Inductive measurements of heavy Fermion superconductors

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Inductive measurements of heavy Fermion superconductors
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INDUCTIVE MEASUREMENTS OF HEAVY FERMION SUPERCONDUCTORS


By


PHILIPPE J.-C. SIGNORE












A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1994












ACKNOWLEDGMENTS


The author wishes to express his sincere gratitude and appreciation to his advisor,

Professor Mark W. Meisel, for his patient guidance and encouragement throughout the

entire course of this work. In addition, the author gratefully acknowledges Professors

G. Bosman, L. E. Seiberling, C. J. Stanton, G. R. Stewart, and N. W. Sullivan for serving

on his supervisory committee, and Professors S. E. Brown and F. Sharifi for their active

contributions to this work.

The faculty, postdoctoral fellows, administrative and technical support of the entire

low temperature group have contributed a great deal to this work, and their efforts are

duly appreciated. Special thanks are expressed to the graduate students of the low

temperature group for their help and for maintaining a camaraderie that contributed to a

pleasant and positive working atmosphere. The author feels proud to belong to such a

group and wishes his colleagues the best of luck for the future.

The author gratefully acknowledges his family for their constant support and their

inspiration throughout the years. The author would like to thank Caroline Cox for her

assistance in the editing and proofreading of the manuscript, as well as for her patience

and faithful encouragements.













TABLE OF CONTENTS

Page

A CKN OW LED G M EN TS ............................................................................................. ii

LIST O F TA B LES ................................................................................................... vi

LIST O F FIG URE S................................................................................................. vii

A B STR A C T ........................................................................................................... xiv

CHAPTERS

1 IN TROD U CTION .................................................... .......... ............................ 1

1.1 Heavy Fermion Systems ....................................................................... 2
1.1.1 Heavy Fermion Compounds................................................................. 2
1.1.2 Heavy Fermion Superconductors ................................................ 3
1.2 Unconventional Superconductivity .................................................. ........... ... 6
1.2.1 D definition ....................................... ................ ............................. 6
1.2.2 Experimental Evidences for Unconventional Superconductivity............. 8
1.2.3 Unconventional Superconductivity in UPt3 ...................................... 11

2 OVERVIEW OF EXPERIMENTAL WORK -- UPt3 AND UBel3..................... 14

2.1 Experim mental W ork -- UPt3......................................................................... 14
2.1.1 Crystal Structure ............................................... ........................ 15
2.1.2 Specific Heat Measurements.................................................................. 15
2.1.3 Ultrasonic Studies........................................... ........................... 21
2.1.4 Phase Diagrams ............................................................................. 22
2.1.5 Antiferromagnetism and Superconductivity...................................... 26
2.1.6 Structure Modifications and Superconductivity................................... 28
2.1.7 Nuclear Magnetic Resonance and Muon Spin Relaxation Knight
Shifts...................................................................... .......................... .. 29
2.1.8 Point Contact Spectroscopy................................................................ 30
2.1.9 Concluding Remarks........................................ .......................... 31
2.2 Experimental Work-- UBe3....................................................................... 32
2.2.1 Crystal Structure ................................................................................ 33
2.2.2 Specific Heat Measurements........................................ .......... ... 33







2.2.3 Penetration Depth, Mean Free Path, and Coherence Length .............. 36
2.2.4 Phase Diagrams............................................ ..................................... 38
2.2.5 Nuclear Magnetic Resonance and Muon Spin Relaxation Knight
S h ifts ......................................................... ........................................ 3 9
2.2.6 Anom aly Around 8 K ........................................................................ 41
2.2.7 Thorium Doped UBel3 Experiments................. ................................... 42
2.2.8 Concluding Remarks.................................................................... 45

3 APPARATUS AND EXPERIMENTAL TECHNIQUES.................................. 46

3.1 Low Tem peratures................................................................................... 46
3.1.1 Dilution Refrigerator ................................. ........................... 46
3.1.2 Therm om etry................................................................................ 49
3.1.3 Temperature Control......................................... .............................. 57
3.2 ac Susceptibility .......................................................... ...................... 60
3.2.1 Hardware Used for the Mutual Inductance Technique........................ 61
3.2.2 Principles of the Mutual Inductance Technique ................................... 64
3.2.3 Peak in X"(T) in the Vicinity ofTc and in Zero dc Magnetic Field....... 72
3.3 Tunnel D iode O scillators ........ .................................................................. .. 77
3.3.1 Hardware Used for the Tunnel Diode Oscillator Technique................. 77
3.3.2 Tunnel Diode Oscillators and Penetration Depth............................... 80

4 INDUCTIVE MEASUREMENTS -- UPt3................................................ ... 85

4.1 Penetration Depth in Superconductors................................. ............. 85
4.1.1 Conventional Superconductors.................... ........ ........... .. 85
4.1.2 Unconventional Superconductors.................................................... 92
4.1.3 Motivation for Studying the Penetration Depth in
Superconductors ................................................................................ 95
4.2 Previous Work on the Penetration Depth of UPt3 .................................... 95
4.2.1 dc Measurements.............................................................................. .95
4.2.2 Radio Frequency Measurements ...................... ........ ........... .. 96
4.2.3 Muon Spin Relaxation Measurements........................................ ......... 97
4.2.4 Motivation for Studying the Penetration Depth in UPt3..................... 97
4.3 Sam ple H stories ................................................................................... 98
4.3.1 Sam ple N o. 1.................................................................................... 98
4.3.2 Sam ple N o. 2................................................................................. 100
4.3.3 Sam ple N o. 3.................................................................................. 104
4.3.4 Sam ple N o. 4.................................................................................. 104
4.3.5 Sam ple N o. 5.................................................................................. 108
4.3.6 Sam ple N o. 6.................................................................................. 108
4.4.7 Sam ple N o. 7............................... ............................................... 108
4.4.8 Sample No. 8........................................................................ 108
4 .4 R results ..................................................................................................... 109
4.4.1 M utual Inductance Results.............................................................. 109







4.4.2 Resonant Technique Results........................................................... 132
4.5 Discussion................................................................................................. 145
4.5.1 Linear Temperature Dependence of X(T)......................................... 145
4.5.2 Quadratic Temperature Dependence of X(T)................................... 146
4.5.3 Double Feature Near Tc .............. ................................................... 147
4.5.4 High Frequency Effects..................... .......................................... 148
4.5.5 Upturn in x'(T) for Bdc > 1.2 T ....................................................... 154
4.6 Conclusions............................................................................................... 159

5 INDUCTIVE M EASUREM ENTS -- UBe13 ................................................ 160

5.1 Upper Critical Magnetic Fields in Superconductors ................................... 160
5.1.1 Type I, Type II, Bcl(T) and Bc2(T)...................................... 160
5.1.2 Bc2(T) for Type II Superconductors................................................ 161
5.2 Previous W ork on the Upper Critical Field of UBel3 ................................ 165
5.2.1 Bc2(T) of Single Crystal UBel3........................................................ 165
5.2.2 Anisotropy of Bc2(T) for T/Tc 1................................................ 167
5.2.3 Our M otivation for Studying Bc2(T) in UBel3 ................................. 169
5.3 Sample Histories ................. ....................................................................... 169
5.4 Results ...................................................................................................... 175
5.4.1 Normal State Susceptibility............................................................... 175
5.4.2 Phase Diagrams...................................................... ....................... 180
5.4.3 Temperature Dependence of the Penetration Depth........................... 204
5.5 Discussion................................................................................................. 222
5.5.1 Isotropic Bc2(T) in the limit T/Tc -+ 1............................................. 223
5.5.2 Anomaly in Sample No. 3 ................................................................ 228
5.5.3 Temperature Dependence of the Penetration Depth......................... 230
5.6 Conclusions............................................................................................... 230

6 CONCLUSION S ............................................................................................. 233

6.1 UPt3....................................................................................................... 233
6.1.1 Conclusions..................................................................................... 233
6.1.2 Future W ork..................................................................................... 234
6.2 UBe 3 ....................................................................................................... 236
6.2.1 Conclusions...................................................................................... 236
6.2.2 Future W ork..................................................................................... 237

APPENDIX ........................................................................................................... 238

REFERENCES...................................................................................................... 242

BIOGRAPHICAL SKETCH.................................................................................. 256













LIST OF TABLES


Tables page


1-1 Ground state of various heavy fermions systems..........................................


2-1 Summary of lattice parameters of UPt3..................... ..................... 15


2-2 Summary of lattice parameters of UBel3 .................................................33


3-1 Diagnostic thermometers................................................... 51


3-2 Coefficients for LS-Burns, 350 mK < T < 2 K.................... ........... ......... 55


3-3 Coefficients for LS-Burns, 2 K < T 4.2 K........................... ........... 55


4-1 Predicted X(T) for the axial state and the polar state...................................... 93


4-2 Characteristics of the eight UPt3 samples investigated in this work .................. 99


4-3 Summary of .(T) obtained from this work................................................ 144


5-1 Characteristics of the three UBel3 samples investigated in this work.............. 170












LIST OF FIGURES


Figure page


1-1 Definition of unconventional superconductivity ............................................. 7


2-1 Atomic configuration in the hexagonal unit cell of UPt3............................... 16


2-2 Specific heat of UPt3 in the vicinity of Tc ...................................... 20


2-3 B-T phase diagram of UPt3 for the magnetic field parallel to the c-axis............ 24

2-4 B-T phase diagram of UPt3 for the magnetic field perpendicular to the
c-axis ..................... ....................... ............................................... 25


2-5 P-T phase diagram of UPt3.............................................................................. 27


2-6 Crystal structure of UBel3....................................................................... 34


2-7 Upper critical field in UBe13 for a single crystal and a polycrystal .................. 40


2-8 Superconducting Tc-x phase diagram for UlxThxBel3 ..................................43


3-1 Determination of the circulation rate of the dilution refrigerator....................... 50


3-2 Calibration curve for thermometer No. 26 ................................... ............ 53


3-3 Calibration curve for thermometer LS-Burns.............................................. 54


3-4 Magnetoresistance of RuO2 thick chip thermometer .................................... 56

3-5 Resistance ofRuO2 thick chip thermometer mounted on the
mixing chamber and on the cold finger...................................... ............ ... 58







3-6 Circuit used for temperature control............................................................ 59


3-7 Circuit used for the mutual inductance technique............................................ 62

3-8 Geometry and dimensions of coils used for the mutual inductance
technique...................................................................................................... 63


3-9 C' and X" as a function ofrs/8 for a normal metal ........................................... 68


3-10 x'(T) and X"(T) for aluminum at 317 Hz..................................................... 71


3-11 X'(T) and X"(T) for zinc at 317 Hz.................................................................72

3-12 X'(T) and X"(T) calculated for a normal metal exhibiting a sharp drop in
resistivity............................... ................... 75


3-13 Circuit used for the tunnel diode oscillator technique..................................... 78


3-14 Typical I-V characteristic of tunnel diodes.................................... ............... 79


3-15 TDO results on aluminum at 10 MHz........................... ............................ 82


3-16 TDO results on zinc at 2 MHz............................................................ 84


4-1 Temperature dependence from BCS in the local limit....................................... 90


4-2 SEM pictures of UPt3 sample No. 1 ........................................................... 101


4-3 Resistivity as a function ofT2 for UPt3 sample No. 1................................... 102


4-4 Specific heat in the vicinity of Tc for UPt3 sample No. 1.............................. 103


4-5 SEM pictures of UPt3 sample No. 2 ............................................................ 105


4-6 SEM pictures of UPt3 sample No. 3 ........................................................... 106


4-7 SEM pictures of UPt3 sample No. 7 ........................................................... 107


viii







4-8 X'(T) for unannealed and annealed UPt3 sample No. 1 ................................... 111


4-9 x'(T) in the vicinity of Tc for UPt3 sample No. 1............................................ 112

4-10 x'(T) and X"(T) for UPt3 sample No. 1, as grown......................................... 113


4-11 X'(T) and X"(T) for UPt3 sample No. 1, annealed, etched, unpolished............. 114

4-12 X"(T) in the vicinity ofTc for UPt3 sample No. 1,
before and after polishing ....................................... ................................... 115


4-13 X'(T) and X"(T) for UPt3 sample No. 1, annealed, etched, polished................. 117


4-14 X'(T) in the vicinity ofTc for UPt3 sample No. 1 for 48 Hz < f< 32 kHz....... 118

4-15 X'(T=80 mK, 600 mK) as a function of frequency for
U Pt3 sam ple N o. 1 polished........................................................................... 119

4-16 B-T phase diagram for UPt3 sample No. 1, polished.................................... 121


4-17 x'(T) for UPt3 sample No. 1, polished and with B < 0.5 T.............................. 122


4-18 X'(T) for UPt3 sample No. 1, polished and with 1 T < B < 1.6 T.................. 123


4-19 x'(T) and X"(T) for UPt3 sample No. 2......................................................... 124


4-20 X'(T) and X"(T) for UPt3 sample No. 3...................................................... 125


4-21 X'(T) and X"(T) for UPt3 sample No. 4...................................................... 127


4-22 x'(T) and x"(T) for UPt3 sample No. 5......................................................... 128


4-23 x'(T) and X"(T) for UPt3 sample No. 6......................................................... 130


4-24 X'(T) and x"(T) for UPt3 sample No. 7...................................................... 131


4-25 C'(T) and X"(T) for UPt3 sample No. 8......................................................... 133


ix






4-26 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 1,
annealed, etched, unpolished ......................................................................... 134

4-27 [f(Tmi) f(T)] / f(Ti) for UPt3 sample No. 1,
annealed, etched, polished.......................................................................... 136

4-28 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 2 ....................................... 137

4-29 [f(Tm) f(T)] / f(Tmi) for UPt3 sample No. 3 ......................................... 138

4-30 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 4 ........................................ 140

4-31 [f(Tmi) f(T)] / f(Tmin) for UPt3 sample No. 5 ....................................... 141

4-32 [f(Tin) f(T)] / f(Tmin) for UPt3 sample No. 6 ........................................ 142

4-33 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 7 ........................................ 143

4-34 Inductive response in the vicinity ofTc for UPt3 sample No. 2 and
for low and high frequencies.......................................... ........... 150

4-35 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 1,

annealed, etched, polished at 6 MHz and 33 MHz ....................................... 153

4-36 Upturn in x'(T) for UPt3 sample No. 1 and Bdc = 1.4 T .......................... ....... 155

4-37 Upturn in X'(Time) sample No. 1 and Bdc = 1.2 T and Bdc = 1.4 T................. 158

5-1 Upper critical field of niobium ................................................................... 163

5-2 Upper critical field ofa UBel3 single crystal............................................... 166

5-3 Anisotropy ofBc2(T) in UBel3............................................................................... 168

5-4 SEM pictures of UBel3 sample No. 1.......................................................... 171

5-5 SEM pictures of UBel3 sample No. 2......................................................... 172








5-6 SEM pictures of UBe13 sample No. 3.......................................................... 173


5-7 SEM pictures of UBe13 sample No. 3, after polishing.................................... 174

5-8 X'(T) for UBe13 sample No. 1, in the normal state.......................................... 176


5-9 X'(T) for UBe13 sample No. 2, in the normal state.......................................... 177

5-10 X'(T) for UBel3 sample No. 3, in the normal state........................................ 178

5-11 X'(T) in the vicinity of Tc, for UBe13 sample No. 1, Bdc 1| [100], with
B dc 0.5 T .......................................................................................... .... 18 1

5-12 X'(T) in the vicinity of Tc, for UBel3 sample No. 1, Bde II [110], with
B dc < 0.5 T .................................................................................................... 182

5-13 x'(T) in the vicinity of Tc, for UBel3 sample No. 2, Bdc | [100], with
B dc < 0.5 T ....................................... .......................................................... 183

5-14 X'(T) in the vicinity of Tc, for UBel3 sample No. 2, Bdc I| [110], with
B dc 0.5 T .......................................................................................... .... 184

5-15 X'(T) in the vicinity of Tc, for UBe13 sample No. 3, Bdc 11 [100], with
B dc < 0.5 T .................................................................................................... 185

5-16 X'(T) in the vicinity of Tc, for UBel3 sample No. 3, Bdc | [110], with
B dc 5 0.5 T ................................................................................................... 186

5-17 X'(T) in the vicinity ofTc, for UBel3 sample No. 1, Bdc 1| [100], with
1 T < B dc 8 T .................................................. ....................................... 187

5-18 X'(T) in the vicinity ofTc, for UBel3 sample No. 1, Bdc I| [110], with
1 T B dc < 8 T ............................................................................................. 188

5-19 x'(T) in the vicinity of Tc, for UBel3 sample No. 2, Bdc || [100], with
1 T < B dc < 8 T ............................................................................................. 189

5-20 X'(T) in the vicinity of Tc, for UBel3 sample No. 2, Bdc |I [110], with
1 T B d 8 T .................................................... ...................................... 190

5-21 x'(T) in the vicinity of Tc, for UBel3 sample No. 3, Bdc 11 [100], with
ST B dc 7.5 T ................................................ ....................................... 191






5-22 x'(T) in the vicinity of Tc, for UBel3 sample No. 3, Bdc |1 [110], with
1T < Bdc < 7.5 T ....................................................... ..................... 192

5-23 X'(T) in the vicinity of Tc, for UBel3 sample No. 1, Bac || [100], for two
different runs.................... ...................................... 194

5-24 Complete B-T phase diagram for UBe13 sample No. 1................................. 195

5-25 Complete B-T phase diagram for UBel3 sample No. 2................................. 196

5-26 B-T phase diagram for UBel3 sample No. 3 ........................ ...................... 197

5-27 x'(B) and X"(B) for UBel3 sample No. 1, for T = 200 mK and
T = 600 m K ................................................................. ..................... 199

5-28 X'(B) and X"(B) for UBel3 sample No. 2, for T = 500 mK.............................. 200

5-29 x'(B) and X"(B) for UBe13 sample No. 3, Bdc | [100], for T = 100 mK.......... 201

5-30 x'(B) and x"(B) for UBel3 sample No. 3, Bdc | [100], for T = 175 mK .......... 202

5-31 x'(B) and X"(B) for UBel3 sample No. 3, Bdc 1 [100], for T = 250 mK.......... 203

5-32 x'(B) and X"(B) for UBel3 sample No. 3, Bdc | [100], for T = 400 mK......... 205

5-33 X'(T) for UBei3 sample No. 3, Bdc I| [100], for Bdc = 3 T ......................... 206

5-34 X'(B) and X"(B) for UBe13 sample No. 3, Bdc 1 [110], for T = 100 mK .......... 207

5-35 x'(B) and X"(B) for UBel3 sample No. 3, Bdc 1 [110], for T = 250 mK......... 208

5-36 X'(B) and x"(B) for UBe13 sample No. 3, Bdc [110], for T = 400 mK.......... 209

5-37 X'(T) for UBel3 sample No. 3, Bdc || [110], for Bdc = 3 T .............................. 210

5-38 X'(T) for UBe13 sample No. 3, Bdc II [110], for Bdc = 2 T ........................... 211







5-39 x'(T) for UBel3 sample No. 3, Bde I [110], for Bde = 1 T ........................ 212

5-40 x'(T) for UBel3 sample No. 3, Bdc II [110], for Bdc = 0.5 T ......................... 213

5-41 X'(T) for UBel3 sample No. 3, B, I [110], for Bdc = 0 T.............................. 214

5-42 Complete B-T phase diagram for UBe13 sample No. 3................................. 215

5-43 x'(T) for UBel3 sample No. 1, Bac 11 [100], in the superconducting state........ 216

5-44 x'(T) for UBe13 sample No. 1, Bac I [100], T/Tc < 0.5 ................................ 217

5-45 x'(T) for UBe13 sample No. 1, Bac II [110], in the superconducting state........ 218

5-46 x'(T) for UBe13 sample No. 1, B acI [110], T/Tc < 0.5 ................................ 219

5-47 X'(T) for UBe13 sample No. 3, Bac I| [100], in the superconducting state........ 220

5-48 x'(T) for UBe13 sample No. 3, Bac 1 [100], T/Tc < 0.5 ............................. 221

5-49 Effect of demagnetization factor on Bc2(T).................................................... 227

5-50 Phase diagram of UBel3 proposed by Ellman et al. (1991)............................ 231












Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


INDUCTIVE MEASUREMENTS OF HEAVY FERMION SUPERCONDUCTORS

By

PHILIPPE J.-C. SIGNORE

December, 1994

Chairman: Mark W. Meisel
Major Department: Physics


Experimental results are presented on the temperature dependence of the

penetration depth, X(T), in UPt3, and on the upper critical field, Bc2(T), of UBel3. The

objective of this work was to obtain a better understanding of the unusual

superconductivity in these materials.

The inductive response of eight UPt3 samples obtained from seven different

materials fabrication groups was measured from 50 to 700 mK at frequencies varying

between 32 Hz and 33 MHz. The low frequency (< 4.7 kHz) data suggest a linear

temperature dependence of X(T/Tc < 0.5) for the samples possessing a double feature near

the superconducting transition temperature, Tc. We have verified that this double feature

present in X(T) for some of the samples corresponds to the double jump observed in the

specific heat. On the other hand, X(T/Tc < 0.5) was found to have a quadratic

temperature dependence for unannealed specimens which exhibit only a single transition at

Tc. The linear temperature dependence in X(T) is consistent with the presence of line

nodes in the basal plane, while the quadratic dependence found for other samples indicate







that impurity scattering dominates in these specimens. The double transition in X(T) has

been studied in magnetic fields up to 1.6 T for one of the specimens, and the resulting

phase diagram is consistent with the one constructed from thermodynamical

measurements. The high frequency (> 3 MHz) data suggest X(T/Tc < 0.5) oc TI, where

2 < r5 < 4. The possible origins of the frequency dependence of X(T) are discussed, as

well as the effect of the surface quality on i.

Using sensitive mutual induction techniques, we have systematically measured

Bc2(T) for three UBe13 single crystals in fields up to 8 T oriented along the [100] and

[110] directions, with particular emphasis on the region near the zero field critical

temperature. In that low field regime and within our experimental uncertainties of 0.5 mK

and 0.5 mT, no anisotropy in Bc2(T) was observable for any of the samples. For one of

these specimens, an anomaly in X'(B,T) was observed below the superconducting

transition. The complete Bc2(T) phase diagrams are presented and compared to the

results of other workers. The temperature dependence of the penetration depth down to

50 mK and in zero field is measured on two of the crystals.












CHAPTER 1
INTRODUCTION



One of the exciting challenges in condensed matter physics is the search for

evidence of superconducting phases whose theoretical descriptions deviate from the

conventional Bardeen-Cooper-Schrieffer (BCS) model. Due to the great number of

unusual properties they exhibit, heavy fermion (and high temperature) superconductors are

considered good candidates for being non-BCS-like superconductors, and are often

referred to as "unconventional superconductors." In section 1.2, the meaning of

"unconventional superconductivity" will be more precisely defined. Prior to this

discussion, the seven heavy fermion superconductors known to date, along with several

other heavy fermion systems, are overviewed in section 1.1. The rest of the dissertation is

outlined as follows. Chapter 2 reviews the experimental work already published on UPt3

and UBe13. These two materials are the focus of this dissertation. Because of their rather

low critical temperatures, Tc, (Tc ; 0.5 K for UPt3 and Tc 0.9 K for UBe13), our

experiments utilized several standard low temperature physics tools, which are described

in Chapter 3, along with the details of our measuring techniques. Our results on UPt3 are

presented and discussed in Chapter 4. Our study of UPt3 centered around measuring the

temperature dependence of the penetration depth, X(T), which is a measure of the

superconducting electron density. The motivation behind these measurements is

established in section 4.1. Chapter 5 treats the UBel3 experiments, which were

concentrated around measuring the upper critical field, Bc2(T), in the low field region

(B < 0.5 T), in an effort to search for crystalline anisotropy ofBc2(T) that would suggest

the existence of an anisotropic order parameter. The motivation behind these








measurements is given in section 5.1. Finally, the conclusions of this work are

summarized in Chapter 6, where future directions are also suggested.


1.1 Heavy Fermion Systems

Heavy fermion compounds are characterized by a large specific heat electronic

coefficient, y, compared to ordinary metals, such as lead (y z 10 mJ/(K2 mole)). As will

be shown in subsection 2.1.2.a, within the Fermi liquid theory, y is directly proportional to

the density of states per unit volume at the Fermi surface, which, in turn, is proportional to

the effective mass, m*, of the quasiparticles. Therefore, a large y is indicative of a large

m*, justifying the designation of the materials as heavy fermion (or electron) systems.

Although the absolute definition of a "heavy" fermion compound is not clear cut, a

material with m* > 100 me, where me is the mass of the bare electron, is usually

considered a heavy fermion system. In subsection 1.1.1, a brief description of the heavy

fermion state is given, in addition to examples of heavy fermion compounds. For a more

comprehensive discussion of the heavy fermion state, see, for example, Stewart (1984),

Lee et al. (1986), Ott (1987), Sigrist and Ueda (1991), Grewe and Steglich (1991),

Steglich et al. (1992), and Proceedings of the International Conference on Magnetism

(1992).


1.1.1 Heavy Fermion Compounds

Heavy fermion compounds contain lanthanide (predominantly Ce) or actinide

(predominantly U) ions, that possess an unfilled electronic f-shell, which gives rise to a

magnetic moment at the ion site. Above a characteristic temperature, referred to as T*

(typically of the order of tens of kelvin), the systems can be described best by conduction

electrons with conventional masses moving through a lattice of well localized magnetic

moments. This picture is supported, for example, by a Curie-Weiss type susceptibility,

which is consistent with a set of noninteracting local moments. Below T*, by a







mechanism that is not yet fully understood, but most likely related to the interaction

between the "light" conduction electrons and the localized f-electrons, the system loses its

local moments and thef-electrons become part of the Fermi surface. The system is then

described best as a Fermi liquid with an enhanced effective mass, i.e. linear specific heat

with a large y and a temperature independent Pauli spin susceptibility. One of the

experimental results that has motivated a great deal of research on these materials is the

puzzling fact that heavy fermion systems possess a wide variety of possible ground states,

some of which are listed in Table 1-1. In the next subsection, we take a closer look at the

heavy fermion systems possessing a superconducting ground state.


1.1.2 Heavy Fermion Superconductors

The first report of superconductivity in a heavy fermion compound was published

by Steglich et al. (1979), who reported resistivity, ac susceptibility, dc Meissner effect and

specific heat measurements in CeCu2Si2 consistent with a superconducting transition

around 0.65 K This result was somewhat of a surprise since rare earth ions (Ce), through

their 4f electron magnetic moments, were known to break the superconducting paired

electrons. For example, the transition temperature of LaAl2 is suppressed when doped

with Ce ions. Another unexpected result was that, while CeCu2Si2 was superconducting,

LaCu2Si2 was not. Since the only difference between the two compounds is the 4f

electrons at each Ce site for CeCu2Si2, it became evident that these electrons played a key

role in generating the superconductivity. This evidence is further supported by the large

jump in the specific heat at the transition temperature.

The second heavy fermion superconductor to be discovered was UBel3 (Ott et al.,

1983). A review of the experimental results reported for this material is given in

section 2.2. One aspect that sets this material apart from the other heavy fermion

superconductors (except for Th doped UBe13) is its cubic crystal structure. Because of

this higher symmetry, anisotropies in the normal state properties are minimized, and this









Table 1-1 Ground state of various heavy fermion systems, as determined experimentally.
Values for y are given in (mJ/K2 mole).

Compounds Ground State Ordering y References
Temperature

CeCu6 Paramagnet no spontaneous 1300 Satoh et al. (1988)
ordering Jin et al. (1991)
T>3mK
CeRu2Si2 Paramagnet no spontaneous 600 Gupta et al. (1983)
ordering Steglich (1985)
T>40mK
UCdll Antiferromagnet TN z 5.0 K 840 Fisk et al. (1984)

NpBe13 Antiferromagnet TN 3.4 K 900 Stewart et al.
(1984a)

U2Zn17 Antiferromagnet TN 9.7 K 550 Ott et al. (1984a)

CeCu2Si2 Superconductor Tc z 0.65 K 1000 Steglich et al.
(1979)
UBel3 Superconductor Tc =0.9 K 1100 See section 2.2


Ul.xThxBel3 Antiferromagnet & Tc 0.6 K 2300 See section 2.2.7
Superconductor TN 0.4 K (x = 3%)
UPt3 Antiferromagnet & TN z 5.0 K 430 See section 2.1
Superconductor Tc 0.5 K

URu2Si2 Antiferromagnet & TN 17.0 K 180 Palstra et al. (1985)
Superconductor Tc 1.5 K
UNi2A13 Antiferromagnet & TN z 4.6 K 120 Geibel et al.
Superconductor Tc 1.0 K (1991b)
UPd2Al3 Antiferromagnet & TN z 14.0 K 150 Geibel et al.
Superconductor Tc 2.0 K (1991a)
Ce3Bi4Pt3 Semiconductor Gap 35.0 K 75 Hundley et al.
(1990)
CeNiSn Semiconductor Gap 6.0 K 210 Takabatake et al.
(1990)





5


fact facilitates the study of the anisotropies of the superconducting properties. As will be

discussed in section 1.2, measurements of the anisotropy of the superconducting

properties can lead to the determination of the symmetry of the superconducting order

parameter, which establishes whether or not a superconductor is conventional or

unconventional. In fact, measurements of the anisotropy in the upper critical field are

central to our experiments on UBe13, which are presented and discussed in Chapter 5.

Another interesting compound is Ui.xThxBel3. For 0.019 < x < 0.045, this

material exhibits a superconducting transition around 0.6 K, followed by a second

transition (Ott et al., 1984b). The nature of this second transition is controversial,

although muon spin relaxation data suggest magnetic ordering (Heffner et al., 1990). A

brief discussion of this heavy fermion system is given in subsection 2.2.7.

The next heavy fermion superconductor to be discovered was UPt3 (Stewart et al.,

1984b). For the past decade, this system seems to have become the most studied heavy

fermion compound. Its unusual superconducting phase diagram, and the occurrence of

power law dependence for various thermodynamic properties at low temperatures have

made this material the best candidate for unconventional superconductivity. In fact, one

of the main results from our experiments on UPt3, namely the linear temperature

dependence of the penetration depth (see Chapter 4), offers a strong piece of evidence for

non-BCS-like superconductivity. A review of the experimental results reported on UPt3 is

given in section 2.1, and two (of the many) theoretical descriptions proposed for UPt3 are

discussed in subsection 1.2.3.

The three other heavy fermion superconductors listed in Table 1-1, namely

URu2Si2, UNi2Al3, and UPd2Al3, have a common feature of possessing a relatively low
specific heat electronic coefficient, y < 200 mJ /(K2 mole), and are sometimes referred

to as semi-heavy fermion materials. In addition, all three exhibit, above the

superconducting transition, an unambiguous antiferromagnetic ordering, identified via

neutron experiments (Broholm et al., 1987; Schroder et al., 1994; Kita et al., 1994) and








also observed in specific heat and susceptibility data (Palstra et al., 1985; Geibel et al.

1991a, 1991b). This relative ease in observing the signature of the antiferromagnetic

transition is not shared with UPt3, where the antiferromagnetic transition around 5 K has

only been observed through neutron scattering and muon spin relaxation experiments (see

section 2.1.5).


1.2 Unconventional Superconductivity

In this section, the definition of the term "unconventional superconductivity" is

given (subsection 1.2.1), followed by a discussion of various experimental signatures

expected from unconventional superconductors (subsection 1.2.2). In subsection 1.2.3,

two of the theoretical "unconventional" models, proposed to describe superconductivity in

UPt3, are discussed.


1.2.1 Definition

The term "unconventional superconductivity" is often associated with the presence

of an anisotropy in the energy gap and/or the presence of nodes in the energy gap

structure. As will be shown below, these two conditions are not appropriate criteria for

unconventionality.

The definition that we adopt has been given, among others, by Rainer (1988) and

Fulde, Keller, and Zwicknagl (1988), who stated that the symmetry of the energy gap (or

order parameter) of an unconventional superconductor is lower than the symmetry of the

Fermi surface (or underlying crystal). This definition has been widely adopted by many in

the field, for example Choi and Sauls (1991), and Sauls (1994a). The concept is

illustrated in Fig. 1-1. In this figure, a cubic crystal structure is assumed. Clearly, the

Fermi surfaces in drawings (a) and (b) have the same symmetry as the energy gap, i.e. a

7t/2 rotation gives the same picture. However, the energy gaps in drawings (c) and (d)

























(a) (b)
















(c) (d)





Fig. 1-1. Examples of conventional superconductors (a) and (b) : the Fermi surfaces
(inner lines) have the same symmetries as the energy gaps (outer lines). Examples of
unconventional superconductors (c) and (d) : the Fermi surfaces (inner lines) have higher
symmetries than the energy gaps (outer lines). Drawings are from Fulde, Keller, and
Zwicknagl (1988).








have lower symmetries than the Fermi surfaces, so that a i/2 rotation gives the same

Fermi surface, but not the same gap.

A different definition of unconventional superconductivity that is sometimes put

forward is related to the nature of the pairing interaction. In this case, phonon mediated

pairing is considered as conventional, while any other pairing mechanisms are

unconventional.

A third definition sometimes found is the statement that superconductors

exhibiting anisotropic properties are unconventional. A caveat associated with this

assertion is the fact that conventional superconductors also possess anisotropic properties,

for example Nb and V have upper critical fields with orientation dependence, which are

non-negligible at higher fields (Williamson, 1970; Butler, 1980). These anisotropies are

caused by Fermi surfaces of cubic symmetry (non-spherical).

For the purposes of this dissertation, we chose to use the definition given by

Rainer (1988) and Fulde, Keller, and Zwicknagl (1988). Given this definition, the type of

experimental results that can be taken as evidence for unconventional superconductivity

need to be discussed.


1.2.2 Experimental Evidence for Unconventional Superconductivity

1.2.2.a Multiple phase diagrams

It is well known that the magnetic field temperature, B-T, phase diagram of a

conventional type I superconductor exhibits one phase (the Meissner state), while the B-T

phase diagram of a conventional type II superconductor features a Meissner state plus a

vortex lattice state (Tinkham, 1975). One of the most convincing experimental results,

that can be put forward as a signature of unconventionality, is a superconducting state

possessing additional phases. For example, superfluid 3He, the only unambiguous

unconventional superfluid known to date, exhibits an A-phase and a B-phase in zero field

and an additional A1 phase in a magnetic field (see, for example, Tilley and Tilley, 1990).








The existence of these multiple phases is related to the fact that the Cooper pairs in 3He

are in a triplet (S = 1, where S is the spin angular momentum of the Cooper pair) and a

p-wave (L = 1, where L is the orbital angular momentum of the Cooper pair) state, which

implies several possible superfluid phases (Vollhardt and Wi6fle, 1990). As can be

expected, the two phases with lowest energy are observed in zero field. In comparison, a

superconductor, whose Cooper pairs are in a singlet (S = 0) and an s-wave (L = 0) state,

has only one zero field phase. On the other hand, a superconductor in a singlet and d-

wave (L = 2) state possesses several zero field superconducting phases. When 3He goes

from one phase to the other, the transition can be observed, for example, with specific heat

measurements. For instance, at constant pressure, specific heat data show two transitions,

one between the normal fluid and the A-phase, and one between the A-phase and the B-

phase (Halperin et al., 1976). Similar double transitions in the specific heat have also been

observed in UPt3. In fact, various techniques (such as specific heat, ultrasonic attenuation,

thermal expansion) probing the magnetic field-temperature-pressure phase diagram of

UPt3 have revealed the existence of at least four different superconducting phases (see
section 2.1.4). Of all undoped heavy fermion superconductors, such a multiple

superconducting phase diagram has only been reported for UPt3.

1.2.2.b Power law temperature dependence

A second hint for unconventionality is the non-exponential temperature

dependence of various properties, such as specific heat, ultrasonic attenuation, and

penetration depth. A conventional superconductor possesses a finite energy gap

surrounding its entire Fermi surface, so that the thermally activated quasiparticles lead to

an exponential temperature dependence of various properties such as specific heat,

penetration depth, and ultrasonic attenuation. On the other hand, if the energy gap goes

to zero somewhere on the Fermi surface, then the properties related to the excitation

spectrum will follow power laws. At sufficiently low temperatures, the exponent of the

power laws depends on the types of nodes (lines or points) and on the manner in which the









gap goes to zero (linearly or quadratically). Several of the heavy fermion superconductors

exhibit such power law dependence. For example, as discussed in subsection 2.2.2, the

specific heat of UBel3 follows a T-2.750.25 dependence, and the penetration depth of

UPt3, discussed in detail in Chapter 4, is linear in temperature. The observation of non-
exponential temperature dependence is therefore often taken as evidence for

unconventional superconductivity. As an aside, we note that in Fig. 1.1, a conventional

superconductor with nodes in its energy gap structure is shown schematically. This

condition is theoretically possible, but in reality, no such material has been identified.

Although for some superconductors the gap is reduced along a particular direction, it

never actually goes to zero, with the exception of a few two-dimensional systems (e.g.

some organic and high temperature superconductors).

1.2.2.c Anisotropic properties

A third experimental clue for unconventional superconductivity is a strong

anisotropy in the superconducting properties. For example, measuring the angular

dependence of the slope of the upper critical field, Bc2(T), as T/Tc -> 1 has been

considered a "crucial" test for unconventional superconductivity (Gor'kov, 1987; Rainer,

1988). Of course, all solids have a crystal structure and therefore a non-spherical Fermi

surface, so that an energy gap of the same symmetry as the Fermi surface (conventional) is

never isotropic. However, in numerous metals, the anisotropy of the Fermi surface is

rather small, so that conventional superconducting properties are usually isotropic. For

that reason, large anisotropies, not accounted for by the anisotropy in the normal state

properties, are usually taken as evidence for unconventional superconductivity. A typical

example is the ultrasonic attenuation in UPt3 discussed in subsection 2.1.3. However, it

should be emphasized, that when suggesting unconventionality on the basis of anisotropic

properties, one has to check first that the orientation dependence is not simply related to

Fermi surface effects. This test is not always feasible if the Fermi surface is not well

characterized.









A special case is the upper critical field as T/Tc -> 1 which is always isotropic for

conventional superconductors with a cubic crystallographic symmetry. Therefore,

observing an angular dependence in Bc2(T/Tc -- 1) for UBel3 (cubic) would provide

strong evidence for unconventional superconductivity in this material, while an isotropic

behavior would be an inconclusive result. Our measurements of Bc2(T) in UBel3 are

discussed in detail in Chapter 5.

1.2.2.d Concluding remarks

Very few experimental measurements offer unambiguous proof for unconventional

superconductivity. The three tests just discussed only give suggestive evidence.

However, if an overwhelming majority of experimental results are consistent with

unconventionality, then one can legitimately believe in unconventional superconductivity

for a particular material, such is the case for UPt3. Other experimental measurements,

such as photo-emission spectroscopy, Josephson junction tunneling (Tsuei et al. 1994), or

non-linear Meissner effect (see section 6.1.2), have been proposed to directly test the

conventionality of superconductors. On the other hand, these experiments are usually

technically difficult, and can still lead to ambiguous results (Rainer 1988).


1.2.3 Unconventional Superconductivity in UPt3

Several theoretical models have been proposed for UPt3 (see, for example, Blount

et al., 1990; Joynt et al., 1990; Machida and Ozaki, 1991; and Chen and Garg, 1993), and

in this section, we treat two of them (Sauls, 1994a and 1994b; Putikka and Joynt, 1989).

The states that are briefly reviewed here, and that lead to unconventional

superconductivity as defined in subsection 1.2.1, are two of the most successful in

accounting for the various experimental results found in UPt3, and for this reason, they

have become two of the most popular.

The two states reviewed here, named E2u and Elg, represent two of the basis

functions of the irreducible representations of the symmetry group for hexagonal crystals.








These basis functions, which were established for various symmetry groups by Volovik

and Gor'kov (1985), correspond to the superconducting phases that can form at the

transition temperature (Yip and Garg, 1993). This approach is only valid if spin-orbit

coupling is strong, so that the electron spins are "frozen in the lattice," i.e. the rotations of

the crystal also rotate the order parameter (Volovik and Gor'kov, 1985). It turns out that

for the hexagonal symmetry, the basis functions can be divided into two categories. The

first includes the wavefunctions that transform into themselves under all symmetry

operations and are nondegenerate. These functions are called one-dimensional and are

labeled by A (even), and B (odd). The second includes wavefunctions that require a linear

combination of two basis functions in order to transform under symmetry operations.

These functions, which have a degeneracy of two, are called two-dimensional, 2D, and are

denoted by Eu (even) and Eg (odd). The E2u wavefunction refers to a triplet state, while

Elg refers to a singlet state.

Both of these 2D models exhibit only one phase transition in zero field. In order to

be consistent with the double transition observed experimentally in UPt3, they require a

symmetry breaking field to lift the degeneracy. Such a symmetry breaking perturbation

may originate from the antiferromagnetic ordering at higher temperature (Aeppli et a.,

1988), or possibly from another mechanism such as the structure modulation reported by

Midgley et a. (1993).

Another similarity shared by these two states is that they both lead to gap

structures with line nodes in the basal plane. This feature is important since a number of

experimental works, including the data presented in Chapter 4, suggest the presence of

such nodes for UPt3.

The two models differ on two important points. First, the E2u can account for a

tetracritical point in the magnetic field-temperature diagram, while the EIg cannot (Sauls,

1994a, 1994b). Although the presence of a tetracritical point has not been established

experimentally with certitude, at least for all orientations of the field (Adenwalla et al.,





13


1990), the inability of the Eig wavefunction to account for this possibility has been

considered a major obstacle for this state in describing UPt3. The second difference

between the two states is that while the E2u wavefimction can explain the observed

anisotropy of the upper critical field in terms of anisotropic Pauli limiting, the singlet Elg

state cannot. Although more experiments are needed to determine with confidence the

nature of the superconducting state in UPt3, the E2u state seems to be a good candidate.

In Chapter 6, several future experimental tests are proposed to answer this issue.













CHAPTER 2
OVERVIEW OF EXPERIMENTAL WORK -- UPt3 AND UBel3


In this chapter, the major experimental results involving UPt3 and UBe13 are

reviewed. Emphasis is placed on the measurements directly related to the

superconducting state, although some normal state results are discussed as well. One can

consult the numerous reviews available in the literature for a more exhaustive summary

(Stewart, 1984; de Visser et al., 1987a, 1987b; Fisk et al., 1986, 1988, 1993; Ott, 1987;

Taillefer et al., 1990; von Lohneysen, 1994).


2.1 Experimental work-- UPt3

Superconductivity in UPt3 was discovered by Stewart et al. (1984b) who reported

resistivity, specific heat, and susceptibility measurements consistent with a

superconducting transition around 0.5 K This discovery sparked great excitement for the

experimental and theoretical study of this material. For the past decade, several

experimental findings have established various unusual phenomena related to the

superconducting state of UPt3, suggesting that this material is a non-BCS-like

superconductor.

In the next subsections, the aforementioned results are presented and compared to

the behavior expected for a BCS superconductor. Measurements of the penetration depth

are reviewed in detail in Chapter 4 and are not discussed here. After a review of the

crystal structure in UPt3, normal and superconducting state specific heat measurements

are presented and followed by a brief summary of ultrasonic studies. The all important

phase diagrams are then reviewed prior to a discussion of two possible symmetry breaking

fields, namely antiferromagnetism and structure modulations. Two short status reports on








the Knight shift measurements and point contact spectroscopy results are then given.

Brief conclusions mark the end of the section.


2.1.1 Crystal Structure

UPt3 has a hexagonal closed-packed (hcp) structure, with two formula units in the

primary cell (Heal and Williams, 1955). Figure 2-1 shows a schematic representation of

the atomic configuration in the hexagonal unit cell. Many measurements on UPt3 are

taken with respect to two principal symmetries which are defined by the c-axis and the

basal plane, respectively. The lattice constant and other crystallographic parameters are

summarized in Table 2-1.


Table 2-1. Summary of lattice parameters of UPt3. Values for a and b are from
Chenetal. (1984).
a c c/a Volume of unit Mol. weight Density Tmelt
(A) (A) cell M (kg) p (kg/m3) (C)
___ (m3)
5.764 4.899 0.850 1.41 x 10-28 0.8233 1.94 x 104 1700


2.1.2 Specific Heat Measurements

Historically, normal state specific heat measurements played an important role in

the understanding of UPt3. In this subsection, these measurements are considered first and

are followed by a survey of results obtained in the superconducting state.

2.1.2.a Normal state

Between 1 K and 20 K, the specific heat as a function of temperature, c(T), can be

fitted to the expression (Stewart et al. 1984b; de Visser et al., 1984; Frings et al. 1985;

Brodale et al. 1986)















































Fig. 2-1. Atomic configuration in the hexagonal unit cell ofUPt3. The a, b, and c vectors
define the primary cell, and their values are given in Table 2-1. This figure was taken from
de Visser et al. (1987b).










c(T) = y T + 3 T3 + 8 T3 In(T / TSF). (2.1)


The first term, yT, corresponds to the electronic contribution. The linear coefficient, y, is

related to the density of states per unit volume at the Fermi surface, N(&F), by


S= -k N(&F), (2.2)
3


where kg is the Boltzmann constant (Ashcroft and Mermin, 1976). For a Fermi liquid,

N(EF) is given by

4 m* kF
N(sF) 4 (2.3)
h2


where kF is the Fermi wave vector, m* is the effective mass of the electrons

(quasiparticles), and h is Plank's constant. The Fermi wave vector can be estimated for

UPt3 by assuming threef-electrons per U-atom, or six per unit cell (Z = 6), contribute to
the itinerant band and by using the following expression for a spherical Fermi surface,



kF = 2 ] (2.4)



From the value of rm given in Table 2-1, one finds kF 1.08 x 1010 m-1. Substituting

Eqs. (2.3) and (2.4) into Eq. (2.2) and using the value y = 0.430 J / (K2 mol UPt3)

reported by Fisher et al. (1989), one may estimate the effective mass of the quasiparticles

in UPt3 to be m* 180 me, where me is the mass of the bare electron. Compared to the

effective mass in ordinary metals (1- 3 me), m* in UPt3 is very large, justifying the








designation of this material as a heavy fermion (electron) system In addition, this large

effective mass gives a relatively low Fermi temperature, TF z 300 K for UPt3, compared

to ordinary metals for which TF z 105 K Different values for y in UPt3 have been

reported and vary between 0.413 (de Visser et al., 1984) and 0.450 J/ (K2 mol UPt3)

(Stewart et al. 1984).

The second term in Eq. (2.1) corresponds to the phonon contribution to the

specific heat. The phonon coefficient, 0, can be used to estimate the Debye temperature,

OD, since these two parameters are related by the expression


p= 234 n kB D, (2.5)


where n is the number of atoms per unit volume (in UPt3, n = 8 / Qm) (Ashcroft and

Mermin, 1976). De Visser et al. (1987a) reported a value of 0 near

8.5 x 10-4 J/ (K4 mol UPt3) giving OD = 420 K This value is of the same order as the

Debye temperatures found for ordinary metals (Al: OD z 400 K; Cu: OD z 315 K). One

might expect that below 1 K (<< OD), all real phonon excitations are minimal, so that the

properties measured below this temperature, particularly in the superconducting state, are

electronic in nature.

The third term in Eq. (2.1) is generally attributed to the presence of spin

fluctuations (SF) in analogy to 3He, where a T31nT term is also observed and considered

to be a general property of an interacting Fermi liquid (Stewart et al., 1984; de Visser

et al., 1984; Frings et al., 1985). It is precisely the coexistence between magnetic spin

fluctuations and superconductivity that motivated a great deal of experimental and

theoretical work on UPt3.

2.1.2.b Superconducting state

Specific heat measurements in the superconducting state have revealed a number

of interesting results. For example, Fisher et al. (1989) reported two jumps separated by








60 mK in the vicinity of the critical temperature, Tc, as an intrinsic feature of UPt3. This

result is shown in Fig. 2-2. Critics have argued that the double jump could come from the

presence of a second crystallographic phase with a different critical temperature. This

argument was later weakened by the fact that, in some cases, "as grown" samples, initially

showing a single broad transition, exhibited two sharp jumps near Tc after annealing

(Vorenkamp et al., 1990; Midgley et al., 1993). The double feature was eventually

reported in results for many different samples (single crystal and polycrystalline

specimens) and was observable in other physical properties, such as ultrasonic attenuation

(as discussed below), thermal expansion (Hasselbach et al., 1990), and susceptibility (this

work). It is important to note that one of the original samples studied by Fisher et al.

(1989) is sample No. 4 described in this work, for which the results are presented and

discussed in Chapter 4.

Magnetic field studies of the specific heat showed that the separation between the

two peaks became smaller with field, ultimately vanishing near B = 0.5 T for fields

perpendicular to the c-axis (Hasselbach et al. 1989; Bogenberger et al. 1993). The

multiple phase diagram in the B-T plane drawn from such measurements and others (see

section 2.1.4.b on the upper critical field below) contributes the strongest piece of

evidence for unconventional superconductivity in UPt3 as defined in Chapter 1.

Another interesting finding is the temperature dependence of the specific heat

below Tc, cs(T), which varies as c,(T) = yT + 6 T2 between 100 mK and 400 mK

(Hasselbach et al., 1989). This result is in contrast with the exponentially decaying cs(T)

predicted by the BCS theory in the limit T -+ 0. The non-exponential decay of cs(T) in

UPt3 is taken as evidence for the presence of nodes in the energy gap structure. More

specifically, a quadratic temperature dependence is consistent with lines of nodes, while a

T3 dependence, as observed in UBel3 (see section 2.2.2.b), indicates the presence of point

nodes. In addition, the residual linear specific heat, YT, of UPt3 is evidence that the







800


S400- -
CL '0.49 K

o
E
N 200
A. Sample 2
E

O _0.36 K
I -0
0 4 430 :0

400-
0.42 K



200 /
UPt3


0 I I
0 0.2 0.4 0.6 0.8 1.0
T (K)

Fig. 2-2. Specific heat of UPt3 in the vicinity of the superconducting transition indicating
the double transitions at zero field. Measurements were performed by Fisher et al. (1989)
on two polycrystalline specimens. Sample labeled No. 2 on the figure is sample No. 4
used in this work (see Chapter 4). For that specimen, the lower transition is Tci = 0.36 K,
while the higher transition is Tc2 = 0.42K, so that the splitting ATc z 60 mK.








superconductivity does not cover the entire Fermi surface and that nodes in the energy gap

are present. However, one should keep in mind that, first, these power laws are extracted

for very limited ranges of temperature, i.e. at most one decade (50 mK to 500 mK), and

second, the exponential dependence predicted from the BCS theory is expected only for

T << Tc. Typically, for UPt3, the lowest temperature achieved by specific heat

measurements corresponds to about T/Tc 0.1.

Using a nuclear demagnetization cryostat, Schuberth et al. (1990 and 1992) were

able to measure specific heat down to 10 mK. The extraction of the power law

temperature dependence of cs(T) was difficult because their data showed a large anomaly

around 18 mK. The experiment are conducted on two different samples, one of which is

sample No. 2 of this work (see Chapter 4). These authors suggested that this result was

the signature of a new phase transition taking place in the superconducting electronic

system. To clarify this issue, Jin et al. (1992) measured sound velocity and attenuation of

a third sample down to 5 mK and reported no sign of an anomaly near 18 mK The

specific heat anomaly near 18 mK is, therefore, sample dependent and could be related to

the presence of impurities.


2.1.3 Ultrasonic Studies

Measurements of sound velocity and sound attenuation are powerful techniques

because they probe the properties of the crystal along specific axes. This feature, shared

also with penetration depth and thermal conductivity measurements, is especially useful

for studying the energy gap structure. A very important result, establishing the anisotropic

nature of superconductivity in UPt3, came from transverse sound attenuation measured by

Shivaram et al. (1986a) for sound propagation within the basal plane and polarization, E,

parallel and perpendicular to the plane. The attenuation in the superconducting state

increased linearly with temperature for E parallel to the basal plane, while the increase was

quadratic for E perpendicular to the plane. These results are very different from the








attenuation expected for a BCS-like superconductor which exhibits an isotropic

attenuation, aBcs, with a temperature dependence given by the expression


2 a~
aBCS- eA(T) k (2.6)



where a. is the normal state sound attenuation and A(T) is the energy gap (Tinkham,

1975). Attenuation of longitudinal sound along the c-axis has been measured by Bishop

et al. (1984) who reported a quadratic temperature dependence.

The results from these two groups are consistent with an energy gap possessing

line nodes in the basal plane and point nodes where the z-axis intersects the Fermi surface.

As discussed in Chapter 4, the penetration depth measurements presented in this

dissertation are consistent with this picture.


2.1.4 Phase Diagrams

2.1.4.a Lower critical magnetic field

Early measurements of the lower critical magnetic field, BcI(T), in UPt3 were

reported by Shivaram et al. (1989) who used a tunnel diode oscillator technique similar to

the one used in this work (see Chapter 3). Keeping the temperature constant and

sweeping a dc magnetic field, they extracted the lower critical field and reported a kink in

Bci(T) around 0.7 T/Tc for fields perpendicular to the c-axis. This kink is thought to
correspond to the lower transition observed in the specific heat. The scatter in the data

for the fields parallel to the c-axis did not allow one to unambiguously establish the

existence of a kink for this orientation. Later measurements of Bci(T) by Vincent et al.

(1991) showed a kink in both directions. From these measurements, one can estimate the

lower critical field as T -> 0 to be around 10 mT for both directions, although Wiichner

et al. (1993) reported a lower value of 6.5 mT.








2.1.4.b Upper critical magnetic field

The upper critical field, Bc2(T), in UPt3 has been extensively studied with various

techniques which include specific heat (Hasselbach et al., 1989; Bogenberger et al., 1993),

ultrasound velocity (Bruls et al., 1990), resistivity (Chen et al., 1984; Hasselbach et al.,

1990), thermal expansion (Hasselbach et al. 1990; de Visser et al. 1993), magnetostriction

(de Visser et al., 1993), and susceptibility (this work). The most comprehensive study on

Bc2(T) was performed by Adenwalla et al. (1990) and Lin (1993), who measured sound
attenuation and velocity in fields parallel, perpendicular and at 45 with respect to the

c-axis. The two samples investigated in their study are samples No. 5 and No. 6 of this

work (see Chapter 4). Two of the phase diagrams (parallel and perpendicular to the

c-axis) constructed from these sound measurements are shown in Figs. 2-3 and 2-4 (Lin,

1993). It is clear that there exist three distinct phases (A, B, and C) in the mixed state of

UPt3. As mentioned previously, these multiple superconducting phases set UPt3 apart
from any other superconductor and represent the most convincing argument for

unconventional superconductivity in this material.

The field at which the transition lines cross or merge (see Figs. 2-3 and 2-4) is

sometimes referred to as B*. An important point to notice from these phase diagrams is

their anisotropy. The value of B2(T) as T -+ 0 is Bc2(0) 2.0 T for the field parallel to

the c-axes, while Bc2(0) 2.5 T for the field oriented perpendicular. In addition,

B* = 0.9 T for B parallel to the c-axis, while B* ; 0.4 T for B perpendicular to the c-axis.

Furthermore, Bc2(T) has a clear kink at B* for fields perpendicular, while the kink may

not be present for fields parallel to the c-axis. Finally, the four transition lines seem to

come to a tetracritical point at B* for B perpendicular to the c-axis, while the resolution

for the other orientation does not allow a distinction between a single tetracritical point

and two tricritical points (Lin, 1993).











25 .

UPt3 HIIc


20



15 B 0

0 Y o 0

" H.(T) 0 (T)
10 T0




5 o o
o o

C oA0
0 0

0 100 200 300 400 500


T(mK)




Fig. 2-3. The complete superconducting B-T phase diagram of UPt3 for the magnetic
field parallel to the c-axis. The open circles are determined from temperature sweeps and
open squares are determined from field sweeps. Data taken from Lin (1993) (sample
No. 1). This specimen is sample No. 5 of this work.













25

UPta Hic

20
20 -




15- o

0
-- 0
-- B
10 -
SH.2(T)
O3
0 0 D3 0
5 HRF(T) 0 00 00



C H(T.) bI A 00
0 '- --'- -
0 100 200 300 400 500

T(mK)



Fig. 2-4. The complete superconducting B-T phase diagram of UPt3 for the magnetic
field perpendicular to the c-axis. The open circles are determined from temperature
sweeps and open squares are determined from field sweeps. Data taken from Lin (1993)
(sample No. 1). This specimen is sample No. 5 of this work.









2.1.4.c Critical pressure

Behnia et al. (1990) reported upper critical magnetic fields measured by resistivity

as a function of temperature and pressure. Their results showed that Bc2(T) behaves very

differently below and above B*, in a manner suggesting that the high temperature, low

field phase is rapidly suppressed by pressure.

The double jump in specific heat has been studied as a function of hydrostatic

pressure, P, by Trappman et al. (1991) who reported that the two transitions observed for

P = 0 merged at a critical pressure P* = 3.7 kbar. The phase diagram that they

constructed is shown in Fig. 2-5.

Recently the complete B-T-P phase diagrams for fields parallel and perpendicular

to the c-axis were reported by Boukhny, Bullock and Shivaram (1994) from ultrasonic

studies.


2.1.5 Antiferromagnetism and Superconductivity

One of the most difficult challenges facing the various superconductivity models

for UPt3 is to account for the unusually rich phase diagram presented in the previous

subsection. Most models call for a combination of a nonzero angular momentum pairing

and a symmetry breaking field responsible for lifting the degeneracy, thereby splitting the

transition in B = 0 and creating a multiple phase diagram. To date, there exist two strong

candidates for this symmetry breaking field. One of them, discussed in this subsection, is

an antiferromagnetic transition with a N6el temperature, TN, near 5 K, while the other,

discussed in the next subsection, is related to a structure modulation.

Aeppli et al. (1988, 1989) performed neutron diffraction measurements which

suggested that UPt3 was an antiferromagnet with ordered moments of (0.02 0.01)I'B

lying in the basal plane and a NMel temperature of 5 K The first evidence for the coupling

between the antiferromagnetic moments and superconductivity was given by the

temperature dependence of the magnetic Bragg intensity in zero field, which increases














0.5


I-J


0.45


0.4' 1 1-
0 2 4 6
P (kbar)




Fig. 2-5. The temperature-pressure phase diagram derived from specific heat
measurements by Trappman et al. (1991).








linearly with decreasing temperature from 5 K down to Tc, but then decreases from Tc

down to the lowest measured temperature (" 0.1 K).

Further evidence for the strong correlation between the order parameter of

antiferromagnetism and superconductivity in UPt3 came from neutron diffraction studies

under hydrostatic pressure (Hayden et al., 1992). These experiments showed that

antiferromagnetism is suppressed at roughly the same pressures at which the two specific

heat jumps merge (Fig. 2-5).

One of the problems associated with considering antiferromagnetism as the

symmetry breaking field is the short magnetic coherence length of approximately 200 A

(Aeppli et al., 1988, 1989; Hayden et al., 1992), which is of the same order of magnitude

as the superconducting coherence length (Shivaram et al., 1986b). It is unclear how such

a small magnetic coherence length, in addition to the tiny antiferromagnetic moment, can

produce a sizable splitting of the superconducting transition (von Ldhneysen et al., 1994).

Furthermore, specific heat, sound velocity, and magnetization measurements have failed to

unambiguously identify this antiferromagnetic transition around 5 K (Fisher et al., 1991;

Adenwalla et al., 1990). In view of these unanswered questions, alternative possibilities

for the symmetry breaking field have been put forward, one of which is discussed in

subsection 2.1.6.


2.1.6 Structure Modifications and Superconductivity

Recently, transmission electron microscopy was performed by Midgley et al.

(1993) to investigate the structure of annealed and "as grown" single crystals of UPt3 cut

from the same rod. In the annealed sample, they found a well-defined incommensurate

lattice distortion extending over domains of approximately 104 A (two orders of

magnitude longer than the zero temperature superconducting coherence length). On the

other hand, the "as grown" sample exhibited some superstructure which lacked the long

range coherence of the annealed specimen. Moreover, the annealed sample showed a well








defined double transition in the specific heat (Tc2 0.52 K and Tcl 0.47 K), while the

"as grown" crystal exhibited a single broad transition with a much lower onset

temperature, Tc 0.41 K These observations suggest that the splitting of the transition

in UPt3 may be related to these structural domains which clearly break the hexagonal

symmetry over long distances. However, the microscopic details of this structure

perturbation on the superconducting state have not yet been described.


2.1.7 Nuclear Magnetic Resonance and Muon Spin Relaxation Knight Shifts

Information about the type of pairing present in a superconductor can be obtained

by measuring the electron spin susceptibility, Xsc, in the superconducting state. In an

s-wave superconductor, the electrons are paired in states with opposite spins. Thus, the

spin susceptibility falls from its normal state value to zero at T = 0. In the BCS theory, the

temperature dependence of X, is given in terms of the Yosida function, Y(T), as

Xsc = XnY(T), where Xn is the normal state susceptibility. On the other hand, an odd parity
pairing, with finite angular momentum, will give a different functional form for Xsc. The

spin susceptibility can be studied by measuring the Knight shift, Ks(T), in nuclear magnetic

resonance (NMR) (Slichter, 1978) and muon spin relaxation (iSR) experiments (Knetsch,

1992).

The Knight shift in UPt3 has been measured by Kohori et al. (1987) who

performed NMR on 195Pt. The results showed a temperature independent Knight shift

above and below the transition temperature indicating that the spin susceptibility did not

change. The authors suggested that these results were consistent with an odd parity

pairing (p-wave) or a singlet superconductor with significant spin-orbit scattering from

impurities. It is important to note that, because of the Meisner effect (the expulsion of the

magnetic field from the sample except within a distance given by the penetration depth),

NMR must be performed on powdered samples or on a great number of small whiskers.

After obtaining a powder of their sample, Kohori et al. (1987) remeasured the ac








susceptibility and found a very broad transition with a lower Tc compared to the original

bulk specimen, indicating the presence of impurities or defects in the powdered sample.

The Knight shift in UPt3 was also measured using tiSR by Luke et al. (1991) who

reported results consistent with the findings of the NMR study.

Lee et al. (1993) measured the Knight shift on small, single crystal whiskers. The

whiskers (approximately one thousand of them) were aligned together using a magnetic

field method at room temperature. The normal state measurements, performed between

4 K and 50 K, were consistent with the results obtained by Kohori et al.


2.1.8 Point Contact Spectroscopy

A point contact (PC) can be obtained by pressing a normal metal needle or wire

against the surface of a superconductor. A simplified view of a point contact is a tunnel

junction in parallel with a shunt resistance, Ro. The current consists of two components: a

small tunneling current and a direct part (through the shunt). As the temperature drops

below the critical temperature of the superconductor, the magnitude of the tunneling

current changes due to the gap structure. An I-V characteristic of a PC can, therefore,

give information about the value of the energy gap. The details of point contact

spectroscopy analysis are discussed by Blonder et al. (1982). In practice, since the

majority of the total current flows through the shunt, the relative changes in the I-V

curves, as the bias voltage is increased above the gap, are very small, so that only rough

estimates of the gap values are possible (usually after plotting dV/dI vs. V).

Results of point contact spectroscopy on single crystals, reported by Goll et al.

(1993), indicated some anisotropy in the energy gap structure. They found that the

magnitude of the minimum in the dV/dI curves for current flowing parallel to the c-axis

was 2% greater than that for currents in the basal plane. Furthermore, they reported the

presence of a double minimum in the dV/dI curves for currents flowing parallel to the

c-axis and only in the low field, low temperature superconducting phase, suggesting








different superconducting order parameters for the different phases. Recently,

de Wildeetal. (1994) have also reported point contact results inconsistent with an

isotropic energy gap.

It is important to keep in mind that point contact spectroscopy does not allow the

clear identification of the direction of current flow, since the exact geometry of the contact

between the electrode and the sample is not known. A much more powerful technique is

to measure tunnel junction spectroscopy, but to date, these types of experiments have not

lead to conclusive results for UPt3.


2.1.9 Concluding Remarks

The results discussed above represent only a fraction of the overall pool of data

available on UPt3. For instance, thermal conduction (Benhia et al., 1991, 1992) and

expansion (de Visser et al. 1990, 1993; Hasselbach et al., 1990) measurements were not

presented in this section, although they contain useful information. For instance, thermal

conduction measurements can be performed to study the B-T phase diagram and its

anisotropy. Experiments probing the Fermi surface, such as the de Haas-van Alphen

effect (Taillefer et al., 1988a) and quantum oscillatory magnetoresistance (Julian et al.,

1992, 1994), are also available in the literature. The effects of doping UPt3 with various

elements such as B, Y, and Pd have been studied extensively (de Visser et al., 1987, 1993;

Aronson et al., 1991; Knetsch et al., 1992; Bakker et al., 1992; Vorenkamp et al., 1993a,

1993b). Measurements of resistivity, including magnetoresistance (Remenyi et al., 1987;

Taillefer et al., 1988b) and surface resistance (Grimes et al., 1991) (discussed in Chapter

4), have also been reported. As mentioned above, a number of experiments measuring the

penetration depth have been performed and will be discussed in great detail in Chapter 4.

Although this section treated only a limited portion of the total experimental work

on UPt3, several important points can clearly be made from the observations described

above. First, UPt3 is not a conventional BCS-like superconductor. This assertion is








especially supported by the unusual phase diagrams (Figs. 2-3 and 2-4). Second, the

energy gap is anisotropic and possesses nodes on the Fermi surface where it goes to zero.

The exact structure (points or lines) and location of the nodes remains uncertain, although

there is strong evidence (including the results of this work) that lines of nodes lie in the

basal plane. Third, some of the experimental results are sample dependent, which means

that specimen quality and characterization become very important. This sample

dependence justifies the need for systematic studies of the same type of measurements on

many different samples. For this reason, one of the original goals of the work described in

this dissertation was to provide a comprehensive and systematic study of penetration

depth measurements in UPt3.


2.2 Experimental work -- UBe13

Superconductivity in UBel3 was first observed in 1975 by Bucher et al. (1975),

who reported a superconducting transition at 0.97 K. This transition was argued to be

extrinsic to UBe13 and to be due to precipitated U filaments. Superconductivity, as an

intrinsic property ofUBel3, was first reported in 1983 by Ott et al. (1983), who reported

resistivity, specific heat, and susceptibility results consistent with a superconducting

transition below 0.85 K. Since then, a number of experimental results have suggested that

superconductivity in UBe13 is unconventional (as defined in Chapter 1), although none of

them offer definite proof.

In the next subsections, after describing the crystal structure of UBel3, a number

of the measurements supporting the unconventional superconductivity in this material are

reviewed. First, a brief summary of the specific heat results and a discussion on the

various length scales of the superconducting state are given. The phase diagrams are then

presented, before a short status report on NMR and pSR measurements. A subsection on

the possible magnetic transition near 8 K is then included. Finally, experiments on








thoriated UBel3 are presented before giving some concluding remarks. The anisotropy of

the upper critical field is discussed in detail in Chapter 5 and is not presented here.


2.2.1 Crystal Structure

The crystal structure of UBe13 (shown in Fig. 2-6) is the cubic NaZn13

configuration (Baenzinger and Rundle, 1949) with eight formula units in the primary cell.

The lattice constant and other parameters are summarized in Table 2-2. Eight of the Be

atoms (referred as Be I) and the eight U atoms form a sublattice of cubic CsCl structure

with lattice constant equal to a/2, while the remaining 96 Be atoms (Be n) surround the U

in an icosahedral formation, and the Be I atoms in a polyhedron (snub cube) arrangement.

The NaZn13 structure is described in detail by Shoemaker et al. (1952).

For the purpose of this work, it is important to define two particular orientations

with respect to the crystal structure. In Chapter 5, where the anisotropy in the upper

critical field is discussed, measurements are presented for which a magnetic field has been

applied parallel to the [100] and the [110] directions. The [100] direction refers to the

direction parallel to the side of the cube defining the unit cell, while the [110] direction

refers to the plane diagonal of the cube. Finally, the [111] direction refers to the body

diagonal of the cube.


Table 2-2. Summary of lattice parameters of UBel3_
a Volume of unit Mol. weigth Density Tmelt
(A) cell M (kg) p (kg/m3) (C)
fm (m3)
10.257 1.08 x 10-27 0.355 4.368 x 103 2000


2.2.2 Specific Heat Measurements

2.2.2.a Effective mass

Ott et al. (1983) reported a value for the electronic coefficient of the specific heat,

y, of 1.1 J/(K2 mol of UBel3). Using Eqs. (2.2) and (2.3), one can estimate the effective







Be(I)


( -



I --4-- (a)












(b)














Fig. 2-6. Crystal structure of UBe13. (a) Unit Cell showing the Cs-Cl sublatice. The Be n
are not shown for clarity. (b) Section of the unit cell showing the Be n (open circles) in an
icosahedral formation, and surrounding the U atom (not shown) located at the center of
the cube Also shown is the Be 1 polyhedron (snub cube) arrangement surrounding the
Be I atoms (closed circles). This figure was taken from Knetsch (1992).








mass of the quasiparticles, m*. The Fermi wave vector, kF, can be calculated using

Eq. (2.4) with Z = 24 (3 electrons per U atom and 8 U atoms per unit cell) and

0m = a3 = 1.08 x 10-27 m3, giving kF = 8.68 x 109 m-1. These values give m* ; 300 me
which is greater than m* = 200 me reported by Ott et al. (1983) who used a slightly

different value for Z.

2.2.2.b Superconducting state

Early specific heat measurements in the superconducting state by Ott et al.

(1984b) and Mayer et al. (1986) showed a strong deviation from the BCS theory and

suggested a temperature dependence of cs(T) close to T3. This result was interpreted as

evidence for the presence of point nodes on the Fermi surface. It is important to stress

that these T3 power laws were extracted from a limited temperature range : 0.08 < T/Tc <

0.9 for Ott et al., and 0.22 < T/Tc < 0.98 for Mayer et al.. Since the BCS theory predicts

an exponential dependence only for the limit T/Tc -+ 0, the statement that the T3 power

law is a signature of point nodes on the Fermi surface is not conclusive. Furthermore, as

the results discussed below will show, this power law dependence of the specific heat

might be sample dependent.

Measurements by Brison et al. (1988a) indicated that an upturn in c,(T) / T arises

at low temperatures with a minimum around 90 mK and that the T3 dependence was

hardly achieved even at intermediate temperatures (150 mK 500 mK). The authors

suggested that these results could be explained by the presence of a small amount of

impurities whose influence is amplified by the high sensitivity of anisotropic

superconductors to them.

Later measurements of the specific heat in magnetic fields (1.89 T to 7.88 T) on

the same sample used for the measurements just mentioned, indicated that, first, the power

law dependence of cs(T) in fields was close to c,(T) = yT + 1T2; second, the upturn in

cs(T) / T became greater with field; and third, the minimum rose in temperature with
increasing fields (Brison et al., 1988b). On the basis of entropy balance arguments, they








concluded that the upturn was intrinsic to UBe13, and furthermore, that it corresponded to

a magnetic transition.
The motivation for searching for a second transition below the superconducting

one is partially based on the results obtained on Th-doped UBel3 as will be discussed in

greater detail in section 2.2.7. Briefly, for Ul.xThxBel3, with 0.017 < x < 0.04, a second

transition below Tc has been observed, the nature of which has not been clearly identified.

One of the models attempting to explain these results predicts a second superconducting

transition below Tc in the pure system as well. Other measurements, in addition to the

specific heat data from Brison et al. (1988a, 1988b) have hinted at the existence of such a

transition in pure UBe13 (Rauchschwalbe et al., 1987; Ellman et al., 1991).


2.2.3 Penetration Depth. Mean Free Path, and Coherence Length

2.2.3.a %(0), f. and E

An important property of a superconductor is the zero temperature value of the

penetration depth, X(0). Several groups have reported values for X(0), which range from

2000 A as determined by NMR results (MacLaughlin et al., 1984) to 4000-5000 A

(Gross et al. 1986; Alekseevskii et al., 1986) and z 11000 A (Gross et al., 1988; GroB-

Altag et al., 1991) as extracted from magnetization measurements.

The electronic mean free path, R, can be estimated using the expression


e=VF hkF I (2.7)
(27x m* n e p(T)



where vF is the Fermi velocity, T is the average time between scattering events, n = Z / nm

is the number of electrons per unit cell contributing to the conduction band, and p(T) is

the resistivity (Ashcroft and Mermin, 1976). From the resistivity data of Maple et al.

(1985), one can estimates p(Tc) z 125 pocm which gives e 13 A. It is important to








note that near Tc, the resistivity is still changing with temperature, so that p(Tc) is not the

residual resistivity, and the calculated e may not reflect the residual mean free path.

Furthermore, Eq. (2.7) is based on the free electron model, which may not be reliable for

UBe13. Taking these points into consideration, Brison et al. (1989) suggested that the

mean free path may reach several hundred angstroms.

From the slope of the upper critical field near Tc, Maple et al. (1985) have

estimated the superconducting coherence length, 4, in UBe13 to be 140 A. However, by

using the BCS formula hvf / 27tA(0), they find 2 ; 50 A. From the above discussion of

the mean free path, it is not clear whether UBel3 is a superconductor in the clean limit

(e > 4 ) or in the dirty limit (f < 4 ). Assuming k(0) = 5000 A, we can estimate the

Ginzburg-Landau parameter K = 0.96 X(0)/4 ; 100 for the clean limit and K = 0.715 h(0)/l

S300 for the dirty limit, so UBe13 may be considered a strong type II superconductor.

2.2.3.b K(T)

The motivations for measuring the temperature dependence of the penetration

depth, X(T), are discussed in detail in Chapter 4. Briefly, X(T) is related to the

temperature dependence of the density of superconducting electrons, ns(T), which

depends upon the energy gap structure. The fact that X(T) can be measured along

different crystallographic directions, makes the study of the penetration depth a very

powerful tool for investigating the gap structure.

The temperature dependence of the penetration depth on UBe13 single crystals has

been reported by Grof-Alltag et al. (1991). These authors reported a quadratic

dependence for the excitation field parallel to the [100] and [110] directions. The results

on our single crystals are described and discussed in Chapter 5, and are in conflict with the

results of GroB-Alltag et al.

Measurements on polycrystalline specimens performed by other groups also

indicated a quadratic temperature dependence of X(T) (Gross et al., 1986; Einzel et al.,

1986). Gross et al. argued that these results were consistent with the existence of a gap








with linearly vanishing point nodes located on the Fermi surface. However, the same

authors acknowledged the fact that any dirty superconductor could give a quadratic

temperature dependence. The lack of conclusive evidence from penetration depth

measurements motivated our studies of X(T) for our high purity single crystals described

in Chapter 5.


2.2.4 Phase Diagrams

2.2.4.a Lower critical magnetic field

Rauchschwalbe (1987) measured Bci(T) from magnetization studies and reported

Bcl(0) 4.6 mT and d(Bcl)/dT near Tc a 10 mT/K. Although the author plotted the data
against T2, Bci(T) did not quite follow the empirical quadratic dependence. This result

cannot be taken as a signature for unconventional superconductivity since the BCS

prediction and several conventional superconductors (such as aluminum, tin, indium, lead

and mercury) deviate from this quadratic temperature dependence. In that regard, it

would be interesting to fit Rauchschwalbe's data to the BCS prediction.

2.2.4.b Upper critical magnetic field

The upper critical field of polycrystalline UBel3 has been measured by several

groups (Chen et al., 1985; Rauchschwalbe et al., 1985, 1987; Remenyi et al., 1986;

Schmiedeshoffet al., 1988, 1992; Brison et al., 1989). The results indicated an unusually

large slope ofBc2(T) in the limit T/Tc -+ 1. For instance, Rauchschwalbe (1987) reported

values near 200 T/K for 0.25 T < B < 2 T, although values near 35 T/K are more common

(Chen et al., 1985). Above 2 T, Bc2(T) changes slope to a lesser value

(; 11.5 T/K) until about 6 T where it becomes steeper again (a 15 T/K) (Brison et al.,

1989). The reported values for the upper critical field in the T/Tc -> 0 limit range from

t 10 T (Remenyi et al., 1986) to 13.5 T (Brison et al., 1989). Various models have

been used to explain the temperature dependence of Bc2(T), but the data has not been








fitted successfully over the entire temperature range for any of them (Rauchschwalbe et

al., 1987; Brison et al., 1989).

Results ofBc2(T) on single crystals will be discussed in detail in Chapter 5, with an

emphasis on the earlier work performed on the anisotropy in Bc2(T) in the limit T/Tc 1.

Briefly, the behavior ofBc2(T) is similar in single and polycrystalline samples for B < 6 T,

with a large initial slope (, 40 T/K for B < 0.5 T) and a change around 2 T to a smaller

value (; 9 T/K) (Maple et al., 1985). Above 6 T, the slope stays constant (as opposed to

increasing as in the polycrystalline samples) and eventually levels off to Bc2(0) ; 9 T.

Figure 2-7 gives an example of Bc2(T) for single and polycrystalline samples, the data is

from Schmiedeshoffet al. (1992).

Finally, specific heat measurements as a function of magnetic field by Ellman et al.

(1991), have suggested the presence of a second transition line below Bc2(T). The

existence of a second transition, below the superconducting one, has been postulated by

Rauchschwalbe (1987) from his analysis of the thorium doped UBel3 results, which are

discussed below.


2.2.5 Nuclear Magnetic Resonance and Muon Spin Relaxation Knight Shifts

2.2.5.a NMR

Early measurements of the temperature dependence of the spin-lattice relaxation

time, TI, by MacLaughlin et al. (1984) indicated a T3 dependence for 0.2 K< T < Tc, in

contrast to the exponential behavior expected for a BCS superconductor at low

temperatures. The authors suggested that their results were consistent with an energy gap

possessing lines of nodes. However, they reported a strong deviation from the T3 power

law for 0.06 K < T < 0.2 K, and this observation was unexplained. Clearly, this T3

dependence of TI, extracted for such limited temperature range, cannot be considered as

strong evidence for unconventional superconductivity.















UBe13


0.


~.,
S


.. I I I i 't


0.2


0.4 0.6

T (K)


0.8


Fig. 2-7. Upper critical field in UBe13 for single crystal (lower curve) and polycrystal
(upper curve). Data from Schmiedeshoffet al. (1992).


0)
Og
0


C'J
U


I I I I .' I i I l I I Il I I I I








The NMR Knight shift was reported as changing by less than 0.01% below Tc for

an applied field ofB = 1.5 T, consistent with an odd parity pairing (Heffier et al., 1986).

However, as shown in subsection 2.2.5.b, these results were not always reproducible.

2.2.5.b Muons

Heffner et al. (1986) reported a muon Knight shift change of 40% below Tc (in

contrast to the NMR results just mentioned), and suggested that this result was most likely

consistent with an even-parity pairing state. The same authors suggested that the

difference between the NMR and the muon experiments was related to differences in

sample qualities.

Luke et al. (1990) confirmed a very large muon Knight shift and reported a

temperature dependence similar to that given by the Yosida function for a spin

susceptibility of a conventional superconductor below Tc. However, the situation was

further complicated when Luke et al. (1991) reported a temperature independent Knight

shift in UBe13, consistent with an odd parity pairing. As the above review showed, the

inconsistency among the various experimental results makes it difficult to conclude on the

nature of the superconductivity in UBel3 based on the NMR and pSR data.


2.2.6 Anomaly Around 8K

In the previous section on UPt3, it was stated that a candidate for the symmetry

breaking field, responsible for the unusual phase diagram in this material, is the presence

of antiferromagnetic ordering around 5 K. Another heavy fermion superconductor with

non-BCS-like properties is URu2Si2. In this material, an unambiguous magnetic ordering

is observed at 17.5 K (Palstra et al., 1985). The coexistence of magnetism and

superconductivity in UPt3 and URuzSi2 motivated numerous studies of the normal state of

UBel3 in search of a magnetic transition. Magnetostriction measurements by Kleiman

et al. (1990) showed evidence of magnetic ordering around 8 K It later appeared that

this transition might be sample dependent. Several groups conducting similar experiments








did not observe the magnetic transition (de Visser et al., 1992a, 1992b; Clayhold et al.,

1993). The discrepancy between these results motivated measurements of the normal

state susceptibility of our three UBel3 single crystals.


2.2.7 Thorium Doped UBe13 Experiments

Many different elements have been used to dope UBe13. A comprehensive study

of doping experiments was performed by Kim (1992). This section treats only the work

performed on Ul-xThxBel3 for which a great deal of experimental results have been

published since Ott et al. (1984a) reported a non-monotonic Tc-x curve. A complete

Tc-x phase diagram, shown in Fig. 2-8, was established using a variety of measurement

techniques (Ott, 1989; Heffner et al., 1990). In particular, specific heat measurements

indicated the existence of a second transition for 0.019 < x < 0.045 (Ott etal., 1984a,

1985). The nature of this second transition has been the focus of numerous experimental

and theoretical work, for it might hold the key to understanding superconductivity in pure

UBel3. An excellent guide to the literature on this subject has been given by Knetsch

(1992).

There are several possible explanations for the second phase transition in thoriated

UBel3, three of which are briefly presented here. The first was proposed by Batlogg et al.

(1985), who, from their ultrasound experiments, suggested an antiferromagnetic

transition. Their hypothesis was supported by a well defined ultrasound attenuation peak

near Tc2, 200 times larger than the expected total ultrasonic attenuation due to scattering

from conduction electrons. Since s-wave superconductivity is known to be weakened by

the presence of magnetism, the coexistence of superconductivity and antiferromagnetism

suggests that superconductivity in U1.-ThxBel3 is unconventional.

An alternative interpretation was later proposed by Joynt et al. (1986) who

attempted to explain the ultrasound result in terms of a transition in which one







1.0





U--xThxBe13








'-c 0.5




I, I

MAGNETIC






0.0 I I II I
0.0 2.0 4.0 6.0

X(%)


Fig. 2-8. Superconducting Tc-x phase diagram of U.-xThxBel3 taken from Heffier et al.
(1990). The symbols represent transition temperatures determined by different
measurement techniques. 0: Tci from susceptibility, 0: Tcl from Bci(T), A: Tcl, Tc2
from specific heat, V: Tc2 from Bcl(T).








superconducting order parameter is replaced by another. Such a first order transition

occurs in 3He between the A- and the B-phases (Vollhardt and Wolfle, 1990).

A third explanation for the second transition in Ul.xThxBel3 has been put forward

by Rauchschwalbe et al. (1987) who suggested the existence of two different

superconducting phases originating from two distinct parts of the Fermi surface. The

authors took their analysis a step further and claimed that these two phases also coexisted

in pure UBel3, so a second superconducting transition near Tc2 ; 0.6 K should be

observed below the first one at Tel 0.9 K. Rauchschwalbe et al. suggested that this

second transition in UBel3, less pronounced than in the thoriated system, was supported

by specific heat data which showed an enhancement below Tcl compared to the

predictions obtained from two different models (a strong coupling BCS and a strong

coupling triplet state). The existence of a second phase transition below the first

superconducting one will be further discussed in Chapter 5 where the phase diagram for

one of our single crystals of UBel3 (No. 3), exhibiting signs of an anomaly below Bc2(T),

is presented. This discussion is also relevant to the specific heat results on pure UBe13 by

Brison et al. (1988b), presented earlier, which suggested the existence of a transition

below Tc, and to the phase diagram proposed by Elman et al. (1991) with the second

transition line below Bc2 (see section 2.2.4). It is important to note that the picture

presented by Rauchschwalbe et al. is independent of the type of pairing existing in the

superconducting states.

Several other models have been proposed for the second transition in thoriated

UBe13 (Kumar and Wolfle, 1987; Sigrist and Rice, 1989; Langner et al., 1988). A

scenario for the complete Tc-x diagram in Ul.xThxBel3 has been given by Knetsch (1992).

The model assumes a magnetic transition at Tc2 and spin fluctuation mediated

superconductivity.









2.2.8 Concluding Remarks

While some of the results presented here suggest that UBel3 is an unconventional

superconductor, there exists no strong, unambiguous evidence supporting this assertion.

Several of the experimental measurements such as specific heat, Knight shift, observation

of second transitions, values of Bc2(0), the slope of Bc2(T/Tc -* 1), and 8 K anomalies

give sample dependent results. Therefore, it seems important to perform measurements

on several different samples and to characterize the specimens as much as possible. This

approach has been adopted for the work presented in Chapter 5.

This sample dependence has made the overall interpretation of the data on UBel3

(and UPt3) very difficult and may be related to the fact that unconventional

superconductors are very sensitive to impurities, more so than their BCS counterparts

(assuming that UBe13 is indeed unconventional). Sample fabrication is, therefore, an

extremely important aspect in this field. As different samples are grown using various

techniques and starting materials, their impurity concentrations may vary, resulting in

some inconsistencies in the experimental results. Sample histories, such as annealing,

etching, polishing, and cutting also become relevant issues. In fact these sample

fabrication problems are common to most superconductors possessing short coherence

length and small energy gaps (e.g. organic, high temperature, Chevrel phase, and A15).













CHAPTER 3
APPARATUS AND EXPERIMENTAL TECHNIQUES


This chapter is divided into three parts. First, the dilution refrigerator, and its

auxiliary equipment, used to achieved the low temperatures (from about 40 mK to 4 K),

are described. In the second section, our technique to measure ac susceptibility, namely

mutual inductance, is presented. In the third section, the tunnel diode oscillator technique,

used to measure the inductive response of samples at radio frequencies (rf), is described.


3.1 Low Temperatures

This section deals with the cryogenic aspects of the experiments conducted for this

dissertation. The topics include the dilution refrigerator, the thermometry, and the

temperature control.

3.1.1 Dilution Refrigerator

Because the history and the principles of the dilution refrigerator are well covered in

the literature (Lounasmaa, 1974; Betts, 1976; Pobell, 1992), this subsection covers only

the aspects which are specific to our refrigerator. We perform our low temperature work

using a homemade, continuous flow dilution refrigerator built in 1975 by R. M. Mueller

under the supervision of Professor Dwight Adams. The entire refrigerator, with the

exception of the gas handling system, fits inside an rf shielded cage made of copper screen.

This cage protects the experiments and the electronics from potential noise and heating

effects of external radio frequency radiation. The dilution unit is suspended from a one

inch thick, triangular aluminum plate which is supported at its three corners by pneumatic

isolation mounts from Newport Research Corporation. This setup provides isolation from

mechanical vibrations. The gas mixture (about 20 liters at STP) has a 3He/4He ratio of








about 1/3 by volume. A rotary vacuum pump (model Ed330 from Edwards) and a

diffusion pump (NRC model VHS-4, from Varian) make up our pumping system. It takes

about 10 hours to cool from room temperature (RT) to the lowest temperature, Tmn, and,

if necessary, the entire system can be turned around (Tm -> RT -- Trn) in 24 hours.

Our 5 feet long dewar is a custom-built Cryogenic Associates "superinsulated" model with

a 6-inch diameter fiberglass neck, a 12-inch diameter aluminum "belly", and a 6-inch

diameter aluminum "tail". Due to the fact that fiberglass is permeable to helium at room

temperature, exposure to concentrated helium gas (> 30 min) could compromise the

vacuum inside the dewar. A 120 W bath heater, located at the bottom of the helium bath,

facilitates rapid warming of the cryostat. The heater is equipped with a thermistor that

shuts the heater off when it is just above room temperature (Mueller et al., 1982).

In addition to the inner vacuum chamber (IVC), the dilution unit is surrounded by

two copper radiation shields anchored at 0.6 K and 40 mK. The six step heat exchangers

are made of copper and hold copper sinter. The mixing chamber is a 12 cm3 copper

cylinder holding approximately 6 m2 of copper sinter, and is equipped with a 21.7 kM

resistor that is used as a heater. The various experiments are mounted on two copper

plates screwed into the mixing chamber. These two plates and the three copper legs

connecting them have been electroplated with gold. In the next subsection, the

electroplating procedure is presented. The cooling power of the dilution refrigerator is

130 gW at 100 mK with a 3He circulation rate of 100 nmol/sec (see subsection 3.1.1.b).

The minimum temperature measured outside the mixing chamber is Tmin ; 40 mK.

3.1.1.a Gold electro-plating

The following procedure was performed on several parts attached to the mixing

chamber of our dilution refrigerator. The plating prevents the copper from oxidizing,

which decreases its thermal conductivity. This procedure, which is standard in the micro-

electronics industry, has been available at the University of Florida's Department of

Physics through the help of Larry Phelps. Mr. Phelps supervised the plating of our parts









and assisted in the compilation of these notes. This method works exclusively for gold

plating on copper.

Parts must be clean if plating is to be successful. If the parts have been machined,

grease must be removed. A first wash in hydrochloric acid is followed by a dip into a

Loncoterge solution made by the London Chemical Company (sold by Kepro). The parts

are then washed in deionized water and finally rinsed with distilled water. The second step

of the process involves estimating the surface area to be plated. This area determines the

current to be used in the electro-plating. The current is calculated using the formula

I = 12.5 mA/in2 x (surface area). Once the power supply is set for the right current, the

part is immersed into an Orosene 999 Gold Plating solution made by Technic, Inc. (sold

by Kepro). The object to be plated is the cathode and the stainless steal bucket holding

the solution is the anode. As the plating occurs, cyanide bubbles emerge from the

solution. It is therefore best to perform the plating under a hood, but a well ventilated

room is acceptable. The current density mentioned above was empirically found to give

fairly slow outgassing. During plating of parts with complex geometries, the bubbles

released may be trapped temporarily on the surface of the pieces. This effect may result in

uneven plating. For this reason, it is important to keep the current low to minimize the

degassing rate, giving the bubbles a chance to escape. Of course, lower currents mean

slower plating rates. The above current density gives approximately 50 millionth of an

inch every 40 minutes. For very flat parts, which would not trap bubbles, higher current

densities can be applied. While the reaction takes place, one should occasionally agitate

the parts gently to release any bubbles from the surface. After the plating has been

stopped, the parts are washed with deionized water. The main plating solution can be

recycled -- its cost, which follows the market price of gold, can be quite high.

3.1.1.b Measuring the 3He circulation rate

In this subsection, the method used to estimate the 3He circulation rate of our

refrigerator is discussed. After cooling to 46 mK, the refrigerator was placed into a "one









shot" mode, i.e. the gas recovered from the still was not recondensed, but stored into the

cold trap (T = 77 K). The still was maintained at a constant temperature of about

650 mK. This still temperature was previously determined by maximizing the cooling

power of the refrigerator. The pressure, P, of the cold trap was monitored as a function of

time. From the volume of the trap (V = 2.64 x 10-3 m3), the number of recovered moles

was estimated from the ideal gas relation


PV
N (moles) (3.1)
RT


where R is the universal gas constant (8.314 J mole-1 K-1). This method does not account

for the amount of gas absorbed onto the surface of the charcoal present in the trap. Since

the vapor pressure of 4He at 650 mK (constant temperature of the still monitored through

the experiment) is negligible compared to that of 3He, one can assume that the gas

recovered from the still is mostly 3He. The two assumptions just mentioned, namely the

ideal gas law and a 100% 3He circulation, constitute the main source of errors for this

estimation of the 3He circulation rate, and it is important to keep in mind that the result is

only a coarse estimate. Figure 3-1 shows N vs. time obtained from the above procedure.

The line is the result of a linear fit, and indicates that the data fall on a straight line of slope

95 pmol / sec, which corresponds to the circulation rate of 3He for our refrigerator.


3.1.2 Thermometry

In this section, the different thermometers used on our cryostat are discussed.

First, the "diagnostic" thermometers are described. These carbon resistors may be used to

locate a problem, in the event that the refrigerator is malfunctioning. In the second

subsection, the data acquisition thermometers are characterized. Finally, work on the

















0.3





0.2
0
0)
o
E
0.1





0.0


1000


2000


3000


time (sec)







Fig. 3-1. Number of 3He atoms (moles) as a function of time (sec) for our dilution
refrigerator operated in a "one shot" mode at 46 mK. The line is the result of a linear fit.
The slope of the line is 94 imol/sec, and corresponds to the 3He circulation rate. After
2500 sec, the temperature started increasing.









magnetoresistance of thermometers and measurements of heating effects due to eddy

currents are reported in the third subsection.

3.1.2.a Diagnostic thermometers

We use a total of eight diagnostic thermometers. Their labeling, location and

resistance values at different temperatures are given in Table 3-1. Except for thermometer

#8, all diagnostic thermometers are carbon resistors purchased from Speer. They were

ground flat on two sides to expose the carbon core, and to increase the thermal contact

area. The two surfaces were glued with stycast 2850 epoxy to a copper heat sink.

Thermometer #8 is a RuO2 thin film resistor. The resistances of the diagnostic

thermometers are read with a Linear Research picowatt ac resistance bridge (LRl10)

operated in a two wire measurement mode.


Table 3-1. Labeling, location and resistance values of the eight diagnostic thermometers.
All resistances are given in ohms. H. Ex. stands for heat exchanger, and M. C. stands for
mixing chamber.


Labels #1 #2 #3 #4 #5 #6 #7 #8

Location->
1K Still 1st 3rd 5th 6th M. C. M. C.
Temperature pot H. Ex. H. Ex. H. Ex. H.Ex.


300 K 913 771 348 336 356 346 345 1124

77 K 890 890 371 361 387 372 367 1127

4.2 K 1510 1280 410 410 420 420 410 1300

Tmin 40 mK 2860 4990 3280 6850 7850 14000 18800 10250



3.1.2.b Data acquisition thermometers

We measure the temperature of our samples with two germanium resistive

thermometers. The names "No. 26" and "LS-Burns" were adopted for them. We use









No. 26 between Tmn and 400 mK, while LS-Burs is used between 350 mK and 4.2 K

We measure the resistance of these two thermometers with an Linear Research ac

resistance bridge (LR-400) in a four wire measurement configuration.

Other researchers, who have used thermometer No. 26, have calibrated it

extensively against a 3He vapor pressure thermometer, Cd, Zn, Al superconducting fixed

points, and a 3He melting curve thermometer. Figure 3-2 shows some of the calibration

points, and the solid line represents the result of a fit to Eq. (3.2). This expression is used

to determine temperatures from No. 26 resistance values, for temperatures between

40 mK and 400 mK, namely


T(mK) = exp ( 22.188 6.243 x LR + 0.7558 x LR2 0.03333 x LR3), (3.2)


where LR = In (R), and R is in ohms.

Thermometer LS-Burns (used for temperatures above 350 mK) was purchased by

Professor M. Burns in 1987 from LakeShore Cryotronics, who calibrated it against a

cerium magnesium nitrate magnetic thermometer as well as the National Bureau of

Standards superconducting fixed points. Our laboratory inherited the use of the device

after Dr. Burns joined Conductus Inc. in 1989. Figure 3-3 shows the calibration curve for

LS-Burns, for which two different fits were performed. First, the calibration data were

fitted to Eq. (3.3) for temperatures between 350 mK and 2 K:


T (K) = I Ai x Cos ( i x Arccos(X)), i= 0 to 6 (3.3)


where X = ((Z ZL) (ZU Z))/(ZU ZL), Z = loglo(R(n)), ZL = 2.27231411327,

ZU = 4.78304279264, and the coefficients Ai's are given in Table 3-2. The result of the fit

is shown in Fig. 3-3.















104






103


102






0.02 0.05 0.1


0.2
T


0.5
(K)


1 2


Fig. 3-2. Calibration curve for thermometer No. 26. The points, (o), were obtained from
various primary thermometers (see text). The solid line represents the result of a fit to Eq.
(3.1), which is used to determine temperatures from No. 26 resistance values, for
temperatures between Tm. and 400 mK.


I a I I I I I t I I I I


I lrl 117) 1 1 1 11117




















104




103





102




0.2 0.5 1 2 5 10
T (K)





Fig. 3-3. Calibration curve for thermometer LS-Burns. The points, (o), were obtained
from a cerium magnesium nitrate magnetic thermometer (LakeShore Cryotronics). The
solid lines represent the results of fits which are used to determine temperatures from LS-
Burs resistance values. For temperatures between 350 mK and 2 K, the data are fitted to
Eq. (3.2); for temperatures above 2 K, the data are fitted to Eq. (3.3).








Table 3-2. Coefficients Ai's for Eq. (3.3), which is used to fit the calibration data for
LS-Burns between 350 mK and 2 K


AO Al A2 A3 A4 A5 A6

0.812184 -0.764916 0.282624 -0.095017 0.030477 -0.009407 0.002965



The second fit for the calibration data of LS-Burns was performed to Eq. (3.4) for

temperatures between 2 K and 4.2 K:


T(K)= Bi x Ri,


i= 0 to 8


(3.4)


where R is in ohms, and the coefficients Bi's are given in Table 3-3. The result of the fit is

shown in Fig. 3-3.


Table 3-3. Coefficients Bi's for Eq. (3.4), which is used to fit the calibration data for
LS-Burns between 2 K and 4.2 K


BO B1 B2 B3 B4

14.42179 -0.398553 7.02877 x 10-3 -7.9265 x 10-5 5.7991 x 10-7


-2.7254 x 10-9 7.9050 x 10-12 -1.284 x 10-14 8.9113 x 10-18


3.1.2.c Thermometry in magnetic fields

We have measured the resistance as a function of temperature of a RuO2 thick chip

resistor in zero field and in fields up to 8 T. Some of the results of this work are shown in

Fig. 3-4. This thermometer (named D-lk3) was manufactured by Dale Electronics, and is

the RC-550 type with a nominal room temperature value of 1 kO. The

magnetoresistance, R(B), of this unit was first studied by Meisel, Stewart, and Adams

(1989), who reported [{R(8T) R(0)}/R(0)] = 8.8 % at 80 mK Using this value, and our
















50



Ru02 thick chip
20



O 10


o : zero field
5 0:8T




50 100 200 500
T (mK)





Fig. 3-4. Resistance of a RuO2 thick chip thermometer (D-lk3) at zero field (o), and
8 T (0). The lines are the results of fits : loglo(R) = 5.77 3.266 loglo(T) +
0.497 [loglo(T)]2 for zero field, and loglo(R) = 5.25 2.785 loglo(T) + 0.391 [loglo(T)]2
for 8 T.








observed value [{R(8T) R(0)}/R(0)] = 3.3 %, we estimate the heating generated by our

experiments (mutual inductance, see Chapters 4 and 5) to be about 3 mK at 8 T and

80 mK. Similar calculations give a temperature rise of about 1 mK at 8 T and 100 mK

These results indicate that heating does not play an important role in our field experiments,

which do not extend above 8 T, or below 150 mK.

For the aforementioned magnetic field experiments, the sample was mounted on

the end of a copper finger located at the center of the superconducting solenoid providing

the field. The temperature was regulated with thermometers No. 26 and LS-Burns, which

were mounted on the mixing chamber. To determine if a thermal gradient existed between

the mixing chamber (thermometer) and the end of the copper finger (sample), we

measured the resistance of another Dale thick chip when mounted on the mixing chamber

(first run), and on the end of the finger (second run). The results are shown in Fig. 3-5,

and indicate that there was a negligible temperature difference present between the two

locations, and that the temperature read by the thermometers was that of the sample.


3.1.3 Temperature Control

The circuit used to control the temperature is shown in Fig. 3-6. A Quick Basic

routine, which includes the thermometer calibration fits (Eqs. 3.2, 3.3, and 3.3), sends a

command to the digital-analog converter (DAC, model 59501B from Hewlett Packard),

which sends a constant voltage to the ac resistance bridge (LR400 from Linear Research).

This voltage corresponds to a resistance value, which corresponds to a temperature. The

relation between the voltage and the resistance is given in the manual of the LR400, and is

entered in the computer routine, in addition to the relation between the resistance and the

temperature (calibration fits). The LR400 compares the resistance set by the computer,

via the DAC, and the resistance read from the thermometer (No. 26 or LS-Burns). A

constant voltage, corresponding to the difference between the two resistances, is then sent

by the LR400 to the temperature controller (LR130 from Linear Research). If the















50



RuO2 thick chip

20



Ol 10

o mounted on
mixing chamber
5 : mounted on
copper finger


50 100 200 500
T (mK)




Fig. 3-5. Resistance of a RuOz thick chip thermometer (D-lk "B7p21") mounted on the
mixing chamber (o), and on the end of the copper finger used for the magnetic field
experiments (0). The line is the result of a fit : loglo(R) = 5.50 3.06 loglo(T) +
0.46 [loglo(T)]2.











Computer


Thermometer

4-wire


Heater

r^V---


DAC
HP 59501B







LR400


LR130


Fig. 3-6. Circuit used for temperature control.









temperature of the refrigerator is too low compared to the set temperature, the LR130

sends a voltage to a heater (21.7 kQ resistor). If the temperature is too high, the LR130

reduces the voltage sent to the heater, and lets the refrigerator decrease the temperature.

For 5 mK steps, temperature stability (within 0.5 mK) can be achieved in about 2 min.

In practice, the method just described leads to oscillations around the set

temperature on the order of 3 mK. This effect can be problematic for experiments

demanding a temperature to be reached without actually going above it. For example, in

Chapter 5, we determine the critical temperature, Tc, of UBe13 samples as a function of

magnetic field. In these experiments, it is important to reach Tc from the superconducting

state only. An overshooting of the temperature above Tc would trap flux inside the

sample and affect the determination ofTc. For these experiments, the temperature control

was performed by replacing the HP 59501B by a computer-controlled, home-made DAC

(built by Jeff Legg of the electronics shop) giving a constant voltage that can be ramped

by steps corresponding to less than 0.05 mK The temperature was increased very slowly

(10 mK/hour) while the data was taken continuously. Using this method, the only

temperature fluctuations that the sample experiences come from the instabilities of the

refrigerator, and are typically less than 0.5 mK in magnitude.


3.2 ac Susceptibility

One of the experimental techniques utilized in the experiments discussed in this

dissertation is the (standard) mutual inductance technique. In the first subsection, the

hardware used for this technique is described. As will be shown in subsection 3.2.2, the

mutual inductance technique permits the measurement of the ac susceptibility of materials.

In the case of superconducting materials, the ac magnetic susceptibility of the sample is

related to the penetration depth. The study of the penetration depth as a function of

temperature for UPt3 and UBel3 is central to this dissertation, as is discussed in Chapter 4

and Chapter 5. Therefore, it is important to derive the relationship between the detected









signal from a mutual inductance experiment and the penetration depth of a

superconducting sample. This derivation is given in the second subsection.


3.2.1 Hardware Used for the Mutual Inductance Technique

The circuit used for the mutual inductance measurement is shown in Fig. 3-7. An

important feature of our measurement is the use of two PAR 124A (from EG&G) lock-in

amplifiers to detect the real and imaginary components of the signal We also attempted

to carry out our measurements with an SR530 from Stanford Research Systems and a

5302 from EG&G, but the PAR 124A provided the highest signal to noise ratio (typically

5000 compared to about 50 for the other two devices). However, two disadvantages with

using the PAR 124A exist. First, these lock-in amplifiers can only detect one phase of the

ac signal at a time, so two of them had to be used, one for each phase. Second, they are

not IEEE interfaced, so an HP voltmeter was introduced, between the lock-in amplifiers

and the computer, for data acquisition. Furthermore, the PAR 124A lock-in amplifiers are

constant voltage sources; thus, a 47 k. resistor was placed in front of the primary coil to

insure that the current, and ac magnetic field produced inside the primary, stayed constant.

The drift in the excitation current was monitored by taking data as a function of time and

at constant temperature and was found to be negligible.

Another important aspect of our measurements is the fabrication of the coils

(primary and secondary). The geometry and dimensions of a typical set of coils are shown

in Fig. 3-8. The materials used for these coils are either Vespel Sp-1 polymide (from

Du Pont de Nemours) or phenolic, (available from McMaster-Carr). These materials are

easily machinable and give a relatively low background contribution, possessing a weak, if

any, temperature dependence. (We note that Vespel Sp-22 was also used, but had a

susceptibility that was strongly temperature dependent, most likely due to the graphite

present in it.) The primary coil is wound with approximately 500 turns of niobium-

titanium superconducting wire (from Niomax Superconductors, wire CNA36/05).




62



Computer




Multimeter : HP 3457A




EG&G EG&G
124A Ref, 124A
in out of
phase phase


47 KOhms
Cryostat





Sample






Primary Secondary


Fig. 3-7. The circuit used for the mutual inductance technique.












//-


primary


sampLe


Cu wires


0,2"


A


f-


secondary






0.75"


0,50"


Fig. 3-8. The geometry and dimensions of the primary and secondary coils used for the
mutual inductance technique. The primary coil is located inside the secondary coil.








This wire (0.002 inches in diameter) is multifilament and possesses a Cu-Ni matrix. With

the excitation level on the lock-in amplifier set on 10 Vms, the ac magnetic field produced

at the center of the primary coil is about 10 pT. The secondary coil is wound with

approximately 10,000 turns of copper wire (5000 on each half). Because the net currents

flowing through the secondary coil are on the order of 1 nA, and the resistance of the coil

is about 1 kh, the power dissipated in the coil is negligible (1 x 10-15 W).


3.2.2 Principles of the Mutual Inductance Technique

In this section, we show how the mutual inductance technique allows one to

measure the penetration depth of a superconductor, X(T). In particular, the relationship

between the magnetic susceptibility of a sample, X(T) = X'(T) + i X"(T), and X(T) is

reviewed.

When a specimen is located in one of the two counterwound pick-up coils

(Fig. 3-7), the signal detected by the lock-in amplifiers corresponds to the difference

between the voltage due to d(/dt (( is the magnetic flux) in the two pick-up coils, given

by


V=27t d(B(r) rdr) iCtr2 Bac (3.5)
dt


where rs is the radius of the sample, o = 27tf, and fis the probing frequency. In addition,

we have assumed a circular sample cross-section and B(r,0,z) = B(r). We may define the

magnetic susceptibility x by


2 B(r) rdr = go(l + )Hac (3.6)
SO








where Bac = Po Hac, with to being the permeability of free space. Substituting Eq. (3.6)

into Eq. (3.5) yields


V = io 7 rs2 Bac X (3.7)


In other words, from Eq. (3.7), the detected voltage is directly proportional to the

magnetic susceptibility of the sample.

For a normal metal, we know from Maxwell's equations that the field inside the

sample must satisfy


V2 B + K2 B = 0, (3.8)


where K2 = i 47xtno/c2 and an is the normal metal conductivity. Using the definition of

the normal skin depth:



5(T) = 2(, (3.9)
p Cn(T) o



we obtain

(1 + i)
K(T) (3.10)
6(T)


For a cylindrical sample of radius rs with the field parallel to its axis, Eq. (3.8) can be

solved explicitly, giving the textbook result


Jo(Kr)
B(r) = B,, J(K (3.11)
a o(Kr,)








where Jo is the 0th order Bessel function (Landau and Lifshitz). Substituting Eq. (3.11)
into Eq. (3.6) gives the following expression for the magnetic susceptibility:


X = X' + i" + (Kr (3.12)
4n K rs J(Krs)'


where J1 is the 1st order Bessel function. The Bessel functions Jo and J1 for imaginary
arguments (peiO) can be written as


Jo(pei) = Uo(p,4) + i Vo(p,o) (3.13)


0( / 2)2S
where Uo(p,4) = (-1) (p/2) cos(2S) (3.14)
s=o (S!)2


Vo(p,4) = (-1) (/2)2S sin(2Sd) (3.15)
s=-o (S!)2


and Jl(peiO) = Ul(p,4) + i VI(p,d), (3.16)



dp


and Vi(p, ) [Uo(p,o) sin(o) Vo(p,4) cos(4))]. (3.18)
dp


Equations (3.13) through (3.18) were obtained from National Bureau of Standards
(1943). The real and imaginary parts of the susceptibility can then be written as


X = _1 + 2 cos(O) (U1 U + V Vo) + p sin() (VI U U1Vo) ] (3.19)
47 [-2 (Uo2 + V2)(
4x P2 (Uo + V02)





67



and X" 1 p cos() (V1 Uo U V) p sin(4) (V Vo + U1 Uo) )
4x p2 (Uo2 + Vo2)



where, for the normal state, p and 4 are obtained from Eq. (3.10), giving


p= rs and =tan- (1)- (3.21)
8 4


In Fig. 3-9, X' and X" are plotted as a function of rs/8. As will be shown in the next

section, the shapes of X'(rs/) and X"(rs/8) are of significant importance. By measuring

both the in-phase component of the output signal and its quadrature, we obtain x'(T) and

X"(T) independently. From Fig. 3-9, we note that for rs/6 < 0.5, X'(rs/6) and X"(rs/6)
approach zero. This result will be verified in our experiments presented in Chapter 4. For

example, sample No. 1 (see section 4.3.1), with rs z 0.20 mm and 8(Tc, 317 Hz) = 2.7 mm

gives rs/8 ; 0.075 << 0.5, so that no significant signal is observable above Tc (see

Fig. 4-13 for example).

Below Tc, the above analysis still holds, but an is replaced by as = a1 + ia2, where

ao and a2 are the real and imaginary parts of conductivity respectively, and Eq. (3.10)
becomes


2(T) i 47 a(T)o i 47rta(T)o i 47a2(T)o 2i 1
K2(T) (3.22)
Kc2 2 c2 6(T)2 X(T)2



where X(T) is the penetration depth and is given by


S(c2
(2 T) = (3.23)
41 02(T)















0.4


0.2

S0.0


-0.2


< -0.4


-0.6


-0.8


-1.0


2 4 6 8


rs /





Fig. 3-9. X' and X" as a function ofrs/8, and calculated from Eqs. (3.19) and (3.20) for the
normal state, i.e. p = rs/8 and 4 = x/4.








Just below Tc (T/Tc 1), both components of the complex conductivity contribute to

X'(T) and X"(T), so that one cannot directly measure X(T) from either X'(T) or X"(T). In
other words, both the magnitude, p, and the phase, 4, of the imaginary argument in

Eqs. (3.19) and (3.20) are functions of X(T) and 6(T). This effect makes the analysis of

the data around Tc difficult. In fact, several different theories exist regarding the

microscopic mechanism responsible for the peak in X"(T) in the vicinity of Tc, observed in

some materials (Maxwell and Strongin, 1963; Khoder, 1983; Hein, 1986). These theories

will be discussed in the following subsection. At lower temperatures (T/Tc << 1), where

the real part of the complex conductivity, Co, drops to zero, Eq. (3.22) takes the simple

form:


K (3.24)
X(T)


r.
so that p and = tan-1 () (3.25)
X(T) 2


Substituting Eq. (3.25) into Eqs. (3.14), (3.15), (3.17), and (3.18) gives


= (r/2h)2s
Uo = ,(rX (3.26)
sIo S!2


Vo 0, (3.27)


U =0, (3.28)


V = 2S (r/2)2S-1'
S=S!2 (3.28)
s=o S!









Substituting these results into Eqs. (3.19) and (3.20) gives


X'(T) I (-1 + 2(T) (3.29)
47c rs


and X"(T) = 0. (3.30)


Equations (3.29) and (3.30) are the central results of this section. They show that when

the quadrature of our experimental output signal, X"(T), is zero, the in-phase signal, x'(T),

is directly proportional to the penetration depth of the superconductor. This result can

also be derived by assuming a field distribution inside the superconductor given by


B(r) = Bace-(rs-r)/X (3.31)


and substituting it in Eq. (3.6). From Eq. (3.7), we know that the output signal is also

proportional to the applied ac field, Bac, and to the radius of the sample, rs. The

uncertainties in the exact values of Bac, the gain of the lock-in amplifier, and more

importantly rs, keep us from extracting precise absolute values for X(T). Consequently,

our mutual inductance results only enable us to measure X(T) in arbitrary units.

In order to validate our technique, we have measured a number of well

characterized superconductors such as Al and Zn. Figure 3-10 shows x'(T) and X"(T) for

a cylindrical specimen (1 = 2.5 mm, rs = 0.2 mm) of 6N purity aluminum (from Morton

Thiokol). The onset transition temperature was observed at 1.180 K which is in excellent

agreement with earlier work on aluminum by other groups (McLean, 1962; Tedrow et al.,

1971). An important aspect of the data is that x'(T) appeared to reach its minimum value

just below Tc, i.e. X'(T) was temperature independent, within our experimental resolution,

from 1.1 K to Tm. This result can be explained by an exponentially decaying X(T), with




















XI


Aluminum

1 f = 317 Hz
-d




->< 3 X
-a2







I ,I I I I

0.2 0.4 0.6 0.8 1.0 1.2 1.4
T (K)





Fig. 3-10. X'(T) and X"(T) for Al at 317 Hz. The onset transition is 1.180 K Within our
experimental uncertainties, x'(T) was temperature independent below 1.1 K









changes in X(T < 1.1 K) smaller than our experimental resolution. We also performed a

measurement on a cylindrical (1 = 3.8 mm, rs = 0.95 mm) zinc sample (a standard from

National Bureau of Standards). The data on zinc, shown in Fig. 3-11, exhibited a "step-

like transition", similar to the aluminum data, Fig. 3-10.


3.2.3 Peak in y"(T) in the Vicinity ofTc and in Zero de Magnetic Field

A great number of measurements of X'(T) and X"(T) on various superconductors

and in zero dc field have indicated the presence of a peak in x"(T), located at the midpoint

of the transition of X'(T). The peak has been observed in conventional superconductors,

such as tin, lead-bismuth alloys, and niobium (Maxwell and Strongin, 1963; Strongin et

al., 1964, Hein, 1986), but also in organic (Ishida and Mazaki, 1981; Kanoda et al.,

1990), heavy fermion (Koziol et al., 1992; Koziol, 1994; Signore et al., 1992; and this

work), and high-Tc superconductors (Mazaki et al., 1987; van der Beek and Kes, 1991).

The possibility of studying superconductivity through the presence of this peak (as well as

its magnitude, anisotropy, temperature and frequency dependence) has motivated a great

deal of work on this topic (Maxwell and Strongin, 1963; Gregory et al., 1973;

van der Klundert et al., 1973; Khoder, 1983, Hein, 1986; Brandt, 1991; Itzler et al.,

1994). The difficulty in understanding this peak comes from the fact that, as was shown in

the analysis presented above, both the real and the imaginary parts of the conductivity

contribute to x'(T) and X"(T). In this subsection, three possible explanations for this effect

are reviewed.

A simple approach was described by Gregory et al. (1973), who considered the

peak in X"(T) as a purely normal metal effect. In the previous subsection, expressions

for x'(T) and X"(T) were given in Eqs. (3.19) and (3.20), which were used in conjuncture

with Eq. (3.21) to plot the susceptibility as a function of rs/8 in normal metals (Fig. 3-9).

From this figure, we see that X"(rs/6) goes through a peak near rs/6 z 1.8. Given a sample

of radius rg, the ratio rs/6 is determined by 6 only, namely the normal metal skin depth,
























nrne mmm _________________ _OOOOOOa0OOSLA&M


Zinc

f = 317 Hz


0
0
0
0
0
0
0
0

0
0

0

0
0


-10


0.2


ocDoDooD0 4o-mooooooooco


I I I I


0.4


0.6

T (K)


0.8


1.0


Fig. 3-11. X'(T) and X"(T) for Zn at 317 Hz. The onset transition is 0.9 K Within our
experimental uncertainties, X'(T) was temperature independent below 0.8 K


I x"


0








-5


I I 1 1


q








defined in Eq. (3.9). For example, a UPt3 cylindrical sample, of radius 0.2 mm (sample

No. 1 in Chapter 4), and of resistivity 0.8 pDcm (just above Tc), has a skin depth of

1.79 mm at 317 Hz (a typical value in our low frequency work), giving rs/6 = 0.11, to the

left of the peak in X"(rs/8) shown Fig. 3-9. As the resistivity drops to zero below Tc, the

skin depth goes to zero and the ratio rs/8 approaches infinity. In other words,

Gregory et al. argued that the peak in X"(T) observed in superconductors was simply the

peak in X"(rs/6) as the ratio rs/5 goes from a small value (less than 1.8) to infinity, see

Fig. 3-9. This scenario is illustrated in Fig. 3-12, where x'(T) and X"(T) were calculated

for the above example and from a resistivity curve (shown in the figure) that one might

expect for UPt3. This explanation for the presence of the peak in X"(T) is based solely on

normal state electrodynamics and the sample size, but does not require the material to

become superconducting. In fact, a few non-superconducting materials exhibit x'(T) and

X"(T) behaviors which look similar to a superconducting transition. For example, it was
noted by Gregory et al. that Ga-In possesses a peak in X"(T) around 25 K, associated with

a drop in x'(T). However, specific heat and dc magnetization data clearly show that this

material is not superconducting at 25 K This example reminds us that it is prudent to

study a material with several different probes before claiming superconductivity. One

major argument stands against Gregory et al.'s explanation: the peak in X"(T) is developed

after the onset of superconductivity, so the electrodynamics of the superconducting state

(namely the complex conductivity) must play a role.

Maxwell and Strongin (1963) proposed a different interpretation, although their

explanation is still based on the normal state behavior of '(rs/8) and X"(rs/S), and Fig. 3-9.

They suggested that the electrodynamics of a superconductor could not produce a peak in

X"(T), and that both X'(T) and x"(T) change monotonically through the superconducting
transition. To account for observed peak, they proposed that some superconductors

(labeled as filamentary superconductors) exhibited traces, or filaments of

superconductivity in the normal state, just above Tc. These superconducting inclusions





75












0.5 2.0



X" : A

1.5
A A& ^ A" AAAAAA AAAAAAA
0.0 ""AA oooooooooooooooooooooooooooooooooohAOOooOOOOoo
0

u 0
1.0 "
c 1.0

P .++++++++++

-0.5 -
0 +
0.5


0 +
0 +
0
0 +
0 +
000000 +
-1.0 .... ****'-+ 0.0

0.4 0.5 0.6 0.7

T (K)





Fig. 3.12. x'(T) and is X"(T) calculated from Eqs. (3.19) and (3.20) using the normal state
relation Krs = rs(l+i)/6 and the resistivity values shown here. Based on these kinds of
plots, Gregory et al. suggested that the peaks observed in X"(T) of superconductors, in the
vicinity of Tc, can be simply explained with the electrodynamics of the normal state.









would decrease the average resistivity of the material, thereby increasing the ratio rs/6

above 1.8 and creating a peak in X"(T). Physically, the decrease in resistivity leads to an

increase in the current density, which in turn increases dissipation inside the sample so that

X"(T) increases. As the temperature is lowered further, eddy currents become significant,
and through their shielding, decrease the current density and the losses, i.e. x"(T)

decreases. Thus, this explanation gives a means to differentiate between bulk and

filamentary superconductors, provided that the ratio rs/6 is lower than 1.8 before the

appearance of the superconducting inclusions.

Khoder (1983) proposed a third interpretation, which was not based on the normal

state properties of X'(T) and x"(T). He calculated X'(T) and x"(T) directly from the values

of a and 02 predicted by the weak coupling BCS theory. Khoder showed that coherence

effects, which cause a peak in oa below Tc (Tinkham, 1975), could account for the peak

observed in superconductors. Within this picture, the peak in X"(T) arises from the

competition between two effects. The first is the ability of the supercurrents to be

accelerated, which is represented by ac(T). The second is the Meissner effect, represented

by G2(T), which reduces the field amplitude in the superconductor and thus prevents the
energy absorption. Khoder's conclusion was that all bulk superconductors should exhibit a

peak in x"(T), although it might be too small to detect experimentally for some materials.

The question on the origin of the peak in X"(T) has not been settled yet. It is

possible that a combination of the above scenarios actually takes place. The presence of

this effect in high-Tc materials has created new interest for this topic. In these materials,

dissipation due to flux flow plays an important role, and is most likely related to the

observation of the peaks. For superconductors possessing nodes in their energy gap, the

finite quasiparticle density below Tc contributes to X"(T) and must also be taken into

account in the analysis. In conclusion, the behavior of X'(T) and x"(T) just below Tc is a

complex, unsolved problem. Understanding the peak in C"(T) should provide information

about superconductivity, and this topic should be studied further in the future.









3.3 Tunnel Diode Oscillators

Tunnel diode resonating circuits have been used to study penetration depth, X(T),

of various superconductors for a number of years (Tedrow et al., 1971; Varmazis and

Strongin, 1974; and Varmazis et al., 1975). In this section, the hardware associated with

this technique is presented first, followed by a discussion on how the penetration depth of

a superconductor, X(T), can be measured using this method.


3.3.1 Hardware Used for the Tunnel Diode Oscillator Technique

3.3.1.a Circuit

The circuit used for the tunnel diode oscillator (TDO) technique is shown in Fig.

3-13. The voltage divider is comprised of two 1.35 V mercury batteries, and two variable

resistors (0-lkQ and 0-10k) connected in parallel. The 10 kM resistor is connected in

series with the load. The tunnel diode is connected in parallel with a capacitor, C2 = 5 pF,

which contributes to the stabilization of the oscillations. The tank circuit is comprised of

an inductor, L, inside which the sample is placed, and a capacitor, Cl, the value of which

can be varied to change the resonance frequency of the circuit. The rf signal is amplified

(using a model W500K from Tron-Tech) before being read by a frequency counter (model

5385A from Hewlett-Packard), which is IEEE interfaced.

3.3.1.b Coils

The inductors are fabricated from phenolic (available from McMaster-Carr) and

are wound with several hundred turns of copper wire. A number of coils, with

inductances on the order of 1 p.H, were fabricated in order to maximize the packing factor

for the differently sized samples.

3.3.1.c Tunnel diodes

A tunnel diode is a heavily doped p-n junction, which combines a tunneling current

with a regular diode current to give its unusual I-V curve, shown in Fig. 3-14. The diode















F
C


Fig. 3-13. The circuit used for the tunnel diode oscillator technique.


frequency Voltage
ounter
IP 5385A Divider






amplifier

BD-5 ( C2






,Sample

L C 1

p-
0=


/^


rf







0 200 400 600 800 1000


40


30


C',
a


20


10


0



, 10
E

- 5



0


0 200 400 600
V (mV)


800 1000


Fig. 3-14. Typical I-V characteristic of tunnel diodes at room temperature (0), and at
77 K (0). (a) model BD-5 and (b) model BD-6, measured in our laboratory.








can be biased so as to possess an effective "negative resistance". Because of this property,

tunnel diodes can be used to build oscillating circuits, amplifiers and other devices, which

are discussed in detail by Chow (1964). The tunnel diodes that we use are models BD-5

and BD-6 from Germanium Power Devices (~$50 each).


3.3.2 Tunnel Diode Oscillators and Penetration Depth

The tunnel diode oscillator operates on the basis of a small bias voltage
( 150 mV) being applied across a tunnel diode to keep it within its negative resistance

region. The diode then generates a small oscillating current (~ 100 pA) through a tank

circuit connected in series. The specimen to be studied is placed inside the inductor, L, of

the circuit. A change in either the penetration depth for T < Tc, or the skin depth for

T > Tc, causes a change in inductance which in turn results in a shift in the resonant

frequency. It can be shown that the change in frequency of the oscillator is given by

Af AL 2 AR 1
=- (1- Q)- R( Q)' (3.32)
fo 2Lo Q2 R Q2


where R is the resistance of the tank circuit, Q = 27fL/R, Af= f(Tm) f(T), fo = f(Tm),

AL = L(Tnn) L(T), and Lo = L(T ) (Chow, 1964). Thus, if Q is large enough, the

relative change of frequency is simply given by

Af AL
f L-- (3.33)
fo 2Lo


Tedrow (1971) measured the Q of tunnel diode oscillating circuits similar to the one used

in our work, and found that if the Q is large enough (> 100) to allow oscillations, then it is

large enough to make the terms in 1/Q2 in Eq. (3.32) negligible, so that Eq. (3.33) can

indeed be applied. We qualitatively verified this assertion by performing TDO








measurements on Cu with various size inductors and samples, and found that for small L

(low Q) the circuit did not oscillate, but for larger L (high Q) oscillations were stable.

Equation (3.33) can also be written as


Af t rs AA
f A (3.34)
fo A


where rs is the radius of the sample, A is the cross-sectional area between the coil and the

sample, and A is either the skin depth 6 when T > Tc, as defined in Eq. (3.9), or the

penetration depth for T < Tc, as defined in Eq. (3.23). Consequently, the experiment

consists of monitoring the frequency as a function of temperature.

The disadvantage of this technique, compared to a mutual inductance

measurement, is its inability to measure the resistive contribution. On the other hand, one

advantage comes from the fact that, in the normal state, Af/ fis proportional to AM(T). If

the resistivity of the sample just above Tc is known, then one can use the normal state data

to calibrate the coil and estimate absolute values for X(T).

An additional advantage to this technique is its ability to achieve frequency stability

on the order of 5 parts in 106 or better. The radio frequency field, Bf, generated inside

the inductor was always less than 10 p.T.

To check our method, we have performed several runs on Al and Zn. The data for

a cylindrical specimen (1 = 6.3 mm, rs = 0.5 mm) of 6N purity Al is shown in Fig. 3-15.

The onset transition temperature was observed at 1.175 K, in good agreement with the

results of other groups (Behroozi et al., 1974), and with Tc observed in our low frequency

measurements, Fig. 3-10. The data fit well the BCS temperature dependence for a non-

local superconductor



(T) = (0) tanh 353 ( (3.35)
A(0) 4t A(0)



















15 *"
Aluminum
10 MHz
10 -

I
0

5







0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
T (K)





Fig. 3-15. The relative change in frequency, Af / f = [f(Tm) f(T) / f(Tn)], as a
function of temperature for Al at 10 MHz. The solid line is a nonlinear least squares fit to
the BCS temperature dependence given in the text, (assuming X(0) = 500 A,
(Raychaudhuri et al., 1983)). The transition temperature is 1.175 K, in good agreement
with the 317 Hz data (Fig. 3-10).





83


where 2A(t) is the BCS energy gap function, t= T/Tc, and we have assumed X(0) = 500 A,

which is the estimated value for aluminum (Raychaudhuri et al., 1983). The data for zinc

is shown in Fig. 3-16, and these results support the proceeding discussions.





















10 .-

Zinc
2 MHz
14-


,O 5



o

...- *, ,__ si .




0.0 0.2 0.4 0.6 0.8 1.0
T (K)





Fig. 3-16. The relative change in frequency, Af / f = [f(Ti) f(T) / f(Tmn)], as a
function of temperature for Zn at 2 MHz. The solid line is a nonlinear least squares fit to
the BCS temperature dependence given in the text, (assuming X(0) = 290 A,
(Raychaudhuri et al., 1983)). The transition temperature is 0.84 K.













CHAPTER 4
INDUCTIVE MEASUREMENTS -- UPt3


In this chapter, the inductive measurements performed on UPt3 are presented. In

section 4.1, a brief theoretical description of the penetration depth, ,(T), is given for both

conventional and unconventional superconductors. In section 4.2, the results of previous

experimental investigations of ,(T) for UPt3 by other groups are reviewed. Next, the

fabrication and treatment histories of the various samples used in our study are given

(section 4.3). Our results are presented and discussed in sections 4.4 and 4.5. Finally,

conclusions from this investigation are drawn.

4.1 Penetration Depth in Superconductors

Since the bulk of the work presented in this chapter relates to measuring the

penetration depth of UPt3, it is important to understand what one learns from such

measurements. In the first subsection, the relationships between penetration depth and

microscopic quantities, such as the density of superconducting electrons and the energy

gap, are presented. The different temperature dependence of the penetration depth

predicted by the BCS theory are then given. In the second subsection, the temperature

and frequency dependence of ,(T) in unconventional superconductors are discussed.

Finally, the motivation for studying the penetration depths of various superconductors is

summarized.


4.1.1 Conventional Superconductors

A good starting point is to consider the response of superconducting electrons to

an external electric field, E. The electrons will experience a force equal to -eE, such that