migdal, 1958]. This is often referred to as “Migdal's theorem.” Using Migdal’s theorem as a starting point, Eliashberg developed a theoiy of strong coupling superconductivity based on what are now known as the Eliashbeig equations [Eliashbeig, I960]. W. McMillan solved the Eliashberg equations for a set of strong coupling materials and derived an approximate equation for Tc [McMillan, 1965]: -1.04(1 + Kep) T = exj (n.6) 1c 1.45% L^-/i*(l- 0 .62 ^ p)J coin is the logarithmic average of the phonon frequency, p* is the effective coulomb repulsion and Xep is the electron phonon coupling parameter. For more on strong coupling effects in superconductors see McMillan and Scalapino [McMillan, 1994; Scalapino, 1994]. IIC. Alkali Fulleride Superconductors HC1. Fullerenes and Solid C$o: The discoveiy of a new stable form of pure carbon, C 6 0 > in 1985 by Kioto et al. [Kioto, 1985] and the subsequent discovery of superconductivity in alkali metal doped C6o by Hebard et al [Hebard, 1991] prompted so much excitement in the scientific community that whole new subfields in chemistry and solid state physics were created devoted to the study of C The molecule C60 is actually just one member of a large family of carbon clusters C„ (n = 30 and up) which form closed-cage polyhedra. These carbon clusters had actually been observed prior to 1985 in laser ablation studies of graphite by Rohlfing et al [Rohlfing, 1985] but it was Kroto and his coworicers who observed that the most stable phase was for n = 60. Kroto et al suggested that the geometric form of Cqo was that of a truncated icosahedron or, in less technical terms, C6o had the same hexagonal and pentagonal faces as that of the pattern on a soccer ball. The molecular diameter of C 60 is 7.1 A and the average C-C distance is 1.44 A. For this new form of carbon Kroto adopted the name buckminsterfiillerene (or simply fullerene) from the famous architect Richard Buckminster Fuller, whose geodesic domes reminded the scientists of the carbon cages of these molecules. The fullerene C6o is often affectionately referred to as a “buckyball.” Figure 2.1 shows a buckyball. After the Ceo molecule had been successfully identified and isolated by Kroto and his colleagues, the next major breakthrough was the discovery of a technique for the production of macroscopic quantities of the material in solid form by Kratschmer et al [Kratschmer, 1990]. Kratschmer and his coworkers found that C 60 in crystalline form could be extracted from carbon soot (which is produced by the spark erosion of graphite) by first dissolving the soot in a nonpolar solvent such as benzene and then distilling the liquid from the remaining soot. After allowing the distilled liquid to evaporate Kratschmer et a l were left with small crystallites of Ceo- X-ray studies and mass spectroscopy showed that at room temperature C <50 crystallizes into a face centered cubic (fee) structure with a lattice constant of 14.2 A. This lattice of buckyballs is held together by weak van der Waals attraction between the molecules. The significance of the discovery of Kratschmer et a l was that it was now possible for researchers to obtain C6o in macroscopic quantities from a relatively straightforward fabrication method. These Ceo 13 Diamond Graphite Figure 2.2. The three forms of solid carbon: diamond, graphite, and C 6 0 - Figure reprinted with permission from A. P. Ramirez, Superconductivity Review 1,1 (1994). 14 crystals can be characterized as a new form of solid carbon. Figure 2.2 shows solid Ceo and the two other forms of pure carbon: graphite and diamond. One feature of solid C60 that can be easily seen with NMR is that at room temperature the C 60 molecules rapidly rotate at each site in the fee lattice |Tycko, 1991]. This is shown in Figure 2.3 by the “motional narrowing” of the 13C NMR line in solid C60 (measurement done by our group). The 13C lineshape in a sample where the molecules are frozen into position is a broadened powder pattern reflecting the anisotropy of the chemical shifts of the 13C nuclei. However, because of the rapid reorientations of the fullerenes at room temperature, the magnetic environment surrounding the nuclear spin becomes averaged. This averaging is reflected in the 13C NMR by a sharp and narrow lineshape. More detailed 13C NMR measurements by Tycko et a l [1991] suggest that the frequency of the C6o reorientations at room temperature is faster than 20 kHz and motion slower than this NMR time scale is not “frozen in” until the temperature is reduced below approximately 133 K. Low temperature 13C NMR spectra reflecting this “freezing” of molecular motion is also shown in Fig. 2.3. 13C spin-lattice relaxation time measurements of Tycko et al and x-ray diffraction results of Heiney et al. [Heiney, 1991] show that solid C 6 0 undergoes a structural phase transition from an fee structure to a simple cubic (sc) structure at To - 249 K. Above To the fullerenes have random orientations and are freely rotating. Below To the Cgo molecules at each fee lattice site become frozen into the symmetry equivalent orientations required to form an sc lattice and, because motional narrowing of the NMR line is seen to persist down to 133 K, the molecules “ratchet” between these orientations. IIC2. A3C 6 0 : Like graphite, solid C 60 exhibits insulating or semiconducting behavior. The energy gap between the lowest unoccupied molecular orbital (LUMO) and 15 13 C Spectra of C _0 290 K -- in Benzene 290 K -- Solid C. 290 K - 155 K _ 140 K - 120 K -50 0 50 100 150 200 250 300 ppm (from TMS) Figure 2.3. 13C spectra of C 60 dissolved in benzene at room temperature and 13C spectra of solid Cgo at indicated temperatures. The 13C spectrum of solid C60 is expected to be a broad “powder pattern” resonance (width » 200 ppm) reflecting the chemical shift anisotropy of the 13C nuclei averaged over the ciy stallite orientations. At room temperature the NMR line is “motionally narrowed” to a single resonance at 143 ppm (similar to that of C60 in solution) due to the rapid rotations of the C 60 molecules with respect to the NMR time scale of « 10"4 s. As the temperature is reduced the powder pattern is recovered as the molecular motion decreases (All 13C lines are measured relative to TMS). 16 the highest occupied molecular orbital (HOMO) of C6o is 1.5 eV. Due to the small overlap of the molecular orbitals in the solid C6o, the band structure of the solid will be similar to that of an individual molecule. Thus, a gap similar in magnitude exists in the solid. One feature that distinguishes Ceo from graphite is the curvature of the molecule's surface. This curvature causes hybridization between the carbon p and s orbitals which gives solid Cgo a greater electron affinity than graphite. This enhanced electron affinity prompted Haddon et a l to try doping solid C 60 with alkali metals which are highly electropositive [Haddon, 1991]. They obtained a maximum conductivity of 2 (Ocm)'1 for potassium doping with the stoichiometry K 3C60- Like solid C60 A3C60 has a face centered cubic structure. The lattice constants for A3C60 range from 14.24 A for K 3C60 to 14.56 A for RbCs 2C«) (see the reviews on C 60 and A 3C60 listed in the references). The alkali metal ions reside at either tetrahedrally (7) or octahedrally ( O) coordinated sites in the lattice of molecules. For each C^o there are two T sites and one O site. Figure 2.4 shows one unit cell of A 3C60. The ratio of occupations of the two inequivalent lattice sites in A 3C60 by the alkali ions can be seen beautifully with the 39K NMR of KsCgo- In Figure 2.5 the 39K spectrum is shown at room temperature (measured in units of ppm from a 2M KNO 3 solution). We can immediately see that there are two distinct peaks in the spectra. Closer examination reveals that the ratio of the peak areas is 2:1. With this area ratio 2 to 1 we can assign the peaks to the potassium ions in the T and O sites respectively. Thus, we can see that NMR provides an excellent probe of the structure of A3C60 materials. More NMR lines of A 3 C60 compounds will be presented later. IIC3. Superconductivity in A 3 C6 0 : In 1991 Hebard et a l at Bell Laboratories made an exciting discovery [Hebard, 1991]. They found that K 3C60 was a superconductor which had a transition temperature of 19 K. This transition temperature 17 Unit Cell of A3C60 fee Figure 2.4. Unit cell of A3C60- The large spheres represent C60 molecules and the small spheres the alkali metal ions. For each buckyball there will be two alkali ions with tetrahedral spatial coordination (shaded) and one with octahedral coordination. Figure reprinted with permission from D. W. Murphy et al, Journal o f the Physical Chemistry o f Solids 53, 1321 (1992). 18 K Spectra of K3C60 at Room Temperature Tetrahedral Octahedral -200 -150 -100 -50 0 50 100 150 200 Shift (ppm) Figure 2.5. 39K spectrum of K 3C60 at room temperature (measured in units of ppm from K+ ions in a 2M aqueous solution of KNO 3). The peak at 8 ppm corresponds to the K ions with tetrahedral spatial coordination and peak at -49 ppm the ions with octahedral coordination. 19 might be compared with the superconducting graphite intercalation compound KCs which has a Tc of .55 K. Further research by the group at Bell Labs and by Holczer et al. uncovered the compound Rb 3C60 which had Tc = 29 K [Holczer, 1991]. The current record for the highest Tc (33 K) for an alkali fulleride superconductor is held by RbCs 2C6o [Tanigaki, 1991]. A simple clue into the mechanism of superconductivity in A 3C6O can be seen with the increase of Tc with increase in lattice constant a. By doping with larger alkali ions (Rb instead of K for example) the transition temperature increases (19 K to 29 K) [Rosseinsky, 1991]. This can be seen theoretically with Equation QI.3). As the distance between C 60 molecules increases the overlap of the molecular wavefimctions will decrease. This will produce an increase in the density of states at the Fermi level and therefore an increase in Tc by Eqn. (0-3). Thus, it is intriguing to think that because the BCS theory predicts such straightforward behavior that the BCS framework is appropriate for AsCgo- HC4. Phonons in A 3 C 5 0 : The vibrational frequencies of A 3 C60 are very important because within the BCS theory the phonons are the instigators of electron pairing. Essentially the phonon modes can be divided into two categories: those that are confined to one individual molecule, the intramolecular phonon modes, and those between molecules or between the molecules and the alkali ions, the intermolecular phonon modes. The intermolecular (termed rather loosely here) modes can be subdivided into the low energy librational or rocking modes (10 - 20 cm-1), the translational modes (30 - 45 cm-1), and the optical modes (approximately 50 -110 cm*1). These modes are depicted schematically in Figure 2.6. The intramolecular vibrations fall into two categories: radial or breathing modes (300 - 800 cm*1) and tangential modes (> 800 cm*1). These are also shown in Fig. 2.6. 20 Phonons in A3C6O ______1 1__ 1__ 1__ » « » 1 1______1______1______1____ (0 20 SO >00 200 500 1000 2000 FREQUENCY (cm '*} Figure 2.6. Phonon spectrum of A 3 C6O obtained by neutron scattering methods. The modes are classified into two categories: the intermolecular modes, 10 - 110 cm-1, and the intramolecular modes, > 300 cm-1. Figure reprinted with permission from A. F. Hebard, Physics Today 45 (11), 26 (1992). 21 Because A3C60 has two distinct classes of phonon modes (i.e. intermolecular and intramolecular) one can obtain a simple prediction about which of the two phonon classes are relevant to superconductivity if one knows the strength of the electron-phonon coupling Xep = N(0)V. We can see this from Eqn. (H.4): for a given Tc, if XeP is “small” (i.e. in the weak coupling limit) one would need a higher o)c than if Xep were “large” (strong coupling). Thus, a weak coupling result for A 3C60 would imply that the high frequency intramolecular modes are important for pairing and a strong coupling result would implicate the low frequency intermolecular phonon modes. More detailed analysis using Migdal-Eliashbeig methods [Akis, 1991] for a sharply peaked spectral density a2F(oJ) at a single phonon frequency allows for a more quantitative comparison between phonon frequency and superconducting state parameters. Table 2.2 lists strong coupling parameters and the related phonon modes in A 3C60- IIC5. The Energy Gap in A 3 C60 : One of the important issues in the study of alkali fulleride superconductivity is the measurement of the zero temperature energy gap A(0) which is directly related to the coupling strength Aep. This issue has been one of great debate as a result of conflicting values obtained for A(0). Tunneling measurements of Zhang e ta l on K 3C60 and Rb3C6o yield values for 2A(0 )/ksTc of 5.2 and 5.3 respectively (see Table 2.1) [Zhang, 1991]. These numbers are indicative the strong coupling regime of the BCS theory. Weak coupling values of 3.1 and 4.0 for K 3 C60 and Rb3C60 were obtained from the early NMR data of Tycko et al [Tycko, 1992]. Far ER measurements of Digiorgi et a l yielded 3.6 (K3C60) and 3.0 (Rb3C6o) [Digiorgi, 1992]. pSR results of Kiefl et a l produced 2A(0)/^TC = 3.6 for Rb 3C6o [Kiefl, 1992]. The early NMR and the |iSR experiments also point towards the weak coupling picture. Thus, we can see from this body of data that there is some doubt to the magnitude of A(0). 22 Table 2.2. Table of superconductivity theoiy parameters and their relation to phonon modes in A 3C60 obtained from Migdal-Eliashberg calculations of Akis et al [1991]. Within the context of the theoiy of AMs et al one may infer the pairing phonon frequency by obtaining values for TJu)in or A(0) from experiment. ______ Phonon Mode: Frequency (cm-1) TcIWin K p (M*=0) 2A(0)/kBTc Molecular 20 1.0 - - Reorientations Intermolecular 40 0.5 - - Vibrations Optical (Qo- Alkali 10O 0.2 3.1 5.2 Vibrations) Intramolecular: 500 0.04 .53 3.72 Radial Intramolecular 1406 0.015 .35 3.56 Tangential Al (7V=1.2K) 2 0 6 .004 .43 5.54 Pb(7V=7.2K) 39 .128 1.55 4.5 Chapter III Nuclear Magnetic Resonance IIIA. Chapter Overview This chapter introduces the reader to some of the fundamentals of nuclear magnetic resonance and its application to the study of metals and superconductors. Section IHB presents a discussion NMR in normal metals where we introduce two important results: the Knight shirt and the Korringa relation. Section m e extends the normal state results to the superconducting state. In Section HID we present NMR data on A 3 C60 from other research groups. Much of the discussion regarding NMR in metals and superconductors will be along the lines of Slichter and MacLaughlin [Slichter, 1989; MacLaughlin, 1976]. MB. NMR of Metals UIB1. The Nuclear-Spin Hamiltonian: A bare nuclear magnetic moment p in an applied field H= H 0k experiences a torque which will make |i precess about H. The frequency of this precession 0)0 = Yvflo is called the Larmor frequency. yn is the gyromagnetic ratio of the nucleus. The quantum mechanical analog to this precession is 23 that the component of the nuclei's angular momentum operator hi along the axis of the applied field direction (z-axis in our case) becomes quantized into 2 /+ 1 levels with energy separation AE= hYnH 0= hco0. (m.1) For an ensemble of N independent nuclear spins at thermal equilibrium with a temperature reservoir (in a metal the reservoir is the conduction electrons) the ratios of the populations of the 21+1 levels will be given by the Boltzmann distribution: In an NMR experiment one induces a resonant absorption between these energy levels with radiation of frequency too. The frequency range of the radiation in typical laboratory fields (1-10 Tesla) is in the 10's to 100's of MHz or at radio frequencies. Nuclear spins in a metal are of course not independent, they couple with sources of magnetism in the sample. One such source is the spin of the conduction electron and the dominant mechanism of this coupling is the hyperfine interaction’. (m.3) It is through this hyperfine coupling that NMR methods probe the electronic properties of metals and superconductors. The first term in (HI. 3) is the Fermi contact interaction and the second term is the dipolar interaction between the electron spin S and nuclear spin I. 25 We will see for 87Rb and 133Cs in Rb 2 CsC60 that the contact term will be dominant because the conduction electron wavefiinction is largely s in character at these nuclei. For 13C there is a large admixture of p orbital in the electronic wavefunction so the dipolar term will prove to be dominant. If the nucleus possesses a quadrupole moment Q(I > 1/2) it will interact with the strong electric field gradients in the sample. The contribution to the hamiltonian from the quadrupole interaction is given by A f A* A« A»i Aa 1 Hq = w k i) lv«(3r* " /)+ (y~_ w ~ fti- (h1-4) The fxtj ^yy> terms are the principle components of the electric field gradient tensor. We will see for 87Rb that the effects of HQ need to be taken into consideration when interpreting the relaxation and lineshape results. For 133Cs we will see that HQ is negligible because the pulses cover the whole 133Cs lineshape. For 13C we know that Hq = 0 because 13C is spin 1/2. It is convenient to write the total Hamiltonian H, ignoring nuclear spin-spin interactions, the following way: H = -h YnH6k • [1 + K] • I + Hq . (m.5) «« K is called the magnetic shift tensor which includes the effects of the hyperfine interaction. K typically consists of a sum of three different terms in the normal state: K= Ks+Kl + u 26 a is the magnetic shift tensor of the core electrons about the bare nucleus and can be taken to be zero because all shifts are measured relative to a nucleus with core electrons. For 13C the we use TMS as the reference, aqueous Rb+ for 87Rb, and aqueous Cs+ for 133Cs. k L is the shift resulting from the fields produced by the orbital motion of the conduction electrons and is called either the orbital or chemical shift. a is the shift produced by the interaction of the nuclear spin with the spin of the electron (the hyperfine interaction) and is called the Knight shift. It is now convenient to give an expression for the center of mass frequency for an NMR lineshape. For a polycrystalline sample, such as the A 3C60 samples we studied, HQ (to first order) and the off-diagonal terms in k will only broaden the NMR lineshape as one averages over the crystallite orientations. The center of mass position will be unaffected. If we consider only diagonal terms in K (which we will call just K) then the center of mass frequency w is given by u ^ Y M - K ] . (m.7) K = Ks + Kl is measured in parts per million (ppm) of the reference frequency and is the experimentally derived quantity. We will see in the analysis of the shift data in the superconducting state of Rb 2CsC6o that we have to add one more term to K to represent the shift of the NMR frequency due to Meissner screening effects. The Isotropic Knight Shift: The Larmor frequency of a nuclear spin in a metal (placed in an external field) will be shifted due to the average magnetic field at the site of the nucleus produced by the spin of the conduction electrons. This the isotropic Knight shift, Ks. In zero external field the spin of the electrons will have no preferred 27 average orientation and Ks = 0. In an applied field, however, the electron spins are polarized giving a non vanishing Ks. The interaction responsible for the Ks is the Fermi contact term in (DL3) (the dipolar term in a polycrystalline sample only broadens the line) and it can be shown that K? is proportional to the electron spin susceptibility / \ KS = - * - XS. (m.8) The constant of proportionality A is the hyperfine coupling constant of the nucleus. Thus, if the hyperfine coupling constant is known, ^ can be determined by measuring the Knight shift. We will see that the temperature dependence of Xs in the superconducting state provides a means of measuring the superconducting energy gap A. 111B 3. The Korringa Relation: In a metal the hyperfine interaction is also the principle means by which the nuclear spins arrive at thermal equilibrium with the lattice. A nuclear transition from a state mton will involve the simultaneous transition of a conduction electron with wavevector k and spin s to a state k1 and s'. The relaxation rate can be expressed with the “golden rule:” $\M\2N\E)f(E)[l-f(E)\dE. (m.9) Here N(E) is the density of states, M is the matrix element of the electron-nuclear interaction, and J(E) is the Fermi-Dirac distribution function. The temperature dependence of l f l i can be seen easily if one notes that th e /£ )[l -J(E)] term in (m.9) can be rewritten as -k,BT{dfldE). This acts as a 5-function which picks out electron states only with 28 energies within kg T of Ep. Thus, the relaxation rate of nuclear spins in a metal will be proportional to temperature or, equivalently, \IT\T is constant: ^=^aV(0). (m.10) Here AT(0) is the density of states at the Fermi energy. The property of HT\T- const, in a metal is often called the Korringa law and the mechanism of relaxation (scattering of conduction electrons via the hyperfine interaction) is often termed the Korringa mechanism. If we assume that the conduction electrons are a Fermi gas of noninteracting spins then = (K^2/ 2)N 2(0). Using (in.8) for K s we can define the Korringa relation between \!T\Tz.ndI^: (we reserve the term “Korringa relation” to refer explicitly to the relationship given by Eqn. (m .ll)). We will see later that UT\Tin the normal state A 3C60 materials is not constant because of variations in N(0) as a function of temperature. Also, we will see in the next section that l/7i Twill be altered in the superconducting state as a result of the gap opening in N(E). IIIC. NMR of Superconductors m C l. Electron Spin Susceptibility: When a superconductor undergoes the transition from the normal state to the superconducting state a gap A(7) opens in the density 29 of electron states. Electrons below the gap are in the BCS ground state (Cooper pairs) and those above are single electron quasiparticle excitations of the ground state. This has profound effects on both Ks and HT\T. In the normal state Ks is typically temperature independent because the Pauli susceptibility x? is temperature independent. However, in the superconducting state electrons begin to pair with opposing spin orientations (zero net spin). As T -*■ 0 more quasiparticles condense into the BCS ground state and consequently Xs and K s — 0. The Knight shift will follow the Yosida form for the susceptibility in the superconducting state: (HI. 12) Figure 3.1 shows the Knight shift data of Fine et a l for aluminum [Fine et aL, 1969]. Also shown in the figure a the fit to these data using the Yosida function. In equation (in. 12) NS(E) is BCS form for the superconducting density of states: E (m.13) An important feature of NS(E) is the singularity at E = A(T). In (HI. 12) the singularity is logarithmic and becomes overpowered by the rapid increase in A(2). However, for the superconducting state relaxation rates the singularity will be very important because NS(E) is squared in the integral. Figure 3.2 shows a plot of Ns(E)/N(Ep) as a function of ElA(0) for T= 0, .964rc, and Tc. Figure Figure superconducting state. The data are well fit by the BCS Yosida (solid Yosida fiinction line). well are fit BCS the by data The state. superconducting 1. .1 3 Knight Shift (Normalized) - Spin susceptibility data of Fine Fine of data susceptibility Spin 0.2 0.2 0.4 0.6 0.8 1.2 0 1 0 27 A Kih Sit n Superconducting in Shift Knight Al . 04 . 0.8 0.6 0.4 0.2 Aluminum T/T t al. et c [1969] for aluminum in the the in aluminum for [1969] 1 1.2 30 31 = 0,= .964TC, T T=0 E/A(0) T=.964T Superconducting Density ofStates T=T 0 0.5 1 1.5 2 1 1 5 3 7 6 2 0 c 4 u> (0) N/G) n Tc. and Figure 3.2. Plot of superconducting density of states vs. energy for 32 m C2. Hebel-Slichter Peak: Values for A are on the order of meV, which is around two orders of magnitude larger than NMR transition energies. Thus, only the quasipaiticle states above the energy gap of a superconductor can participate in relaxation of the nuclear spins. This has two distinct effects on T\. First, for T « Tc there will be very few quasiparticle excitations and 1/Ti will exhibit activated or Arhenius behavior: ( \ 1 — «ext| - m (m.14) 7i * V kBT The second effect occurs at temperatures slightly below Tc. As the gap opens and the singularity in NS(E) forms there will be a much larger number of states within a small energy range than there was in the normal state. This provides many more states for the nuclei to scatter the quasiparticles into. This produces an increase in 1/TiT. There are also the so called coherence factors in the matrix element |M| which contribute to increasing 1 !T\ T, but it is the singularity in NS(E) that produces the largest effect. This peak is often referred to as the Hebel-Slichter coherence peak after its discoverers and it remains to be one of the most beautiful experimental verifications of the BCS theoiy {[Hebei, 1959]. Figure 3.3 shows the data of Masuda and Redfield for Rs/R,, (ratio of the superconducting to normal state relaxation rates) vs. T!TC for aluminum [Masuda, 1962]. IHC3. NMR of Type II Superconductors: The above discussion pertained to the zero-field state of a superconductor and is valid for both Type I and Type n materials. However, the NMR properties of Type n superconductors in an applied field (as is the case for all NMR measurements discussed in this dissertation) can be altered by the presence of fluxoids in the material. For example, the NMR lineshape will be 33 27 Al Rs /Rn in Superconducting Aluminum 2.5 —1—1—1—i—1—1—1—r 2 - 1.5 in 1 - • • 0.5 , I 0.2 0.4 0.6 0.8 1.2 T/T Figure 3 .3 . Hebel-Slichter coherence peak in the superconducting state of aluminum [Masuda, 1962]. The increase in Rs below Tc is primarily a result of the singularity in the superconducting density of states (see Fig. 3.2). This singularity creates a large number of energetically similar electron states which a nuclear spin can scatter electrons to and from. 34 broadened by the distribution of internal magnetic fields associated with the triangular vortex lattice. The vortex lattice can affect the T\ relaxation rates in a superconductor near Hc2. In the superconducting state if the applied field H is slightly less than Hc2, a significant fraction of the sample will be in the normal state vortex cores. This faction f„ can be shown to be equal to fn 2itHc2{T) ‘ 51V T\ studies in V 3 Sn by Silbemagel et a l showed that for T « Tc the 51V relaxation rates were faster than what was predicted by Eqn. (HI. 14) [Silbemagel, 1966]. The explanation was that spin diffusion was occurring between the rapidly relaxing nuclei in the normal cores and the slowly relaxing nuclei in the bulk of the superconductor. Masuda and Okubo, also studying 51V TVs in V 3Sn, found that large applied fields can suppress the Hebel-Slichter peak in Type n materials [Masuda, 1969]. Figure 3.4 shows the 51V data of Masuda and Okubo. HID. NMR Results of Other Groups on A3C6 O Since the discovery of C 60 and A 3 C60 many different groups have used NMR methods to study these materials. In the last section we saw how the narrowed 13C lineshape told us that the C 60 molecules were rotating at their lattice sites. Further 13C NMR measurements of Barrett and Tycko (13C 2-D NMR) [Barrett, 1992] and of Yoshinari et al (13C T\ and T2) [Yoshinari, 1993] reveal that the dynamics of the C60 35 Normalized 51V Spin-Lattice Relaxation Rate vs. T\TC for Varying Applied Field Strength 2.0 SMHl (44700) 15MHz(I34I0G) . 05 05 Figure 3.4. Rs/Rn data for 51V in the mixed state of VaSn at magnetic fields of 4.5,8.9, and 13.4 kG. For H «Hci (Hc2 = 16 kG in V 3Sn), the Hebel-Slichter coherence peak in Rs/Rn is suppressed. The data also confirms theoretical predictions of Cyrot [1966], which predict complete peak suppression at magnetic fields large enough to suppress Tc by 40% (® 11 kG in V 3 Sn). From this, one would not expect the Hebel-Slichter peak to vanish in the alkali fullerides until fields reached several tens of Tesla. The intrinsic (zero-field) peak in the alkali fullerides, however, is much smaller than it is in V 3Sn. Thus, not much suppression is required to reduce to peak to a size too small to be detected. Figure reprinted with permission from Y. Masuda and N. Okubo, J. o f the Phys. Soc, o f Japan 26, 309 (1969). 36 molecules in K 3C6oare quite complex, involving both slow reorentational “steps” and rapid anisotropic rotations. 87Rb NMR of R 3C60 by Walstedt et a l [Walstedt, 1993] and our group gives evidence for a possible structural distortion in these materials. The rest of this section will present the 13C NMR data of Tycko et al in the superconducting state of RbsCeo [Tycko, 1992]. It is with these important measurements that we motivate our study of the superconducting state of A 3C60. 1111)1. Knight Shift: Figure 3.5 shows the 13C shift data in Rb3C60 of Tycko et al at 9.39 T [Tycko, 1992]. Between 7V = 29 K and 5 K the center of mass position changes by approximately 40 ppm (192 ppm at Tc to 150 ppm at 5 K), The change in line position in the superconducting state is due to both the vanishing of K? and the diamagnetic screening. Tycko and his colleagues calculated that the screening reduced the applied Held by approximately 20 G (200 ppm)--the l3C shift in Rt> 3C6o could be due to the diamagnetic screening alone. A calculation of the 13C hyperfine coupling using the methodology of M. Karplus and G. Fraenkel [Kaiplus, 1961] predicts that Ks = 490 ppm (see Ch. VI). This value is much larger than the observed shift and even the estimated diamagnetic screening. Thus, there is a large uncertainty in both the sign and magnitude of IS5. As a consequence of the uncertainty in determining Ks, it is impossible to use the Yosida function to extract information about superconductivity in these materials. We will show later that with multinuclear NMR methods we are able to extract the temperature dependence of Ks and obtain an excellent fit to the data with the Yosida function in the weak coupling limit of BCS. We will also discuss our attempts to extract the signs and magnitudes of Ks. IIID2. Spin Lattice Relaxation Rates: Figure 3.6 shows the IIT\T measurements for 13C in RI 53C60 by Tycko et al (Tycko, 1992]. The normal state values are not constant as one would predict by Eqn. (HI. 10). Between 300 K and Tc UT\T 37 Line Shift vs. Temperature in Rb3C60 200 100 160 150 140 5 0 100 150 T (K) Figure 3 .5 . 13C shift data in Rb 3 C6o taken by Tycko et al The total shift of the center of mass of the 13C spectrum in the superconducting state is approximately 40 ppm. It is not possible from this data to determine how much of this shift results from the vanishing of the Knight shift and how much from Meissner screening effects. Figure reprinted with permission from R. Tycko et a l, Phys. Rev. Lett. 68, 1912 (1992). 38 1IT±T vs. Temperature for 13C in RbaCtfo 0.02Q 0.015 0.010 0.000 50 100 150 200 250 300 T (K) Figure 3 .6 . 13C \IT\T data in RbsCeo (Tc “ 29 K) of Tycko et a l at approximately 9 Tesla. We note two features of interest: (1) there is no Hebel-Slichter peak in the superconducting state and (2) the temperature dependence of \!T \T 'm the normal state (\IT\T is expected to be a constant in a normal metal). Figure reprinted with permission from R. Tycko et a l, Phys. Rev. Lett. 68, 1912 (1992). 39 decreases by about 30 %. Tycko et al suggest that the relaxation is due to the Koninga mechanism but thermal contraction of the lattice makes N(E f ) decrease with decreasing temperature. We will show this same trend exists in the 13C 1/TiTin all A3C60 samples and it will be more proper to think of \IT\T as being a function of the lattice constant a rather than temperature. The second feature in the relaxation data is the absence of the Hebel-Slichter peak below Tc. It is possible to broaden out the peak with inelastic scattering mechanisms producing finite electron lifetimes or a distribution of Tc's, but such a severe suppression of the coherence peak is usually indicative of the strong coupling limit of the BCS theoiy. This would be in marked contrast with the bulk of other measurements which point towards the weak coupling regime in the alkali fulleride materials. The presence or lack of a Hebel-Slichter peak in the NMR data is a very important issue that needs to be addressed in these materials Chapter IV Experimental Method IVA. Chapter Overview In this chapter we present the details of the experimental apparatus and methods used in this dissertation. Section IVB gives a description of the NMR spectrometer and probe assembled by the author and Section IVC provides a discussion of techniques used including pulse sequences. IVB. Experimental Apparatus IVB1. NMR Spectrometer: All NMR measurements were done using pulsed NMR techniques (see Fukushima [1981]). The static laboratory field was produced by an Oxford 98 mm bore superconducting magnet with a maximum field strength of approximately 9 T (we used a conservative upper limit of 8.8 T because of radio interference at the 9 T 13C frequency). Although a maximum field homogeneity less than 1 ppm was obtainable within 10 cc area, the actual field homogeneity was typically on the order of 1-10 ppm in these experiments due to sample orientation and nominal shimming. The spectrometer consisted of a Tecmag pulse-programmer/signal-averager and commercial 40 41 or home-built rf components. A Macintosh Ilci computer was used to control the experiment and for data analysis and data manipulations. Figure 4.1 is a schematic of the spectrometer. 13C measurements were made at applied field strengths of 8.8, 7, 5, 3, and 1.5 Tesla with resonance frequencies of 94.2, 74.9, 53.5, 32.1, and 16.5 MHz. 87Rb (133Cs) measurements were made at 8.8, 5, and 3 Tesla with resonance frequencies of 122.6 (49.1), 69.6 (27.9), and 41.8 (16.7) MHz. 39K and 85Rb measurements were done at 8.8 Tesla with resonance frequencies of 17.5 and 36.2 MHz respectively. In a typical experiment a suitable sequence is chosen on the computer (we will discuss the actual pulse sequences used in Section IVC2) and compiled to be run by the Tecmag pulse programmer. The Tecmag converts the pulse sequence into series of timed TTL outputs which control the gates, switches and rf phases. Typical 90° and 180° pulse lengths were on the order of 1-10 ps and the phases of the pulses could be adjusted in increments of 22.5°. The ability to control the phases of the rf pulses is very important because it allows us to employ phase cycling techniques to eliminate unwanted artifacts in the NMR signals (examples of phase cycling will be given later). The sequence also has an acquisition event which tells the Tecmag to acquire the signal by converting the analog input into a digitized array. This array can be saved on the computer for later analysis. The sequence is repeated as desired such that the signal is averaged to obtain a suitable signal to noise ratio. The spectrometer employs a quadrature detection technique. In this method the signal from the probe (after some amplification stages) is first split into two channels. These channels are then mixed with two channels of rf from the oscillator that have a relative phase difference of 9 0 °. The result is that one obtains both the x and y components of the magnetization in the rotating frame. The advantage of quadrature detection is that the required excitation bandwidth is halved because the carrier frequency can be placed in the 42 NMR Spectrometer o o o o o o o o o TecMag Pulse Pro. RF Oscillator and Sig. Averager Computer RF Splitter 1 _ Filter _ O O ( Atten. ) Audio 0 o I Amp Switch L_ 0 ) 2 R L 0 + J Mixer + J Amplifier C l t o r Ll_ Q£ 1 90 Mixer Phase Splitter A/4 Cable Doty Low Noise Amp Probe Figure 4 .1 . Schematic of the NMR spectrometer used in this work. 43 center rather than the side of the spectra. This means that there will be less folded in noise (signal to noise increases b y ), less power is needed to excite the spins, and the digitizer only needs to digitize at half the rate as it did without quadrature detection. Low temperature measurements were carried out with an Oxford continuous-flow cryostat insert with both liquid helium and liquid nitrogen capabilities. An Oxford ITC4 with an Au Fe vs. chromel thermocouple was used for temperature control and measurement. Calibration of the temperature controller with a carbon glass resistor indicated that temperature measurements were accurate within .1 K. High temperature (above room temp.) measurements were executed with a high temperature probe designed and built by Pennington group member Chris Hahm. IVB2. NMR Probe: The NMR sample probe used in this work were designed and built by the author and Richard Kindler at the Low Temperature shop at The Ohio State University Physics Dept. The sample was held in a brass rf coil which was part of tuning tank circuit. Brass was used because its electrical resistivity is dominated by impurities rather than phonons (as in copper) leaving its resistance fairly independent of temperature. This is very important because it is convenient to have the tuning characteristics of the tank circuit to be temperature independent. The circuit was sealed in a brass can to shield the circuit from any external rf pickup. Figure 4.2 shows the inside of the probe can and the equivalent tuning circuit used. The circuit employed two different adjustable Polyflon teflon capacitors for matching and tuning purposes. The capacitors were machined down to prevent locking at low temperatures and low thermal conductivity fiberglass "screwdriver" handles were used to adjust the capacitance in situ. Also, low thermal conductivity stainless steel coaxial cable was employed to connect the tuning circuit with the spectrometer. NMR Probe Equivalent Circuit Teflon Sample Figure 42. Schematic of NMR probe and the equivalent circuit diagram. The sample oriented in the rf coil perpendicular to direction of the static Held. 45 For a given rf choke of inductance L and resistance R it can be shown with some algebra that the value for the matching capacitor is approximately (IV.l) Here 0)0 is the resonance frequency, Q is the quality factor o)oL!R , and Zq is 50 Q. Typical values for L, Q, and Cm are on the order of 1 pH, 100, and 10 pF respectively. Peak to peak rms voltages of the pulses (Vo) into the probe were approximately 100 V but because of the nature of the tank circuit the voltages across each of the components get typically multiplied by a factor of ten. For example the voltage across the matching capacitor Vm is (TV.2) So at 100 MHz a circuit with Q = 100 and L = 1 pH will have 1.4 kV across its matching capacitor for a Vo of 100 V. As a result of these high voltages significant capacitor breakdown and arcing occurred which limited pulse lengths (and the effective bandwidths) to values larger than 5ps. I VC. Experimental Techniques IVCl. The Bloch Equations: In the simplest pulsed NMR experiment one observes the free induction decay (FID) of the nuclear spin magnetization. An FID is produced when the equilibrium spin magnetization along the static field (z-direction) is flipped into the x-y plane by means of a 90° pulse (y„Bi = nil, where B\ is the transverse field in the rotating frame). The “free” magnetization (that is in the presence of no rf field) precesses about the static field and eventually decays to zero as the spins dephase as a result of their distribution of Larmor frequencies. The time dependence of the x and y components of the magnetization ( Mx and My) can be described by the phenomenological differential equations: dMx Mx , dM M I 2 is the characteristic time that it takes for the spins to dephase in the transverse plane and is called the transverse relaxation time. One often distinguishes between transverse relaxation processes that result from field variations between spin isochromats that are static (i.e. field inhomogeneity) and those that are time-dependent (i.e. dipolar couplings). The relaxation time which has contributions from both processes is T2 and the relaxation time which has contributions from only time-dependent processes is T2 (T2 is sometimes called the spin-spin relaxation time). Experimentally, T2 characterizes the decay of an FID and T2 characterizes the decay of a spin-echo (see below). The magnetization is also being simultaneously restored to equilibrium along the z-direction. The time rate of change of the z-component of the magnetization Mz can be described by the following equation: 47 The characteristic time T\ for Mz to return to the local equilibrium magnetization Mo is called the longitudinal or spin-lattice relaxation time. In a spin-lattice relaxation process energy is transferred between the nuclear spin system and its surroundings. Equations (TV.3) and (IV.4) are called the Bloch equations. IVC2. The Spin Echo: In most solid state NMR experiments the NMR signal is obtained by means of a spin echo rather than an H D because in a solid an HD typically falls to zero in a time shorter than the amplifier's dead-time r*. Another way of saying this is that T2* < Tdt- Thus, it is necessary to refocus the transverse magnetization by a spin echo. A spin echo pulse sequence consists of a 90° pulse, a free precession period of time r, and the application of a 180° pulse. Immediately after the application of the first pulse (a 90° pulse along the Jt-axis for example) all the magnetization will be pointing along the y-axis in the rotating frame. As the spins precess the magnetization decays because the spins with Larmor frequencies above the oscillator frequency will precess counterclockwise and those below will precess clockwise in the rotating frame. After a time r a 180° pulse is applied along the jc-axis and the whole “fan” of spin magnetization is flipped 180° such that spins that were aligned in the y direction will now be aligned along -y direction, etc. Now, instead of the spins processing apart they will refocus and a “spin echo” will appear a time 2 r after the 90° pulse. Figure 4.3 illustrates the formation of a spin echo in the rotating frame. IVC3. Pulse Sequences and the Config.con File: Figure 4.4 shows a pulse sequence (this one is for an FID) as it appears in the MacNMR software package that came with the Tecmag pulse sequence generator. Each pulse sequence creates a series of timed TTL outputs from the Tecmag and digitizations of analog signals into the Tecmag. 48 Formation of a Spin Echo z T = 0 Spins dephase 90°Pulse 4 \ Z ■*y x ^ __ T= x Spins start T = 2x to refocus 180 Pulse Figure 4.3. Diagram showing the formation of a spin echo. Immediately after the application of a 90° pulse along x at T = 0 all the magnetization is along y-axis in the rotating frame. The spins then dephase. At T= r a 180° pulse along x is applied and flips the fan of spins 180°. The spins will then refocus and an echo forms at T= 2r. FID EUENT 1 6 N aae: wai t igate 190 Iwaittacq luiait D e Ia u : 10u |5u Mu j60u JphficAlOnt Pulse(l) ‘ Gate(2) Tr i ggen(3)* Sflreset Phase PuIse2 RCQ Loopl Loop2 EH ES 0 Figure 4.4. Pulse sequence for the formation of a free induction decay or FID. = I _ J = m config.con !E3i| |Pulse(1) 1 40 1 9 0 1 99 TX Gate(2) 2 41 1 9 0 1 99 TX e = Trigger(3) 3 42 1 9 0 1 99 TX jjijji SAreset 4 87 1 9 0 1 99 TX Phase 5 80 49 0 1 99 PH Pulse2 6 44 1 9 0 1 99 TX ill Figure 4.5. Configuration file for pulse sequences in the MacNMR application. 50 The column specifies the time window of these “events” and the row specifies the type of event that is taking place in the window of time. Row 1 specifies the length of the rf pulse by creating a 1 bit TTL output from the Tecmag to open the rf switch. Row 2 is a 1 bit TTL output that gates the rf amplifier, row 3 is a 1 bit TTL output to trigger the oscilloscope, and row 4 is a 1 bit TTL output that allows the signal averager to be reset within a single scan. The phase of the rf pulses is controlled by row 5. An event in row 5 consists of a 4 bit word (a number between 0 and 15) which allows the phases to be changed in 22.5° increments (the PTS frequency synthesizer understands negative logic such that “0” =0°, “12” =90°, “8” = 180°, “4” =270°, etc.). Row 6 is a row for another rf pulse if desired, row 7 is the acquisition window (the detection phase is specified here), and rows 8 and 9 are for the creation of loops within the sequence. The file which designates the types of possible events in a particular pulse sequence is called the “config.con” file and is located in the MacNMR folder on the computer harddrive. Figure 4.5 shows the config.con file used for the pulse sequences in this thesis. Each row in the config.con file represents a row in the pulse sequence. For example the first row in Fig. 4.5 is the “pulse” row. The first number (1) is the order this row will appear in the pulse sequence, the second (40) is the TTL output channel used by the tecmag (or the first in a consecutive series if more than one bit is specified), and the third (1) specifies the number of bits (this row just opens and closes the rf switch so only “0” or “1 ” for closed or open is required). Rows 6 and 7 give the upper and lower limits in ps of the time window for an event (1 to 99 ps) and the last two letters specify the type of icon that will appear in the sequence to represent the event. IVC4. FID and Spin Echo Sequences: Figures 4.4 and 4.6 show the pulse sequences for the observation of an FID and for the observation of a spin echo respectively. In the FID sequence there is a 90° pulse (1 ps), a delay (60 ps), an 51 Echo EUENT 1 8 Naae: wait [gate 190 iTau igate 1180 Iwaitlacq lacq Delau: 10u |5u j4.7u JlQ5u jSu |9.4u j40u jphea8j500m Pulse(l) ' Gate(2) Tr igger(3)- Sflreset Phase Pulse2 flCQ Loopl Loop2 EE ECU 3 Figure 4.6. Pulse sequence for the formation of a spin echo. 52 acquisition event, and an event called the “last delay” (10 ms). The 60 ps delay after the 90° pulse is to prevent unwanted acquisition of “ring down” from the probe circuit or amplifier recoveiy. The last delay time is representative of how fast the pulse sequence will be repeated (it is typically much longer than any of the other times in the sequence). The length chosen for the last delay is dependent on what state one wishes to observe the spin system in. For example (this is typically the case), if one wishes to observe the magnitude of the equilibrium magnetization, then the last delay needs to be longer than T\ (approx. 5Ti) to allow for adequate relaxation of the spin system back to equilibrium between pulses. In Fig. 4.6 we show a spin echo pulse sequence. There is a 90° pulse (of 4.7 ps) followed by a delay r (of 200 ps) and then a 180° pulse (9.4 ps). The acquisition begins 40 ps after the 180° pulse for reasons mentioned above (the echo will have its peak be at 160 ps in the acquisition window). There will also be FID's from the 90° and 180° pulses (if T2 * is long enough and if the length of the 180° pulse is incorrect) which can run into the echo and produce systematic errors. These unwanted FID’s (and also the recovery/ring-down to a great extent) can be removed phase cycling methods. IVC5. Phase Cycling: Figure 4.7 shows a diagram depicting the “add/subtract” phase cycling method. The first scan produces an echo by applying a 90° and then a 180° pulse along the x-axis in the rotating frame (written as (90°)* and (180°)* respectively). One then detects this echo with the receiver phase along the x-axis. The second scan applies a 90° pulse along the -x-axis ((90°)-*) and then a (180°)x pulse. This produces an echo that is 180° out of phase with the first. To make the echoes add coherently we set the detection phase to be along the -x-axis. In other words, we “add” the first scan and “subtract” the second. The beauty of this technique is that we cancel out the unwanted FID's and ringdown from the 180° pulses. 53 “Add/Subtract” Phase Cycling Add \l o n (180°) (Acq) Sub (1 8 0 °), (Acq) (9 0 ) -X Add (Acq) Sub (9 0 I (ISO0)., (Acq) Figure 4.7. Add/subtract phase cycling. The successive application of all four scans will cancel FID's and transients from the pulses and coherently add the signals from the spin echoes. 54 The third and fourth lines are similar to the first two except we use 180° pulses that are along the j-axis. When we add/subtract the echoes from the third and forth scans the FID's from the 90° pulses will cancel with the FID's from the 90° pulses from the first two scans. These four scans combined will coherently add the echoes but cancel out the FDD's and other transients produced by the pulses. The quadrature detection circuit in the spectrometer uses two receiver channels. These channels may have different dc offsets and gains. This will produce zero frequency spikes and aliasing. These errors can be removed if one takes the above four scans (or any phases cycling scheme) and rotates all the pulse and detection phases by 90° producing four more scans. If one takes these four scans and adds them to the first four (a total of eight scans), the differences in gain or dc offset between the two receiver channels will be removed. This eight pulse sequence is what was employed in this work in the detection of spin echoes. IVC6. Saturation/Inversion Recovery: Spin-lattice relaxation times {T\) were obtained by using either saturation or inversion recovery methods. In a saturation recovery experiment one applies a series (“comb”) of 90° pulses with spacing t such that T2*< t< T\. This puts the spin system in an initial state with Mz = 0 (it is saturated). One then waits a time rand observes the recovery of Mz to the equilibrium value Mo with an FID or spin echo. Repeating this for various values of r one will obtain Mz{r). Solving the Bloch equations we get that Mz(f) goes like (IV.5) 55 Figure 4.8 shows the saturation recovery pulse sequence as it appears in the MacNMR software. Saturation recovery methods were used for 13C and 133Cs T\ measurements in A3C60- The second method, inversion recovery, utilizes a 180° pulse to first invert the magnetization such that Mz ~ - Mo . The magnetization is then allowed to relax back to equilibrium for a time r. Solving the Bloch equations for these conditions gives the following expression for Mz(t)\ Mz(t) = M0 (1 - 2e"T,r' ). (IV.6) Figure 4.9 shows the inversion recovery pulse sequence used in this work. Inversion recovery was used in the determination of 87Rb T\ values in A 3 C60. There are advantages and disadvantages to both the above methods. In an inversion recovery experiment one must wait a time much longer than T\ in between scans because it is required that Mz = Mo at the beginning of each scan. This is not the case for saturation recovery because all that is required is that Mz = 0 before each scan. Thus, saturation recovery experiments are much faster than inversion recovery experiments (one can obtain more signal to noise in the same allotted time). A disadvantage of saturation recovery is that it is very sensitive to pulse length which can distort the recovery curve and produce inaccurate T\ values. This is also a problem for inversion recovery but because Mz(0) = - Mz(t » T\) one can readily see in the experimental data if the pulse lengths are correct. 56 eP e Sat. free. Echo T 1---- i -----1------r — i------1----1----- 1----- r----- 1----- 1----- 1-----r EUElff 1I 2-> 1i •> 3 1 < • r. 1 7 ] 8 i 9 I 10 I 11 ! 12 I 13 1 14 ! 15 1 16- I I I JLJ I __ L5 I ___ IL I I I I I I I I I Kc i i c : ifgj i [go tc lOO lug it Igata IPO lleopliou Igatc 108 lag it {gstc MSP iwo i K Cocq ILO D e lo n ; ICki |9u |22u jiOOu |5u |22u Mu ’tou !ju [22u |250uj2 |5u !50u khaoSiis i i Pube( I) J ------L Gnte(2) - Ti'igger»(3)- Sflneset P h ase Pu13?2 HCQ L aopl Loop? i ■ . . a WT T»l3 Figure 4.8. Pulse sequence for a saturation recovery experiment. The first two 90° pulses saturate the magnetization, “tau” is the wait time, and the magnetization is observed with a spin echo. P inn. Rec,Echo □ r i i i i i i r l r l i i i EUENT » 1 j 2 | 3 j 4 j 5 j 6 I 7 j 8 j 0 j 10 j 11 j 12 j 13 ! 14 j 15 1 1 I 1 I 1 I I I 1 1 I I 1 H o a e ; vai 1 1 IvaI t tea4c 1190 I Itau Icale I9D Ivai 1 lantc 1180 IwaiLlcicq ILD De 1 a il i lOu jlu |30s j*u jt.au 1 feu j3u |l45u jsu jfcn jsOu Jphqr jis 1------1 1------1 J------1 P u ls e ( l) G dte(2> lY ig g e ^ O ) Sflneeet r = r ( = r rp r f= r i« r P hase j y J s J i d j = i j y j y i P u 1se2 PCP yiftf i Loopl i Loep2 0 Figure 4.9. Pulse sequence for an inversion recoveiy experiment. The 180° pulse inverts the magnetization, “tau” is the wait time, and the magnetization is observed with a spin echo. Chapter V Sample Characterization VA. Chapter Overview This chapter describes the synthesis and characterization of the materials used in this woric. First, in Section VB we discuss the preparation of A 3 C60 samples fabricated by a liquid-ammonia-route synthesis technique. Section VC presents NMR and dc magnetization measurements characterizing the purity, structure and superconducting fractions of these materials. VB. Sample Preparation VBl. Synthesis of A 3 C6 0 : A3C6O and other fullerene materials were fabricated in the laboratories of Dr. R. P. Ziebarth at The Ohio State University by his graduate students D. R. Buffmger and S.-M. Lee. To distinguish between samples created by these researchers we adopted the convention of using a prefix “DR” or “L” respectively in the sample's identification number. For example, “DR60B” is a sample synthesized by D. R. and “L-100” is a sample synthesized by Lee. 57 58 The preparation of high quality A 3C60 samples is essential if one wishes to obtain a clear understanding of these materials. The usual means of fabricating alkali-fulleride materials is quite simple: stoichiometric amounts of A = alkali metal and C60 or equal amounts of A6C60 and C 60 are sealed in a tube and the tube is subsequently heated [Hebard, 1991]. This method typically leads to a mixture of products with low superconducting fractions (« 10 - 30% and 50% for the first and second combinations of starting materials respectively). Also, this procedure is applicable only to metals with fairly high vapor pressures (10‘2 and above) at temperatures below the point at which they attack the quartz reaction container (® 600 °C). These problems are circumvented by employing a solution route synthesis technique using liquid ammonia. This method produces high purity A 3C60 compounds with superconducting fractions of approximately 100%. Also, this procedure allows for the fabrication of new C 60 compounds containing elements soluble in liquid ammonia such as the alkaline earths Yb and Hu. Briefly (a detailed account of the procedure can be found in Buffinger et a l [Buffinger, 1993]), C 60 was prepared the usual way from the soot generated by a dc carbon arc (4.0 kW ) using a graphite/silica gel column for purification with toluene as the eluent. The C 60 was then reacted with a stiochiometric amount of A (A = K, Rb, and Cs) in liquid ammonia all held in a quartz tube at -78 °C for 30 - 60 min. The sample was then slowly heated (< 100 °C) to remove the ammonia and subsequently annealed 24 - 48 hours at 375 °C in a sealed quartz tube. X-ray powder diffraction patterns and measurements of Tc by magnetic susceptibility confirmed the identity of all the materials (magnetic susceptibility measurements were done on a quantum design MPMR2 SQUID magnetometer). VB2. Samples Used: Table 5.1 lists the primaiy A 3 C6O samples used in this thesis. This table is to allow the reader to find the chemical formula for an A 3 C60 compound given only the sample's identification code. In the text both sample ID number and chemical formula will be given when possible or deemed relevant. In this thesis, the majority of the NMR measurements studying A 3 C60 superconductivity were done on Rb2CsC60 samples DR60B and DR168A. The bulk of the Knight shift results were obtained with DR60B and the T\ measurements with DR168A. Knight shift and T\ measurements performed on both DR168A and DR60B yielded the same results. VC. Sample Characterization VCl. SQUID Magnetization Curves: Superconducting transition temperatures Tc and superconducting shielding fractions Fsh were obtained by measuring the sample's magnetic moment p as a function of temperature. The measurements were performed on a Quantum Design MPMR2 SQUID magnetometer with applied fields of 10 gauss. A3C60 samples were first pressed into pellets with masses of approximately 5-10 mg and then sealed in quartz ampoules. Measurement of the susceptibility of an empty quartz tube showed that it was negligible when compared to the susceptibility of the A 3C6O materials. Fsh for a sample of mass m in a field H was determined by the following formula: Table 5.1. A 3 C 60 compound and the related sample identification code used by the Ziebarth and Pennington research groups ______ Sample Identification Chemical Code Form ula DR65B K3 C60 DR77B K2RbC6o DR87B KRb2C6o DR26A, DR30B, DR50A, RbsCeo DR50B DR60B, DR168A Rt^CsCgo DR66B RbCs2C6o Table 5.2. NMR shifts of aqueous alkali ions that of the same ion in free space. ______ Ion in Solution Shift from the Free Ion 39K in 2 M KNO 3 107 ppm 87Rb in .1M RbCl 2 1 2 ppm 133Csin.l MCsCl 330 ppm 61 For the density p of A3C60 we used a generic value of 2.2 g/cc. Figure 5.1 shows zero- field cooled and field cooled Fsh vs. temperature curves for four different A 3C6O materials: DR26A, DR30B, DR60B, and DR168A. DR26A (Rb 3C6o) was produced using a solid state reaction route starting with stiochiometric amounts of rubidium metal and Ceo- DR30B (RI33C60) was also produced using a solid state reaction route but starting with equal amounts of Rb6C60 and Ceo* These methods are discussed by Hebard et al [1991] and McCauley et al [1991] respectively. Both methods yield superconducting materials but low values of Fsh (w 35% and 65% respectively). DR60B and DR168A (both Rb2CsC6o) were prepared by the aforementioned liquid ammonia synthesis route with anneal times of approximately 48 hrs. The transition widths (0 - 90% of max. shielding fraction) are around 5 - 8 K with maximum shielding fractions of 100% and the r c's of 30 and 31 K, respectively. Samples of K 3 C6O, K2RbC 6o, KRb2C6o> Rb 3 C6 0 , and RbCs 2C6o were also created by this method with similar dc magnetization results and Tc's comparable to those found in the literature. Thus, we see from the dc magnetization data that the liquid ammonia route (with annealing) produces higher quality samples than the more traditional solid state reaction methods. VC2. NMR Lineshapes: Another tool used to aid in sample characterization was NMR itself. In Ch. n we already gave a few examples of how NMR lineshapes alone can add much insight into the structural properties of A 3C60. This subsection is intended to expand on those examples by presenting a series of NMR spectra from which we can offer a “picture book” analysis of A 3C60 structure. Representative room-temperature spin-echo Fourier transform lineshapes for 13C, 87Rb, 133Cs, and 39K in K 3C60 , K2RbC6o, KRb2C 60, Rb3C6o, Rb 2 CsC60, RbCs2C6o, and Ceo are shown in Figure 5.2. The trend in the figure is with the top row being 62 DC Magnetization vs. Temperature 0.2 Field Cooled c “ - 0.2 o Zero Field 4- > □□ o -0.4 03 Cooled v_ L i. O) -0 .6 c |6 □ DR26A 2 "°-8 !c a DR30B CO O DR168A J ♦ DR60B 0 10 15 20 26 30 35 405 Temperature (K) Figure 5.1. Zero-field cooled and field cooled Fsh vs. temperature curves for four different A 3C60 materials: DR26A, DR30B, DR60B, and DR168A. DR26A (Rb 3C6o) was produced using a solid state reaction route starting with stiochiometiic amounts of rubidium metal and C60- DR30B (Rb3C6o) was also produced using a solid state reaction route but starting with equal amounts of Rb6C6o and C 60- Both methods yield superconducting materials but low values of Fsh (w 35% and 65% respectively). DR60B and DR168A (both Rb2CsC6o) were prepared by the liquid ammonia synthesis route with anneal times of approximately 48 hrs. The transition widths (0 - 90% of max. Fsf,) are around 5 - 8 K with a maximum Fsh of 100% and Tc's of 30 and 31 K, respectively. We see from this data that the liquid ammonia route (with annealing) produces higher quality samples than the more traditional solid state reaction methods. 63 9 J oP ©dr- 3 o o o o ro o u o o u Oo o «k <11 O o o CO o CO 7 s _ 1 o o ro o © o ro« © o O’3D O o o o0> o ro o © ro Oo 9 Ol o o 1 1 Figure 5.2. Spectra for 13C, 87Rb, 39K, and 133Cs at room temperature for indicated 4 3 C60 compounds. 64 undoped C60 and each successive row an A 3C60 compound with increasing lattice constant (or larger alkali dopant radius). The reference for 13C is tetramethylsilane (TMS), for 87Rb it is .1 M aqueous RbCl, for 133Cs it is .1 M aqueous CsCl, and for 39K it is 2 M aqueous KNO3 . One may refer the shifts of these alkali ions to that of the respective free ion by using Table 5.2 [Akitt, 1987; Lindman, 1978]. We first consider the 13 C lineshapes. As previously mentioned, the 13C lineshape of C60 is completely motionally narrowed as a result of the rapid orientations of the buckyball in its lattice site on time scales on the order of the NMR time scale (« 20 (is). As larger intercalant ions are introduced into the C60 lattice the 13C lineshapes become increasingly broadened into “powder pattern” lineshapes as the rotations of the molecules become inhibited by the alkali metal ion. These powder pattern lineshapes result from the 13C anisotropic chemical and Knight shift tensors combined with an averaging of crystallite orientations with respect to the applied field. The resonance frequency of 13C in pure C60 is 143 ppm from TMS. We can also see from this figure that the center of mass resonance frequencies of the alkali doped materials are all within 180 to 190 ppm. It is therefore tempting to conclude that the difference of approximately 40 ppm between 13C in C 60 and that in A 3C6O is a result of the isotropic contribution to the Knight shift. Tycko et a l, however, have measured the line position of 13C insulating Rb4C60 and observed that it 37 ppm from C 60 [Tycko, 1993]. Thus, this difference may be all from chemical shift contributions. The 13C lineshape results for Ceo, K3C6O, and Rb 3C6o are in agreement with those of Tycko et a l [Tycko, 1992], and the motion of the fullerene molecules in solid C<5o has been the subject of study by Yoshinari et al [Yoshinari, 1993] and Barrett and Tycko [Barrett, 1992] using NMR techniques. Now we consider the room temperature NMR lineshapes of the alkali intercalates. These spectra provide us with information about the structure of the A 3 C60 materials. 65 Unlike 13C, which is spin 1/2, the 39K, 87Rb, and 133Cs nuclei (spins of 3/2,3/2, and 5/2) all possess quadrupole moments of 0.07,0.14, -0.004 cm2 respectively. By comparing the nutation curves of each of the alkali metal nuclei in A 3C60 to that of the same ion in aqueous solution, we determined that for 39K and 87Rb we are observing only the central (-1/2,1/2) Zeeman transition; the satellite (-3/2, -1/2) and (3/2,1/2) transitions are shifted away to first order by the quadrupole interaction. For 133Cs all Zeeman transitions are observed. We give a more complete discussion of the 87Rb quadrupole shifts in Appendix A. Looking first at the 39K spectrum in K 3C6O (this is the same spectrum as in Ch. II), we see two lines with area ratios of 2:1. From x-ray studies we know that the A 3 C6 0 compound forms an fee lattice with a C 60 molecule at each lattice site. For every C60 molecule there are three interstitials, one with octahedral ( O) symmetry and two with tetrahedral (7) symmetry. Knowing this information, we can quickly assign the peak in the 39K lineshape data with twice as much area to the ions in the T sites and the other to ions in the O sites. Continuing with this idea, we see that for K2RbC«) and KRb 2C6o the 39K spectra have only one peak. This is because now the larger 87Rb ions are occupying all the larger O sites and the 39K ions are in the T sites (the 39K ions and the extra 87Rb ions in KRb 2C6o are equally distributed in the T sites). Thus, the trend is simply that the larger O site will be filled by the larger alkali dopant. By inspection, the 87Rb and 133Cs lineshapes of all the different A 3C50 materials tell similar structural stories. Closer observation of Fig. 5.2 reveals that the room temperature 87Rb lineshape for RbsCeo shows an unusual feature. Naively, one would expect two 87Rb NMR peaks with an area ratio 2:1 corresponding to the two sites of tetrahedral coordination and the one of octahedral coordination per C 60 analogous to the 39K data. However, we find three peaks for 87Rb at room temperature with center of mass positions at 43 ppm, 169 ppm, and 257 ppm and area ratios of 35:50:15. We label these O (octahedral), T, and T (both tetrahedral) according to the assignment of Walstedt et al who have also observed this 66 unexpected result [Walstedt, 1993]. Looking at the room temperature 87Rb lineshapes for the other compounds, we see that the Rb2CsC6o and RbCs2C60 spectra show only the T and T lines. This demonstrates the full occupancy of the octahedral site by the Cs ion, consistent with its larger ionic radius. In the KRb 2C6o sample the T peak is indiscernible, possibly only manifesting itself as a small “shoulder” on the Time. The room temperature 133Cs spectrum in RbCs 2C60 also exhibits a T peak and the area ratios of the octahedral to tetrahedral (Tand T) sites is 1:1 as expected. Walstedt et al. suggest several possible explanations of the extra line T. One possibility they suggest is that T may be the result of a time-dependent structural distortion dependent on the orientations of the C 60 molecules. We note that some of the features of our data support this idea. The room temperature lineshapes presented in Fig. 5.2 show a correlation between the motional narrowing of the 13C spectra and the presence of the T peak in the alkali metal spectra. The T peak is absent in samples such K 3 C60 and K2RbCgo where the smaller alkali metal ions inhibit the reorientations of the C60 molecules to a lesser extent than in compounds with larger alkali intercalants like Rb3C60 or RbCs2Cfio for example. In the latter two materials the T peak is clearly present. Walstedt et aL show that above » 300 K the Tand T lines show spectral diffusion and at high enough temperatures they motionally narrow, or collapse into a single line. Figure 5.3 shows a 443 K and a room temperature (290 K) 87Rb lineshape in a sample of RbsCeo demonstrating this behavior. It is possible that at high enough temperatures the C60 molecule is rotating at a high enough frequency to average out the effect responsible for the splitting. Also shown in Fig. 5.3 is a room temperature 85Rb lineshape which also exhibits the T peak. The separation between the Tand T peaks in the spectra for the two isotopes of Rb is approximately equal in units of ppm. This combined with the fact that the 67 87Rb and 85Rb Spectra in Rb 3 C6o Rb C Rb 443 K Rb 290 K Rb 290 K -500 -300 -100 100 300 500 ppm Figure S 3 . 87Rb and 85Rb spectra in Rb3Ceo at the indicated temperatures. At 443 K the T and T lines are motionally narrowed into one peak. The 85Rb spectra is identical to the 87Rb spectra at room temperature (290 K). 68 magnitude of the 85Rb quadrupole moment (.28 cm2; 85Rb is spin 5/2) is twice that of 87Rb indicates that the separation is not quadrupolar but magnetic in origin, consistent with Walstedt et al Figure 5.4 shows 39K lineshapes in K 3C60 from room temperature down to 150 K. It is not clear whether the T peak is present in the 39K spectra at low temperatures. At 200 K and 150 K a small shoulder grows on the T peak which we may tentatively assign as T. We do note that this shoulder appears within approximately the same temperature range as when the C 60 reorientation correlation time is comparable to the NMR timescale as noted by Barrett and Tycko [Barrett, 1992], Table 5.3 gives the line positions of 13C, 39K, 87Rb, and 133Cs in C 60 and A3C60 materials at 8.8 Tesla for the following temperatures: room temperature, just above Tc (in italics), and near zero Kelvin (in bold). 69 39K Spectra in K 3C6 O 39 290K 250K 225K 200K 150K -250 -150 -50 50 150 250 ppm Figure 5.4. 39K spectra in K 3C60 at indicated temperatures. At room temperature only the T and O peaks are present. As temperature is reduced the T peak may manifest itself as a broadened "shoulder" on the Tpeak. 70 Table 5.3. NMR shifts for several C 60 compounds m order of increasing lattice constant, for several nuclear species and locations within the unit cell, and for the following temperatures: room temperature, just above Tc (in italics), and near zero Kelvin (in bold). 87Rb and 133Cs line positions are given for three positions within the unit cell: the site of octahedral coordination ("O"), and the two sites of tetrahedral coordination, T and T'. The "center of mass" of the T and T1 lineshape peaks for 87Rb and 133Cs lineshapes are also given in the column labeled "CM." ______ 13C 87Rb 133Cs 39K 0 T T CM O T T w CM OT 187 -51 9 K3C60 195* 1 6 2* 188 -147 9 K2RbC60 -242 •2 1 6 KRb2C60 184 -147 -20 -20 16 182 -144 -10 83 -6 Rb3C60 192* 151* 182 -7 86 -3 -300 Rb2CsCgo 186 -134 -420 1 4 7 -9 5 -3 2 1 RbCs2C60 177 11 108 16 -311 -65 67 -45 Tycko e ta l [1992]. Chapter VI The Knight Shift VIA. Chapter Overview This chapter reports the measurements of the Knight shifts Ks for the nuclei in A3C60- In Section VIB we deduce the temperature dependence of the electron spin susceptibility in the superconducting state of A 3 C60 from the K5 data. We find that the temperature dependence of x? can be well understood within the weak coupling limit of the BCS theory. Section VIC discusses the measurements we performed to obtain the signs and magnitudes for Ks for all the nuclei in A 3C60. VIB. Electron Spin Susceptibility This section presents the measurement of the temperature dependence of the electron spin susceptibility x?(T) in the superconducting state of A 3C60- We are able to deduce ^ ( T ) using multinuclear NMR methods: by measuring the total isotropic shifts Ko(7)for 13C, 87Rb, and 133Cs (a = 13, 87, and 133 respectively) in Rt^CsCeo we are able to subtract out unwanted contributions to K a(T) leaving a quantity with the same temperature dependence as x?(T). Fitting the data for ^ (T ) with the BCS theory, we will 71 72 show that it is in excellent agreement with the weak-coupling limit of the BCS theory, independent of the nuclei and of the applied field. VIB1. Subtraction Method: We first discuss the theory behind the multinuclear approach used to measure ^(7 ). In the Superconducting state Ka(T) can be written as the sum of three terms: s i AB(T) JTo(r)-*2(7*)+*£+-“ • (VI.1) These three terms are the Knight shift Kc?(T), the chemical shift Kch> and a term AB(7)/B representing the fractional change in the average internal field in the sample resulting from Meissner screening currents. The subscript a is used to identify the isotope in question (i.e. a = 13 for 13C). The quantity Kc?(T) is proportional to the electron spin (Pauli) susceptibility 2^(7): (VL2) where A a is the isotropic part of the hyperfine coupling tensor. Equation (VI. 1) can now be rewritten in terms of (VL3) We can see from (VI.3) that the temperature dependence of Ka(T) has contributions from both ^ (T ) and AB(T)fB (Koh is typically expected to be temperature independent). In the 73 normal state AB(T) = 0 and the temperature dependence of K q{T) is typically the same as that of X?(T). The difficulty in extracting )^(T) arises from the fact that AB(7)/B is unknown and usually hard to measure. If AB(T)IB is much smaller than Xs(T) then we could assume C K(£T) x CO. However, we will see that from the multi-field measurements that this is not possible (or at least not obvious) in A3C60- If we assume that the local spin susceptibility has the same temperature dependence at each atomic site (a reasonable assumption for alkali fullerides), then we may infer ^(T ) by taking the difference in the fractional shifts for two different nuclear species: K c f n - K p C n - f ^ ] A* 11X ( T ) + co n st (VI-4) lYeYa*) yeyf »J_ Here the subscripts a and /? refer to different isotopes. The constant may be determined C from the data if one assumes that x (O -*■ 0 as T -*• 0, an assumption true for both s and d-wave pairing states (neglecting spin-orbit scattering). We will now introduce the “raw” shift data for Rb 2CsC60- After this we will implement the above subtraction method to obtain the temperature dependence for )^(T) from the raw shift data. VIB2. R b 2 C sC 60 Lineshapes: Figure 6.1 shows representative spin echo Fourier transformed lineshapes for the 13C, 87Rb, and 133Cs isotopes in Rb2CsC60 (sample DR60B) taken at room temperature, at an intermediate temperature, at Tc, and in the superconducting state, all in an applied field of 8.8 T. We note some features of interest. First, the Tand T peaks in the 87Rb lineshapes broaden and then heavily overlap as temperature decreases. Second, the positions for the 87Rb Tand T peaks and the 133Cs 74 C, Rb, and Cs Lineshapes in the Normal and Superconducting States of Rb CsC at 8.8 T 2 60 87, 290 K 292 K 292 K 160 K 160 K 160 K 30 K 30 K 30 K 7 K 16 K 15 K Ll_l I I I I I I I I I I I I I I I I I ) I I 400-400 BOO -800 -400 400 -1000 -600 -200 2 00 ppm ppm ppm Figure 6.1. Spin echo Fourier transformed lineshapes for 13C, 87Rb, and l33Cs in Rb 2CsC60 (DR60B) at indicated temperatures. We note that the Tand T lines broaden with decreasing temperature and eventually become indiscernible at low temperatures. The extra breadth of the superconducting state lines is a result of the magnetic field distribution produced by the flux vortices. Also, the line positions for the 87Rb T and T, and the 133Cs peak move to lower frequencies with decreasing temperature, by approximately the same amount. However, as the temperature is lowered below Tc, the 87Rb frequency decreases, while the 133Cs frequency increases. This latter increase might result from a negative hyperfine coupling for 133Cs, which would make the similarity of the normal state shifts difficult to understand. The contrasting behaviors of 87Rb and 133Cs in the superconducting state is used to extract the superconducting state electronic spin susceptibility. 75 peak move to lower frequencies with decreasing temperature in the normal state by approximately the same amount., however as temperature is lowered below Tc the 87Rb line shifts to lower frequencies while the 133Cs line shifts to higher frequencies. We will talk more about this in the following subsection. VJB3. R b 2CsC60 Line Shifts: Figure 6.2 shows the measured center of mass line positions £o(7) of the 13C, 87Rb, and 133Cs isotopes in Rb 2CsC60 (sample DR168A) at an applied field of 8.8 T. Because the 87Rb T and T peaks start to overlap at low temperatures (as noted above) we take the center of mass of the 7* and T lines for the 87Rb shift £87(7). This is a fair approximation because T and T' appear to have the same temperature dependence within the temperature range that they are discernible. As already mentioned above, the normal state temperature dependencies of £87(7) and £ 133(7) are approximately the same. This is possibly the result of a temperature dependent Pauli susceptibility. However, in the superconducting state the temperature dependencies of the line positions of the two are different: £87(7) decreases and £ 133(7) increases. This is veiy difficult to understand if one uses the above explanation of a temperature dependent X? for normal state temperature dependence. We know from Eqn. (VI-1) that the shifts of the lines in the superconducting state results from two processes: 1) the onset of Meissner screening and 2) the vanishing spin susceptibility (the vanishing of £^(7)). Now, the Meissner screening effects are diamagnetic which produce a reduction of the field at the site of the nuclei, shifting £<*(7) to lower frequencies. Thus, the difference in sign of the temperature dependencies of £a(7) for the two nuclei must be the result of x S(T) -*■ 0. This implies that the sign of the hyperfine coupling for 133Cs (A 133) must be negative and that the similar normal state behavior of Ksi(T) and £ 133(7) is perhaps a coincidence. The 13C data are similar to that reported by Tycko et a t. the 13C line position remains at a 76 13 C, n 8791 Rb, and 133 Cs Line Positions vs. Temperature at 8.8 T in Rb,CsC!Z oU 200 5 o $ O <5 o' o 100 - N E t£ o - £ -100 0 K IE 13 CO • K r 87 ]S -200 - • □ K o 133 h- -300 □ 13- □ □ □ -400 ------0 50 100 150 200 250 300 Tem perature (K) Figure 6.2. Total shifts Ka(T ) for 13C, 87Rb, and 133Cs in Rb 2CsC60 (DR168A) as a function of temperature. The temperature dependencies of the normal state shifts for 87Rb and 133Cs are the same, moving lower frequencies with decreasing temperature. This could possibly result from a changing spin susceptibility. However, as Tis lowered below Tc the 87Rb frequency decreases, while the 133Cs frequency increases. This latter increase might result from a negative hyperfine coupling for 133Cs, which would be in contrast to the normal state behavior. The 13C line position remains at a roughly constant value which reflects the small isotropic hyperfine coupling of the 13C nuclei to the conduction electrons. 77 roughly constant value [Tycko, 1992]. This may reflect the small isotropic hyperfine coupling of the 13C nuclei to the conduction electrons. Figure 6.3 shows the measured center of mass line position K a(T) of the I3C, 87Rb, and 133Cs isotopes in Rb 2CsC6o between 5 K and 50 K for applied fields of 3,5, and 8.8 T. Between 50 K and Tc Ka(T) is approximately temperature independent for all three nuclei. However, as we go below Tc Ka(T) changes dramatically as both the spin susceptibility goes to zero and Meissner screening onsets. AKa = Ka(Tc) - Ka(5 K) « 37±5 (74+75,132±15), 169±5 (223±5, 291±10), and -154±5 (-89±18, 14±22) ppm for 13C, 87Rb, and I33Cs at 3 (5,8,8) Tesla respectively. The difference in magnitudes of A Ka for a given nuclear species for the various field strengths results from larger Meissner screening terms AB(T)IB at the lower fields (Kcf(T) is field independent in units of ppm's). From this figure we can see that the Meissner screening term is indeed a veiy substantial contribution to the total NMR shift in the superconducting state. We have also measured (not shown) 85Rb line positions over the full temperature range, and found them to be in good agreement with those of 87Rb. Since 85Rb has an electric quadrupole moment more than twice that of 87Rb, our measurement confirms that the 87Rb shifts for the measured central transitions are almost completely magnetic in origin. VIB4. K 133Cs nuclei reside in the Rl^CsQo lattice at the octahedral interstitials, it is intriguing to think that their negative hyperfine couplings may be a result of the nuclei's lattice position. To test this hypothesis we measured the shift of 87Rb in the superconducting state of K2RbC 6o (the Rb ion is now in the octahedral site). Figure 6.4 shows the line position of 78 13C, 87Rb, and 1 131JJCs Line Positions in the Superconducting State of Rb2CsC6 (3, 5, and 8.8 T Applied Fields) 360 ■ 1111 ■ I " " I " " 1111 ppm 13, 133- o 8.8 T 300 : 240 • JI 180 a o o ° ®|d • 0 120 60 * -4 2 0 I I .-. i i l l . l-> I I I t . 1-1 -4 8 0 '0 10 20 30 40 SO 10 20 30 40 50 0 10 20 30 40 60 Temperature (K) Temperature (K) Temperature (K) Figure 6.3. Total isotropic shifts K a(T) of the 13C, 87Rb, and 133Cs isotopes in Rb2CsC6o (DR60B) between 5 K and 50 K for applied fields of 3,5, and 8.8 T. Between 50 K and Tc (30,29.5, and 29 K at 3, 5, and 8.8 T respectively) K a(T) is temperature independent for all three nuclei. Below Tc K<^T) changes dramatically as both the spin susceptibility goes to zero and Meissner screening onsets. AK a = Ka(Tc) - K a(5 K) « 37±5 (74+15, 132±15), 169±5 (223±5,291±10), and -154±5 (-89H 8 , 14±22) ppm for 13C, 87Rb, and 133Cs at 3 (5,8,8) Tesla respectively. The difference in magnitudes of A Ka for a given nuclear species for the various field strengths results from larger Meissner screening terms AB(T)1B at the lower fields. As noted in Fig. 6.2, the normal state shifts for 133Cs and 87Rb seem to be identical, however, in the superconducting state the lines shift in opposite directions. If the shifts are resultant from a temperature dependent Knight shift, the contrasting behavior between the normal and superconducting state shifts is difficult to understand. 79 Shift of 87 Rb in K RbC and 133 Cs in RbCsC 2 60 2 60 at 8.8 Tesla -100 133 -150 E Q. -200 Q. 'w ' co -250 8 -300 Q. 0) j= -350 -400 -450 0 50 100 150 200 250 Temperature (K) Figure 6.4. Line position vs. temperature for 87Rb in K 2RbC 6o and 133Cs in Rt^CsCgo at 8.8 Tesla. In both compounds the nuclei in question are in the octahedral interstitials. As the temperature falls below Tc (23 K and 31 K in zero field for K2RbC60 and Rb 2CsC6o respectively) the line positions shift to a higher frequency. This net positive shift implies that the hyperfine couplings of the nuclei in the octahedral sites are negative. 80 the center of mass as a function temperature for 87Rb in K^RbCeo and 133Cs in Rb 2CsC6o at 8 .8 Tesla. As the temperature falls below Tc (23 K and 31 K in zero field for K 2RbC 6o and Rb 2CsC6o respectively) the line positions for both nuclei shift to a higher frequency. From these data we see that the hyperfine couplings of nuclei in the octahedral sites do indeed appear to be negative. VIB5. Xs(T) in Superconducting State of RbzCsCfio: We now focus on the extraction of the temperature dependence of Kc?(T), and hence from the data for Ka(7). From Fig. 6.3 we see from the multi-field measurements that AB(T)/B contributes strongly to Kd(T). However, because the quantity AB(T)/B is unknown, it is not possible to infer x?(T) from measurements of the NMR line position of one nuclear species alone. So, we must resort to our subtraction method outlined in Section VIB1 and, in particular, use Eqn. (VI-4) to deduce ^(T ): KaiT)-Kp(T) = f 40 ] * 11X (T) +const. (VI.4) \je Y a * ) lw#JJ Figure 6.5 shows the spin susceptibility inferred using this technique in an applied field of 8 .8 Tesla. Using the three available nuclei, 87 Rb, 133Cs, and 13C, we have three ways (two of which are independent) to extract x?(T). These are by taking the differences ^87" ATi 33, Km - K\3, and JSTi3 - £133. Each method is shown in the figure. All three methods yield the same result, which demonstrates that x?(T) is the same at all three sites. Within the BCS theory the temperature dependence of x?(T) in the superconducting state is expected to follow the Yosida form given by Equation (HI. 12): 81 Temperature Dependence of the Electron Spin Susceptibility in the Superconducting State of Rb2CsC6Q (8.8 Tesla Applied Field) 1 .2 i fill 1 • 1 1 i| 1I I1 1—■ I I |i 1I 1 I I1 1 I |i 1I 1 I I1 I1 i | 1I I1 1 I ' I i| I1—' I "r l 1 I i_r | l 1 I 1 I 1 l——I—li 1 ■ 1 l—r1 0 K -K D I 1 - 87 133-U133 A K -K 87 1313 I? 0.8 - □□ KK -K -K S 13 87 87 .Q ' ------■P _ _ g- 0 6: O tn 5 0. 4 : / co =« 11400400 cmcm' '1 - D- 0 2Ptonon phonon - -1 ® c oco = 100 1 0 0 cmcm’1 ■ phonon _ 0 2 —i i i—i i_i i i i i i i i i i___ 0 5 10 15 20 25 30 35 40 Temperature (K) Figure 6.5. Spin susceptibility vs. T of Rb 2 CsC60 at 8.8 Tesla as derived from data in Figure VI-3 by subtracting the raw shift data for three pairs of different nuclei (e.g. 13C and 87Rb, or 87Rb and 133Cs) to eliminate magnetic fields produced by Meissner screening currents. All three pairs of nuclei give the same result (after normalization) and can be weft understood within the weak coupling limit of the BCS theory (solid line). Also shown is the strong coupling fit for a single phonon mode at 100 cm-1. 82 m X S(T) - =J100a ( dE (HI. 12) J e 2 - A2(r)V where A(7) is the BCS gap function. Taking a Jc of 28 K in the 8 .8 Tesla field (we note that the transition is broadened by approximate 3 K) and using an inteipolation of the above function from Monien and Pines [Monien, 1990], we find that the three sets of data are well fit the Yosida form for x?{T) in the weak-coupling limit. This fit is also shown in Figure 6.5. For comparison we have also shown a fit using a strong coupling density of states, with 2A(0) =4.8 kgTc and characteristic phonon frequency 100 cm -1 [Aids, 1991]. From fits of shift data we estimate a lower bound of 200 cm *1 ( 2A(0) <, 4knTc) on the characteristic phonon energy. Figure 6 .6 shows Ks7 - £ 1 3 3 vs. TITC for magnetic fields of 3, 5, and 8.8 Tesla. With temperatures scaled to respective Tc's of 29,28.5, and 28 K for fields 3,5, and 8.8 Tesla, the three sets of data are in good agreement with each other, and, again, with the weak-coupling BCS fit. These data also suggest that the method of subtracting shifts to extract Xs (J) is remarkably effective, because while the field correction AB(T)IB is presumably quite field dependent, we show experimentally that the quantity Ksi - £ 1 3 3 is not. This stands in contrast to the field dependence of the Hebel-Slichter coherence peak in the relaxation rate data UT\T. VIC. Normal State Knight Shift We now discuss the signs and magnitudes of the normal state isotropic Knight shifts for 87 Rb, 133Cs, and 13C in Rb 2CsC60- In the first subsections we estimate values for the 13C Knight shift K\^s and the alkali metal Knight shifts K%qs and K \w s. The 83 87 Rb and 133 Cs Shift Difference in the Superconducting State of Rb 2^s^6o for Different Applied Fields 300 ♦ 8.8 Tesla 250 A 5 Tesla 200 □ 3 Tesla Q. CO CO 1 0 0 weak coupling CO 50 w =100cm -50 -100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 T/T c Figure 6.6. Difference in the 87Rb and 133Cs shifts Ks7 -K 133 (proportional to the spin susceptibility) vs. TJTC(H) for Rb 2CsC6o at 8.8,5, and 3 Tesla. The result is independent of magnetic field. The solid curve is the weak coupling BCS prediction and the dashed curve is a strong coupling BCS prediction with characteristic phonon frequency 100 cm’1. The data support the weak coupling limit of BCS theory. 84 remainder of the chapter presents the results of an experiment aimed at extracting the Meissner term AB(T)IB by using a combination of the multi-field NMR data and SQUID magnetometer measurements . VIC1. A Priori Estimate of Ki 3 s: Before embarking on the discussion of the normal state Knight shifts in A 3C60, we can obtain a quick experimental measure of the 13C Knight shift. Glancing at Table 5.3 one can see that the center of mass shift K 13 in C60 is 143 ppm from TMS and, because C60 is an insulator (K 135 =0), this is equal to the orbital shift KisL. If we now assume that the orbital shifts of 13C in C 60 are the same as in Rb 2CsC60, we can take the difference in total shifts (K 13 = 182 ppm in Rt> 2CsC6o) and we will be left with the 13C Knight shift in Rt> 2 CsC60 being equal to approximately 39 ppm. Armed with this simple experimental value, we can now examine what we obtain theoretically. The only interaction that affects the isotropic part of Ks will be from the Fermi contact term in the hyperfine Hamiltonian introduced in Ch. HI: Stt Hpc * - j Y e r r t f t ' S5 (r >- (VI.5) The delta function in the above expression indicates that only electrons with wavefunctions that are non-zero at the site of the nucleus will contribute to Ks, To zero-order this means that only electrons with some s atomic orbital mixed into their Bloch wavefunction are important and that isotropic Knight shifts in metals with electronic wavefunctions that are predominately p (A3 C60) or d in character would be very small. However, it is typically not this simple because paired electron spins in s states or states with j character (sp2 hybridized orbitals for example) can be polarized by unpaired electron spins in the p o v d 85 states (core polarization of closed core s states is an example of this [Slichter, 1989]). Unpaired spins on adjacent atoms to which the nucleus is bonded can also contribute to Ks through spin polarization. One must therefore consider many factors in order to understand the 13C Knight shift in A 3C6O. Karplus and Fraenkel [Karplus, 1961] have developed a formalism that can be used to predict these isotropic couplings for a 13C nucleus to unpaired electrons in both the on-site 2pz orbitals and to unpaired electrons in 2 pz orbitals of nearest neighbor carbons to which the caibon is tt bonded. They express the nuclear spin and electron spin interaction Hamiltonian as (VI. 6) Here I is the nuclear spin, So is the on-site electron spin in the 2 pz oibital, and the Sj are the spins in 2 pz orbitals on neighboring carbon atoms. The terms A, Bi, and B'i are measured in units of eneigy and have the following physical meaning: A: Direct Fermi Contact and Core Polarization. a) Direct Fermi Contact: Coupling of the 13C spin with any 2s spin density that is admixed into the 2 pz wavefunction of the on-site electron spin. In Cgo the 2s admixture would result from the curvature of the molecule. b) Core Polarization: Coupling of the 13 C spin to the on site 1$ core electrons polarized by the on-site 2 pz electron spin. 86 Bj: Spin Polarization of the Coupling of the 13C spin to the on-site 2pz electron through its polarization of the (filled) on-site sp2 orbitals which make up the cr bonds. Through Hund's rule considerations, the “up” spin in the 2pz orbital produces a small “down” spin density in the cbond at the nuclear site. B i Spin Polarization of cr Bonds by Neighbor. Coupling of the 13C spin to the electron spin densities in 2pz orbitals on neighboring carbons through their polarization of the on-site 2s orbitals which make up the a bonds (“transferred” 2s spin polarization). Through Hund's rule considerations, the “up” spin in the neighboring 2pz orbital produces a small “up” spin density in the a bond at the nuclear site. Karplus and Fraenkel find for carbon atoms bonded together in a plane (such as in graphite) that if each carbon is equivalent then B = B 'a 2.6x10'19 erg. Because the B and B' terms produce opposing contributions to the Knight shift they cancel each other. This cancellation result is still true for the curved C 60 surface. The core polarization of the 1 j elections (contained in the A term) is also given by Karplus and Fraenkel and is approximately -2.4x10'19 erg. (Karplus and Fraenkel give this as -12.7 gauss which is the magnetic field "seen" by the electron spin-a common practice for electron spin resonance data. This translates into energy by multiplying by Yeh). In the absence of curvature of the buckyball molecule the direct Fermi contact term in A would make no contribution and we would take A = -2.4xl0*19 erg. Karplus and Fraenkel show that in the presence of curvature (11.6° from planarity for C60 [Haddon, 1988]) the admixture of 2s wavefunction into the 2pz orbital results in an 87 additional direct Fermi contact contribution to A equal to2.21xlO' 17(2tan 20) erg or 1.9X10 - 18 erg. Combining this result with what we obtained for the core polarization contribution we find that A = 1.7xl0 "18 erg. If we use the spin susceptibility measured by Ramirez et a l [Ramirez, 1992] of » 1.3xl0 3 emu/moleCeo, or 3.8xl0 ' 29 emu/cait>on atom, we obtain a Knight shift equal to: X * 490 ppm. (VI. 7) [VeVn^J This value is more than ten times larger than our estimate from experiment of 39 ppm. One possible explanation is that the admixture of s orbital wavefunction into the Bloch wavefimctions near the Fermi energy is several times smaller than is predicted by formalism of Karplus and Fraenkel [Pennington, 1995]. VIC2. Estimates of Alkali Knight Shifts: We now present estimates of the 87Rb and l33Cs Knight shifts. If we use are “simple experimental” value of K\3S « 39 ppm and the line shift data of Fig. 6.3 we estimate that K&7s and K\33s are approximately 169 and -154 ppm respectively (this assumes a negligible field correction at 8 .8 T). So with these “experimental” values we can now estimate values for K ^ s and Kl33S- As with 13C the Fermi contact hyperfine interaction is the source of the isotropic Ks. The Knight shift K3 can be written as k s - y(l'P(of)zs (vi.8) 88 where (|^(0)|2^ is the probability density of the Bloch wavefunction averaged over the Fermi surface evaluated at the site of the nucleus [Slichter, 1989]. The Pauli susceptibility X s in the above equation has units of volume (cm3) and & r/3^|¥(0)f the isotropic part of the hyperfine coupling, has units of 1/volume (cm”3). Radzig and Smimof report a measured value for the hyperfine coupling of the free Rb atom of &r/3^|,P(0)|2^ =26.4xl025 cm”3 [Slichter, 1989]. Satpathy e ta l use local density approximations to estimate the admixture of the tetrahedral and octahedral Rb 5s wavefunction into the tju molecular orbital (LUMO) of C 60 for the compound Rb 3C6o [Satpathy, 1992]. Their fractional admixtures range from « 10”3 for octahedral Rb to » 2xl0'2 (total) for the two tetrahedral Rb. In a unit cell of R^Ceo, containing one C 60 and three Rb ions, the tju orbital should then have * lxlO”2 weighting from the 5s orbitals of the tetrahedral Rb. We will then obtain for one unit cell of RbjQo ATs7 « 1 x 10 ^ 2 .6 4 xlO 26) 2.2 * 10‘27 = 5800 ppm. (VI.9) Here we used the susceptibility of Ramirez et al. of 2.2xl0”27 emu/Ceo- The above estimate is like that for 13C, it is an order of magnitude larger than our simple experimental results. The predicted value forK%js for A3C60 is actually comparable to the shift in Rb metal, 6530 ppm [Knight, 1956]. VIC3. Measurement of Demagnetization Factor: Now that we have both a simple experimental value and a theoretical value for the Knight shifts of the nuclei in Rt^CsCgo we can move on to discuss the experiment we performed aimed at measuring Ks for all nuclei in Rt^CsCgo (we used sample DR60B). In essence, we attempted to measure 89 the field correction A BIB due to the Meissner screening in Equation (VI.3) for the total shift K(T). For this discussion Eqn. (VI.3) can be recast in a more lucid form: ' A N M(T) K(T) = X S(T) + K L +a47T (VI.8) JeY nK H Here ABIB has been replaced with the term a47tM(T)/H. H is the applied external field (which we know), M(T) is the magnetization of the material (which we can readily measure with the SQUID magnetometer), and a is the so-called demagnetization factor (a number which depends on sample geometry-for a sphere a =2/3). Because we have a sample of many amorphous crystallites it is difficult to estimate a and we must resort to measurement technique to extract its value. The demagnetization factor can be obtained by measuring shift K(T) and magnetization M(T) at two different fields and subtracting: fiT , ■3T / M st(T) M3,\T ) K st(T) - K31 (T) = a 4 n (VI.9) Here tf5T(7) (/^T(2)) is the shift at 5 Tesla (3 Tesla) and M5T(T) (M ^T)) is the magnetization at 5 Tesla (3 Tesla). As long as we measure the shifts in ppm the contributions from the Knight shift and from the chemical shift will subtract out. Figure 6.7 shows the magnetic moment p of Rb2CsC60 as a function of temperature in applied fields of 3 and 5 Tesla. The sample was first cooled to 5 K in zero applied field, the field was turned on to the indicated strength, the sample was warmed above Tc and then cooled to 5 K in the applied field. In the NMR experiments we 90 Magnetic Moment v s. Temperature for Rl^CsC^ o ^^isivivvvtfvfvi Field Cool - 0.02 3 | -0.04 >> ' +-» | -0.06 o y -0.08 4-> Zero Field Cool d> §> -0.1 (0 o 3 Tesla - 0.12 • 5 Tesla -0.14 5 10 15 20 25 30 35 40 T (K) Figure 6.7. Magnetic moment vs. temperature for Rfc^CsCfl) (DR60B) in applied fields of 3 and 5 Tesla. The sample was first cooled to 5 K in zero applied field, the field was turned on to the indicated strength, the sample was wanned above Tc, and then cooled to 5 K in the applied field. In the NMR experiments the sample is always field cooled. 91 performed, the sample was always field-cooled so it is the field cooled numbers that are important. We note that the transitions at these high fields arc extremely broad. In order to guarantee that we are measuring the equilibrium magnetization we measured field cooled values for p at 5 K in applied fields of 1,3, and 5 T over a period of 2 hours for each field. These measurements are shown in Figure 6.8. By taking an average of the last half hour's data points at each field, we were able to arrive at estimates for iW^SK) and A f^SK ) of -3.66 and -2.60 emu/cc respectively. Also, from the shift data for 87Rb at 5 and 3 T (shown in Fig. 6.2) we found K ^ ^ K ) - = 77 ppm. Using these values we obtain from Eqn. (VI-7) that a = .46±.07 and the field correction at 5 T is 377±65 ppm. The Knight shifts obtained by this method are all negative'. K&js = -154±68 ppm, Ki3s =-303±80 ppm, and Ki 335= -466+123 ppm. These Knight shift values are obviously in direct conflict to the estimates given in the previous subsections and the reason for the discrepancy is, at present, unclear. The final determination of the signs and magnitudes of the Knight shifts of the nuclei in A 3C60 remains an open issue and is in need of ftirther investigation. We can, however, summarize what we have learned and form two conclusions: (1) The positive frequency shift of the alkali nuclei in the octahedral sites is firm evidence that these nuclei have negative hyperfine couplings; there seems to be no alternative explanation. (2) From the of the “a priori” Knight shift estimates with all of the aforementioned Knight shift measurements , the Knight shifts of all the nuclei in A 3C60 are approximately an order of magnitude too small. 92 Field Cooled Magnetic Moment of Rb CsC at 1, 3, and 5 Tesla M oil -0.015 Tesla - 0.02 © -0.025 2 -0.03 3 Tesla -0.035 O) -0.04 -0.045 5 Tesla . -0.05 2 0 2 4 6 8 10 12 14 Time (hrs) Figure 6.8. Magnetic moment vs. time in a field-cooled sample of Rb 2CsC6o- The Field was first turned on to the indicated strength, the sample was cooled to 5 K, and a data point was then taken at the fixed field and temperature every 10 minutes. To obtain values for /i at each field we took an average of the last half hours data points. Chapter VII Spin-Lattice Relaxation Rates VIIA. Chapter Overview This chapter presents the measurements of the 13C, 87Rb, and 133Cs spin-lattice relaxation rates in the normal and superconducting states of Rb 2CsC60- In Section VHB we present the normal state results including the saturation- and inversion-recovery curves. We will see how l/TiT for all three nuclei have identical temperature dependencies and that the Korringa law is valid in these materials if one considers Stoner enhancement to the electron density of states. Section VIIC discusses the T\ results in the superconducting state of Rb2CsC6o- We show that the Hebel-Slichter coherence peak, suppressed at high applied field strengths, is recovered as the field is reduced and that the suppression can be understood as an effect of the vortex mixed state. The superconducting state T\ data will be shown to be reasonably consistent with the weak-coupling limit of the BCS theory with the intramolecular phonon modes being responsible for pair formation. V1IB. Relaxation in the Normal State VHB1. Recovery Curves: First we discuss the extraction of the T\ values for 13C, 87Rb, and 133Cs in Rb 2 CsC6o- The 13C T\ values were determined by the saturation recovery (see VIC6) technique using a Hahn echo to determine the signal size. Antropov 93 94 et al. [Antropov, 1993] has shown by ab initio calculations that the dipolar hyperfine interaction in Eqn. (Ifi.3) is the dominant interaction responsible for the relaxation of the l3C spins because the carbon orbitals are mainly radial p in character. Thus, the 13C Ti will reflect a distribution 13C relaxation rates as one averages the dipolar hyperfine couplings over the C6o molecule. Like Tycko et a l [Tycko, 1992] we found the recovery curves to be slightly non-exponential reflecting this averaging and will represent this multiexponential behavior by using a “stretched” exponential to fit the recovery data: 4(T)=5t,{l-exp[-(r/r/]}. (W.1) Here S(f) is the magnitude of the spin echo at a given delay time r, 5o is the equilibrium magnetization and P is a parameter indicative of the degree of distribution of relaxation rates. We find that we can fit all temperatures (except those at temperatures much less than Tc where appreciable spin diffusion occurs) and all applied field strengths with p= .85. Figure 7.1 shows the 13C recovery curves and fits in Rb2CsC60 (DR168A) at 292,100, 31, and 10 K. As mentioned earlier, we know that only the central (-1/2, 1/2) transition is observed for 87Rb and 85Rb (spins 3/2 and 5/2, respectively) as a result of first-order quadrupole splitting (see Appendix B) and that the 87Rb and 85Rb lineshapes reveal the existence of a second magnetically inequivalent tetrahedral site T. The existence of large quadrupolar effects raises a question as to whether the relaxation of the rubidium nuclei is quadrupolar in origin instead of magnetic like 13C. This question is answered by comparing the ratio of the T\ values for 87Rb and 85Rb. Using the inversion recovery method (see VIC6) the recovery curves for both isotopes in a sample of Rt> 3C6o (DR30B) at 80 K were measured. We fit the recovery data with the familiar multi-exponential forms 95 Saturation Recovery Curves for 13 C in Rb2CsC60 1.2 i i i hiiij i | I i i iiiiij ii imiij TTTTTITTJ ri IIMHj— rTI I llllj I I llllllj I I 111111] “'I I Jill —e — S 292K 'c 1 Z> -S 100K o-ar%- ° *-• /* □ / £ -s - S 31K (0 0.8 •M - • -S 10K la < 0.6 d) ■O 0 .4 3 E ° ’2 o 0 JZ o I I I mill I I I mill ...... I...... ml___ l - l u r n ] ___ i ...... ill i i m i n i ...... ill I I h i m - 0.2 0.001 0.1 10 1000 10- 1 0 7 Delay Time (ms) Figure 7.1. Saturation recovery curves for 13C in Rt> 2CsC6o at 8.8 T for the indicated temperatures. A “stretched” exponential fit of the form 5(r) = S0{1 - exp[-(r/7’1)‘85]} was used to extract T\. 96 used for the relaxation of the (-1/2, 1/2) transitions in spin 3/2 and 5/2 systems (see Appendix B where we solve the “normal modes” problem): 1= 3/2: S(t ) = S0{1 - 2a[(l/10)exp(-r IT{) + (9/10)exp(-6r ITX)]} 1=5/2: (VH.2) S(j) = iSo(l -2a[(l/35)exp(-r/r1) + (50/63)exp(-15r ITX) + (g/45)exp(-6r/T1)]}. Here a <, 1 is a parameter introduced to compensate for imperfect inversion of the magnetization and for S(t) we took the maximum echo amplitude. Figure 7.2 shows recovery curves for 85Rb and 87Rb in Rb3C60 where T\ = 10 s and .9 s respectively. Assuming that the relaxation mechanisms for both are magnetic, we can compute the ratio of the TVs obtained from the fit (assuming that the hyperfine coupling is the same for both): (r i)87 r i 5 zp\= 2- (vn.3) fa)* rh (This ratio does not hold for quadrupolar relaxation mechanisms). The gyromagnetic ratios are well known quantities which yield a value for (K 85 /K8 7 )2 equal to .087. This is in excellent agreement with the experimentally determined value of .09, giving strong evidence that the rubidium nuclei relax via magnetic interactions. Further verification will be given when the temperature dependencies of the 13C and 87Rb UTxTs are shown to be identical (to be shown later). A second point to be considered regarding the 87Rb relaxation is the existence of the so called T peak in the NMR spectrum. The above analysis assumed that the temperature 97 Inversion Recovery Data for 85Rb and 8 /Rb87i in Rb.C,ft at 80 K and 8.8 T 3 Dll 1.2 (O 1 87, c - • - 8 5 , D /• k 0.8 u5 . as k . !o 0.6 < (1 /T ) =.100 s a> 0.4 ■a 4-J3 Q. 0.2 E < 0 o LU 0.2 -0 .4 10'5 0.001 0.1 10 Delay Time (s) Figure 7.2. 8.8 T 85Rb and 87Rb inversion recovery curves in Rb 3 C60 at 80 K. Also shown are the multi-exponential fits for the relaxation of the central (-1/2,1/2) transitions of a spin 3/2 (solid line) and of a spin 5/2 (dashed line) system. The ratios of the relaxation times (Ti)s7/(Ti)%5 « .09 is in good agreement with the ratio (K 8 s)2/(K87 )2 = -087 as expected for relaxation from a Koninga mechanism. 98 dependencies for T\ of all three lines (O, T, T) in the 87Rb spectrum of Rb3C6o are identical. This assumption was based upon using the echo amplitude in the time domain signal for S(f). A more careful method would be to use the Fourier transform peak areas for all three yielding three signal heights So(t), and Figure 7.3 shows l/T\T as a function of temperature for each of the three lines in Rb 3C60- Within the temperature range shown we can see that each line has a different T\ value (below approximately 80 K the three lines broaden and overlap). The temperature dependencies, however, are the same. Thus, it is sufficient to use the echo height for S(r) and treat the T\ value obtained as a weighted average of the 7Ys for each site. Figure 7.4 shows 87Rb inversion recovery curves in Rb 2CsC For 133Cs (spin 5/2) we are observing all transitions. We found that the recovery of the cesium magnetization, using saturation recovery, was exponential as expected. So we fit the data to a single exponential of the form: S(t) = S0[l - e x p (-r/ri)]. (VH.4) Figure 7.5 shows 133Cs recovery curves in Rt> 2CsC6o (DR168A) at 292, 100,31, and 10 K. VHB2. Normal State Relaxation Rates. Figure 7.6 shows the temperature dependence of \JT\T for 87Rb, 133Cs, and 13C in normal and superconducting states of Rb2CsC6o using an applied field of 8.8 Tesla (although we show the superconducting state data, we will wait until the next section to discuss it). In the normal state 99 T (K) i n R b 3 C 60 1/TjT vs. T for the O, T, and T’, Sites 75 75 100 125 150 175 200 225 250 275 300 0 87Rb 1/717vs. temperature forthe “0 ,” “7,” and “7'” lines in Rb3C6o 0.05 0.04 0.03 0.02 0.01 7 3 . ( W l ) l l / l (DR30B). (DR30B). Withintemperaturethis rangethetemperaturedependencies each theforof lines Figure are identical. Below approximatelytheKbroadenlines and80 overlap. 100 Inversion Recovery Curves for 87 Rb in Rb2CsC60 1.5 in ■M ‘c - - -1 0 0 K ZD 1 (0 12.5 K 0.5 1 MIX 0.01 1 1 0 0 10® Delay Time (ms) Figure 7.4. Inversion recovery curves for 87Rb in Rt>2CsC60 at 8.8 T for the indicated temperatures. A double exponential fit of the form S( t ) = Sq{\ - a 2[(1/10) exp(-r / 7\) + (9/10) exp( -6 r /7])]} was used to extract T\. 101 Saturation Recovery Curves for 133 Cs in Rb2CsC60 1.2 I i1 im i r111111------1 m j i \ ir i 111111 m 111------1 i i i1 i 1 nil] rn i| ------ii1 ii nm rni| i ni j ■ "i—i i m 11 11i mjlllj i—ttttttt]—~ 1 i j urn— i—itttttt i i i inn /■N £ - —e——e — S292K S 292K 13 ; - • -S S 100K/*100K /■ P m /*• : | o.8 - --b- s331Kik/ : g [-•-SI■•••■■ S 10KOK• < 0.6 * 1 ( U ^ ■o 0. 4 • ' - +j 1 0 .2 ** : < o ° - o UJ _ Q 2 '''' i i i i_ Mill" " I ____ .I iI ■I i I Millm i l ____ I__ i I m i l l ____ i i i i m i l ____ i i i i m i l ...... m l ...... 1 100 104 106 Delay Time (ms) Figure 7.5. Saturation recovery curves for 133Cs in Rt^CsCeo at 8.8 T for the indicated temperatures. A single exponential fit of the form S(t) = S0[l - exp(-r /7\)] was used to extract Ti. 102 (30 K ^ T ^ 292 K) results we see that 1/Tirvalues for all three nuclei have the same temperature dependence. This is in contrast to the high Tc superconductor YBa 2Cu3C>7 where the planar 170 and 63Cu 7Ys show drastically different temperature dependencies [Martindale, 1993]. In these materials, the differences in the temperature dependencies possibly reflect antiferromagnetic fluctuations of the Cu d shell spins [Slichter, 1994]. The identical behavior of the nuclei in Rb 2CsC60 indicates that all three nuclei probe a single component conduction band. Hence, we may consider each nucleus to be an equally valid probe of a uniform electronic structure. Also, the identical temperature dependencies of the two quadrupolar nuclei 133Cs and 87Rb with that of the 13C is strong evidence that the relaxation mechanism is magnetic for the former two. VIIB3. Stoner Enhanced Density of States: From Chapter m we know that l/7 i T is expected to be a constant in a normal metal: ± ^ a 2 n \e f). (m.io) However, we can see from Fig. 7.6 that the observed values of \!T\T for all three nuclei actually increase by approximately 80 percent between 30 K and 292 K. The analogous increase for K 3C60 is » 45%6, and for Cs2RbC60 140% (this work). Tycko et al suggest the possibility that the temperature dependence may arise from thermal contraction of the crystal lattice [Tycko, 1992], To test this hypothesis, we use our 13C T\ data for RbK 2C6o> Rb 2KC6o» CsRb2C6o, Cs 2RbC 6o and the data of Tycko et a l for K 3C60 and Rb 3C6o along with the known temperature dependence of the lattice parameter a [Flemming, 1994]) and plot the normal state UT\T versus a in Figure 7.7. Now both temperature and dopant are implicit parameters. It is clear from Figure 7.7 that 1/Ti T is at 103 1/T vs. Temperature for 13 C, 87 Rb, and 133 Cs in Rb2CsC60 at 8.8 T 0.1 □ E □ D □ c C b D D D □ • a 0.01 • • • ^ □ o o O O u G w nOOO o O O O n □ !E 0.001 87Rb i • 13c 0 133c s: 1 0 "4 10 30 100 300 Temperature (K) Figure 7.6. Temperature dependence of l/TiTfor 87 Rb, 13C, and 133Cs in Rb2CsC60 (rc = 31 K) taken at applied field 8.8 Tesla (uncertainty is estimated as ±5%). The temperature dependence for all three nuclei is the same. This is in sharp contrast to the behavior in the high Tc cuprate superconductors, where the NMR behavior of the 170 and 63Cu are quite different, suggesting the importance of antiferromagnetic fluctuations in those materials. 104 1/TjT vs. Lattice Constant for a Series of A3C60 Compounds 0.04 I 0.02 H 0.01 0.005 14.1 14.214.3 14.4 14.5 14.6 Lattice Constant (A) Figure 7.7. (1/TiT) for 13C vs lattice parameter a forA3C60 compounds with temperature and dopant ion as implicit parameters. Materials: (open circles) K 3C60 (between Tc and room temperature, Tycko et a i ); (closed triangles) RbsCeo (between Tc and room temperature, Tycko et aL); (closed diamonds) RbK2C6o (292 K, this work); (open triangles) Rb2KC60 (292 K); (open squares) Rb2CsC60 (between Tc and room temperature); (closed circles) RbCs 2C6o (between Tc and room temperature). The dashed curve is a “Band Theory” fit (see text) using ID A and Huckel calculations. The solid curve is a fit to VT\T proportional to the enhanced density of states squared (see text). 105 least “nearly a universal function of a”, with perhaps a small explicit dependence on 7 and on dopant atom. We conclude that MT\Tvaries predominantly due to the variation of the squared density of states with lattice constant. We interpolate the variation of the density of states (N) with a using N « exp (a/ 1fid ) . To find d we utilize an average of reported local density approximation (LDA) and Huckel calculations [Gelfand, 1992] of the ratio R = N(Rb 3C6o)AZV(K3C6o), which gives R a v g = 1.165 and d = .71 A for low temperature lattice constants 14.304 A (Rb3C60) and 14.150 A (K3C60) [Flemming, 1994]. From this we generate a fit with no adjustable parameters, apart from a constant factor corresponding to the hyperfine coupling (see below), for 1/TiTvs. T of the form: 1 f - \ — 2 OCN o' exp {a/ 1/2 d ). (VII.5) T\T This fit is shown by the dashed line in Fig. 7.7. Clearly, the experimental variation with a is much faster than calculated regardless of what value chosen for the l3C hyperfine coupling. Thus, to fit the data it is necessary to consider enhancement to the density of states. Ramirez et al. [Ramirez, 1992] find that both Stoner enhancement and the weak coupling limit of BCS theory (with 00ph - 1400 cm’1 characteristic of tangential intramolecular vibrations) are uniquely required to explain the spin susceptibility, specific heat jump, and Tc's of K 3C60 and Rb 3C6o- Along these lines, we use an enhanced density of states, Nen, of usual form for Soner enhancement [Narath, 1968], Nen = 2V/(1 - IN) (Iis the enhancement parameter), to generate a fit (solid line) to the TiTdata in Figure 7.7 of the form: 106 1 f i Y- 1 2 exp ( a /j2 d ) {TlT j - i J i a W ( m 6 > In Figure 7.7 we use the parameters of Ramirez et al. for the enhanced density of states: IN = 0.49 and N = 7 states/(eV spin C6o) for K 3C60 at low temperature, and d = 0.74 A. This lit is in excellent agreement with the experimental results. The remarkable success of the fit provides strong support for the conclusions of Ramirez et a l , which include predictions of Tc using the McMillan equation, and which suggest a conservative upper bound for Tc in this family of fee doped Cgo3" compounds of 70 K, obtained in the limit of the magnetic instability where IN-* 1. To check the reasonableness of the inferred hyperfine coupling parameter we take the "pptf' model of Antropov et a l [Antropov, 1993] in which l/T^Tis dominated by dipolar coupling to the electron spin in the pz orbital. The hyperfine coupling parameter for the pp7t model is Our fit implies a value for the coupling parameter of (( a 0 / r f ) =1.7 (normalized to one carbon atom, with a0 equal to the Bohr radius), which is in excellent agreement with the free atom value of 1.6 derived fiom ESR splittings [Weil, 1994]. VIIB4. (U T iT )112 vs. K for ®7Rb and 133Cs: The relaxation rates for the 87Rb and 133Cs isotopes in Rb2CsC6o can be related to their respective Knight shifts via the Koninga relation: 107 1 ( 4 7!kB Yn (vn.8) * y 2e This equation is identical to Eqn. (ELI1) except that in the above expression we have added a phenomenological parameter b to represent the degree to which electron-electron interactions (“strong correlations”) are important [Slichter, 1989]. b= 1 implies that the effects of strong correlations are negligible. Figure 7.8 shows the \IT\T data taken from Fig. 7.6 and the K (total shift) data taken from Fig. 6.2 for 87Rb plotted as (l/T\T)112 vs. K. Also shown is a fit of the linear form (VH.9) If we assume that all the temperature dependence of K comes from Ks then the slope a will be equal to the square root of the proportionality constant between UT\T and K in Eqn. (VH.8). By comparing the ratio of a2 to 4 itkBYnlhY2 we can determine b. We can also estimate a value for the orbital shift from the intercept # when (l/7i7)1/2 = 0 , ^ = 0 and KL = -j6/a. Using this analysis we find from Fig. 7.8 that for 87Rb b = .93 and K siL = -387 ppm, giving a Knight shift at Tc of 271 ppm. Thus, within the experimental error of our measurements, the Korringa relation is well obeyed in the A 3C6O materials (for copper b = .77). Also, the Knight shift obtained by these methods is also much smaller than what we expect from a priori considerations. We note here that a comparison of this sort is not possible for 13C due to the dominance of the dipolar hyperfine mechanism in 13C relaxation (an analysis by Pennington et al. 108 (1/T^T)1'2 vs. K for 87Rb in Rb2CsC6Q 0.23 0.22 CM t 0.21 * CM \ 0 *2 CM ^ 0.19 h* b=.93 t 0.18 0.17 0.16 -120 -100 -80 -60 -40 -20 K (ppm) Figure 7.8. Plot of ( l/T iT )l/2 as function of the total shift K for 87Rb in Rt> 2CsC6o deduced from their respective temperature dependencies. For a normal metal with non-interacting electrons a straight line is expected with a slope a equal to the square root of the Korringa constant: ^ 4 7rkBY*/hYe • The parameter b = a 2/(^7ikBYn/hYe) reflects the degree to which the Korringa relation is obeyed. The above plot shows excellent agreement with the Korringa relation. 109 (1/TjT)1'2 vs. K for 133,*"Cs in Rb2CsC 60 0.075 CM a * CM 0.07 Vo CM b=.82 0.065 0.06 -400 -390 -380 -370 -360 -350 -340 -330 K (ppm) Figure 7.8. Plot of (IIT\T)112 as function of the total shift K for 133Cs in Rt^CsCgo- The Koninga relation is well obeyed for the 133Cs isotope. 110 [Pennington, 1995] of the anisotropic contributions to the Knight shift and to T\ for I3C will be discussed in Chapter VUI). Figure 7.9 shows the same plot as above except the nucleus is now 133Cs. Using the same analysis we obtain b = 82, K m 1 = -661 ppm, and K 1335 = 262 ppm. Again, there is excellent agreement with the Korringa relation. The sign of the 133Cs Knight shift obtained by these methods is positive which is contradictoiy to the negative Knight shift implied by the raw shift data. Given that there is no other explanation of the shift data other than being a result of a negative hyperfine coupling, one must conclude that the above analysis is invalid. It is, however, extremely suprising and, thus, worthy of mention, as to how well the Korringa relation is followed by the alkali nuclei. VIIC. Relaxation in the Superconducting State VHC1. The Hebel-Slichter Coherence Peak: The superconducting state l/7’i7’results at 8.8 T in Figure 7.6 reveal an apparent anomaly first noted by Tycko et al. [Tycko, 1992]: the absence or near absence of the Hebel-Slichter (HS) coherence peak. The HS peak (see IIIC2) appears as an increase in l/TiTas Tfalls below Tc. The absence of the HS peak could suggest that A 3C6O is in the strong-coupling regime of the BCS theory and that the low energy intermolecular phonon modes play an important role in the superconductivity of these materials. This result is in conflict with other data supporting weak or intermediate coupling such as the IR data of Degiorgi et al [Degiorgi, 1992] and the Rb 3C6o pSR results of Kiefl et al. [Kiefl, 1992], taken at an applied field of 1.5 T, which do show a HS peak. The absence of the HS peak in Fig. 7.6 is also in conflict with our Knight shift results which are in excellent agreement with the BCS weak-coupling picture. Ill We decided to investigate what effect the strength of the applied field had on the relaxation data. Upon lowering the magnetic field to strengths below approximately 7 T we noticed a small but measurable increase in 1/T\T for temperatures slightly less than Tc. Figure 7.10 demonstrates this by showing 3 Tesla saturation recovery curves for the 13C spin echo at Tc « 30 K, 28 K, 27 K, and 25 K where the fits are “stretched” exponentials with P = .85 (see above). The data are plotted as Echo Amplitude vs. (Delay Time x Temp.) in order to demonstrate the faster recovery (increase in HT\T) of the 27 and 28 K data compared to the 30 and 26 K data. Thus, between Tc and approximately 83% of Tc the HS peak is clearly present. Figure 7.11 shows field and temperature dependent 13C spin lattice relaxation 1/riTfor Rb 2CsC60, normalized to one in the normal state near Tc to yield the ratio Rs/Rn (assuming 1/T\T is constant for T < Tc) of normal to superconducting state relaxation rates. Also shown (inset) is the maximum value of Rs!Rn reached below Tc as a function of field. (RsIRn)max at the lowest field of 1.5 Tesla is 120 ± .06, and occurs at * 27 K. VDC2. Arhenius Plots of Relaxation Data: From Ch. in we know that in the BCS theory for T « Tc the relaxation rate l/T\ will exhibit activated or Arhenius behavior 1 —• °«ext (m.i4) r, * kBT Figure 7.12 shows the temperature dependence of the low temperature relaxation rates for 13C in Rb2CsC60 (DR168A) at 8.8,5, and 3 Tesla. For comparison are shown Arhenius curves using A(0) = l.76kBTCf 2.0kBTc, and 2.5 kBTc. It is clear from the figure that only the range 1.76 to 2.0kBTc provides a reasonable fit. This result is consistent with the 13C 112 13 C Echo Height vs. (Delay Time x Temperature) in Rb2CsC60 at 3 Tesla (A -M 0.9 'c 3 0.8 0.7 0.6 0) TJ 3 S30K _4J 0.5 Q. E S28K < 0.4 o JC u LLl 0.3 ■ S25K 0.2 ----- 3 x 10 4 10 5 Delay Time x Temperature (ms K) Figure 7.10. Saturation recovery curves for 13C spin echo at 3 Tesla at indicated temperatures. The lines are fits of the data using a “stretched” exponential with (3 = .85 (see text). The data are plotted as Echo Amplitude vs. {Delay Time x Temp.) in order to demonstrate the faster recovery (increase in \!T\T) of the 27 and 28 K data compared to the 30 an 25 K data. iue .1 13C of Maximum 7.11. Figure R /R z 0.2 0.4 0.6 8 .8 0 1.4 1.2 Rs/Rn 0 1 0 5 0 5 0 5 40 35 30 25 20 15 10 vs. applied field. vs. applied Rs/Rn 3 C sR v. eprtr i Rl^CsC^ in Temperature 13 vs. Rs/Rn C vs. vs. o Dfeet ple Fields Applied Different for T of Rb of e eaue (K) perature Tem 2 CsC 6 fr eea mgei fed. Inset: fields. magnetic several for o i 113 114 13 C Tj vs. 1/T in Rl^CsC^ for Various Applied Field Strengths 1000 1 0 0 10 CO — 1.76 kT 1 - - 2.0 kT 2.5 kT 0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 1/T (1/K) Figure 7.12. Relaxation rates for I3C in Rt> 2CsC6o at 8.8,5 and 3 Tesla plotted as T\ vs. 1/T. Arhenius curves using A(0) = \.16kBTc, 2.0kaTc, and 2.5 IcbTc are also shown for comparison. It is clear from the figure that only the range 1.76 to 2.0 knTc provides a reasonable fit. This plot shows that the Arhenius behavior with A(0) = 1.76ksTc is independent of applied field. 115 13 C, 87 Rb, and 133 Cs Tj vs. 1/T in R^CsC^ at 8.8 T 1000 1 0 0 v> I- 133, 0.1 1.76 kT 0.01 1/T (1/K) Figure 7.13. Arhenius plots for 13C, 87 Rb, and 133Cs in Rb 2CsC6o at 8.8 Tesla with weak-coupling BCS fits. This plot shows that the Arhenius behavior with A(0) = l.76ksTc is independent of nucleus. 116 NMR results on K 3C6O and R 3C60 by Tycko et a l [Tycko, 1992]. Figure 7.13 shows Arhenius plots for 13C, 87Rb, and 133Cs in Rb 2CsC6o (DR168A) at 8.8 Tesla with weak- coupling BCS fits. Ideally one would require more decades of data points, however, but we are limited by long T\ values and spin diffusion from the rapidly relaxing nuclei in the vortex cores to the nuclei in the bulk of the superconductor. These plots are still suggestive that the Arhenius behavior with A(0) = 1.76 ksTc is independent of nuclei and applied field. VHC3. BCS Fits to R sIRn Data: The BCS theory predicts a divergent HS peak due to the singular nature of the superconducting density of states. So, if A 3C6O is a conventional superconductor in the weak-coupling limit of the BCS theory, why isn't the HS peak larger than we observe (assuming that the 3 T and 1.5 T results are indeed close the "zero field" result)? In the original experiments by Hebei and Slichter [Hebei, 1959] and Masuda and Redfield [Masuda, 1962] on the weak-coupling superconductor aluminum (R sI R n )m a x was found to be approximately 2. The peak broadening in these materials was attributed to finite quasipaiticle lifetimes effects produced by mechanisms such as inelastic scattering or to gap anisotropy [MacLaughlin, 1976]. Their ( R s /R n ) m a x = 2 is still, however, significantly larger than the modest (R s IR n )m a x “ 1.2 we obtain for A 3C60 (we will show plots comparing these data sets in Chapter vm). In order to investigate what effect finite quasiparticle lifetimes have on the size of the HS peak (using an analysis similar to Kiefl et al [Kiefl, 1992]), we fit the 3 T data to the following: (VH.10) where NS(E) is the superconducting density of states and MS(E) is the "anomalous" density of states (due to coherence effects): 117 (E-iT) A & MS(E)=-Re (Vn.ll) _(E-rT)2-A2_ In the above expressions both NS(E) and MS(E) have been broadened by a phenomenological parameter T(T) = To(T/Tc)n. Figure 7.14 shows fits to the 3 T data for values ToIkgTc of .2 (n = 0,2) and .25 (n = 1,2,3). An alternative approach to understanding Rs/Rn data is to consider explicitly the effects of strong coupling. Aids et al. [Akis, 1991] calculate the suppression of the HS peak using the phonon coupling spectra a2F(ai) of lead characterized by a phonon frequency io/n--the logarithmic average of the phonon frequency. Figure 7.15 shows our 3 Tesla data with fits using their model for coi„ approximately equal to 100 and 130 cm-1. With this analysis the effects of strong coupling to the low frequency intermolecular phonon modes cannot be specifically ruled out: our 3 Tesla data lie somewhere in between the curves, suggesting that coin is in the vicinity of 100 cm-1-th e lower limit of intramolecular phonon frequency spectrum. This strong coupling analysis of the HS peak data seems to be contradictory to our other results: the low temperature T\ data follow an Arhenius law with A(0 )lksTc » 1.76 to 2, which is inconsistent with coin = 100 cm-1 (A(0)lkBTc of approximately 2.5), and the Knight shift measurements provide a seemingly conservative lower limit on toin of 200 cm’1. It therefore seems possible that the "true" coin required to fit the HS peak data is larger than 100 cm-1. There are many mechanisms that can reduce the HS peak [MacLaughlin, 1976]. A spread in Tc values within the sample could broaden and wash out the peak. The Knight shift results, however, suggest a spread in Tc of at most only 2 or 3 K making this mechanism an unlikely candidate. Anisotropy in the energy gap broadens the singularity in 118 R IR o f13 C in Rb,,CsC,ft at 3 Tesla with s n 2 60 Broadened Density of States Fits 25 n=3 .25 n=2 .2 n=2 .25 n=1 .2 n=0 • Rs/Rn 3T 0.2 0.4 0.6 0.8 Figure 7.14. 13C Rs/Rn vs. T of Rb 2CsC<$o at 3 Tesla with BCS weak-coupling fits using a broadened density of states (see text). 119 13 C Rg/Rn vs. Temperature in Rb2CsC6Q at 3 T with Strong Coupling Fits 1.5 I i i i I i i i | i i i | i i i | r 130 cm 1.25 100 cm 1 Rs/Rn 3T c 0.75 oc 0.25 0 -0.25 _i i i I i i i L -I I I I I L. 0 0.2 0.4 0.6 0.8 1 1.2 T/T c Figure 7.15. Rs/Rn vs. TITC for Rb2CsC60 at 3 Tesla together with Eliashberg theory calculations of Akis et al. for two values of effective phonon frequency u)[n, 100 and 133 cm-1 (the characteristic frequency of phonons mediating pairing). The data lie in between the two theoretical curves, suggesting that o)[n is not far from 100 cm-1, which would indicate strong coupling. In the text, however, arguments are given to suggest that the inferred frequency should be considered as only a lower limit for the phonon frequency. 120 the BCS density of states and thus reduces the HS peak Clem, 1966]. This mechanism also seems unlikely as these materials are thought to be in the dirty limit [Dresselhaus, 1994] where elastic scattering of electrons would “average away” any effects of anisotropy. Another possibility is that a more detailed strong coupling approach is required in these materials. Mazin etal. [Mazin, 1993] used Eliashberg theory which incorporated two peaks in the phonon spectral density « 2F(o?)-one high frequency peak at approximately 1 0 0 0 cm -1 representing // 2/ramolecular phonons, and a very low frequency peak characterizing the reorientational or librational phonon modes. These researchers were able to understand both the early high field NMR data of Tycko et a i [Tycko, 1992] and the tunneling data of Zhang et al. [Zhang, 1991], which was indicative of strong coupling, with their model. J. P. Carbotte [private communication] has suggested to us that such an approach might permit one to have both a strongly suppressed Hebel-Slichter peak and a low temperature gap which is close to the weak coupling value. VHC4. Field-Induced Mechanisms of Peak Suppression: For the above analysis we were assuming that the 3 Tesla data was representative of the “intrinsic” or zero-field result. It is possible that a lower field NMR measurement is necessary to find the true height of the HS peak. For the remainder of this chapter we will discuss some possible mechanisms of HS peak suppression associated with the application of magnetic fields, both to understand the effect itself and to consider the possibility that fields lower than 1.5 Tesla may be necessary for an accurate measurement. Field-induced suppression of the HS peak in type II superconductors (in the dirty limit) has been previously observed. One notable result is the 51V relaxation data in V 3Sn reported by Masuda and Okubo [Masuda, 1969]. These researchers saw significant suppression of the HS peaks in situations where the applied magnetic field Ha was close to the upper critical field Hc2. The results of Masuda and Okubo could be understood, for a dirty type n superconductor with Hc2 - Ha « Hci, within the theory of “gapless” superconductivity by deGennes [deGennes, 1964; deGennes, 1989]. DeGennes argued that magnetic fields add a perturbation to the electron Hamiltonian that is not time-reversal invariant. This lifts the degeneracy between the two one-electron states in a pair, producing a finite lifetime ro f the Cooper pair. The gap parameter becomes broadened by this pair- breaking mechanism and even vanishes at the vortex centers (the average value of the gap is non-zero). In the dirty limit, the NMR T\ is also a local property and will take on different values depending upon position relative to the vortex cores. Cyrot [Cyrot, 1966] calculated what effect gapless superconductivity has on the slope of the NMR relaxation rate at Tc: where cr is the electrical conductivity, Pa is a coefficient of order 1, k is the Landau Ginzberg parameter, and t = TC(H)/TC, where TC(H) is the suppressed value of Tc in the applied field H. g(t ) is computed numerically by Cyrot, and for t within approximately 0.8 to 0.95, the region of interest, g is of order one. Equation (VII. 12) has validity only when Ha is close to HC2 . If we explicitly use the theory of Cyrot to explain what we observe in the alkali fullerides, estimates of the quantity d^R3 fR„ )/dT are typically at least an order of magnitude larger than observed. In fact, to completely eliminate the peak within the context of Cyrot's theory we would need an applied field that suppresses Tc by some 40% or more. Thus, it seems that we cannot completely understand the NMR relaxation data with the theory of Cyrot. There are, however, considerable uncertainties in the formula input parameters and in the experimentally measured slope and Cyrot's expression assumes that 122 no other broadening mechanisms are present. The Cyrot model assumes that the intrinsic (zero field limit) Hebel-Slichter peak is quite large. For the case of the alkali fiillerides, however, the intrinsic peak may not be very large (the 1.5 T data suggests (Rs/Rn)max * 1.2). So if we include other broadening mechanisms into the theory it seems plausible that we can understand the field suppression in this context. There is a significant amount of evidence that suggests that we are observing the above effect in the alkali fiilleride relaxation data. The NMR data a t» 9 Tesla by Tycko et al. on 13C in Rb3Cgo and K 3C60 show no HS peak. The upper critical fields at 0 K in these materials are approximately 26 and 55 Tesla for K 3 C 6O and Rb 3 C 6 0 respectively-both of which are laiger than Ha. However, for temperatures close to Tc, Ha » HC2 (T), which is consistent with the effect observed. Tycko et a l also report that relaxation rate of 13C in K 3C60 at approximately 3 T also doesn't show a coherence peak. This can be understood because Hc2 for K 3C60 is around half that of Rt> 2CsC6o (which does show a peak at 3 T). Figure 7.16 shows Rs!Rn for 13C in Rt> 2CsC6o and K 3C60 at 3 Tesla. In order to fit our data we use a cruder model to explain the HS peak suppression (Stenger and Pennington, unpublished). The model was initially by Goldberg and Weger [Goldberg, 1968] for conventional type n superconductors to describe spin diffusion effects on relaxation rates for T « Tc (we also observe spin diffusion in the alkali fullerides at low temperatures but do not address it in this thesis). They estimate that the experimentally derived relaxation rate should be close to a weighted average of the normal rate R„ (in the vortex cores) and the intrinsic rate (Ry)int (in the bulk of the superconductor): & ~ fn + (vn.13) R, exp 123 b CR«s n in Rb,CsC„ 2 60 and K,C 3 60 at 3 Tesla 1.2 • Rb CsC 1 0.8 0.6 0.4 T (Rb CsC ) = 30K c 2 60 0.2 0 0.4 0.60.2 0.8 1 1.2 1.4 T/T Figure 7.16. Normalized relaxation rates as a function of temperature for 13C in Rb2CsC60 and K 3C60 at 3 Tesla. The data for Rb2CsC60 show a clear Hebei Slichter peak. The K 3C60 data show no signs of the peak. This is a result of K 3C6O having a lower second critical field than Rt> 2CsC6o (see text). 124 We take an area of n g 2 at the vortex core to consist of normal metal which to relaxes at the normal metal (Korringa) rate. The fraction f n of normal sample is then given by H/(2ttHC2(T)). f n approaches one at temperatures near TC{H). The remainder of the material of fractional area 1 -fn is assumed to relax at the intrinsic superconducting state rate . Using an estimate of 50 Tesla for HC2 (T=Q) and taking a BCS temperature dependence of HC2 (T), the intrinsic relaxation rate can be extracted from the 3 Tesla data. The 8.8 T result can now be calculated from the extracted (i?s)int and the same values of HC2 (T). Figure 7.17 shows this fit. 125 13 C Rs/Rn at vs. Temperature in Rb2CsC6Q at 8.8 T with Fit from Low Field Rs /R n Data 8 . 8 T z 0.9 vj 0.8 0.7 0.6 20 22 24 26 28 30 32 34 Temperature (K) Figure 7.17. 13C Rs/Rn for Rb 2CsC60 in a field of 8.8 Tesla. The points with error bars are the experimental data. The solid curve is a prediction obtained assuming the total relaxation rate is equal to a weighted average of the intrinsic, zero-field relaxation rate and the normal state rate. Chapter VDI Discussion of Results VIIIA. Chapter Overview This chapter provides a summary of this work. In Section VIIIB we summarize the superconducting state NMR results for A^Ceo- We will also compare our NMR results for A 3 C 6O with what has been obtained previously for other superconducting materials-specifically the weak coupling superconductor aluminum and the high-Tc superconductor YBa 2Cu3 0 7-§. In Section VIHC we will talk about what the normal state NMR data can tell us about strong correlation effects in A 3C60. VIIIB. Superconducting State VIIIB1. Summary of Superconducting State Data: The primary focus of this dissertation was to see what we could learn about the superconducting state of A 3C60 with the use of NMR methods. Assuming that the BCS framework is applicable in A 3C60, we saw that via the hyperfine interaction between the nuclear and electron spins NMR provides a very sensitive probe of the superconducting state order parameter A. A is directly related to strength of pairing interaction (which in turn is related to the frequency of the phonons which mediate the pairing) and the symmetry of the pairing wavefunction. 126 127 The NMR Knight shift data in the superconducting state yield a simple result. The data are consistent with the weak coupling limit of BCS theory, A(0) = 1.76 kgTc, where the intramolecular phonon modes are the dominant pairing agent. The weak coupling fits to the Knight shift data indicate that the high frequency phonons above 1000 cm'1 are probably the dominant modes and certainly the modes below 200 cm-1 are not involved in pairing. We do note that a Stoner enhanced susceptibility which was required to understand the normal state temperature dependence of UT\ T (discussed in Ch. VH) should also have been a requirement in our fits of the Knight shift data (as suggested by A. Leggett in a private communication). Figure 8.1 shows a normalized weak coupling Yosida function and a normalized weak coupling “enhanced” Yosida fiinction: x s(t)= ' x " (s , L - (vm.i) Here Xo(T ) is the unenhanced susceptibility and x S(t ) is the enhanced susceptibility with the enhancement parameter IXoiT) taken to be one half in the normal state for a factor of two enhancement. The inclusion of Stoner enhancement makes the fall-off in superconducting state steeper. This makes a weak coupling result look more like that for strong coupling (in other words, Stoner enhancement would not mask a strong coupling result as a weak coupling one). In light of the normal state l/TiTdata it is surprising that our results agree with the BCS Yosida weak coupling limit so well. Spin-lattice relaxation results are more ambiguous with regards to the strength of coupling and phonon modes in A 3C60. The low temperature superconducting state relaxation rate follows an Arhenius law and suggests a low temperature energy gap consistent with the weak coupling limit. This data is quite consistent with our Knight shift 128 Weak Coupling Yosida Functions with and without Stoner Enhancement 1.2 weak-coupl. 1 weak-coupl. + enhancement 0.8 +■» Q. Q) U to 0.6 3 CO 0 .4 Q. CO 0.2 0 0 0.2 0 .4 0.6 0.8 1 1.2 T/T Figure 8.1. Solid curve: Weak coupling BCS prediction for the temperature dependence of the electron spin susceptibility (Yosida function) in the superconducting state. Dashed curve: Weak coupling BCS prediction for superconducting state electron spin susceptibility plus Stoner enhancement. We saw from Chapter VH that Stoner enhancement to the density of states (i.e. spin susceptibility) was necessary to understand the temperature dependence of HT\T in the normal state. It is, therefore, appropriate to include these effects in the susceptibility fits. We see from this figure that Stoner enhancement could mask the effects of weak coupling, making weak coupling ^ ( T ) data look as if it were indicative of strong coupling. This is, however, not a concern for the A 3C60 X?(T) data which are clearly indicative of weak coupling. 129 results. The data near Tc do show a Hebel-Slichter peak which, although strongly suppressed at high fields, is clearly present at low magnetic fields. The existence of an HS peak alone is very suggestive of the appropriateness of a BCS picture. The size of the peak, however, is not consistent that of a weaker coupling superconductor such as aluminum but rather with a degree of strong coupling similar to that of Hg, Ga, or Bi. At best we can assign a “robust” lower limit (there are other unidentified sources of peak suppression) of » 130 cm-1 on the pairing phonon frequency. Perhaps a more complicated phonon spectrum is required to understand the T\ data: Mazin et al. [Mazin, 1993] suggest that two peaks in the phonon spectral density oc2F( gq) are required. One high frequency peak at approximately 1000 cm-1 representing m/ramolecular phonons and one low frequency peak characterizing the reorientational or librational phonon modes. J. P. Carbotte [private communication] has suggested to the us that such an approach might permit one to have both a strongly suppressed Hebel-Slichter peak and a low temperature gap which is close to the weak coupling limit value. We believe that a calculation demonstrating this possibility would be an important confirmation that the BCS framework can provide a consistent description of the full range of experimental data. V1I1B2. Comparison with other Superconductors: We will now present plots that compare our NMR results for A 3C6O with the conventional weak coupling superconductor aluminum and the high-Tc superconductor YBa 2Cu3 0 7 . This section does not attempt to explain the results of other groups on these materials in any significant detail but rather to serve as a “showcase” to the reader of all these data sets together in one place. Figure 8.2 shows the normalized superconducting state spin susceptibility Xs(T) as measured by the Knight shift for the conventional superconductor Al [Fine, 1969], for Rb2CsC6o, and for the high Tc (Tc = 93 K) cuprate superconductor YBa 2Cu3C>7 130 Knight Shift vs. Temperature in the Superconducting Stai of Aluminum, YBa„Cn.O_, and Rb-CsC,. 2 3 7 2 6U W eak Coupling Limit (0 . =100 cm'1 Rb,CsC ^ 0.8 2 60 YBa Cu O 2 0.6 Aluminum P 0.4 - 0.2 0.2 0.4 0.6 0.8 T/T c Figure 8.2. Normalized superconducting state spin susceptibility ^ ( T ) as measured by the Knight shift for the conventional superconductor Al, for Rb2CsC60, and for the high-rc superconductor YBa 2Cu3 C>7. Also shown is a weak coupling limit Yosida curve and an Eliashberg strong coupling fit for a single phonon frequency a)[n = 100 cm '1. Both the Al and the Rb 2CsC6o )^(T) data are well described by the weak coupling BCS theory. The YBa 2 Cu3C>7 data, however, are unlike the Al and Rb 2 CsC6o results and are more indicative of strong coupling behavior (we assume, for the sake of comparison, that the BCS phonon-mediated interaction is appropriate for high-rc). 131 [Barrett, 1990]. Also shown is a weak coupling limit Yosida curve and an Eliashberg strong coupling fit for a single phonon frequency coi„ = 1 0 0 cm -1. The Ceo superconductor behaves quite conventionally like Al, while the YBa2Cu 3 0 7 shows a much steeper fall-off in X?(T) as T drops below Tc. If we were to assume that YBa 2Cu3C>7 were an 5-wave, phonon-mediated pairing superconductor, we would find that the strong coupling curve for co/„ = 100 cm-1 fits the data reasonably well. However, closer inspection of the high-Tc Knight shift data reveals that not only is the initial fall-off in y?(T) (as T drops below Tc) steeper then the BCS strong coupling prediction but also, for T < temperatures is suggestive of a d- wave pairing state: the nodes in a d- wave gap would allow for low temperature excitations that have a non-zero spin thereby contributing to X?{T) [Thelin, 1993]. No evidence of this kind is seen in the A 3C60 data. VmC2. Relaxation Rates: Figure 8.3 shows the ratios of superconducting to normal state relaxation rates R J R n for the same materials: Al (Masuda, 1962], Rb 2CsC6o, YBa2Cu306.9i (obtained with NQR methods) |Tmai, 1988]. Fits using the theory of Hebei and Slichter (BCS weak coupling fit with broadened density of states) [Hebei, 1959] and the strong coupling Eliashberg theory for c<;/„ = 100 cm-1 and 130 cm-1 are also shown. Although the HS peak in Rb 2CsC6o is less pronounced than that of Al, its suppression is not at all comparable to that of YBa 2 Cu3 (>6.9 i. For example, at T!TC = 0.95, R s / R n is approximately 2.0 and 1.1 for Al and Rb 2 CsC6o respectively, however for YBa2Cu306.9i it is 0.6. Also, for T « Tc,R s/R n YBa2 Cu3(>6.9 i vanishes slower than in Al or A 3C60. Like in the x?(T) data, this is suggestive of low temperature quasiparticle excitations across the nodes of a d- wave gap. The low temperature T\ behavior described above can be seen more lucidly with an Arhenius plot of the same classes materials. Arhenius plots for all three materials are 132 R$/Rn vs. Temperature in the Superconducting States of Aluminum, YBa.Cu.O, , and Rb,CsC.A L j 0«"1 Z oU 2.5 Q R/Rn° Cu(NQR)VBOO □ R/R Aluminum 1.5 Hebel-Slichter Theory c w. =100 c m '1 DC □ □ W — to. = 133 cm DC 0.5 -0.5 0 0.2 0.4 0.6 0.8 1 1.2 T/T c Figure 8.3. Ratios of superconducting to normal state relaxation rates RsfRn for Al, Rb 2CsC6o> YBa 2Cu3C>6 .9 i (NQR) as a function of T!TC. The fits are from the theory of Hebei and Slichter (BCS weak coupling fit with broadened density of states) and from the strong coupling Eliashbeig theory for a = 100 cm’1 and 130 cm-1. We can see that although the Hebel-Slichter peak in the alkali fulleride data is significantly smaller than the Al data, it is definitely unlike the precipitous fall-off in Rs!Rn seen in the high-Tc data. 133 Tls/Tln vs. Temperature in the Superconducting States of Aluminum, YBa2Cu3Ofi91 , and Rb2CsC6() 1000 ° T Aluminum 100 o T YBCO 10 10000 1 100 0.1 A(0)=1.5kT 0.01 0.01 0.5 1 1.52 2.5 3 3.50 4 1/T C Figure 8.4. Arhenius plots for Al, Rb 2 CsC60, YBa2Cu3(>6 .9 i (NQR). Also given in the figure is a straight line representing Arhenius law behavior with A(0) = lJ6 k sT c and 2.0JcbTc. We see from this figure that the TVs for the nuclei in both Al and Rb2CsC60 show Arhenius behavior characterized by a nodeless gap (7) exponentially goes to infinity as temperature goes to zero). The high-7^ data, however, shows T\ vanishing as f 1 or P (the inset shows more decades of data), suggestive of low temperature excitations across the nodes of a d-wave gap 134 shown in Figure 8.4. Also given in the figure is a straight line representing Arhenius law behavior with A(0) = 1.76^7^. For temperatures well below Tc, both Al and Rb 2CsC6o follow the Arhenius law rather well, even taking on the appropriate BCS weak coupling slope. The YBa2Cu306.9i data, however, does not follow the Arhenius dependence over any extended temperature range. VIIIC. Normal State From the above discussion we see that the superconducting state NMR data seems to be reasonably understandable within the context of the BCS theory, although some work is required to tie together all the details. We now turn to normal state spin-lattice relaxation and Knight shift data and ask what can be learned from this data about the normal state electronic structure and superconductivity in A3C60- We focus on two issues: 1) the question of strong electron-electron correlations and 2) the relationship between the electronic densities of states and Tc’s using the McMillan equation. VmCl. Strong Correlations in A 3 C 6 0 : There is some evidence in the literature which suggests that strong correlation effects may be present in doped C 60- Notably, Lof et ai [Lof, 1992] measure a Coulomb interaction U between electrons of 1.6 eV using Auger spectroscopy. They argue that because this value is larger than the LDA calculated bandwidth of .5 eV, strong correlations must be considered in band theory calculations. Signs of strong correlations manifest themselves as “anomalies” in NMR data. That is, the results one expects for a normal metal (i.e. Korringa behavior) are either eliminated or somehow obscured. In the NMR data of the high-Tc superconductors [Slichter, 1993], the signatures of strong correlation effects are plentiful, so it is reasonable to ask if the NMR data for A 3C60 give similar evidence. We will show that they do not. 135 The first strike against strong correlation effects came when we saw that the behavior of l/TiTfor all nuclei situated at differing positions within the unit cell of A 3C60 was the same. This is shown in Figure 7.6 where the temperature dependencies of the 13C, 87Rb, and 133Cs spin-lattice relaxation rates in Rb2CsC60 appear to be identical . This is in sharp contrast to the l/TiTdata in the cuprates (Takigawa, 1992], The results of Takigawa et al for the 170 and 63Cu UT\Tys. temperature in YBa2Cu306.63 are shown in Figure 8.5. The different lattice positions of the planar 170 nuclei and the 63Cu lead to different hyperfine couplings to antiferromagnetic fluctuations of the copper moments which, in turn, lead to very different relaxation behavior for the two nuclei. This kind of behavior is clearly not seen in A 3C60- Although the identical temperature dependencies of \IT\T for the nuclei in A 3C60 does not actually rule out the effects of strong correlation, it may impose some kind of upper limit on their importance. The best indication of strong correlation effects is through direct comparison of the Knight shift and l/T \T using the Korringa relation (Eqn. (ELI 1». This is done for 87Rb and 133Cs in Rb 2 CsC6o in Section VIIB4. We found that the by plotting (\IT\T)m vs. the isotropic shift we obtained straight lines with slopes equal to the Korringa constant. Deviation from this would imply that one needs to consider electron-electron interactions [Slichter, 1989], For 13C, such straight forward comparison between l/TiT and K is not possible due to the dominance of the dipolar hyperfine interaction in the relaxation of 13C [Antropov, 1993]. For 13C one needs to compare 1/T\T with the anisotropic shift tensor •* K (anisotropy in the Knight shift broadens but does not shift the center of mass of the NMR line). Pennington et a l (Pennington, 1995] have found a way to extract the anisotropy in T\ to test the Korringa relation for 13C. The anisotropic shift tensor is given in terms of the angle 6 made between the direction of the applied field, z, and z ’, the direction of the caibon 2p orbital. In terms of the angle 0, the anisotropic Knight shift is given by: 136 170 and 63Cu Spin-Lattice Relaxation Rates vs. Temperature in YBa„Cu O 2 3 0*03 8 0.3 7 0.25 6 0.2 5 o © o CO 4 0.15 3 CO 0.1 2 0.05 1 YBa Cu O 3 6.63 0 0 50 100 150 200 250 300 350 Temperature (K) Figure 8.5. Spin-lattice relaxation rates for 170 and 63Cu in the high-Tc superconductor YBa2Cu30 6.63 for an applied field along the c-axis [Takigawa, 1992]. The contrasting temperature dependencies between the two nuclei and the deviation from \IT\T-const, for both is indicative of the importance of strong correlation effects in the high-Tc materials. This data is in sharp contrast to the similar temperature dependencies of the nuclei in A 3C60 (Fig. 7.6). 137 K {d )=f - ^ s c o A - i ) . (vm.2) The analogous expression for spin-lattice relaxation is (Vffl.3) From these expressions one sees that spins with higher frequencies (larger values of K(8)) should relax 2.5 times as fast as those with lower frequencies. To test this prediction, Pennington et a l measure the 13C T\ in Rb 2 CsC6oas a function of frequency for a fixed applied field. Figure 8.6 shows these results for T= 80 K and an 8.8 Tesla applied field. The relaxation rate at of the 13C spins the low frequency is 1.17 s'1, while the spins high frequencies relax at the rate 0.75 s'1, giving a ratio 1.56 (and a powder average rate of 1.03 s'1). This number is clearly smaller than the ratio of 2.5 predicted by Equations (Vin.2) and (Vin.3). Pennington et al propose that in addition to the dipolar relaxation mechanism of Antropov et aL, there is an additional mechanism associated with the isotropic hyperfine couplings (A, B, and B 1) discussed in section VI.C.l. The measured anisotropy of 1.56 implies an isotropic rate contribution of 0.49 s_1, and the powder averaged anisotropic dipolar rate is then 0.54 s_1. Using this rate and the Korringa relation for dipolar shifts and relaxation averaged over the solid angle Cl for a 2p orbital [Pennington, 1995], (Vm .4) 138 13C Lineshape and 1/T^ as a Function of Frequency in Rb2CsC60 at 80 K and 8.8 T w 0 2 0.8 - 0.2 0.7 -200 -100 0 100 200 300 Shift from C (ppm) 60 Figure 8.6. 13C NMR lineshape of a powder sample of Rb 2CsC6o taken at temperature 80 K in an 8.8 Tesla field [Pennington et a l, 1995]. Also shown is the spin lattice relaxation rate 1 IT\ measured separately at different frequencies within the lineshape. Spin lattice relaxation rates at the low frequency range are found to be approximately 1.56 times as fast as rates at the high range. This ratio can be compared with the predicted anisotropy of 2.5 to 1 for the electron-nuclear dipolar interaction. Pennington et al conclude that there is an additional isotropic relaxation mechanism. They infer the magnitude of both the isotropic and anisotropic parts, and compare the magnitude of the anisotropic part with the measured linewidths using a Korringa relation. These results are shown in Figure 8.7. 139 Pennington et al calculate an anisotropic shift tensor of (-97, -97,194) ppm. This tensor is then added to the anisotropic part of the C^o shift tensor (77,43, -118) ppm, along with the measured isotropic Knight shift. The resulting powder pattern lineshape is compared with the experimental lineshape in Figure 8.7. The overall width of the theoretical powder pattern is in reasonable agreement with experiment. Note, however, that the experimental data do not contain the sharp features of the calculated pattern. It is found that by taking the calculated pattern and convoluting it with a square function of width ±50 ppm, a very reasonable match to the experimental data is obtained. This broadened powder pattern is also given in Figure 8.7. The analysis of Pennington et aL demonstrates two important facts. First, the Korringa mechanism predicts a 13C powder pattern width which is in very reasonable agreement with experiment. Therefore, it is not necessary to invoke effects of strong correlation. Second, the dipolar hyperfine mechanism of spin-lattice relaxation proposed by Antropov et al. does contribute a substantial fraction of the total 13C spin-lattice relaxation, but not all of it. VIIIC2. Inferences about Superconductivity: The identity of the phonons which predominate pairing in fullerene superconductivity can be recognized from the normal state NMR data by using the McMillan equation (Section IIC3): (vm.5) 12k,B V ( l+ 0.62^) Xep is the electron-phonon coupling parameter and is proportional to the electronic density of states at the Fermi energy JV(0). If one can determine from N(0) for alkali fulleiide 140 13 C Lineshape in Rb2CsC60 with Theoretical Fits Experimental Lineshape Data Powder Pattern: (31,-4,125) ppm Shift Tensor Powder Pattern (31,-4,125)ppm, with (+-50ppm square broadening) 1.4 * 0.8 Q. | 0.6 (0 c 0.4 0.2 - 0.2 -200 -100 0 100 200 300 Shift from C6Q (ppm) Figure 8.7. Pennington et al [1995] propose that in addition to the dipolar relaxation mechanism there is an additional, incoherent isotropic mechanism, associated with the isotropic hyperfine couplings. Using the relaxation rate contribution of the dipolar mechanism only appropriate Korringa relation they calculate an anisotropic Knight shift tensor of (-97, -97,194) ppm. This tensor is then added to the anisotropic part of the Qio shift tensor (77,43, -118) ppm, along with the measured isotropic shift, to obtain a shift tensor (31, -4,125) ppm. The resulting powder pattern lineshape is compared with the experimental lineshape in this figure. The overall width of the theoretical powder pattern is in reasonable agreement with experiment. Note, however, that the experimental data do not contain the sharp features of the calculated pattern. It is found that by taking the calculated pattern and convoluting it with a square fiinction of width ±50 ppm, a very reasonable match to the experimental data is obtained; this broadened theoretical powder pattern is also shown (dashed line). 141 compounds with different Tc's, then one can extract coph if an appropriate value for j f is known. This can be done from the normal state NMR data because 1/TjTis proportional to 7V2(0), and hence (l/2 \ t f 2 = cAph, where c is a constant. Tycko et al [1992] used these relationships to extract a value for the phonon frequency (0ph. They found the ratio R of Rb 3C60 (Tc = 29 K) spin-lattice relaxation to that of K 3C60 (Tf = 18 K) to be in the range 1.28 < R < 1.40. Using reasonable values of 11*, they found a phonon frequency of approximately 200 cm-1, a frequency substantially lower than the intramolecular “tangential” modes (1400 cn r1). Tycko et a l , however, emphasize that their analysis neglects the possibility of Stoner enhancement. This could lead to an inaccurate result because KeP in the McMillan equation involves the bare density of states, N^O), while l/T \T is proportional to the enhanced density of states N ev(0). We saw in Chapter VH that Stoner enhancement was necessary to understand the normal state temperature dependence of IIT\T. We found that the temperature dependence of the relaxation data could be understood with the same values used by Ramirez et a l [1992]. Ramirez et al. needed these values to simultaneously understand their spin susceptibility, isotope effect, and specific heat jump data on both K 3C60 and RI 33C60 near the weak coupling limit, with »1400 cm-1. Thus, our result in Section V13B3 demonstrates that the lattice constant dependence of the NMR l/T\ T in the normal state is consistent with weak coupling to the intramolecular phonon modes. Appendix A Nuclear Quadrupole Effects in A 3C6 O In this appendix we examine the interaction between the 87Rb (/= 3/2) nuclear quadrupole moment, Q (equal to ,14e x 10'24 cm-2), and the electric field gradients, Vcca ( a = Jt, y, or z), in the A 3C60 crystal lattice. We find that the nuclear quadrupole effects in A 3C60 are quite paradoxical. Nutation frequency experiments and spectral information reveal that the first-order (±3/2, ±1/2) satellites are outside the excitation bandwidth. However, simple calculations and also a direct measurement of the second-order shift imply that the first-order satellites should be observed. The quadrupole interaction was introduced in Chapter HI as H Q. If the nucleus resides in a position of cubic symmetry (Vxx- Vyy = Vzz), H q is zero. In A 3 C 60 the alkali ions reside in either octahedrally or tetrahedrally coordinated sites in the lattice, both of which possess cubic symmetry, so one would expect that there should be no nuclear quadrupole interactions present in A 3C60- We, however, find that this is not the case for 87Rb as we observe only the central (1/2, -1/2) Zeeman transition (the (±3/2, ±1/2) transitions are shifted away). We demonstrate this by comparing the nutation curve (signal vs. pulse width) for the nucleus in the solid state with that of the nucleus in a solution (where one expects to observe all the Zeeman transitions due to the rapid rotational averaging of any electric field gradients). Figure A.l shows the nutation curves for 87Rb in Rb2CsC6o and in an aqueous solution of RbCl. The nutation frequency of 87Rb in 142 143 Nutation Curves for 87 Rb in RbCl (aq.) and in R^CsC^ RbCl Rb CsC 60 O# .S> 0.5 -0.5 0 5 10 15 20 25 30 35 Pulse Width (p s) Figure A .l. 87Rb “nutation” curves: spin echo height vs. width of the 90° pulse (1/2 the length of the 180° pulse) for 87Rb in Rb2CsC60 and in aqueous RbCl. A nuclear quadrupole moment can interact with the electric field gradients in the crystal. To first order, this interaction can destroy the equal spacing of the Zeeman energy levels, leaving only the central (-1/2,1/2) transition unpertuibed. If the only the central Zeeman transition of a spin 3/2 system is excited in the solid, its nutation period should be 1/2 of the nutation period in the liquid. We see that the nutation period for 87Rb in Rb2CsC60 is approximately half that of RbCl (any deviation we attribute to rf field inhomogeneity), therefore, only the central Rb transition is observed in A jCeo. 144 Nutation Curves for 133 Cs in Cs2C03 (.5 M) and in Rb CsC 2 60 O Cs CO 2 3 • Rb CsC 4-> 0 5 ‘a5 • o x • o (0 • o c • o 0 5 c/5 -0.5 0 10 15 20 25 30 355 Pulse Width ( ms) Figure A .l. 133Cs “nutation” curves: spin echo height vs. width of the 90° pulse (1/2 the length of the 180° pulse) for 133Cs in Rb 2CsC6o and in aqueous CS 2CO3 . A nuclear quadrupole moment can interact with the electric field gradients in the crystal. To first order, this interaction can destroy the equal spacing of the Zeeman energy levels, leaving only the central (-1/2,1/2) transition unpertuibed. If the only the central transition of a spin 5/2 system is excited in the solid, its nutation period should be 1/3 that of the nutation period in the liquid. The nutation frequencies for 133Cs in Rb 2CsC6o and in CS 2CO3 are approximately identical (any deviation we attribute to rf field inhomogeneity), therefore, all the Cs transitions are observed in A 3C60. 145 Rb 2CsC60 is approximately twice that of 87Rb in RbCl, as one would expect for the excitation of only the central transition of a spin 3/2 nucleus. We show the nutation curves for 133Cs in Rt>2CsC60 and in a CS 2CO3 .5 M solution in Figure A.2. Here the nutation frequencies are seen to be identical, implying all Zeeman transitions are observeed for 1 3 3 Cs. We attribute this difference to the larger Q o f87Rb than 133C s (Q = -.004 x 10’24) and the possibility that the Rb ion, due to its smaller size, may be more “off-center” than the Cs. We now provide some theoretical framework to estimate the 87Rb quadrupole shifts. If the effects of HQ are small compared to the Zeeman energy, one can treat HQ as a perturbation to the 2(1 + 1) equally spaced Zeeman energy levels [Bloembergen, 1954], Following the formalism of Bloembergen, we write the NMR frequency v\ for the satellites of a spin 3/2, polyciystalline sample to first-order as u, =u0 ± A,(3cos20 -1); A, =e2qQ I Ah. (A.l) Here h is Plank's constant and eq= o^e/Scos2^ ~l)r/3, where single crystal with its symmetry axis parallel to the applied field Ai = V]. For a powder sample comprised of numerous crystallites, the shifts of the satellites will be distributed as 3cos 2 0 -l where 0is the angle between the applied magnetic field and the symmetry axis of the crystal. An estimate of A\ for an octahedral 87Rb ion in RbjQo (lattice constant * 14.4 A) can be obtained from a simple (possibly oversimplified) calculation assuming that the ion is affected only by the six nearest neighbor C60 molecules, each with a charge -3e. Treating 146 the 87Rb as a hard sphere of radius 1.66 A and the radius of an octahedral interstitial is 2.1 A, the ion could at most be .44 A off-center. This yields a value of 235 kHz (1900 ppm) for the A\ of 87Rb in RbjCeo- Contradictoiy to the nutation data, this value for the shift is well within our excitation bandwidth and should be observed. We now consider the measurement of A\ from the NMR lineshape data. The central (1/2, -1/2) transition will be shifted by a frequency Vi to second-order. Using the difference in second-order shifts between the 87Rb and 85Rb central transitions in Rb3C6o, 87 V2 - 85 V2, we can measure A\ for 87Rb. From Bloembergen we write the second-order quadrupole shifts for these ions as *V *X + 3-^-/(0) /(0)=(l-9cos2(0)Xl-cos2(0)). (A.2) 3 85A2 m 2 0 10 vn Integration of f(0) over the solid angle yields a value of -8/15. Define the difference between these shifts as A K (measured in ppm): i7v2-*7vQ 85l>,-85 vn AK = 87 85 * 10' 6 . (A.3) Ui V n AK is the quantity which we experimentally determine from the lineshape data. Using the equality 85Ai = (85j2/870 87Ai, A K can be approximated by letting 87 vo/85 Vo » 87y/85y where y is the gyromagnetic ratio. The expression for the measured shift difference becomes Using the spectra for the central transitions of 87Rb and 85Rb isotopes in Rb3C60 (DR50B), both measured relative to Rb+ in aqueous RbCl, we obtain A K « 8 ppm. This measured value for A K produces 87Aj =210 kHz. This is in good agreement with the simple theoretical calculation, however, it disagrees with what we see in the spectral and the nutation data. Appendix B The Normal Modes Problem Consider the case of a spin / system (/ > 1/2) at thermal equilibrium in a magnetic field. The populations of the 2Z+1 energy levels will be given by the Boltzmann function. Assume that only the populations of the mj = 1/2 and -1/2 levels are perturbed from equilibrium, leaving the other 2/-1 levels untouched. The system and is then allowed to relax back to equilibrium. Also, assume that the relaxation mechanism is magnetic and the correlation time of the magnetic fluctuations responsible for relaxation is larger than the time scale of the Larmor frequency. This is an example of a whole class of problems that can be solved within the context of a very general theory of relaxation. In this theory the spin system is decomposed into “normal modes” which relax with different rates, not unlike normal modes of oscillation in the “small oscillations” elementary mechanics problem. We will call this the “normal modes” relaxation problem. Solutions to the normal modes problem yield multi-exponential relaxation behavior. This above situation is what we have for the relaxation of the 87Rb (I =3/2) in A3C60. Due to first-order quadrupole effects, the (±3/2, ±1/2) transitions are shifted outside the excitation bandwidth and cannot be perturbed by the NMR pulse. However, the m \ - 3/2 and -3/2 are still thermally coupled to the /«/= 1/2 and -1/2 levels and will, therefore, affect the relaxation of the whole spin system. Thus, if we are to measure the relaxation time 7i, we need to solve the normal modes problem. 148 149 The differential equations for the time rate of change of the population, Nm, of the mth eneigy level of a spin 3/2 system are given by ^ = ^1/2,+3/2 ^+1/2 "W+3/2, +1/2^+3/2 dK ^+3/2,+1/2 ^+3/2 + ^-1/2,+1/2^-1/2 (^+1/2,-1/2 + W+l /2,+3 /2 )^+ l /2 (B.l) ILL dt ~ W - 3 /2 ,- 1 / 2 N - 3 /2 + W+l /2.-1 /2^+l /2 (^-1/2, +1/2 + ^-1 /2,-3/2 )^ -1 /2 dt = ^-1/2,-3/2^-1/2 "^-3/2,-1/2^-3 /2 where is the transition rate from the m to w 1 eneigy level. At thermal equilibrium, each of the above equations can be set equal to zero. The first equation in (B. 1) gives: W*.i±1 /2,t3/212 *.2)0 _ N'v±3/2 + L _ J . AE/k, „T ^±3/2,±1/2 ^±1/2 Here iV° is the thermal equilibrium population of the m energy level. Let the nm be the difference between the pertuibed thermal equilibrium populations, Nm - N °, and assume we are in the high temperature limit, A E «I cbT, such that Wmim< = Wm' m. Equations B.l now become: *fo+3/2 _ „ r ( \ ~ vv+l/2,+3/2V"+l/2 "+3/2/ = ^+l/2,+3/2(W+3/2 “ W+l/2 ) + ^ + l/2 ,-l/2 (W-l/2 — W+1 /2 ) dn ' 3) f a 2 = BC1/2i_3/2(«-3/2 — ^-1/2 ) "^ i^+1 /2,-l/2( ^ + 112 “ ^ - 1 /2 ) ^-3/2 dt ^ - 1 /2,-3 /2 W-3 /2 ) 150 For a magnetic mechanism of relaxation, Ix = (/+ + /_)/2 terms are required to induce transitions. Letting Wm m, = W(m]fx\nt), we rewrite the equations B.3 and also put them in a more convenient matrix form: / " - 3 / 2 -3 3 0 n. ■312 d_ n -ll2 3 -7 4 0 «_1/2 = W (BA) dt n l/2 0 4 - 7 3 n 1/2 or, equivalently, dn(t ) = Wn(t). (B.5) dt We now diagonalize W and find its eigenvalues, A, and eigenvectors, |A). They are (here we ignore the trivial solution corresponding to thermal equilibrium): ' l ' f " 1' -1 . 1 -1 1 3 |A = -2W ) = , |A = -6W ) = — (B.6) 2S 1 -1 -3 , 3 , I* J The corresponding eigenequation is W|A)= A|A>. (B.7) 151 The advantage of this approach is that we can now express any arbitrary state, n(t) , of the spin system as a sum of eigenstates, or “normal modes,” each of which relax with single exponentials, «(0 = 2>/(0)ev |4). (B.8) provided we know the initial conditions (or the coefficients Cj(0)) of the system. First, lets consider the case where we can flip the whole line with a 180° pulse. The spin system in this situation will relax with a single exponential and will be represented by one of the eigenvectors above in (B.6). To identify the eigenvector we explicitly consider the equilibrium population, N ° : e-Emkj 1 - E kBT -E..k f “ N-.------(B.9) £ (1 - E m,k„TY ni m' In the above equation, we again assumed the high temperature limit was applicable for the spin system. Identifying Em with the Zeeman eneigy -h Y„H0m we rewrite (B.9): N (l + am). (B.10) 27 + 1 For our spin 3/2 system we now obtain f \ - 3 a 12s -o N 1 - a 12 (B .ll) T \+ a l2 1 + 3a/2 152 If we flip the whole line by 180° at?= 0 we let a -* -a in(B .ll): ( \ + 3 a /2 ' N l+a/2 (B.l 2) MW)-7 1 - a l l 1 - 3 a /2 The normalized eigenvector is then ' 3a ' ' 3 > N_ a 1 1 (B.13) 4 -a ~ 2 & -1 y. We identify (B.13) as being -1 times one of the eigenvectors in (B.6) (i.e. the vector flipped 180°). From this, we define 1/2Was T\. Now, consider our original problem of the relaxation of a spin 3/2 system where only the m = 1/2 and -1/2 levels are perturbed from equilibrium. The state, n(0), at /= 0 after a 180° pulse can be written: 1 n(0) = (B.14) -1 Using (B.6) we can write (B.14) as a linear combination of eigenvectors: 153 'O ' '- 3 ' ' I ' 1 1 1 -1 3 -3 (B.15) J2 -1 2 jW 1 ~ 2 j\0 3 < 0 , c K At a timer we have ' 3 N '- 1 ' 1 3 1 -2Wi . 3 - 1 2 Wt n(t) = 2Vf0 -1 e 2 jw -3 ,-3 , <1> In an NMR experiment we measure the magnetization, M(t), at a time t after the system is disturbed from equilibrium and compare it with the equilibrium magnetization Mo. For the above problem, the quantity M(r)-Mo is therefore proportional to wi/2(0 ” w-i/2(0- So, if we are monitoring the (-1/2,1/2) transition, using (B.16), the initial condition M(0)-Mo = -2Mb, and MT\ = 2 W we obtain M(r) - M0 = -2(.l This is Eqn.s (VII.2) for the /= 3/2 case. 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