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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 11 Ground state of various heavy fermions systems.......................................... 21 Summary of lattice parameters of UPt3..................... ..................... 15 22 Summary of lattice parameters of UBel3 .................................................33 31 Diagnostic thermometers................................................... 51 32 Coefficients for LSBurns, 350 mK < T < 2 K.................... ........... ......... 55 33 Coefficients for LSBurns, 2 K < T 4.2 K........................... ........... 55 41 Predicted X(T) for the axial state and the polar state...................................... 93 42 Characteristics of the eight UPt3 samples investigated in this work .................. 99 43 Summary of .(T) obtained from this work................................................ 144 51 Characteristics of the three UBel3 samples investigated in this work.............. 170 LIST OF FIGURES Figure page 11 Definition of unconventional superconductivity ............................................. 7 21 Atomic configuration in the hexagonal unit cell of UPt3............................... 16 22 Specific heat of UPt3 in the vicinity of Tc ...................................... 20 23 BT phase diagram of UPt3 for the magnetic field parallel to the caxis............ 24 24 BT phase diagram of UPt3 for the magnetic field perpendicular to the caxis ..................... ....................... ............................................... 25 25 PT phase diagram of UPt3.............................................................................. 27 26 Crystal structure of UBel3....................................................................... 34 27 Upper critical field in UBe13 for a single crystal and a polycrystal .................. 40 28 Superconducting Tcx phase diagram for UlxThxBel3 ..................................43 31 Determination of the circulation rate of the dilution refrigerator....................... 50 32 Calibration curve for thermometer No. 26 ................................... ............ 53 33 Calibration curve for thermometer LSBurns.............................................. 54 34 Magnetoresistance of RuO2 thick chip thermometer .................................... 56 35 Resistance ofRuO2 thick chip thermometer mounted on the mixing chamber and on the cold finger...................................... ............ ... 58 36 Circuit used for temperature control............................................................ 59 37 Circuit used for the mutual inductance technique............................................ 62 38 Geometry and dimensions of coils used for the mutual inductance technique...................................................................................................... 63 39 C' and X" as a function ofrs/8 for a normal metal ........................................... 68 310 x'(T) and X"(T) for aluminum at 317 Hz..................................................... 71 311 X'(T) and X"(T) for zinc at 317 Hz.................................................................72 312 X'(T) and X"(T) calculated for a normal metal exhibiting a sharp drop in resistivity............................... ................... 75 313 Circuit used for the tunnel diode oscillator technique..................................... 78 314 Typical IV characteristic of tunnel diodes.................................... ............... 79 315 TDO results on aluminum at 10 MHz........................... ............................ 82 316 TDO results on zinc at 2 MHz............................................................ 84 41 Temperature dependence from BCS in the local limit....................................... 90 42 SEM pictures of UPt3 sample No. 1 ........................................................... 101 43 Resistivity as a function ofT2 for UPt3 sample No. 1................................... 102 44 Specific heat in the vicinity of Tc for UPt3 sample No. 1.............................. 103 45 SEM pictures of UPt3 sample No. 2 ............................................................ 105 46 SEM pictures of UPt3 sample No. 3 ........................................................... 106 47 SEM pictures of UPt3 sample No. 7 ........................................................... 107 viii 48 X'(T) for unannealed and annealed UPt3 sample No. 1 ................................... 111 49 x'(T) in the vicinity of Tc for UPt3 sample No. 1............................................ 112 410 x'(T) and X"(T) for UPt3 sample No. 1, as grown......................................... 113 411 X'(T) and X"(T) for UPt3 sample No. 1, annealed, etched, unpolished............. 114 412 X"(T) in the vicinity ofTc for UPt3 sample No. 1, before and after polishing ....................................... ................................... 115 413 X'(T) and X"(T) for UPt3 sample No. 1, annealed, etched, polished................. 117 414 X'(T) in the vicinity ofTc for UPt3 sample No. 1 for 48 Hz < f< 32 kHz....... 118 415 X'(T=80 mK, 600 mK) as a function of frequency for U Pt3 sam ple N o. 1 polished........................................................................... 119 416 BT phase diagram for UPt3 sample No. 1, polished.................................... 121 417 x'(T) for UPt3 sample No. 1, polished and with B < 0.5 T.............................. 122 418 X'(T) for UPt3 sample No. 1, polished and with 1 T < B < 1.6 T.................. 123 419 x'(T) and X"(T) for UPt3 sample No. 2......................................................... 124 420 X'(T) and X"(T) for UPt3 sample No. 3...................................................... 125 421 X'(T) and X"(T) for UPt3 sample No. 4...................................................... 127 422 x'(T) and x"(T) for UPt3 sample No. 5......................................................... 128 423 x'(T) and X"(T) for UPt3 sample No. 6......................................................... 130 424 X'(T) and x"(T) for UPt3 sample No. 7...................................................... 131 425 C'(T) and X"(T) for UPt3 sample No. 8......................................................... 133 ix 426 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 1, annealed, etched, unpolished ......................................................................... 134 427 [f(Tmi) f(T)] / f(Ti) for UPt3 sample No. 1, annealed, etched, polished.......................................................................... 136 428 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 2 ....................................... 137 429 [f(Tm) f(T)] / f(Tmi) for UPt3 sample No. 3 ......................................... 138 430 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 4 ........................................ 140 431 [f(Tmi) f(T)] / f(Tmin) for UPt3 sample No. 5 ....................................... 141 432 [f(Tin) f(T)] / f(Tmin) for UPt3 sample No. 6 ........................................ 142 433 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 7 ........................................ 143 434 Inductive response in the vicinity ofTc for UPt3 sample No. 2 and for low and high frequencies.......................................... ........... 150 435 [f(Tmin) f(T)] / f(Tmin) for UPt3 sample No. 1, annealed, etched, polished at 6 MHz and 33 MHz ....................................... 153 436 Upturn in x'(T) for UPt3 sample No. 1 and Bdc = 1.4 T .......................... ....... 155 437 Upturn in X'(Time) sample No. 1 and Bdc = 1.2 T and Bdc = 1.4 T................. 158 51 Upper critical field of niobium ................................................................... 163 52 Upper critical field ofa UBel3 single crystal............................................... 166 53 Anisotropy ofBc2(T) in UBel3............................................................................... 168 54 SEM pictures of UBel3 sample No. 1.......................................................... 171 55 SEM pictures of UBel3 sample No. 2......................................................... 172 56 SEM pictures of UBe13 sample No. 3.......................................................... 173 57 SEM pictures of UBe13 sample No. 3, after polishing.................................... 174 58 X'(T) for UBe13 sample No. 1, in the normal state.......................................... 176 59 X'(T) for UBe13 sample No. 2, in the normal state.......................................... 177 510 X'(T) for UBel3 sample No. 3, in the normal state........................................ 178 511 X'(T) in the vicinity of Tc, for UBe13 sample No. 1, Bdc 1 [100], with B dc 0.5 T .......................................................................................... .... 18 1 512 X'(T) in the vicinity of Tc, for UBel3 sample No. 1, Bde II [110], with B dc < 0.5 T .................................................................................................... 182 513 x'(T) in the vicinity of Tc, for UBel3 sample No. 2, Bdc  [100], with B dc < 0.5 T ....................................... .......................................................... 183 514 X'(T) in the vicinity of Tc, for UBel3 sample No. 2, Bdc I [110], with B dc 0.5 T .......................................................................................... .... 184 515 X'(T) in the vicinity of Tc, for UBe13 sample No. 3, Bdc 11 [100], with B dc < 0.5 T .................................................................................................... 185 516 X'(T) in the vicinity of Tc, for UBel3 sample No. 3, Bdc  [110], with B dc 5 0.5 T ................................................................................................... 186 517 X'(T) in the vicinity ofTc, for UBel3 sample No. 1, Bdc 1 [100], with 1 T < B dc 8 T .................................................. ....................................... 187 518 X'(T) in the vicinity ofTc, for UBel3 sample No. 1, Bdc I [110], with 1 T B dc < 8 T ............................................................................................. 188 519 x'(T) in the vicinity of Tc, for UBel3 sample No. 2, Bdc  [100], with 1 T < B dc < 8 T ............................................................................................. 189 520 X'(T) in the vicinity of Tc, for UBel3 sample No. 2, Bdc I [110], with 1 T B d 8 T .................................................... ...................................... 190 521 x'(T) in the vicinity of Tc, for UBel3 sample No. 3, Bdc 11 [100], with ST B dc 7.5 T ................................................ ....................................... 191 522 x'(T) in the vicinity of Tc, for UBel3 sample No. 3, Bdc 1 [110], with 1T < Bdc < 7.5 T ....................................................... ..................... 192 523 X'(T) in the vicinity of Tc, for UBel3 sample No. 1, Bac  [100], for two different runs.................... ...................................... 194 524 Complete BT phase diagram for UBe13 sample No. 1................................. 195 525 Complete BT phase diagram for UBel3 sample No. 2................................. 196 526 BT phase diagram for UBel3 sample No. 3 ........................ ...................... 197 527 x'(B) and X"(B) for UBel3 sample No. 1, for T = 200 mK and T = 600 m K ................................................................. ..................... 199 528 X'(B) and X"(B) for UBel3 sample No. 2, for T = 500 mK.............................. 200 529 x'(B) and X"(B) for UBe13 sample No. 3, Bdc  [100], for T = 100 mK.......... 201 530 x'(B) and x"(B) for UBel3 sample No. 3, Bdc  [100], for T = 175 mK .......... 202 531 x'(B) and X"(B) for UBel3 sample No. 3, Bdc 1 [100], for T = 250 mK.......... 203 532 x'(B) and X"(B) for UBel3 sample No. 3, Bdc  [100], for T = 400 mK......... 205 533 X'(T) for UBei3 sample No. 3, Bdc I [100], for Bdc = 3 T ......................... 206 534 X'(B) and X"(B) for UBe13 sample No. 3, Bdc 1 [110], for T = 100 mK .......... 207 535 x'(B) and X"(B) for UBel3 sample No. 3, Bdc 1 [110], for T = 250 mK......... 208 536 X'(B) and x"(B) for UBe13 sample No. 3, Bdc [110], for T = 400 mK.......... 209 537 X'(T) for UBel3 sample No. 3, Bdc  [110], for Bdc = 3 T .............................. 210 538 X'(T) for UBe13 sample No. 3, Bdc II [110], for Bdc = 2 T ........................... 211 539 x'(T) for UBel3 sample No. 3, Bde I [110], for Bde = 1 T ........................ 212 540 x'(T) for UBel3 sample No. 3, Bdc II [110], for Bdc = 0.5 T ......................... 213 541 X'(T) for UBel3 sample No. 3, B, I [110], for Bdc = 0 T.............................. 214 542 Complete BT phase diagram for UBe13 sample No. 3................................. 215 543 x'(T) for UBel3 sample No. 1, Bac 11 [100], in the superconducting state........ 216 544 x'(T) for UBe13 sample No. 1, Bac I [100], T/Tc < 0.5 ................................ 217 545 x'(T) for UBe13 sample No. 1, Bac II [110], in the superconducting state........ 218 546 x'(T) for UBe13 sample No. 1, B acI [110], T/Tc < 0.5 ................................ 219 547 X'(T) for UBe13 sample No. 3, Bac I [100], in the superconducting state........ 220 548 x'(T) for UBe13 sample No. 3, Bac 1 [100], T/Tc < 0.5 ............................. 221 549 Effect of demagnetization factor on Bc2(T).................................................... 227 550 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 BardeenCooperSchrieffer (BCS) model. Due to the great number of unusual properties they exhibit, heavy fermion (and high temperature) superconductors are considered good candidates for being nonBCSlike 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 fshell, 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 CurieWeiss 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 felectrons, the system loses its local moments and thefelectrons 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 11. 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 11 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 nonBCSlike 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 11, 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 semiheavy 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. 11. 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. 11. 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 nonnegligible at higher fields (Williamson, 1970; Butler, 1980). These anisotropies are caused by Fermi surfaces of cubic symmetry (nonspherical). 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, BT, phase diagram of a conventional type I superconductor exhibits one phase (the Meissner state), while the BT 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 Aphase and a Bphase 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 pwave (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 swave (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 Aphase, and one between the Aphase 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 fieldtemperaturepressure 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 nonexponential 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 T2.750.25 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 twodimensional 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 nonspherical 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 photoemission spectroscopy, Josephson junction tunneling (Tsuei et al. 1994), or nonlinear 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 spinorbit 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 onedimensional 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 twodimensional, 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 fieldtemperature 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 nonBCSlike 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 closedpacked (hcp) structure, with two formula units in the primary cell (Heal and Williams, 1955). Figure 21 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 caxis and the basal plane, respectively. The lattice constant and other crystallographic parameters are summarized in Table 21. Table 21. 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 1028 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. 21. 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 21. 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 threefelectrons per Uatom, 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 21, one finds kF 1.08 x 1010 m1. 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 104 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. 22. 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 caxis (Hasselbach et al. 1989; Bogenberger et al. 1993). The multiple phase diagram in the BT 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 nonexponential 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. 22. 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 BCSlike 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 caxis 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 zaxis 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 caxis. 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 caxis 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 caxis. 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 caxis) constructed from these sound measurements are shown in Figs. 23 and 24 (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. 23 and 24) 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 caxes, while Bc2(0) 2.5 T for the field oriented perpendicular. In addition, B* = 0.9 T for B parallel to the caxis, while B* ; 0.4 T for B perpendicular to the caxis. Furthermore, Bc2(T) has a clear kink at B* for fields perpendicular, while the kink may not be present for fields parallel to the caxis. Finally, the four transition lines seem to come to a tetracritical point at B* for B perpendicular to the caxis, 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. 23. The complete superconducting BT phase diagram of UPt3 for the magnetic field parallel to the caxis. 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. 24. The complete superconducting BT phase diagram of UPt3 for the magnetic field perpendicular to the caxis. 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. 25. Recently the complete BTP phase diagrams for fields parallel and perpendicular to the caxis 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 IJ 0.45 0.4' 1 1 0 2 4 6 P (kbar) Fig. 25. The temperaturepressure 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. 25). 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 welldefined 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 swave 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 (pwave) or a singlet superconductor with significant spinorbit 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 IV 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 IV 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 caxis 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 caxis 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 BT phase diagram and its anisotropy. Experiments probing the Fermi surface, such as the de Haasvan 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 BCSlike superconductor. This assertion is especially supported by the unusual phase diagrams (Figs. 23 and 24). 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. 26) 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 22. 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 22. 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 1027 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. 26. Crystal structure of UBe13. (a) Unit Cell showing the CsCl 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 1027 m3, giving kF = 8.68 x 109 m1. 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 Thdoped 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 40005000 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 GinzburgLandau 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 GrofAlltag 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 GroBAlltag 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 27 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 spinlattice 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. 27. 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 evenparity 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 nonBCSlike 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 UlxThxBel3 for which a great deal of experimental results have been published since Ott et al. (1984a) reported a nonmonotonic Tcx curve. A complete Tcx phase diagram, shown in Fig. 28, 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 swave 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 UxThxBe13 'c 0.5 I, I MAGNETIC 0.0 I I II I 0.0 2.0 4.0 6.0 X(%) Fig. 28. Superconducting Tcx 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 Bphases (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 Tcx 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 VHS4, 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 custombuilt Cryogenic Associates "superinsulated" model with a 6inch diameter fiberglass neck, a 12inch diameter aluminum "belly", and a 6inch 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 electroplating 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 electroplating. 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 103 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 mole1 K1). 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 31 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. 31. 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 31. 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 31. 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 "LSBurns" were adopted for them. We use No. 26 between Tmn and 400 mK, while LSBurs is used between 350 mK and 4.2 K We measure the resistance of these two thermometers with an Linear Research ac resistance bridge (LR400) 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 32 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 LSBurns (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 33 shows the calibration curve for LSBurns, 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 32. The result of the fit is shown in Fig. 33. 104 103 102 0.02 0.05 0.1 0.2 T 0.5 (K) 1 2 Fig. 32. 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. 33. Calibration curve for thermometer LSBurns. 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 32. Coefficients Ai's for Eq. (3.3), which is used to fit the calibration data for LSBurns 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 LSBurns 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 33. The result of the fit is shown in Fig. 33. Table 33. Coefficients Bi's for Eq. (3.4), which is used to fit the calibration data for LSBurns between 2 K and 4.2 K BO B1 B2 B3 B4 14.42179 0.398553 7.02877 x 103 7.9265 x 105 5.7991 x 107 2.7254 x 109 7.9050 x 1012 1.284 x 1014 8.9113 x 1018 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. 34. This thermometer (named Dlk3) was manufactured by Dale Electronics, and is the RC550 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. 34. Resistance of a RuO2 thick chip thermometer (Dlk3) 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 LSBurns, 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. 35, 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. 36. A Quick Basic routine, which includes the thermometer calibration fits (Eqs. 3.2, 3.3, and 3.3), sends a command to the digitalanalog 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 LSBurns). 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. 35. Resistance of a RuOz thick chip thermometer (Dlk "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 4wire Heater r^V DAC HP 59501B LR400 LR130 Fig. 36. 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 computercontrolled, homemade 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. 37. An important feature of our measurement is the use of two PAR 124A (from EG&G) lockin 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 lockin 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 lockin amplifiers and the computer, for data acquisition. Furthermore, the PAR 124A lockin 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. 38. The materials used for these coils are either Vespel Sp1 polymide (from Du Pont de Nemours) or phenolic, (available from McMasterCarr). These materials are easily machinable and give a relatively low background contribution, possessing a weak, if any, temperature dependence. (We note that Vespel Sp22 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. 37. The circuit used for the mutual inductance technique. // primary sampLe Cu wires 0,2" A f secondary 0.75" 0,50" Fig. 38. 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 CuNi matrix. With the excitation level on the lockin 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 1015 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 pickup coils (Fig. 37), the signal detected by the lockin amplifiers corresponds to the difference between the voltage due to d(/dt (( is the magnetic flux) in the two pickup 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 crosssection 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. 39, 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 inphase component of the output signal and its quadrature, we obtain x'(T) and X"(T) independently. From Fig. 39, 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. 413 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. 39. 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 = tan1 () (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)2S1' 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 inphase 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(rsr)/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 lockin 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 310 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. 310. 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. 311, exhibited a "step like transition", similar to the aluminum data, Fig. 310. 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, leadbismuth 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 highTc 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. 39). 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 4omooooooooco I I I I 0.4 0.6 T (K) 0.8 1.0 Fig. 311. 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. 39. 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. 39. This scenario is illustrated in Fig. 312, 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 nonsuperconducting 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 GaIn 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. 39. 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 highTc 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. 313. The voltage divider is comprised of two 1.35 V mercury batteries, and two variable resistors (0lkQ and 010k) 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 TronTech) before being read by a frequency counter (model 5385A from HewlettPackard), which is IEEE interfaced. 3.3.1.b Coils The inductors are fabricated from phenolic (available from McMasterCarr) 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 pn junction, which combines a tunneling current with a regular diode current to give its unusual IV curve, shown in Fig. 314. The diode F C Fig. 313. The circuit used for the tunnel diode oscillator technique. frequency Voltage ounter IP 5385A Divider amplifier BD5 ( 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. 314. Typical IV characteristic of tunnel diodes at room temperature (0), and at 77 K (0). (a) model BD5 and (b) model BD6, 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 BD5 and BD6 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 crosssectional 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. 315. 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. 310. 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. 315. 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. 310). 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. 316, 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. 316. 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 