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## Material Information- Title:
- Novel Heavy Fermion Behavior in Praseodymium-Based Materials: Experimental Study of PrOs4Sb12
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- 2008
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Electrical resistivity ( jstor ) Fermions ( jstor ) Ground state ( jstor ) Ions ( jstor ) Magnetic fields ( jstor ) Magnetism ( jstor ) Magnets ( jstor ) Specific heat ( jstor ) Symmetry ( jstor )
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- 7/12/2007
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NOVEL HEAVY FERMION BEHAVIOR IN PRASEODYMIUM-BASED MATERIALS: EXPERIMENTAL STUDY OF PrOs4Sbl2 By COSTEL REMUS ROTUNDU 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 2007 Copyright 2007 by Costel Remus Rotundu To my parents, Constantin and Elena Rotundu, for their sacrifices to ensure my education. ACKNOWLEDGMENTS I am dedicating this work to my dearest human beings, my parents Elena and Constantin Rotundu. I cannot find an adequate way to express my love and gratitude to them. I thank them for their infinite support in order to complete my education. I am grateful to my mother and my brother Romulus Neculai Rotundu for believing in me. I owe much to my adviser, Dr. Bohdan Andraka. He has been an incredible source of guidance and inspiration. He was a great adviser with endless patience. My education would not have been possible without his financial help (through DOE and NSF). I am deeply indebted to him. Special thanks go to Prof. Yasumasa Takano for teaching me so many experimental tricks; and for discussions, support, and great collaboration, especially at the National High Magnetic Field Laboratory (NHMFL). He was an endless source of energy. I would like to thank Prof. Gregory R. Stewart for letting me use his laboratory. I thank my other supervisory committee members (Profs. Bohdan Andraka, Gregory R. Stewart, Yasumasa Takano, Pradeep Kumar and Ion Ghiviriga) for reading this work and for their advices. I received help with many experiments at NHMFL and our lab from Dr. Hiroyuki Tsujii. I thank Drs. Jungsoo Kim and Daniel J. Mixson II for their technical advice. Other people in the field I would like to acknowledge are Prof. Peter Hirschfeld, the finest professor I ever had, who gave me insight on the theory of condensed matter physics; and Drs. Eric C. Palm and Tim P. Murphy for their help and support over more than 4 years at the SC\ I / NHMFL. I thank Center of Condensed Matter Sciences for the financial support through the Senior Graduate Student Fellowship. Last and not least, I would like to thank my high school physics teacher (Dumitru Tatarcan), who encouraged and guided my first steps in physics. TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. LIST O F TABLES . . . . . . . . . . LIST OF FIGURES . . . . . . . . . A B ST R A C T . . . . . . . . . . CHAPTER 1 INTRODUCTION .................................. 2 THEORETICAL BACKGROUND ......................... 2.1 The Crystalline Electric Field (CEF) for Cubic Group ............ 2.2 Conduction Electron Mass Enhancement (m*) Mechanism in PrOs4Sb12 . 2.2.1 Quadrupolar Kondo Effect ....................... 2.2.1.1 Thermodynamic Properties of the Quadrupolar Kondo 2.2.2 2.2.3 M odel . . . . . . . 2.2.1.2 Relevance for the Case of Pr3+ Ion in PrOs4Sb12 . Fulde-Jensen Model for m* Enhancement in Pr Metal . Fluctuations of the Quadrupolar Order Parameter ...... 3 PROPERTIES REVIEW OF THE PrOs4Sb12 ............. 3.1 Crystalline Structure .. .. .. .. ... .. .. .. .. ... . 3.1.1 Rattling of Praseodymium Atom .. ............. 3.1.2 V alence . . . . . . . . 3.1.3 Crystalline Electric Fields .. ............... 3.2 Normal-State Zero-Field Properties .. ............... 3.2.1 Specific H eat . . . . . . . 3.2.2 de Haas van Alphen Measurements .. ........... 3.2.3 R esistivity . . . . . . . 3.2.4 DC Magnetic Susceptibility .. ................ 3.3 The Long-Range Order in Magnetic Fields .. ........... 3.4 Superconductivity . . . . . . . 3.4.1 Unconventional Superconductivity .. ............ 3.4.1.1 The Double Transition .. ............. 3.4.1.2 Temperature Dependence of Specific Heat Below Tc 3.4.1.3 Nuclear Magnetic Resonance (Sb NQR) ...... 3.4.1.4 Muon Spin Rotation (pSR) .. .......... 3.4.2 Conventional Superconductivity .. ............. 3.4.2.1 Nuclear Magnetic Resonance (pSR) .. ....... 3.4.2.2 Penetration Depth Measurements (A) by pSR . 3.4.2.3 Low-Temperature Tunneling Microscopy ...... 4 EXPERIMENTAL METHODS ........... 4.1 The Samples: Synthesis and ('!C i o :terization ....... ......... 49 4.1.1 Synthesis ................. . . .... 49 4.1.2 X-Rays Diffraction ('!I i o :terization ...... . ..... 51 4.2 Specific Heat Measurements .................. ..... .. 52 4.2.1 Equipment .................. ............. .. 52 4.2.1.1 Cryogenics. .................. .. .... .. .. 52 4.2.1.2 Sample Platform ................ .... .. 55 4.2.2 Thermal Relaxation Method ................. . .. 55 4.3 Magnetic Measurements .................. ......... .. 57 4.3.1 DC Susceptibility .................. ......... .. 57 4.3.2 AC Susceptibility .................. . .. 58 4.4 Resistivity .................. ................. .. 59 5 MATERIALS CHARACTERIZATION ................ .... .. 65 6 PrOs4Sb12 . . . . . . ... . . . 67 6.1 Investigation of CEF Configuration by Specific Heat in High Magnetic Fields 67 6.2 Magnetoresistance of PrOs4Sbi2 ................... ... 73 7 PrlxLaxOs4Sbl2 ............... ............ .. 93 7.1 Lattice Constant ............... ........... .. 93 7.2 DC Magnetic Susceptibility ............... .... .. 94 7.3 Zero Field Specific Heat .......... . . .... 95 7.3.1 Specific Heat of PrOs4Sb12: Sample Dependence . .... 95 7.3.2 Zero Field Specific Heat of Pr1_-LaOs4Sb12 . . 96 7.3.2.1 Evolution of T, with the La Doping . . ... 97 7.3.2.2 The Discontinuity in C/T at T . . ...... 98 7.3.2.3 The Schottky Anomaly .............. .. .. 99 7.4 Specific Heat in Large Fields .................. ..... 101 7.5 Magnetoresistance of Pr1_-La1Os4Sb12 .............. .. .. 103 7.6 Upper Critical Field H,2 .................. ....... .. 107 7.6.1 AC Susceptibility ....... . . ........ 107 7.6.2 Determination of Hc2(T) by Specific Heat Measurements in Small M agnetic Fields .................. ........ .. 108 8 CONCLUSION .................. ................. .. 136 REFERENCES .................. ................ .. .. 139 BIOGRAPHICAL SKETCH ................... . ... 145 LIST OF TABLES Table page 2-1 The relevant states for the quadrupolar Kondo effect ...... . .... 23 3-1 The a values reported by different groups, extracted from fits of specific heat below T,. ....... .............. .............. .. .. .. 38 3-2 The a values reported by different groups from other measurements than specific heat. . . . . . . .. . . . . 38 LIST OF FIGURES Figure page 2-1 Cubic point group symmetry Th. .................... ...... 27 2-2 Lea, Leask, and Wolff's representation of CEF for J 4 (Lea et al., 1962). . 28 2-3 Representation of the U4+ ions in cubic symmetry undergoing quadrupolar Kondo effect . . . . . .. . . . 28 2-4 Mapping of the quadrupolar Kondo Hamiltonian onto the two-channel Kondo m odel. .. ... .. .. .. ... . . .. .. .. ....... .. .. 29 2-5 S, C, C/T, and x versus T/TK of the quadrupolar Kondo model (Sacramento and Schlotmann, 1991). .................. .. ........ 29 3-1 Crystal structure of PrOs4Sb12. ................ ....... 41 3-2 Fits of X(T) to either F3 or F1 CEF ground state, and C fitted by a two-level Schottky anomaly (Bauer et al., 2002). .................. .... 42 3-3 Fits of X(T) to either F3 or F1 CEF ground state model, calculated S(T) in both F3 and F1 CEF ground state models (Ti 11i i et al., 2003), and the measured S(T) (Aoki et al., 2002). .................. .. ....... 43 3-4 p(T), x(T) and C(T) of PrOs4Sb12 (Bauer et al., 2002). .. . ..... 44 3-5 Fermi surface of PrOs4Sb12 (Sugawara et al., 2002) ............... ..45 3-6 H-T phase diagram of PrOs4Sb12 by dM(T)/dT and dM(H)/dH measurements (T iv ii i et al., 2003) .................. .... 46 3-7 C(T) of PrOs4Sbl2 (Vollmer et al., 2003; M6asson et al., 2004) and the real part of the ac susceptibility (\ !' i.- .. et al., 2004) presenting double SC transition.. 46 3-8 The two superconducting phases of PrOs4Sb12: phase A and phase B (Izawa et al., 2003). The plot of the SC gap function with nodes for both phases (\! ,.1: et al., 2003) ................... ................... .. 47 3-9 Two SC transitions in 3(T) of PrOs4Sb12 (Oeschler et al., 2003) . .... 47 3-10 T dependence of the rate 1/Ti at the 2vQ transition of 123Sb for PrOs4Sb12 and LaOs4Sb12 (Kotegawa et al., 2003). .................. .... 48 3-11 Tunneling conductance between PrOs4Sb12 and an Au tip (Suderow et al., 2004). The gap is well developed with no low-energy excitations, sign of no nodes in the Fermi surface gap .............. ............ .. 48 4-1 Picture of PrOs4Sb12 large ( i-- .I (about 50 mg). ............... 60 4-2 PrOs4Sb12 samples prepared for (left panel) resistivity and (right panel) specific heat measurements. .................. ............ ..60 4-3 Schematic view of the 3He cryostat used in the measurements performed at University of Florida. ............... ............ .. .. 61 4-4 Schematic view of the calorimeter used in the Superconducting Magnet 1 (SC' \ 1), National High Magnetic Field Laboratory. .................. 62 4-5 The sample-platform/Cu-ring assembly. .............. ... 63 4-6 Specific heat C measurement process using the relaxation time method. . 64 5-1 y(T) of PrOs4Sb12. The high temperature effective moment is 3.65PB, very close to the one corresponding to free Pr3+, which is 3.58PB. . . ..... 66 5-2 x(T) of the non-f equivalent LaOs4Sb2. ................ ..... 66 6-1 C of PrOs4Sb12 in fields up to 8 T for H//(1 00) (upper panel), and H-T phase diagram in fields up to 8 T for H//(1 00) (lower panel) (Aoki et al., 2002). .. 78 6-2 C of PrOs4Sbl2 in 10 and 12 T in the vicinity of FIOP transition for H//(10 0). 79 6-3 C of PrOs4Sbl2 in magnetic fields 13, 13.5, and 14 T, for //(10 0). . 80 6-4 C of PrOs4Sbl2 in 16, 20, and 32 T, for //(10 0). .. . . ....... 81 6-5 H-T phase diagram of PrOs4Sbl2 for H//(1 00) (H>8 T). . . .. 82 6-6 Zeeman effect calculations for PrOs4Sbl2 in the F1 CEF ground state scenario. .83 6-7 Zeeman effect for PrOs4Sb12 in the F3 CEF ground state scenario. . ... 84 6-8 C of PrOs4Sb12 for f//(1 10). .................. ......... .. 85 6-9 C of PrOs4Sbl2 in H=12 T, for H//(1 00) (upper panel), and H//(1 10) (lower panel) ...................... ........ ... .... ... 86 6-10 H-T phase diagram of PrOs4Sb12 for H//(1 1 0) (H>8 T). . . ... 87 6-11 p versus T2, and p versus T for PrOs4Sbl2 .................. .. 87 6-12 p(T) of PrOs4Sbl2 in 3, 10, 15, 16, 17, and 18 T, between 20 mK and about 0.9 K. 88 6-13 p versus T2 of PrOs4Sbl2 for 3.5, 5.5, 7, 10, and 13 T . . ...... 89 6-14 a (p po+aT") versus H for PrOs4Sbl2 fields up to 18 T(upper panel). The residual resistivity po(H) (lower panel). .................. .... 90 6-15 A (p=po+AT2) versus H. .................. .. ....... 91 6-16 The calculated p(H) of PrOs4Sb12 (Frederick and Maple, 2003), for both F3 and FL scenarios .................. ................... .. 92 7-1 X-ray diffraction patterns of Pr1_-La0Os4Sb12 versus La content x for x=0, 0.1, 0.2, 0.4, and 1 .................. ................. .. 113 7-2 Lattice constant a of Pr1_-LaOs4Sb12 versus La content x. . .... 114 7-3 x(T) of Pr1_-LaOs4Sb12 normalized to Pr mole between 1.8 and 10 K, measured in 0.5 T ..................... ........ ... .... 114 7-4 X(T) of Pro.33Lao.670sSb12 versus T. The Curie-Weiss fit at high temperature (T>150 K) gives pMff 3.62/B/Pr mole. ................ ...... 115 7-5 C/T versus T for three different PrOs4Sb12 samples from different batches. 116 7-6 C/T versus T2 above T, of LaOs4Sb12 .................. .. 117 7-7 C/T versus T near T, for Pr1_-LaxOs4Sb12 for x=0, 0.05, 0.1, and 0.2. ..... ..118 7-8 C/T versus T near T, of Pr1_-La0Os4Sb12 for x>0.3 .............. .119 7-9 T, versus x of Pr1_-La0Os4Sbl2 .................. ........ .. 120 7-10 Total A(C/T) at T, and Xo versus x of Pr1_-La0Os4Sbl2 for 0>x>l. Xo is X at 1.8 K from Fig. 7-3. .................. ............. 120 7-11 C/T versus T of Pro.33Lao.670s4Sb12 fitted by F1 F5 Schottky. . ... 121 7-12 C/T versus T of Pro.33Lao.670s4Sb12 fitted by F3 F5 Schottky. . ... 121 7-13 C/T versus T of Pro.33Lao.670s4Sb12 fitted by singlet-singlet Schottky. ..... .122 7-14 C for x=0 (10 T) and 0.02 (8 and 9.5 T). .................. ... 122 7-15 f-electron specific heat of Pro.9Lao.1Os4Sb12 in magnetic fields. . ... 123 7-16 f-electron specific heat of Pro.sLao.20s4Sb12 in magnetic fields. . ... 123 7-17 f-electron specific heat of Pro.4La0.60s4Sb12 in magnetic fields. . ... 124 7-18 H-T phase diagram from C measurements for x=0, 0.02, 0.1, and 0.2. ....... .124 7-19 p(H) of Pro.95Lao.o50s4Sb12 at T 20 mK for H//I and HII (I//(00 1)). ... .125 7-20 p(H) of Pro.95Lao.osO4Sbi2 for HlI//(O0 1) at T 20 and 300 mK. ...... .125 7-21 p(H) of Pro.95Lao.0osO4Sb12 for Hil//(0 0 1) at T-20, 310, and 660 mK. . 126 7-22 p versus T2 of Pro.7La0.30s4Sb12 in 0 and 0.5 T. ................ 127 7-23 p versus T2 of Pro.7La0.30s4Sb12 in 9 and 13 T. ...... . . 128 7-24 p(H) of Pro.7Lao.30s4Sb12 for H parallel with all three ( i --1 1,graphic directions and 1//(001) ................... .... ...... ...... 129 7-25 p(H) of Pro.7Lao.3OS4Sb12 when H//I//(O0 1) at 20, 310, 660, and 1100 mK. 129 7-26 p(H) of Pro.33Lao.670s4Sb12 at 0.35 K. ............. .... 130 7-27 AC susceptibility versus T/T, of Prl_-LaOs4Sbl2, for x=0, 0.05, 0.4, 0.8, and 1. 131 7-28 C/T versus T near T, for two PrOs4Sb12 samples from different batches in low magnetic fields .................. ................. .. 132 7-29 C/T versus T, near Tc, of Pro.95Lao.o50s4Sb12 in low magnetic fields ....... .133 7-30 C/T versus T, near Tc, of Pro.9Lao.lOs4Sb12 in small magnetic fields. ..... ..133 7-31 C/T versus T, near Tc, of Pro.7Lao.30s4Sb12 in magnetic fields. . ... 134 7-32 -dH2V/dT versus x. .................. ............ 134 7-33 -dH21/dT/Tc versus x with the critical x,,r0.25. The inset shows A(C/T) at T, with x,,r 0.25-0.3. .................. ........... 135 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 NOVEL HEAVY FERMION BEHAVIOR IN PRASEODYMIUM-BASED MATERIALS: EXPERIMENTAL STUDY OF PrOs4Sb12 By Costel Remus Rotundu May 2007 C'!h ,i: Bohdan Andraka Major Department: Physics PrOs4Sb12 is the first discovered Pr-based heavy fermion metal and superconductor. Our high magnetic field specific heat measurements provided clear evidence for the non-magnetic singlet < iv-i iii., electric field (CEF) ground state. This CEF ground state precludes the conventional Kondo effect as the origin of the heavy fermion behavior. The superconductivity in PrOs4Sb12 is unconventional, as inferred from the double superconducting transition in the specific heat. Prl_-La1Os4Sb12 (0 crystals were synthesized and investigated in order to provide additional evidences for a postulated CEF configuration, to discriminate between different conduction electron mass enhancement (m*) mechanisms proposed, and to provide insight into the nature of the superconductivity. Lanthanum doping induces anomalously small increase of the lattice constant. The specific heat results in high magnetic fields indicated that CEF scheme is unaltered between x=0 and at least 0.2, followed by an abrupt (but small) change somewhere between x=0.2 and 0.4. Magnetoresistance measurements on La-doped samples were consistent with a singlet CEF ground state of Pr. Investigation of the specific heat discontinuity at T, and of the upper critical field slope at T, indicated that the electronic effective mass, m*, is strongly reduced with x, between x=0 and xce0.2-0.3, followed by a weak dependence on x for x>xr. Therefore, we have postulated that single-impurity type models cannot account for the heavy fermion behavior of PrOs4Sb12. Investigation of the magnetic phase diagram and magnetoresistance provided strong correlations between a closeness to the long-range order (antiferroquadrupolar type) and m*, -ii-_-, -ri-; a possibility of fluctuations of the antiferroquadrupolar order parameter responsible for m* enhancement. Lanthanum has very weak effect on the superconducting transition temperature in a stark contrast to other known heavy fermion superconductors. The study of superconductivity provided constraints on proposed theoretical models, including the two band model. CHAPTER 1 INTRODUCTION In the rare earth (Ce,Yb)- and actinide (U,Np)-based alloys the electronic states have an energy orders of magnitude smaller than in ordinary metals, and since c(k) h2k2/2m*, the effective mass m* is orders of magnitude larger than the free-electron value, hence the term heavy fermion. There are several excellent experimental and theoretical reviews [1-5] on heavy fermions. One hallmark of the heavy fermion character is the large Sommerfeld coefficient 7 of the specific heat. The specific heat of metals in the normal state at low temperature is approximated by C= T+3T3, where 7T is the electronic specific heat and 3T3 is the lattice (Debye) contribution. For a normal metal 7 is of order of 1 mJ/K2 mol, and for heavy fermion is from several hundred to several thousand mJ/K2 mol. The magnetic susceptibility X at high temperatures follows the Curie-Weiss form X=C/(T+Ocw), where C is a constant, and 6cw is the Curie-Weiss temperature. At low temperatures X(0) ranges from ~10 to 100 memu/mol. In the 1ni iii lly of heavy fermion metals, the electrical resistivity p at very low temperatures has a T2 dependence: p po+AT2, where po is the residual resistivity and A is on the order of tens of /pcm/K2, much larger than that of normal metals. There are about 20 heavy fermion systems that are superconductors and almost all of them are Ce- or U-based (there is one Pu-based heavy fermion superconductor: PuCoGas [6]). The filled skutterudite PrOs4Sb12 is the first discovered Pr-based heavy fermion compound that is a superconductor [7]. In the conventional heavy fermions, the only microscopic theories somewhat successful in accounting for the effective mass enhancement (m*) as measured by the specific heat are the S=1/2 and S=3/2 Kondo models. These models were initially proposed for Ce-based systems, whose effective degeneracies of f-electrons in ( i-I ,11 iiI" electric fields are either 2 or 4. The Kondo effect in these systems is anomalous because of strong spin-orbit coupling. There is one f electron per Ce atom and according to Hund's rules the total angular moment is J=5/2, which corresponds to the degenerate 6 level case. Crystalline electric fields split this multiple either in a) 3 doublets or b) one doublet and one quartet. Case b) can be only for cubic symmetries. Thus Ce-based heavy fermions with a doublet CEF ground state of Ce are described by the S=1/2 Kondo model while those with Ce in a quartet CEF ground state are described by S=3/2 Kondo model. Thus any understanding of heavy fermion behavior needs clarification of the CEF ground state. Unfortunately, CEF scheme is not known for U-based heavy fermions. Furthermore the valence of U (i.e., whether the electronic configuration is f2 or f3) is not known. The high temperature effective moments for f2 and f3 configurations that could be extracted from the high temperature susceptibility are almost identical. Studies of CEF's done directly by inelastic neutron scattering [8, 9] are more consistent with the f2 configuration, allowing for a similar CEF scheme as that for Pr. Therefore, the investigation of PrOs4Sb12 with Pr having 2 f-electrons might be relevant and help to the understanding of the large class of U-based heavy fermions, since CEF configurations are usually known for Pr. The non-magnetic < i lv-I !i i electric field ground state (thought as either singlet or doublet [7]) excludes the conventional Kondo effect as the origin of the l. i, -i-rermion behavior in PrOs4Sb12, which is considered to be the source of heavy fermion behavior in Ce- and U-based metals. The superconductivity in PrOs4Sb12 is unconventional, but different from that in Ce- and U-based materials. Marks of the unconventionality of superconductivity can be inferred from the double superconducting transition and power low dependence of the specific heat below the transition. The main goals of this work are: to settle the crystalline electric field ground state in PrOs4Sb12, to bring further evidences of the heavy fermion state in PrOs4Sb12, to differentiate between several models proposed for the conduction electron mass enhancement (m*) and to study the relationship between the correlation and the superconductivity. The outline of the dissertation is as follows: Chapter 2 presents the theoretical framework of this thesis. This C'!i lpter begins with the theory of crystalline electric field for the point group symmetry Th, followed by a presentation of the quadrupolar Kondo effect. Other proposed models of the conduction electron mass enhancement are also discussed. ('!i lpter 3 review the essential properties of PrOs4Sbl2. C'!i lter 4 gives a brief description of the apparatus and experimental methods used. A characterization of the materials synthesized and measured is given in Chapter 5. The experimental data are presented and discussed in ('!i lpters 6 and 7. ('!i lpter 6 focus on PrOs4Sbl2 itself (specific heat and resistivity in high magnetic fields). ('!i lpter 7 presents the study of Prl_-LaOs4Sbl2, 0 measurements are discussed. Finally, ('! Ilpter 8 summarizes the main findings and contributions to the field of Pr-based L i,',-i--Fermions. CHAPTER 2 THEORETICAL BACKGROUND 2.1 The Crystalline Electric Field (CEF) for Cubic Group In rare earth compounds, the < ,i-- i i.: electric fields are responsible for a wide v ii. I v of strongly correlated electron behaviors. The 4f-electrons in a rare earth ion experience an electrostatic crystal field potential created by the surrounding electric charge distribution (of the neighbor ions). The potential reflects the local point symmetry of the site of the rare earth ion. In the point-charge ionic model the CEF potential at position F due to the surrounding atoms is VCEF (r) q (2-1) where qj is the charge at the j"h neighboring ion, at Rj. If the magnetic ion has charge q, at r, then the 1 HCEF q- (2-2) i j i RjI The sum Yi is taken over electrons in unfilled shells [10]. The CEF potential can be evaluated in terms of Cartesian coordinates or in terms of spherical harmonics. Hutchings [10] evaluated the potential (2-1) for the simplest 3 arrangements of charges giving a cubic crystalline electric field. The three cases analyzed were when the charges are placed at the corners of an octahedron (sixfold coordination), at the corners of a cube (eightfold coordination), and at the corners of a tetrahedron (fourfold coordination). In Cartesian coordinates the potential (2-1) can be written as [10] V(x,y,z) = C4 + 4 + z4) 4] + D[(6 + + 6) + 5 2 4 2+ X4 2 X4 2+ 4 2+ Z24 24 6, (2-3) + -(xy +z +y +y +x +z )-y]), (23) 4 14 where d is the distance of the point charge q from the origin in each 3 cases. C4 and D6 are -70q/(9d5) and -224q/(9d7) for the eightfold coordination, +35q/(4d5) and -21q/(2d7) for the sixfold coordination, and -35q/(4d5) and -112q/(9d7) for the fourfold coordination respectively. In the spherical coordinates the same potential is written [10] as V = D4{Y + [Y44( Y-4(, )]} + D'{Y6~ ( ) [Y ( ) + -4(0 )]}, (2-4) where D' and D' are -56q v/(27d5) r4 and +32q /13/(9d7).r6 for the eightfold coordination, +7q //(3d5).r4 and +3q V7/13/(2d7)r6 for the sixfold coordination, and -28qv//(27d5).r4 and +16q G\ /13/(9d5).6 for the eightfold coordination respectively. The potential contains therefore terms of order 4 and 6 in coordinates. In general, the less symmetric is the site, the more potential terms occur in the expansion. There are 2 general rules that can tell us the number of nonzero terms in the CF potential. If there is a center of inversion at the ion site there will be no odd-n terms. Secondly, if the z axis is not an m-fold axis symmetry, the potential will contain V,, [10]. However, calculating the potential terms in Cartesian coordinates and even in spherical coordinates is tedious. A more convenient method is the so called operator equivalent or Stevens' operator technique [11, 12]. The Hamiltonian (2-2) is of form HcEF=- i elV(xi, Yi, Z). If f(x,y, z) is a Cartesian function, in order to find the equivalent operator to such terms as 3 f(xj,yz,zt), the coordinates x, y, and z are replaced by angular momentum operators J, Jy, and J respectively, taking into account the non-commutativity of Jjs. This is done by replacing products of x, y, and z by combinations of JJs divided by the total number of combinations. As an example we can consider (x- 6xf + ) [(x ,)4 (:/ -)4 ,, )4]/2 3j(,r4 + J4] i ii i Yi = p(r4)O{ (2-5) where J6=J JiJ,. The Hamiltonian is 34 HCEF C4[(4 +Y+ z4) r4] + D6[(6 +6 + z6) 5 15(2 4 +2z4 y2 4 y2z4 z24 2 14)_ 6]. (2-6) 4 14 Using the equivalent operator representation, the Hamiltonian will be [10] HCEF (C4/20)3j(r4) [0 + 50 (D6/224),(r6) [O 2110], (2-7) or HCEF = B[O + 50] + B [O 210o], (2-8) where B and BO are +71elq3j(r4)/(18d5) and -lelq'y(r6)/(9d7) for eightfold coordination, -7|1elqj(r4) /(16d5) and -31elqyj(r6)/(64d7) for sixfold coordination, and +71|elqj3(r4)/(30d5) and -lelq'j(r6) /(d7) for fourfold coordination respectively. Also, (r4) and (r6) are the mean fourth and sixth power of the radii of the magnetic electrons, and the multiplicity factors aj, 3j, and 7j are for Pr3+ (2) -22 13/(32 5 112), -22/(32 5. 11 2), and -22 17/(34 5 7.112 13) respectively [10]. Also, o0 35J4 [30J(J + 1) 25]J 6J(J + 1) + 3J2(J + 1), (2-9) O = 1/2(J +4), (2-10) 0 = 231J6 105[3J(J + 1)- 7]J4 + [105J2( + )2- 525J(J + 1) + 294]J, 5J3(J + )3 + 40J2(J + )2 60J(J + 1), (2-11) O4 1/4[11J J(J + 1) 38](J4 4) 1/4 4) x [11J J(J + 1)- 38]. (2-12) In order to keep the eigenvalues in the same numerical range for all ratios of the fourth and sixth degree terms, F(4) and F(6) are introduced [13]. The Hamiltonian is written as 4+5044 4 21 04 HCEF = BF(4) 4 + B F(6)o6 (2-13) F (4) F (6) In order to cover all possible value of the ratio between the fourth and sixth degree terms are introduced the scale factor W and the parameter x, proportional to the ratio of the two terms B2F(4) Wx, (2-14) BF(6) W(1- xl), (2-15) where -l HCEF = W x + (t |x|) (2-16) F(4) F(6) The term in the square bracket is a matrix whose eigenvectors and eigenvalues (crystal field energies levels) are determined by usual diagonalization. Praseodymium ion Pr3+ in PrOs4Sb12 has a 4f2 configuration, and then the total angular momentum is J=4. The site symmetry of Pr ions is Th (Fig. 2-1). This is differentiated from the cubic Oh symmetry by the fact that do not contain two types of symmetry operations of Oh: C4 (rotations through 7/2 about the fourfold symmetry axis) and C' (rotation through 7 perpendicular to the principle rotation axis [14]). We recall that if the z axis is not an m-fold axis symmetry, the potential will contain VJ [10]. Therefore, the Hamiltonian will a contain new term: HCEF W x+ (1 6- 216 0- o (21 7) F (4) F(6) Ft+ (6) where the coefficients F(4), F(6), and Ft(6) are 40, 1260, and 30 respectively [15]. The new term Oj-06 has the following symmetry in Cartesian coordinates: 2Y2(x2 2_ 2) + y2 (Y2 ) + 22(z2 2). (218) When parameter y is zero, the system reduces to Oh symmetry. The eigenvalues are tabulated for x=0.6 and several values of y by Shiina et al. [15] Fr: (I + 4) + 4)) + 10) (2-19) 12 6 F23 : 4) + 4)) 0) (| + 2) +-2)) (2-20) 1F) : 1 4|)-a2 -- 2) +a2l+ )+a +4) biT F3) + b2| T1) + b3 1) + b4| 3) (2-21) rF ) 2 4) ali- 2) + ai +2) a2 + 4) b2| T 3) bil T 1) b4l 1) + b3 3) (2-22) If y 0, the eigenstates are those for Oh symmetry [13] Fr: (I + 4) + 4)) + v 0) (2-23) 12 6 3 : ( +4)+ -4))- 0) 1(I 2)+ 2)) (2-24) 2 F4 TV/I T3)T /i+F) 2t( +4) -- 4)) (2-25) F5 : + / :| 3) T Ti ) (I + 2) -I- 2)) (2-26) Depending on the sign of the W parameter (or E), the ground state can be thought as either F23 or F1 (Fig. 2-2). The eigenfunctions and eigenvalues of Fi(Th) and F23(Th) are the same as those of Fi(Oh) and F3(Oh), therefore are not affected by O from the Hamiltonian. When y=0, FP1)(Th) has the same eigenfunctions and eigenvalues as F4(Oh), and F(2) as those for F5(Oh). When yO/, F4 and F5 mix resulting in two F(1'2)(Th) [15]. Therefore, the eigenfunctions and eigenvalues of CEF for Th and Oh are different. The O0 term in Hamiltonian affect some eigenfunctions and eigenvalues, resulting in a change of the transition probabilities of neutron scattering in Pr3+ 2.2 Conduction Electron Mass Enhancement (m*) Mechanism in PrOs4Sbl2 This Section gives a review on the mechanisms believed to be responsible for the conduction electron mass enhancement: the single-ion models, such as quadrupolar Kondo effect, or virtual CEF excitations (Fulde-Jensen model for m* enhancement in Pr metal) and comment on the cooperative model invoking proximity to the long-range order. 2.2.1 Quadrupolar Kondo Effect The quadrupolar Kondo (QK) model of HF was initially proposed by Cox to explain weak field dependence of the specific heat of UBe13 [16, 17]. Barnes [18] found that Cu2+ ions in the cuprate superconductors could lead to such a Kondo effect as well. Later, new evidence believed to be hallmarks of a quadrupolar Kondo effect has been found in the alloys Yl-,UPd3 [19-22] for x=0.1 and 0.2. In UBel3, the total angular momentum of U4+ (5f2 configuration) is J=4. This leads [13] to a F3 CEF ground state for about half the ( i -I I field parameter range (Fig. 2-2). The f2 configuration is expected also for Pr3+ in PrOs4Sb12, and, according to CEF calculations of Lea, Leask, and Wolf [13], a F3 doublet CEF ground state is very probable (Fig. 2-2). Therefore, the heavy fermion behavior in PrOs4Sb12 could be in principle due to a QK effect. Also, the physical properties of UBe13 (and U-1_Th1Bel3) are highly Table 2-1. The relevant states for the quadrupolar Kondo effect for U4+. The last two columns are the projections of the magnetic and quadrupolar moments respectively (Reprinted with permission from Cox and Zavadowski [23]). Config. State J Eigenstate (J) (3J J(J + 1)) f PI3+) J 4 7[14)- 4)] |0) 0 +8 f2 I3-) J4 2)+ 2)] 0 -8 fP \- J_ | +_ 0 f7+) j 5 1) 0 l IF7_) j 5 V/-I 5\ V/I 8\ 5 0 2 6 2/2 6 CI |I8 + 2) J 5 5) I 3 ) 11 +8 S|2 2 6 C1 I|F 2) J il 5 |3l 11 +8 1 Ir + 1) J 11) + -8 c IFs 1) J5 |- 1) -18 _2 2 2 reminiscent of those of PrOs4Sb12. Thus, since the discovery of the HF state in PrOs4Sb12 its normal properties have been associated with the QK effect. The states involved in the quadrupolar Kondo effect for U4+ are given in Table 2-1. The doubly degenerate ground state can be treated as a two-level system (a manifold with a pseudo-spin of '). The projected value of the electric quadrupole moment onto the F3 basis is IQzzI=| 3J J(J + 1)| = 8 and the projected value of the magnetic dipole moment is zero, i.e. J\ = 0 (Table 2-1). Therefore, the coupling is between the electric quadrupole moment and the conduction electrons. The Anderson model for the relevant states of the quadrupolar Kondo effect (since it considers only F3, F7, and Fs) is called the 3-7-8 model. Figure 2-3 shows a schematic representation of the Anderson model relevant for U4+ ions in the cubic symmetry. The ground state F3 (J=4, 4f2) and first excited F7 (J=5/2, 5f') mix only via the conduction partial waves Fs (J=5/2, c'). The transition fl f2 is done by removing a conduction electron and the transition f2 f1 is done by emitting a conduction electron. It can be shown that only Fs conduction quartet partial waves (Table 2-1) may couple to the impurity through hybridization [23] (or, in the group theory framework, F3 0 F7 Fs). Applying a canonical transformation (Schrieffer and Wolff [24]) to the 3-7-8 model, the hybridization term can be eliminated. Also, the transformation yields to an effective exchange interaction between pseudospin- and electric quadrupole moments of the form Hexchange = -2Jexchange3 [a8(0) + a8(0)], (2-27) where r3 is a pseudospin- matrix for the F3 quadrupole, a(s(as) are the pseudospins formed from the Fs + 2, Fs + 1 (Fs 2, Fs 1) partial waves (Table 2-1). The exchange integral Jexchange is proportional to F/TrcfN(0) and is negative. The Hamiltonian has a two-channel Kondo form; two degenerate species of conduction electrons couple with identical exchange integrals Jexcha"ne to the local a3 = object. The channel indices are the magnetic indices of the local conduction partial wave states. Figure 2-4 shows schematically the mapping of the quadrupolar Kondo to the two-channel Kondo model. The two-channel quadrupolar form of the Hamiltonian tell us that the conduction electron orbital motion can screen the U4+ quadrupole moment equally well for magnetic spin-up and magnetic spin-down electrons. 2.2.1.1 Thermodynamic Properties of the Quadrupolar Kondo Model Figure 2-5 shows the thermodynamic properties of the quadrupolar Kondo model [25-27]. The susceptibility (Fig. 2-5, lower panel) diverges logarithmical at T=0, x=-(eTTH)-lln(H/TH), where TH (T/e)TK [28, 29]. Here, e is the base of In, i.e. 2.71... In the quadrupolar Kondo model this corresponds to a divergent quadrupolar susceptibility. For T-0, the free energy in zero field is F -Tln2. Therefore, the zero-temperature zero-field entropy is equal to ln2 [25]. The non-zero entropy at T --0 is consistent with the divergence in susceptibility and argues in favor of a non-singlet ground state (a singlet is the ground state for the standard Kondo model). As expected, the entropy increases monotonically with temperature and reaches .,-ii!.1 .i, ically the ln2 value (free spin) at high T. Also, the T=0 entropy increases with the field. Since the S(T = 0) decreases with H the specific heat increases with H at intermediate T resulting in large values of 7, common for heavy fermion systems. At high T the pseudo-spin is free, therefore S= ln2. The entropy change AS(H)S(T = oo, H)-S(T = 0, H) increases with H from 1 ln2 to ln2 for large H. In the C/T plots the Kondo peaks can be seen. The initial measurements of specific heat [16] were not conclusive for a quadrupolar Kondo effect in UBe13. Also, more recent measurements of nonlinear susceptibility [30] are inconsistent with the quadrupolar (5f2) ground state of the uranium ion, indicating that the low-lying magnetic excitations of UBe13 are predominantly dipolar in character. 2.2.1.2 Relevance for the Case of Pr + Ion in PrOs4Sbl2 PrOs4Sbl2 has been initially reported to have a P3 doublet CEF ground state [7]. Later experiments [31 33] established the < i iiii. electric field (CEF) ground state of the Pr3+ ion in the cubic symmetry environment of PrOs4Sbl2 (Th point group symmetry) as the non-magnetic singlet Fi. The consequence of this is that the original formulation of the quadrupolar Kondo effect cannot be applied to the conduction electron mass enhancement in PrOs4Sbl2. F1 is nearly degenerate with the F(2) triplet. Though Fi itself doesn't carry any degrees of freedom, the pseudo-quadruplet constituted by Fi and F2) is speculated to have magnetic and quadrupolar degrees of freedom [34], and therefore a magnetic or quadrupolar Kondo effect is invoked to explain the enhancement of the effective mass of the quasi-particles. On the other hand, the model does not seem to be relevant since the predicted properties of the quadrupolar Kondo effect are in disagreement with the measurements. But this is a single-ion model. Possibly, intersite effects are responsible for the disagreements. There is no lattice quadrupolar model. 2.2.2 Fulde-Jensen Model for m* Enhancement in Pr Metal The mass of the conduction electrons can be enhanced by the interactions with various low-lying excitations in the solid. This explains the strong dependence of the specific heat of the Pr metal in magnetic fields found by Forgan [35]. Goremychkin et al. [32] proposed that the mass enhancement in PrOs4Sb12 can be explained by a balance between two types of interactions, magnetic dipolar and quadrupolar between conduction and the f electrons of Pr. The theory of Fulde and Jensen [36] of conduction electron mass enhancement ascribes this to the inelastic scattering by crystal field transitions in a singlet ground-state system. The mass enhancement of the conduction electrons are due to their interaction with the magnetic excitations. The relevant Hamiltonian describing the interaction between the conduction electrons and the rare-earth localized moments is [36] Hi = Isf{9L t1) J(Ri) J, (2-28) where If is the exchange integral, gL is the Land6 factor, J is the total angular momentum of a rare-earth ion at site R., and a are Pauli matrices. The mass enhancement due to the inelastic transition at energy A between two levels, |i) and Ij), is mn* 21 (i J 2 (229) 2 S1+ ( )2 N(0) (2-29) where gj is the Land6 factor, Isf is the exchange integral coupling the conducting electrons to the f-electrons, N(O) is the conduction electron density of states at the Fermi level, and (i IJ Ij) is the magnetic dipole matrix element calculated using the derived crystal field parameters. This formula shows that for a small excitation energy A leads to a large enhancement in m*. 2.2.3 Fluctuations of the Quadrupolar Order Parameter In general, models involving spin fluctuations in heavy fermions around their antiferromagnetic instability were considered by authors such as Hertz [37], Millis [38], Moriya and Takimoto [39] (a complete review is given by Stewart [40, 41]). All these models exhibit divergence of the low temperature specific heat. By analogy, in PrOs4Sbl2, the quadrupole fluctuations of Pr ions are believed to pl iv an important role in the HF-SC properties. Therefore, another model proposed (a collective-type model) for the mass enhancement mechanism are due to the fluctuations of the antiferroquadrupolar order parameter due to the proximity to the AFQ ordered phase. PrOs4Sbl2 exhibits an antiferroquadrupolar ordered phase in fields between about 4.5 and 14 T. For fields 5-13 T the two lowest CEF levels are sufficiently close to form a pseudo-doublet with quadrupolar and magnetic degrees of freedom, resulting in a long range order. There is no theory (to this moment) that describes the mass enhancement due to the fluctuations of the quadrupolar order parameter. Our magnetoresistivity data of PrOs4Sbl2 and La alloys presented in C'!i lters 6 and 7 seem to support this mass enhancement mechanism. 21/3, I H ------ -- ------ 4- I 4 -- --I--------- ----- -- ----- I- .. a) b) Figure 2-1. Rotational symmetry Th. In the left (a), the small bold blue segment is assimilate with the distance between two ,i 1inl lv7 atoms belonging to the same icosahedra. A 7 rotation with respect to (100) and a 2 rotation with respect to (111) are allowed. The rotation with respect to (1 00) will not turn the structure into an equivalent one. Therefore, the axes x (or (1 00)) and y (or (0 1 0)) are not equivalent (Reprinted with permission from D. Vu Hung [42]). Figure 2-2. Lea, Leask, and Wolff's representation of CEF for J=4 (Redrawn with permission from Lea et al. [13]). ff F 7 F3(E) Figure 2-3. Representation of the U4+ ions in cubic symmetry undergoing quadrupolar Kondo effect. The model involves a doublet ground state in each of the two electronic lowest-lying configurations: f2 having the quadrupolar or non-Kramers F3 doublet, and fl configuration having the magnetic or Kramers F7 doublet. The conduction electrons mix the two configurations through a hybridization process. The Fs conduction state couples these two doublets (Redrawn with permission from Cox and Zavadowski [231). SIL SO SI so- spin a) Figure 2-4. quadrupole b) Mapping of the quadrupolar Kondo Hamiltonian onto the two-channel Kondo model. a) The standard two-channel Kondo model in spin space: two conduction electrons s,+ and s,_ couple antiparallel to the impurity spin SI. b) In the quadrupolar Kondo case, the spin is due by the quadrupolar or orbital deformations. The two channels come from the real magnetic spin of the conduction electrons. The orbital motion of the electrons produces the screening of the U4+ orbital fluctuations (Redrawn with permission from Cox and Zavadowski [231). nD t2 - 0 30 0.25 o.20 O. IS 0, 10 0.1 0.05 7 0.3 to3 10-1 1 -I 0 1 10 -' 10- 10- 10- 10-' 13' "C' 10 101 Figure 2-5. S, C, C/T, and X versus T/TK of the quadrupolar Kondo model with permission from Sacramento and Schlottmann [27]). (Reprinted CHAPTER 3 PROPERTIES REVIEW OF THE PrOs4Sb12 3.1 Crystalline Structure PrOs4Sb12 praseodymiumm osmium antimonide) is a filled skutterudite that into a LaFe4P12-type body-centered cubic structure with the lattice parameter a 9.3068 A[43](a 9.30311 A[44] after more recent measurements), space group Im3, and Th point group symmetry. The < i i-- 1 .)graphic arrangement of the atoms is given in Fig. 3-1. The mass of a mole of PrOs4Sb12 is 2363.1 g, the molar volume is 242.3-10-6 m3/mol Pr and the mass density is 9.75 g/cm3 [45]. LaOs4Sb12 is the non-fequivalent of PrOs4Sb12 with a similar crystal structure. All the exotic phenomena of PrOs4Sb12 are thought to be associated with its unique crystal structure. In particular, the large coordination number of Pr ions surrounded by 12 Sb and 8 Os ions leads to strong hybridization between the 4f and conduction electrons [46]. This strong hybridization results in a rich variety of strongly correlated electron ground states and phenomena. 3.1.1 Rattling of Praseodymium Atom In CoAs3-type skutterudites whose name come from the cobalt arsenide mineral that was first found in Skutterud, Norway, an alkali metal, alkaline earth, rare earth, or actinide ion, occupies an atomic cage. The icosahedron shaped atomic cage made of Sb atoms accommodates a rare earth ion, and the size of this cage is bigger than the radius of the ion. Therefore, the rare earth ion will vibrate as a result of the weakly bounded rare-earth ion in the oversized cage made of Sb ions. This anharmonic oscillation is called rattling [47]. The consequence is a reduction of the thermal conductivity. The filled skutterudites with the cage are favorable for a thermoelectric device possessing a high coefficient of merit [48]. The amplitude of this vibration of the Pr ion in PrOs4Sb12 is about 8 times 'i.-.-r than the amplitude of Os. EXAFS data [49] supports the idea of a rattling Pr filler ion (based on the low Einstein temperature OE ~75 K) within a fairly stiff cage in this material. Besides the dynamic movement, a static displacement was detected in which there are two equilibrium positions for the Pr ions. The Pr ions can freeze in one of these two equilibrium positions, and at low temperature they can pass from one position to another through tunnel effect. It has been estimated that this displacement is about 0.07 A [49]. Goto et al. [50], based on a theory of Cox et al. [23], -,:-:-, -1. that the tunnel effect between the two positions of the Pr ions could be linked to the appearance of the superconductivity. 3.1.2 Valence At high temperature (above 150 K) the X(T) of PrOs4Sb12 can be described by a Curie-Weiss law with an effective moment peff=2.97pB as reported by Bauer et al. [7], or peff 3.5/B as reported by Ti- ii, et al. [51], and a Curie-Weiss temperature Ocw=-16 K [7]. The effective moment found is somewhat lower than the moment of a free ion Pr3+ which has peff=3.58/PB [52]. X-r-rv--.-..-rption fine-structure (XAFS) measurements [49] carried out at the Pr LIII and Os LIII edges on PrOs4Sb12 sirl:., that the Pr valence is very close to 3+. Each Pr ion has two electrons on the f shell (4f2 electronic structure). 3.1.3 Crystalline Electric Fields In an ionic (localized) model, the cubic crystalline electric field of PrOs4Sb12 environment splits the J=4 Hund's rule multiple of non-Kramers Pr3+ into a singlet (F1), a doublet (F3), and two triplets (F4 and s5) (in the Oh symmetry notation). The CEF Hamiltonian in cubic symmetry was written [13] in terms of the ratio of the fourth and sixth order terms of angular momentum operator of the CEF potential, x, and an overall energy scale factor W. For more than two decades, the symmetry was thought as Oh, instead of Th. Bauer et al. [7] fitted the magnetic susceptibility data (see Fig. 3-2) by a CEF model in which the ground state was chosen to be either the non-magnetic PF singlet (W>0) or the non-magnetic F3 doublet (W<0). The peak present in the x(T) data was thought to be produced when the first excited state is a triplet F5 with a energy <100 K above the ground state and corresponds to a position x close to the crossing points on the LLW diagram (see Fig. 2-2) where F1 or F3 are degenerate with Fs. The results are presented in the upper panel of Fig. 3-2. The notation of the CEF energy levels corresponds with the initially assumed Oh symmetry by Maple et al. [53] and Bauer et al. [54] These authors use the conventional cubic < i --I I1 field model which is applicable to the O, Td and Oh symmetries. In the Th symmetry, the non-Kramers doublet 3F corresponds to the degenerate F2 and F3 singlet states (denoted as F23) and F4 and F5 states coincide with F'1) and F 2) triplet states, respectively, when last term is zero in the < i -I I1 field Hamiltonian (2-17). The singlet state F1 is the same for both cases. The last term of equation (2-17) is unique to the Th symmetry of this material coming from the atomic configuration of Sb ions in the crystal [15] and is absent in the conventional cubic crystal field Hamiltonian that Maple et al. [53] and Bauer et al. [54] used. The omitting of the last term in the Hamiltonian (see equation 2-17) has implications in the interpretation of the inelastic neutron scattering data. The above mentioned fit reproduces the overall shape of the low temperature peak, and also the value of the van Vleck paramagnetic susceptibility with an effective moment close to, but somewhat lower than that, of the free Pr3+ ion. Bauer et al. [7] fitted C assuming a degenerate spectrum Specific heat data was fitted [7] by a system with two levels of equal degeneracy split by an energy 6=6.6 K (it has been assumed that the degeneracy of any level is lifted by CEF when the local site symmetry of the Pr3+ ions is not cubic as a result of some kind of local distortion). The entropy in the 13-F5 case was found to be Sr3-_5=Rln2m7.6 J/(mol K) [7]. The total entropy of the broad peak just above the transition is S=f(C(T)/T)dT ,10.3 J/(mol K). The closeness in values made Bauer et al. [7] to favor the 3 ground state scenario. Initial inelastic neutron scattering measurements [53] considering Oh symmetry -,i--.- -1 that F3 is the CEF ground state in PrOs4Sbl2. The resistivity data measurements were also interpreted in the framework of a F3 CEF ground state [55]. In contrast, Tayama et al. [51] obtained a somewhat better fit of the magnetic susceptibility x(T) data by a F1 CEF ground state model (Fig. 3-3 (a)). Also, the theoretical curves of S(T) based on F1 ground state model show increase of entropy with fields (lower panel of Fig. 3-3 (b)). This trend is confirmed by magnetic field specific heat measurements (Fig. 3-3 (c)) by Aoki et al. [56]. Therefore, zero or small magnetic fields data are contradictory, more experiments are to be done in order to establish the true CEF ground state in PrOs4Sbl2. 3.2 Normal-State Zero-Field Properties 3.2.1 Specific Heat PrOs4Sb12 was synthesized for the first time by Jeitschko et al. [43], and then by Braun et al. [57]. It was in 2002 when Bauer et al. [7] discovered superconductivity in PrOs4Sbl2. Since the discontinuity in specific heat is of the order of 7, this large value discontinuity (A(C/T) IT=1.85K -500 mJ/K2 mol [7]) implies the presence of heavy fermions both in the normal and superconducting states. There is no consensus regarding the precise value of 7, but all the reported values imply heavy fermion behavior. Actually, this is perhaps the strongest evidence for HF states in PrOs4Sb12. Considering the relation A(C/7TT)=1.43 the Sommerfeld coefficient 7 is found to be -350 mJ/K2 mol. The phonon (lattice) contribution to the specific heat C data can be described by 3T3 that is identified with specific heat of LaOs4Sb12 with OD 304 K. 3 is related to OD by /3(1944x103)n/O(, where n is the number of atoms in the formula unit (e.g., n=17 in LaOs4Sb12). 3.2.2 de Haas van Alphen Measurements The Fermi surface (FS) as reconstructed by de Haas van Alphen (dHvA) measurements [44] comparative with the bands structure (LDA+U method [58]) are presented in Fig. 3-5. The topology of the FS of PrOs4Sb12 is very similar to that of the reference compound LaOs4Sb12 [44] (which leaks 4felectrons). This indicates that the 4f2 electrons in PrOs4Sb12 are well localized. The similar topology of the FS for the two compounds is supported also by similar angular dependence of the dHvA. Three Fermi surface sheets, including two closed (practically spherical shaped) and one multi-connected, were identified in agreement with the calculations. The effective masses measured by dHvA are between 2.4 and 7.6mo (mo is the free electron mass). These values are well below the ones reported from the specific heat measurements. These low values have been explained [59] in the framework of the two-band superconductivity model in which band 2 corresponds to the light band detected by dHvA measurements. Band 1 is a heavy band having most of the density of states. The heaviest quasiparticles are seen in thermodynamic measurements (C or H,2) only. However, the applicability of the two-band model to PrOs4Sb12 is not established. Further more, our results presented in section 7.6 sheds some doubts in the interpretation. 3.2.3 Resistivity Additional evidence for heavy fermion behavior in PrOs4Sb12 is provided by an analysis of the slope of the upper critical field Hc2 near Tc. The upper panel of Fig. 3-2(a) shows resistivity versus T. From the resistivity data in small magnetic fields (data not shown, fields up to about 30 kOe) and from the fit of the linear part of -(dHc2/dH)T, curve, the initial slope has been measured using the BCS relations and shown to be ~19 kOe/K [7]. This implies o ~116 A, vF=1.65x106 cm/s, and m* ~50mo. This calculation assumes a spherical Fermi surface. The resistivity data between 8 and 40 K revealed a T2 dependence po+AT2, with A=0.009 pQcm/K2 [7]. The A coefficient is about two orders of magnitude smaller than the value expected for a heavy fermion compound. Considering the Kadowaki-Woods universal relation [60] between A and 7, A/72 = 1 x 10-5/Qcm mol2K2mJ-2. The 7 value is only ~6.5 mJ/mol K2 [7], and this is a typical value for normal metals and is much smaller than 7 of LaOs4Sbl2. 3.2.4 DC Magnetic Susceptibility The X(T) data (Fig. 3-2(a), lower panel) exhibits a peak at ~3 K and saturates to a value of about 0.1 emu/mol [7] as T-+0. This is the hallmark of a nonmagnetic ground state. Above 150 K, X(T) of PrOs4Sbl2 can be described by a Curie-Weiss law. There is a large discrepancy between the high temperature effective moment reported by various research groups. The effective moment according to Bauer et al. [7] is pIff 2.97PB, and qeff 3.5/B is the value reported by Tli-v i, et al. [51] The free ion Pr3+ has a high temperature effective moment of 3.58/B [52]. The Curie-Weiss temperature is Ocw=-15 K [51]. From the diamagnetic onset (inset (ii), Fig. 3-4(a)) it is found that the temperature of the superconducting transition Tc is equal to the value found from the specific heat measurements. 3.3 The Long-Range Order in Magnetic Fields Measurements of specific heat [56] in fields up to 8 T and resistivity [55] in magnetic fields up to about 10 T revealed the existence of a field induced ordered phase (FIOP) above 4.5 T. In this ('! lpter a discussion of the nature of the FIOP will be presented along with the specific heat data that completes the magnetic phase diagram. A similar phase diagram has been obtained later by magnetization [51, 61](see Fig. 3-6) and by thermal expansion and magnetostriction measurements [62]. 3.4 Superconductivity Experiments on PrOs4Sbl2 imply the possibility of unconventional superconductivity (i.e., the existence of nodes in the gap of the Fermi surface). There is still some other evidence that -,-.-.-I -I- an isotropic SC gap. We present below experimental evidence favoring either unconventional or conventional superconductivity. 3.4.1 Unconventional Superconductivity 3.4.1.1 The Double Transition Initial specific heat measurements [7] showed a single superconducting transition at T, of 1.85 K. Higher quality materials revealed actually two superconducting transitions (Vollmer et al. [63], Maple et al. [53], Oeschler et al. [64]). In Figure 3-7 panels (a) and (b) are shown specific heat of PrOs4Sbl2 presenting two superconducting transitions, Tc2=1.75 K and Tci=1.85 K by Vollmer et al. [63], and T,2=1.716 K and Tci=1.887 K by M6asson et al. [59], respectively. Two superconducting transitions at the same temperatures have been reported by Cichorek et al. [65] along with a speculation for a third superconducting transition at ~0.6 K inferred from Hc measurements. It is believed that inclusions of the free Os in the single crystal cannot be responsible for the enhancement of HaI, though T, of pure Os is 0.66 K [66] based on sensitive X-ray and electron microprobe studies [65]. There are two classes of explanations of the nature (intrinsic or not) of the double transition. One argues in favor of two different parts of the sample with two different superconducting phases, and therefore with different T,'s. Thus, the quality of the samples is crucial. For instance it has been considered [59] that despite the sharp specific heat transitions, the samples still present spatial inhomogeneities. One possibility would be an inhomogeneous coexistence of two electronic configurations of Pr, 4f1 and 4f2. The high temperature magnetic susceptibility measurements are in favor of 4f2, since they have found [51] an effective moment Peff 3.6pB/Pr (the expected value for 4f1 is 2.54pB and for 4f2 is 3.58/B). Another possible scenario that is presented in this dissertation is the existence of inhomogeneities due to the closeness of the system to a long range antiferro-quadrupolar order: clusters with a short-range order would have different superconducting parameters than the remaining part of the sample. Figure 3-7 (c) shows ac susceptibility after M6asson et al. [59]. The nature of the two transitions is not yet established. The width of the transition as measured by specific heat and ac-susceptibility is the same, about 0.2 K. The superconducting gap structure investigated using thermal transport measurements in magnetic field rotated relative to the crystal axes by Izawa et al. [67] provides another evidence for the unconventional character of superconductivity in PrOs4Sbl2. The change in the symmetry of the superconducting gap function that occurs deep inside the superconducting state gives a clear indication of the presence of two distinct superconducting phases with twofold and fourfold symmetries (Fig. 3-8). The gap functions in both phases have a point node singularity which is in contrast to the line node singularity observed in almost all unconventional superconductors. The two-band superconductivity (similar to that observed in MgB2) is observed in newer thermal conductivity measurements [68]. A double transition can be seen in the thermal expansion [64] experiment (Fig. 3-9). The two transitions are at the same temperatures at which the specific heat discontinuities occur. Using the Ehrenfest equation OTc/9P = VTcA//AC, where Vm is the molar volume, calculations show that the superconducting transitions T2a is decreased two times faster under pressure than T1i. This is in favor of intrinsic nature of the two superconducting transitions. 3.4.1.2 Temperature Dependence of Specific Heat Below T, In general, the specific heat C below T, exhibits different temperature dependence according to the topology of the superconducting gap As(k). For an open gap the specific heat dependence is the well known exponential e-Ao/T. Nodes in the gap, or zero points in the gap, will be reflected in the T dependence of the specific heat as a power T3 dependence. And for zero line in the gap, the temperature dependence of C is T2 Table 3-1. The a values reported by different groups, extracted from fits of specific heat below T, (Reprinted with permission from Grube et al. [70]). Specific heat data a = A(0)/(kBT,) Grube et al. [70] 3.70.2 Vollmer et al. [63] 2.60.2(F3) Frederick et al. [71] 3.10.2(F1) Frederick et al. [71] 3.60.2(F3) Table 3-2. The a values reported by different groups from measurements other than specific heat (Reprinted with permission from Grube et al. [70]). Experiment a = A(0)/(kBTe) Gap Function Tunneling spectroscopy [72] 1.7 Nearly isotropic pSR [73] 2.1 Nearly isotropic A(T) [74] 2.6 Point nodes Sb NQR [75] 2.7 Isotropic In all reported data the specific-heat measurements exhibit a rapid decrease of C below the superconducting transition. This points to pronounced strong-coupling superconductivity. The so called a-model [69] assumes that the superconductive properties which are mainly influenced by the size of the gap and the quasiparticle-state occupancy could be approximated by simply using the temperature dependence of the weak-coupling BCS gap. The size of the gap in the Fermi surface is a freely adjustable parameter a=A(0)/kBTc, where A(0) is Fermi-surface averaged gap at T=0. Table 3-1 presents comparative a values obtained by different groups. An analysis using the a-model results in an extremely large gap ratio of a=A(0)/kBT= 3.7 and a huge specific heat jump of C/(c)>5 [70]. A summary of the published superconductive gap ratios and gap anisotropy of PrOs4Sb12 from other measurements than specific heat are presented in Table 3-2. Frederick et al. [71] succeeded in making a better fit for the specific heat data of PrOs4Sb12 using a power-law function below the superconducting temperature. The fits, using both power-law and exponential functions, cannot be considered by themselves as proof of the superiority of one fit over the other. 3.4.1.3 Nuclear Magnetic Resonance (Sb NQR) The 121,123Sb Nuclear Quadrupole Resonance (Sb NQR) experiment [75] in zero field shows a heavy fermion behavior and controversial conclusions regarding the nature of the superconductivity in PrOs4Sbl2. In the SC state, 1/Ti shows neither a coherence peak just below T, nor a T3-like power-law behavior observed for anisotropic HF superconductors with the line-node gap. The absence of the coherence peak in 1/Ti supports the idea of unconventional superconductivity in PrOs4Sbl2 (Fig. 3-10). The isotropic energy gap with its size A/k=B4.8 K seems to open up across Tc below T*=2.3 K. The very large and isotropic energy gap 2A/kBTc ~5.2 indicates a new type of unconventional strong-coupling regime. 3.4.1.4 Muon Spin Rotation (pSR) The broken time reversal symmetry has been reported in later muon-spin relaxation measurements. The results [76] reveal a spontaneous appearance of static internal magnetic fields below the superconducting transition temperature, providing unambiguous evidence for the breaking of time-reversal symmetry in the superconducting state. This will favor the multiple superconducting phase transitions observed by specific heat and thermal conductivity studies and support therefore the unconventionality of superconductivity. Magnetic penetration depth data in single crystals of PrOs4Sb12 down to 0.1 K, with the ac field applied along the a, b, and c directions was successfully fitted [74] by the 3He A-phase-like gap with multidomains, each having two point nodes along a cube axis, and parameter A(0)/kBT,= 2.6, -Ii.:.: -Iii-; that PrOs4Sb12 is a strong-coupling superconductor with two point nodes on the Fermi surface. These measurements confirmed the two superconducting transitions at 1.75 and 1.85 K seen in other measurements. 3.4.2 Conventional Superconductivity 3.4.2.1 Nuclear Magnetic Resonance (pSR) The temperature (T) dependence of nuclear-spin-lattice-relaxation rate, 1/TI, and NQR frequency unravel a low-lying CEF splitting below To~10 K. In addition, the temperature dependence of 1/Ti in PrOs4Sbl2 is an exponential one [75] (Fig. 3-10, full symbol), which is the signature of a conventional type of superconductivity. Figure 3-10 (open symbols) also plots the data for the conventional superconductor LaOs4Sbl2. For an s-wave case that is actually seen in the T dependence of 1/Ti for LaOs4Sbl2 with T,-0.75 K, in the SC state, 1/Ti shows the large coherence peak just below Te, followed by an exponential dependence with the gap size of 2A/kBTc, 3.2 at low T. This is a clear evidence that LaOs4Sbl2 is the conventional weak-coupling BCS s-wave superconductor. 3.4.2.2 Penetration Depth Measurements (A) by pSR The transverse-field muon-spin rotation measurements in the vortex lattice of the heavy fermion superconductor (HFSC) PrOs4Sb12 yields [73] an exponential temperature dependence of the magnetic penetration depth A, indicative of an isotropic or nearly isotropic energy gap, indicating a conventional superconductivity mechanism. This is not seen, to date, in any other HF superconductor and is a signature of isotropic pairing symmetry (either s- or p-wave, indistinguishable by thermodynamic or electrodynamic measurements), possibly related to a novel nonmagnetic quadrupolar Kondo HF mechanism in PrOs4Sb12. Also, the estimated magnetic penetration depth A 3440(20) A [73] was considerably shorter than in other HF superconductors. 3.4.2.3 Low-Temperature Tunneling Microscopy The spectra of a direct measurement of the superconducting gap through high-resolution local tunneling spectroscopy [72] in the heavy-fermion superconductor PrOs4Sb12 demonstrates that the superconducting gap is well developed over a large part of the Fermi surface. The conductance has been successfully fitted by a s-wave superconductivity model. The presence of a finite distribution of values of the superconducting gap over the Fermi surface argue in favor of isotropic BCS s-wave behavior. Q Sb 001 Q Os S[100]' Figure 3-1. Crystal structure of PrOs4Sb12 (Reprinted with permission from Aoki et al. [77]) 0.08 0.06 PrOs4Sb12 0.:04: I 0 10 20 30 0.04 x =0.50. W = 1.9 K X =-0.72, W =-5.4 K S 3 (111 K) r, (313 K) Sr,,65 K) r (130 K) 0.02 r (6 K) r. ( 1 K) r, (O K) T,(OK) 0 50 100 150 200 250 300 3.0 (b) t 1-=g0 12 2.5 S 6.6 K 2,0 I 4 S1.5 0 5 10 15 20 S10 o A_ AC(T)= C +(T)+y'T 0.5F PrOs Sb1 0 5 10 15 20 T (K) Figure 3-2. (a) Fits of the magnetic susceptibility x(T) of PrOs4Sb12 to CEF model with either F3 (solid line) or F, (dashed line) ground state. The same symbols are used in the inset, which shows x(T) bellow 30 K. In the inset, the solid line fit saturates just above X 0.06 cm3/mol. (b) C fitted by a two-level Schottky anomaly (Reprinted with permission from Bauer et al. [7]). =3 F 0 4- 12 0 U, 64 2 Figure 3-3. (a) Fits of the magnetic susceptibility x(T) of PrOs4Sb12 to CEF model with either F3 (dashed line) or Fi (doted line) ground state. The solid lane represents the experimental data (taken from Tayama et al. [51]). (b) The calculated entropy S(T) for H//(10 0) in both F3 and Fl CEF ground state models (taken from T.i ,ii i et al. [51]). (c) The measured entropy S(T) for H//(1 00) (Reprinted with permission from Aoki et al. [56]). T (K) T (I) 200 f 150 (a) 100- 010 0.06 PrOs Sb,, o 5 1 15 0.04 - -0.5 H=200e (I) - 0.02- -1.0 .. o 1,8 2.2 286 3.0 0 50 100 150 200 250 300 S2.50 3.0- (b) 2.25. (iii) 1`2 2.25 1 S2,000 2.5- 7, I::;- 1,75 - 2.0 1,50 T., 1.76 K ACITr 500 mJmol K .51.25 1.- 1.0 1.5 2.0 2,5 - 1.0 0.5 O 5 10 15 20 T(K) Figure 3-4. (a) Resistivity p(T) and susceptibility x(T) of PrOs4Sbl2 (b) Specific heat C(T) up to 20 K [7] (Reprinted with permission from Bauer et al. [7]). (a) band 48-hole (b) band 49-hole Y Figure 3-5. Fermi surface of PrOs4Sb12 (Reprinted with permission from Sugawara et al. [44]). , I, 0 0.5 1 T(K) z=0 plane P C- 1.0.5, 0.5, 0.5) Pr 1.5 2 Figure 3-6. H-T phase diagram of PrOs4Sb12 (Reprinted with permission from T i,. iI et al. [51]). Open and closed symbols were determined by the dM(T)/dT and dM(H)/dH data, respectively. Right panel, the Pr charge distributions induced in the antiferroquadrupolar ordered phase in magnetic field (Reprinted with permission from MWasson [45]). 7o I1.2 0.6 Figure 3-7. 1.8 T (K) (C) -0.02 / S-0.03 -0.04 -0.05 .2 -0-06 -0.07 -0-08 1.5 1,8 1.7 1. TK) (a) C(T) of PrOs4Sb12 presenting double superconducting transition (Reprinted with permission from Vollmer et al. [63]) (b) C(T) of PrOs4SbI2 presenting two superconducting transitions (Reprinted with permission from M6asson et al. [59]) (c) The real part of the ac susceptibility of PrOs4Sb12 presenting two distinct superconducting transitions (Reprinted with permission from MWasson et al. [59]). "/ I >I PrOs4Sb12 1 1I 2 I I I 2.0 A- 1.5 ^1.0 - :z. 0.5 0.0- 0.0 Figure 3-8. The two superconducting phases for PrOs4Sb12 (Reprinted with permission from Izawa et al. [67]). The gap function has a fourfold symmetry in A phase and twofold symmetry in B phase. Right: The plot of the gap function with nodes for A phase and B-phase (Reprinted with permission from Maki et al. [78]). -0.5 b-1.0 i- 1.4 1.4 1.6 1.8 2.0 T (K) Figure 3-9. Two superconducting transitions in the thermal expansion coefficient 3 of PrOs4Sb12. The two transitions are visible for the same temperatures of the two transitions in specific heat (Reprinted with permission from Oeschler et al. [64]). iionnnl state 0.5 1.0 T(K) 1.5 2.0 B-phase T e1 1 -; 102 gi LaOs4Sb12 100 10-1 10-2 10 I ** 0.1 1 10 100 Temperature (K) Figure 3-10. Temperature dependence of the rate 1/Ti at the 2VQ transition of 123Sb for PrOs4Sbl2 (closed circles) and LaOs4Sbl2 (open circles) (Reprinted with permission from Kotegawa et al. [75]). -1000 -500 0 500 Bias voltage (IV) 1000 Figure 3-11. Tunneling conductance between PrOs4Sb12 and an Au tip. The gap is well developed with no low-energy excitations. The line in figure is the prediction from conventional isotropic BCS s-wave theory using A=270 peV and T=0.19 K (Reprinted with permission from Suderow et al. [72]). CHAPTER 4 EXPERIMENTAL METHODS This chapter describes the sample synthesis, characterization and the experimental procedures used: dc and ac susceptibilities, resistivity, and specific heat measurements. A brief description of the performed measurements is given. 4.1 The Samples: Synthesis and Characterization 4.1.1 Synthesis The filled skutterudite antimonides studied in this dissertation are prepared using a molten-metal-flux growth method with an excess of Sb flux [54, 57]. Since the flux is one of the constituent elements of the compounds (i.e. Sb) the method is called self-flux growth. High-purity starting elements (Pr and La from AMES Laboratory, 99.9', . purity powder Os from Colonial, Inc., and 99.9'' '. purity Sb ingot from Alfa AESAR) are used in the proportion R:Os:Sb=1:4:20, when the rare-earth element R is Pr and La in various proportions. The R alloys used as components in the flux growth were synthesized eventually by melting its constituent elements in an Edmund-Biihler Arc Melter under a high purity argon atmosphere. First, small chunks of Sb were placed inside of a quartz tube. Above that were placed the Os and the R components that were pre-melted separately to eliminate any trace of oxide from the surface of the elements. The Os powder was pressed in small pellets and then melted. The quartz tube was sealed under low pressure Ar atmosphere (~20 mTorr) after the tube is pumped and flushed 3 to 5 times. The tube with the mixture was placed in a Lindberg 51333 programable furnace (digital controlled, Tma=12000C) using the following heat treatment sequence: temperature ramping to 9800C with a rate of 2000C/h followed by T=9800C for 24 h, then cooling at a rate of 3C/h down to 650C. The last step was a fast cooling in the furnace to room temperature at a ~ 2000C/h rate. The single crystals were then removed from the .irilrilir flux excess by etching in aqua regia (HCl:HNO3 1:1). The crystals were cubic or rectangular up to 50 mg in weight (up to ~3 mm in size) depending on the amount of the starting elements and the cooling rate. For instance, using ~ 1 g of Os and a cooling rate of lC/h the single-crystal mass was about 50 mg (Fig. 4-1). This large < i--I I1 later proved to be very useful for the dc-susceptibility measurements. In the case of R being actually an alloy, such as Prl_-La1, Pr and La are previously melted together using the arc-melter as further described. The poly-< i --i 11,ii: R alloys (used as one of the starting components in the synthesis of the single crystals) were prepared by melting its constituent elements in an Edmund-Biihler Arc Melter AM under a half atmosphere high purity Ar. The apparatus consists of a stainless steel vacuum chamber which sits tight on a water cooled groove crucibles in a copper base plate and with an electrode at the top. The tungsten electrode is motor driven which can be moved freely above the crucible. The melting process can be observed through a dark glass window. All important control functions are integrated in the head of the electrode and ensure safe and convenient operation. When fed at the maximum current the temperature of the electric arc in the melter can go as high as 4000C and melts ~500 g of metals. The arc melter has a flipper, a manipulator for turning the samples in situ. This gives the possibility to flip and again melt the sample, ensuring its homogeneity, without opening the chamber. Before operating, the copper base plate was thoroughly cleaned with acetone to avoid any contamination of the sample with impurities. Right at the beginning, each of the constituent elements were well cleaned to eliminate the oxide 1. -r on the surface. The precision in mass measurements was 0.02 mg. Starting with the radioactive or the hardest element we can adjust the relative masses of the other components to gain the wanted stoichiometric ratio. The total mass was from teens of milligram to ~1 g, the size of the sample bead was up to 1 cm. Right before the elements were melted together, a zirconium button which was also used for ignition of the arc, was melted just to ensure a even higher purity of the Ar, which was filtered through a purifier before entering into the arc chamber. Zirconium is well known as a oxygen absorber. The element with the highest vapor pressure was then placed on the copper plate right below the elements with lower vapor pressures. The aim of this was to not strike with the arc on the element with a high vapor pressure resulting in uncontrolled vaporization of the material. The element with the lowest vapor pressure was melted first. This reduced the mass loss, and the discrepancy between predicted and actual stoichiometries of the synthesized alloys. To ensure an even better control of the temperature at which the elements were melted, the copper plate in the immediate vicinity of the place where all elements were together was first heated with a slowly increasing current. This was done until the element with the highest vapor pressure started to melt and to suck all the other elements. Then flipping the resulting bead and remelting it ensured its homogeneity. This process was repeated several times. Also, for dilute concentrations, such as Pro.98Lao.020s4Sb12, Pro.98La0.02 was needed first. Therefore we started with master alloys (Pro.gLao.1) in order to avoid handling of very small amounts of materials. 4.1.2 X-Rays Diffraction Characterization X-ray diffractions of the materials verified whether the arc melting plus annealing or the flux growth processes led to the formation of the desired <( i--I I1 structure. From the diffraction pattern it is possible to determine the lattice constants and the presence of the secondary phases in the material (if present in a proportion greater than 5'.). The measurements were performed using a Phillips XRD 3720 machine at the AT ij r Analytical Instrumentation Center ( \ AIC) at the University of Florida. Single < iv-i ,1- and poly-< i -i~1 ,11 !i. samples were crushed and ground out into a fine powder using a ceramic mortar. On a glass slide, about 1 cm2 of powder was glued using 7:1 .irnil acetate-collodion mixture. The machine uses two wavelengths in the measurements: Cu K,,11.54056 A and Cu K2 ,1.54439 A. The intensity of the ac beam is twice as great as the a2 beam. All the measurements were taken in a 20 angle range from 200 and 1200 with a 0.020 step and a scan speed of 6/min, the machine recording 1000 counts/sec. 20 is the angle between the incident beam and the reflected one. All measurements were performed at room temperature. The computer controlled X-ray machine records the relative intensities of the peaks which will be plotted/di- 11 i, iI (X-ray pattern) when the scan is completed. Also, the precise angles corresponding to specific peaks were listed. The angles corresponding to the peaks were found from Bragg's law: n = 2d sin 0, (4-1) where 0 is half of the reflection angle, n is an integer (n= 1 for the first order spectrum), d is the inter-plane distance, and A is the wavelength of the incident radiation. The lattice constants are then calculated from d and the intersection points of the lattice planes from the desired space group number is given in terms of the Miller indices (hkl). For a cubic symmetry the same Bragg equation can be rewritten as sin2 0 h2 + k2 2) (4-2) 4 which is derived from d(hkl) = a//h2 + k2 + 2, where a is the lattice parameter. Using a least-squares fitting program with the wavelength, structure type, (hkl) indices and the angles 20 of the narrowest intensity lines as input, the lattice constant can be found. All the X-ray diffractions were taken at room temperature. 4.2 Specific Heat Measurements 4.2.1 Equipment 4.2.1.1 Cryogenics This sub-section describes the probes used in the specific heat, in house resistivity and ac-susceptibility measurements. In house (Stewart Lab., Physics Department at University of Florida) measurements of specific heat were performed in the temperature range of 0.3 to 2 K usually, and in some cases up to 10 K. A home made 3He cryostat was used. The schematic drawing is given in Fig. 4-3. This probe was used for the measurements of specific heat in magnetic fields as well. A specially designed dewar from Cryogenic Consultants Limited was also used. The superconducting magnet reached 14 T at 4.2 K bath temperature. For the ac susceptibility and resistivity measurements, another 3He cryostat was used. The resistivity was measured between the lowest temperature of 0.3 K and the room temperature, while the ac-susceptibility was measured between 0.3 K and about 2 K. The difference from the one used for specific heat measurements is that this probe has no 4He pot, the pumping being performed on the Dewar in order to reach 1.1 K for the use of the 3He cooling system. Specific heat measurements at lower temperature and higher magnetic fields were performed at the Millikelvin Facility (Superconducting Magnet 1-SC'I 1), High Magnetic Field National Laboratory, Tallahassee, Florida using a top loading dilution refrigerator which is permanently installed in a 18/20 T superconducting magnet. The measurement temperature range was 20 mK to 2 K combined with a magnetic field of up to 20 T. The small home made calorimeter (Fig. 4-5) was connected to the general purpose sample mount provided by the facility. Resistivity measurements performed at the same facility were done in the temperature range of 20 mK to 0.9 K. Another sample holder, a so-called 16 pin ample rotator, was used. This allows the change of orientation of the sample in field during the experiment. This holder has 16 pins (16 connection wires to the top of the probe) that allows up to a maximum of four different samples to be measured without pulling out the probe from the dilution refrigerator, saving precious time. Takes up to 6 hours to insert the probe into the refrigerator and cool the sample to 20 mK. Specific heat measurements in magnetic fields up to 32 T were performed at the 33 T, 32 mm bore resistive magnet (Cell 9), at the same National Laboratory. Another home made 3He probe similar to the one mentioned earlier but with slightly different dimensions in order to fit into the magnet and also to accommodate the sample in the maximum field strength region was used. Right before the insertion of the probe into the magnet an electrical check was done on wire connections. The quality of vacuum and sealing was checked also using an Alcatel ASM 10 Leak Detector. For both 33 T and 45 T measurements, a special positioning system made it possible to center the probe inside the magnet such that it did not touch the inner walls of the magnet. The probe was to be perfectly centered into the maximum strength field region. Because of the large amount of heat that had to be removed, the probe was cooled in liquid nitrogen (LN) down to the boiling point (77.35 K). After about 2 hours, when the probe was at thermal equilibrium with the liquid nitrogen, it was transferred quickly into a dewar in which it fits tight. The dewar was cooled in advance in LN as well. The dewar (with the probe inside) was filled with liquid 4He (LHe) and after several hours (depending on the volume of the can) the temperature of the probe reached 4.2 K. After 4.2 K was attained following the procedure described above, the 4He pot was filled with LHe from the bath by opening the needle valve, and 3He gas (a lighter isotope of He) was transferred into the 3He pot+probe line using a home-made 3He handling system. The handling system consists of a tank filled with 3He, a pump which helps to transfer to and back from the 3He pot line, and pressure gauges to display the amount of 3He left in the tank and in the transfer lines. After closing the needle valve and pumping in the 4He line a temperature between 1 and 2 K was obtained. It was necessary to refill the 4He pot by opening the needle valve once in several hours. In order to attain 0.3 K a completely contained 3He cooling part using a sorption pump was required. When cooled, gases generally adsorb to solid surfaces. The sorption pump is based on the idea that at ~10 K almost all of the 3He gas molecules are adsorbed, whereas at ~35 K all of the molecules desorb. The sorption pump consists of a Cu cylinder that contains activated charcoal, which has an enormous surface area (tens of square meters per gram). The cylinder is attached to the lower end of a metallic rod. The whole system, rod+cylinder with charcoal, was placed inside the 3He-gas enclosure. As the charcoal was lowered toward the 3He pot, the 3He was absorbed by the charcoal reducing the vapor pressure and lowering the temperature of the 3He pot. After the charcoal became saturated with 3He, the charcoal was warmed up (by raising the rod with the charcoal), and the gas was released. In about 15 minutes the gas condensed and dripped into the 'He pot again. Then the whole process was repeated. 4.2.1.2 Sample Platform The sapphire platform is attached to the bottom of the probes to the 3He or 4He pot, depending on the probe used (Fig. 4-5 (a)). The sample is attached to a small piece of sapphire disc using H31LV silver epoxy cured at 150C for 1/2 h. The new assembly of sample+sapphire sitting on the sapphire platform and attached by Wakefield grease is shown in Fig. 4-5 (b). This ensured a good thermal contact between the sample and the platform. The platform is thermally linked to a copper ring (silver in the case of the platform of the calorimeter used at the SC\ I NHMFL) as schematically drawn in Fig. 4-5. Two types of platforms were used. Each platform has four wires soldered to silver pads attached to the ring by thermally-conductive Stycast. The two pairs of wires are connected to the platform heater and thermometer, respectively, using EpoTek H31LV silver epoxy. The wires ensure the mechanical support of the platform and the thermal contact with the ring and the 3He or 4He pot by the case. They also provide the electrical contact to a heater and a thermometer on the platform. The platform heater is an evaporated 1li-. of 7'. Ti-Cr alloy. For measurements between 1-10 K, the platform thermometer used was an elongated piece of doped Ge, and the platform wires were made of a Au- 7' Cu alloy. A thin piece of Speer carbon resistor and Pt-i10'. Rh platform wires (more mechanical resistant than the ones made from the Au-alloy) were used for measurements between 0.4 and 2 K. 4.2.2 Thermal Relaxation Method The specific heat was measured using the probes described earlier, employing the thermal relaxation method [79-81]. The thermal relaxation method consists of measuring the time constant of the temperature decay of the sample connected to the heat bath by a small thermal link. A power P is applied (Fig. 4-6) (thermal power by a small current of the order of pA) to the platform-sample system. The temperature of the sample, initially at To, increases by a small amount, AT. When the current is turned off, the system temperature T(t) decays exponentially to the base temperature To: T(t) = To + ATexp (-t/ri). (4-3) The time constant T1 is proportional to the Ctotai (sample+platform): (Ctotal T- Ctota, (4-4) where K is the thermal conductance of the wires linking the sample+platform at T=To+AT and the ring at T=To. The block temperature is regulated by a block heater (a bundle of manganin wire) and measured by a thermometer attached to the block. The time constant is obtained by measuring the time decay of the off-balance voltage signal from a Wheatstone bridge using a lock-in amplifier. Two arms of the Wheatstone bridge are a variable resistance box and the platform thermometer. The bridge is balanced by adjusting the resistance of the resistance box. This made it possible to find the resistance of the thermometer. From an initial calibration of the thermometer R versus T: 1 (45) S A (In R)', (4-5) i=0 it is possible to find the temperature corresponding to the platform thermometer resistance. The thermal conductance is given by: P AP= (4-6) AT' where P=IV is the power applied to the platform heater. Equation (4-3) is valid if the thermal contact between sample and platform is ideal (i.e., sample oc). If the contact is poor (i.e., sample ~ e), then T(t) = To + Aexp (-t/71) + Bexp (-t/7), (4 7) where A and B are measurement parameters and T2 is the time constant between sample and platform temperatures. Ctotla can be calculated from TI, T2, and t. The thermal conductivity is measured by applying a power P=IV and calculating the AT as a result of the power applied to the heater. The specific heat of the sample can be calculated by subtracting the addenda contribution from the Ctotl. The result is multiplied by the molecular weight and divided by the mass of the sample. 4.3 Magnetic Measurements Magnetic susceptibility measurements were made in order to characterize the magnetic properties of PrOs4Sb12 and its La alloys. The direct current (dc) magnetic susceptibility was measured using a Superconducting Quantum Interference Device (SQUID) made by Quantum Design which can perform measurements in magnetic fields up to 5 T, and a temperature range from 1.8 K to 300 K (350 K with special preparations). The alternating current (ac) susceptibility was measured using a home made apparatus. The temperature range can run from about 0.3 K to 10 K, although normally all the measurements were done up to 2 K. Both dc- and ac-magnetic susceptibility methods are discussed in the next subsection. 4.3.1 DC Susceptibility All the measurements were performed in 1 or 5 kOe magnetic fields. For small samples (mass approximately a few mg), 1 T magnetic field was used since the signal of the sample was comparable with the signal of the plastic straw holder. In any case, in order to avoid the straws signal subtraction, the samples were kept tightly between two drinking straws. The principle behind the magnetic susceptibility measurements is the Lenz's law. The magnetic moment is measured by induction: the sample moves 4 cm through a set of superconducting pickup coils and the SQUID instrument measure the current induced in the pickup coils. The SQUID voltage is proportional to the change in flux detected by the pickup coils. In order to get the magnetization data curve a set of 48 points are taken during the movement of the sample. At a given temperature this is repeated 4 times and the signals are averaged for a better accuracy. The magnetic susceptibility for a fixed field is X = M/H (in emu/mol), where M is the magnetic moment, and H is the magnetic field. This is obtained from the signal measured at a fixed field by multiplying with molecular weight of the alloy, and dividing by mass and the applied field. Beside the magnetization at fixed field and various temperatures the SQUID can perform measurements of magnetization in different magnetic fields at constant temperature. 4.3.2 AC Susceptibility The apparatus consists of a primary coil of NbTi superconducting wire, 90/10 CuNi of 0.004" with insulation, 185 turns [82] and two secondary coils made from copper wire, wound in both sides in opposite directions of 2700 turns. The coils are attached to the Cu block (which is in thermal contact with the 3He pot). The apparatus uses the mutual inductance principle. The sample is subject to an alternating magnetic field of 0.1 Oe produced by the primary coil (and also the Earth's magnetic field). The resulting electromotive force (EMF) induced in the secondary coil is detected. The background signal is nulled by the identical secondary coil, connected in series opposition. For the same reason the two screws are identically built. The sample is glued to one screw with General Electric (GE) varnish 7031 which ensures a good thermal and mechanical contact at low temperature and also can be removed easily using acetone. The ac susceptibility measurements were performed at two different frequencies: 27 Hertz (Hz) and 273 Hz. It was deliberately used these frequencies (not integer multiples of 60 Hz) in order to avoid the noise coming from the common electrical outlet. In general, B=/o(H + Mv)=loH(1 + X), with H the magnetic field, My the volume magnetization and = My/H is the magnetic susceptibility. If the applied field H has a sinusoidal form, the time dependent magnetization Mv(t) can be expressed as a Fourier series of the non-linear complex ac susceptibility. Applying the inverse Fourier transform to My(t) it can be found the nh harmonic of both real and imaginary ac susceptibility. The fundamental real component is associated with the dispersive magnetic response which is in phase with the ac applied magnetic field, and the fundamental imaginary component is associated with absorptive or irreversible components which arise from energy dissipation within the sample, or in other words the energy absorbed by the sample from the ac field. The induced EMF in coils V(t)=-dl(t)/dt (complex, i.e. V=V'+iV") is proportional to the X = X' + iX". Therefore, if the reference signal of the lock-in amplifier is derived from the primary driving signal, then V' oc X", and V" oc X'. The superconducting transition temperature is determined by a midpoint of the inductive signal deviation associated with the superconduction transition. 4.4 Resistivity The resistivity measurements used the same probe used for the ac susceptibility. The sample was mounted to a sapphire disk. Four platinum wires (0.002" diameter) were attached to the sample using silver paint (whose resistivity is much lower than that of the sample itself) and then for a good mechanical contact with EPO-TEK H31LV silver epoxy. The extra resistance introduced by the silver-epoxy contacts is avoided by the use of the silver paint for electrical contacts and the epoxy just only for a good mechanical contact between the wires and the sapphire disk. Then, the disk with the sample was glued to the 3He block using GE varnish 7031 ensuring a good thermal contact. At each temperature, the resistivity was obtained by averaging both absolute values for each polarity of the current. The temperatures between 77.4 K and room temperature were covered by measurements in liquid nitrogen (LN). The probe was immersed in LN and the program starts collecting data, while the sample cooled towards 77.4 K. Thereafter, as described in the previous section, the LN was removed, and liquid He was transferred. The temperature dropped further toward 4.2 K. Further down, the resistivity was measured to approximately 0.3 K, making use of 3He gas as described. Figure 4-1. PrOs4Sb12 large crystal, about 50 mg (right). In the left, an Os ball with PrOs4Sb12 single crystals attached, waiting to be etched out. t t -I Figure 4-2. PrOs4Sb12 samples prepared for (left panel) resistivity specific heat measurements. and (right panel) Pumping line for He3 pot----- Pumping line for He4 pot Cu ring with connection pins He3 pot Cu block - Platform with sample Cu cylinder with charcoal Pumping line for vacuum can and tube for wires -Needle valve SCapillary connecting He4 pot and He4 bath --He4 pot Block heater SHeat sinking and wire connection pins _ Vacuum can ~Block thermomet Figure 4-3. Schematic view of the 3He cryostat used in the measurements performed at University of Florida. Indium ring for sealing 2 ___ Flange to connect to the general purpose sample holder (SCM1) -Block heater Ag block with .::::::::::..::::: Block thermomet connection pins ....... T Vacuum can Platform with - sample --___ ----~ _&Agrng Figure 4-4. Schematic view of the calorimeter used in the Superconducting Magnet 1 (SCi\l 1), National High Magnetic Field Laboratory. H31LV Silver Epoxy 3/8 in. sapphire disc I\h Wires: Au-7%Cu or Pt-10%Rh Sample / Wakefield tease - Sapphire \neater Figure 4-5. (a) Top view of the sample-platform/Cu-ring assembly. (b) Lateral view of the sapphire platform and sample. P T Sample Sapphire CTotal = Ge thennometer T + AT Silver epoxy CSa mle+ CAddenda Wakefield grease S= The four AT An-Cu (or Pt-Rh) wires Cu (Ag) ring T, heat reservoir Figure 4-6. Specific heat C measurement process using the relaxation time method (Redrawn with permission from Mixson [831). CHAPTER 5 MATERIALS CHARACTERIZATION All samples were synthesized using the self-flux growth method, described in C! Ilpter 4. The samples are cubic shaped and of sizes ranging from 1/2 mm to 3 mm and weighting from 1 mg to about 50 mg. X-ray diffraction was performed to verify the desired i -I I1 structure. From the diffraction pattern it was also possible to determine the lattice constants. In addition to this, the X-rays confirmed that the samples were single-phase within an accuracy of 5'. The quality of the sample is also given by the sharpness of the transition in the specific heat. A more quantitatively measure of the quality of the sample is the residual resistivity ratio RRR=p(300K)/p(T-+O). This ratio ranges from 50 to about 170 (PrOs4Sbi2 samples studied by M6asson et al. [59] have RRRw40.) Due to the very small size of the samples used, the susceptibilities measured for all concentrations and the background (susceptibility of the sample holder consisting from a plastic drinking straw) were comparable at 10 K. At room temperature the susceptibility was even smaller than the background, especially for dilute concentrations. In order to avoid this background contribution, magnetic susceptibility were remeasured (for x=0, 0.05, 0.3, 0.67, 0.8 and 0.95) using bigger samples. Also, in these measurements, the material was pressed in between two long concentric tubes such that no background subtraction was needed. All these additional measurements yielded to a Curie-Weiss temperature dependence above 150 K, corresponding to an effective magnetic moment close to the one expected for Pr3+ (Fig. 5-1), much closer to the expected value for Pr3+ than the initially reported e/ff 2.97PB [7] for PrOs4Sbl2. The effective moment of the free Pr3+ is /eff 3.58/B [52]. New measurements by Tayama et al. [51] revealed an effective moment close to this value. This supports the notion of an essentially trivalent state of Pr in all PrlxLaxOs4Sbl2 alloys. 0.12 0.10 0.08 0.06 0.04 0.02 H=b KUe 0.00 0 50 100 150 200 250 300 T(K) Figure 5-1. X(T) of PrOs4Sb12. In the inset is the Curie-Weiss fit of high temperature (T>150 K). The high temperature effective moment is Peff 3.65pB, very close to the one corresponding to free Pr3+, which is 3.58pB. 0 50 100 150 T(K) 200 250 300 Figure 5-2. X(T) of the non-f equivalent LaOs4Sb12. 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 - LaOs4Sb12 - * U" U* U* U* U* U* CHAPTER 6 PrOs4Sb12 Any understanding of heavy fermion (HF) behavior requires knowledge of the crystalline electric field (CEF) configuration. Therefore, one of the main objectives of this thesis was to establish the CEF scheme for PrOs4Sb12 and to see how the CEF ground state is reflected in low temperature properties of this material. The C!i lpter starts with results on the specific heat of PrOs4Sb12 in high magnetic fields. The magnetic phase diagram (i.e. phases that exist at a given temperature and field) will allow us to determine the CEF scheme of Pr. High magnetic field low temperature resistivity measurements will be used to argue for a HF state in PrOs4Sb12. 6.1 Investigation of CEF Configuration by Specific Heat in High Magnetic Fields The initially proposed CEF schemes [7] (either 3F or F1 CEF ground state) for PrOs4Sb12 imply non-magnetic ground states and exclude a conventional Kondo effect, believed to be the source of HF behavior in Ce- and some U-based metals. The controversy between the two schemes was brought about by different experiments that seem to favor either configuration. As presented in C(i lpter 3, the first published results such as the zero field specific heat, magnetic susceptibility data [7], resistivity in small magnetic fields [55], inelastic neutron scattering data interpreted using Oh symmetry [53] favored the F3 doublet as the CEF ground state. On the other hand, magnetic susceptibility data of Tayama et al. [51] and entropy changes in small magnetic fields measured by Aoki et al. [56] were better fitted by a F1 CEF ground state model. The zero field Schottky anomaly occurring at 3.1 K can be related to the P3-P5 model, assuming these two levels are split by 6.5 K, or F1-F5 model with the splitting of 8.4 K. The difficulty in interpreting these low temperature, low field results is related to a strong hybridization of 4f and conduction electrons, inferred from the large electronic specific heat coefficient and the size of the discontinuity in the specific heat C at Tc. The idea behind specific heat measurements in high magnetic fields was to suppress this coupling between f and conduction electrons to reveal the ionic character of Pr. In order to present our results in a proper perspective we start from recalling the specific heat data for fields smaller than 8 T obtained by Aoki et al. [56] Figure 6-1, upper panel, shows the low temperature specific heat to 8 T obtained by Aoki et al. [56], the lower panel a comprehensive phase diagram known before our measurements. 4.5 T is the lowest field at which a signature of FIOP is detectable as a small kink (at ~0.7 K). This kink evolves into a sharp peak at 0.98 K in 6 T. The C(T) peak grows and moves also to higher temperatures for higher fields. The FIOP was confirmed by specific heat of Vollmer et al. [63] and magnetization study of T li- i,, et al. [51]. A number of observations brought forward the interpretation of FIOP in terms of antiferroquadrupolar (AFQ) order. These observations included a large anomaly in the specific heat (corresponding to a large entropy removed by the transition) and the very small value of the ordered (antiferromagnetic) moment (about 0.025/B at 0.25 K in 8 T [84]) measured by neutron diffraction, and also similarities to systems displaying quadrupolar order (e.g., PrPb3 [85]). Figures 6-2, 6-3, and 6-4 show the specific heat in fields ranging from 10 to 32 T. The specific heat measurements in fields up to 14 T were done using Cryogenic Consultant Limited superconducting magnet at the University of Florida. Measurements in fields larger than 14 T were carried out at the National High Magnetic Field Laboratory, Tallahassee, Florida using a resistive Bitter magnet. The field was applied along the crystallographic (10 0) direction. The specific heat data in all three figures are after subtracting the phonon background (fiT3 with P=(1944x 103)n/(3 [1]) corresponding to a Debye temperature (GD) of 165 K, proposed by Vollmer et al. [63] This value of HD obtained from the temperature dependence of the specific heat of PrOs4Sb12 is somewhat controversial. Other estimates of the Debye temperature: 304 K (Bauer et al. [7]), 320 K (Aoki et al. [56]), and 259 K (\! i!i. et al. [53]) are based on specific heat measurements of LaOs4Sb12. The lowest temperature of the heat capacity measurements, actual value, is chosen relatively high in order to avoid complications associated with a nuclear contribution of Pr. This contribution is strongly enhanced by coupling with orbital moments of f electrons [86, 87]. It is difficult to measure specific heat by a conventional relaxation method at temperatures where nuclear degrees of freedom dominate because of additional the time scale entering the experiment, nuclear spin-lattice relaxation time T1 [88]. Strongly non-exponential temperature decays at the lowest temperatures (e.g., below 0.5 K in the field of 10 T and bellow 1.5 K in the field of 32 T) indicate the importance of nuclear degrees of freedom and cannot be analyzed using the so-called 72 correction. Therefore, these lowest temperature points carry large uncertainty. When the magnetic field applied along the (1 00) crystallographic direction is 10 T, the temperature of the sharp FIOP peak appears at 1 K (Fig. 6-2). When increasing the field from 10 T field to 12 and 13 T (Figs. 6-2 and 6-3) the ordering temperature T, decreases only slightly but C(T,) is suppressed in a strong manner. The results presented here [31] combined with those of Aoki et al. [56] and Vollmer et al. [63] show that T, (peak position in C) reaches a maximum value around 9 T. Also, C at T, is maximum somewhere between 8 and 10 T. In 13 T a shoulder appears on the high temperature side of the FIOP anomaly. The specific heat value at this shoulder is about 3400 mJ/K mol. This shoulder evolves into a broad maximum for H=13.5 T. Above 13.5 T the FIOP cannot be observed anymore in the specific heat. Thus, these results strongly imply the disappearance of FIOP before T, reaches 0. The broad maximum that appears in 13 T exists at all fields studied up to at least 32 T. The temperature of the maximum increases with the strength of the field (Fig. 6-4). The magnitude of this anomaly, in fields of 13 T and larger, li,,-; between 3300 and 3500 mJ/Kmol and it is field independent. These values are within about 10''. of the maximum value for a Schottky anomaly of a two level system with identical degeneracies [89]. The uncertainty of the specific heat measurements in these fields (and at temperatures where nuclear contribution is small) is about 1C'. Increasing 0D from 165 K, used in the subtraction of the phonon term, to the other extremal value proposed, 320 K would raise the estimate of the electronic part of C by about 290 mJ/K mol at 3.5 K. Thus, the extracted values at the maximum are well within the realistic error bar of the theoretical 3650 mJ/K mol for the two-level Schottky anomaly. The highest field used of 32 T is large enough to split any degenerate levels, therefore the observed Schottky anomaly is due to the excitations between two singlets. Tm is related to the energy separation of the two levels 6 by Tm=0.4176 [89]. An extrapolation of Tm to T=0 (Fig. 6-5) determines the field at which the two levels cross, which is somewhere between 8 and 9 T. These result can be used to infer new information regarding the plausible (i --I I1 field configuration of Pr. Pr can be modeled by the following single-site mean-field Hamiltonian [84]: = -CEF gJ pBJ H J(J') J S- QL(O')O,' (6-1) where -CEF, J and Oi represent the CEF Hamiltonian for the cubic Th symmetry, the total angular momentum, and the i-th quadrupole moment of Pr in a sublattice, respectively, where there are five types of quadrupolar moment operators: O0, O2, Oxy, Oyz, and Oz j and Qi are the inter-sublattice molecular field coupling constants of spin (exchange) and quadrupolar interactions, respectively. The thermal averages of the angular momentum and quadrupole moment of the Pr in the counterpart sublattice are (J') and (0O). Using the CEF parameters proposed by Kohgi [84] for the 1F-F5 CEF configuration, Tm (with Qi=0) and the Oy,-type quadrupolar ordering temperature T, were calculated for (1 00) direction by Aoki et al. [90] As it is demonstrated in Fig. 6-5, the measured phase diagram and the theoretical one (in the insets) expected for the Fj F5 model for H//(1 0 0) are in very good agreement. In both diagrams, the crossing field is very close to the one at which the transition temperature of the FIOP becomes maximum. Thus, the observed correlation between the two characteristic fields constitutes a very strong argument for the F1 singlet being the lowest CEF level. However, the level crossing for field (1 0 0) direction is also expected for F3-F4 model, although at somewhat different field, as demonstrated by Vollmer et al. [63]. More conclusive arguments regarding the CEF configuration can be obtained from the study of the anisotropy of the Zeeman effect. Results of our calculations for the Zeeman effect for 1//(100), f//(110), and Hf//(111) are shown in Fig. 6-6 for FP CEF ground state. The plots show only the four lowest CEF levels. The higher levels are at above 100 K and 200 K from the ground state, and therefore p1 i, no role in the low temperature properties. The calculations were done neglecting exchange and quadrupolar interactions and considering the Th symmetry. Neglecting or retaining the last two terms in (6-1) for the (10 0) direction lead to almost identical results for eigenvalues (Aoki et al. [90] and our results). There is a crossing between FP and the lowest F5 level (split by magnetic field) at about 9 T when H//(1 00) or H//(1 1) and anti-crossing when H//(1 1 0) around the same field. Therefore, the crossing field, extrapolated from the temperature of the Schottky anomaly at high fields should be independent of the field direction. Figure 6-7 shows the same calculations for the F3 CEF ground state model. For H//(1 0 0) there is a crossing between the two lowest CEF levels, although at a field somewhat larger than the one expected for the fl CEF ground state. However, there is no crossing expected involving the lowest CEF levels when the field is applied along the (1 0) or (111) direction in the 73-75 model (Fig. 6-7). Therefore, measurements of specific heat when magnetic field is applied in any direction different than (1 00) differentiate between the two scenarios. Measurements of the specific heat in fields to 14 T were done for H//(11 0) and are presented in Fig. 6-8. The inset to Fig. 6-8 shows the specific heat in fields between 8 and 11 T around the AFQ transition. The specific heat at the AFQ transition and the temperature at which AFQ occurs are maximum for 9 T. Figure 6-10 and the inset to Fig. 6-8 -ir--.- -1 that between 9 and 12 T both the specific heat maximum and the temperature at which this maximum occurs decrease. In H 12 T both the AFQ transition and Schottky anomaly are visible. In fields higher than 12 T the AFQ transition is completely suppressed. The broad anomalies from Figure 6-8 at 12, 13, and 14 T are Schottky type. The H-T phase diagram is presented in Figure 6-10. For H//(11 0) direction we observe a decrease of T, values with respect to the (1 00) direction for the corresponding fields, consistent with the previous magnetization measurements [51] (Fig. 3-6). On the other hand, within the uncertainty of the measurement, there is no change in the position of the Schottky anomaly at 13 and 14 T, as expected for the Fl CEF ground state and inconsistent with the F3 scenario. Moreover, for the (1 10) orientation the Schottky anomaly can be clearly seen already at 12 T. This lower field limit for the Schottky maximum is probably due to competition between the two types of anomalies and lower values of T, for the (1 0) direction (Fig. 6-9). A straight line fit for the three Tm points results in the crossing field value of 91 T. This value agrees, within the error bar, with the estimate for the (10 0) direction. The existence of the crossing field for the (1 10) direction provides an unambiguous evidence for the F1-F5 model. A small misalignment of the sample with respect to the field in either of the measurements cannot explain essentially identical crossing fields for both directions. In fact, the measured difference in T, values for (1 00) and (1 0) directions provides an additional check of the alignment. Similar to the (10 0) direction, there seems to be a close correlation between the crossing field and the field corresponding to T, maximum. Figures 6-5 and 6-10 imply a strong competition between the field-induced order and the Schottky peak. The FIOP transition in the specific heat abruptly disappears before T, reaches zero. Precise magnetization measurements [44, 51], on the other hand, were able to map T, as a function of the magnetic field all the way to Tz0. This apparent contradiction can be explained by a very small entropy available for the FIOP transition above 13 and 12 T for fields parallel to the (10 0) and (110) directions, respectively. Specific heat, being a bulk measurement, can be less sensitive than magnetization techniques in this situation. A strong competition is to be expected in the F1-F5 scenario. The ground state pseudo-doublet formed at the level crossing carries both magnetic and quadrupolar moments. Since a quadrupolar moment operator does not commute with a dipolar one, the quadrupolar interactions leading to FIOP compete with the magnetic Zeeman effect. Therefore, the high magnetic fields measurements of specific heat [31] provided the first unambiguous evidence for the singlet CEF ground state of Pr in PrOs4Sb12. This result was confirmed by recent inelastic neutron scattering experiments [32] analyzed in the Th symmetry, and our magnetoresistivity results described in Sections 6.2 and 7.2. 6.2 Magnetoresistance of PrOs4Sb12 Magnetoresistance of PrOs4Sb12 was measured to search for further experimental evidences of the proposed CEF scheme and for possible signatures of heavy-fermion behavior. The main indication of heavy electrons in PrOs4Sb12 is the large discontinuity in C/T at T,. The mass enhancement inferred from specific heat measurements is of the order of 50 [7]. This value is an estimate and there is no consensus on a precise value. An uncertainty exists in evaluation of the effective mass directly from the low temperature zero-field specific heat, because there is no straightforward method of accounting for the CEF specific heat. The corresponding Schottky anomaly is strongly modified because of the hybridization between the f and conduction electrons. The zero-field specific heat just above T, is dominated by CEF effects. Several other estimates of m* have been proposed. For instance, Goremychkin et al. [32] -,r-.-. -I. m* enhancement to be about 20. However, their estimate was based on the Fulde-Jensen model, which we do not believe is relevant to PrOs4Sb12. This enhancement is 3-7, according to the de Haas-van Alphen measurements [44]. However, dHvA effect was analyzed over a wide range of fields 3-17 T and did not take into account m* being dependent of H [44]. The residual resistivity ratio RRR=p(300K)/p(T-+O) of the investigated sample was about 150. This value is among the highest reported, implying high quality of our sample. Both the current and the magnetic field were parallel to the (1 00) direction (longitudinal magnetoresistance). The measurements were done using the 18 T/20 T superconducting magnet at the Millikelvin Facility, National High Magnetic Field Laboratory, Tallahassee, Florida. The temperature range was 20 mK to ,0.9 K, the maximum field used 20 T. Measurements at the University of Florida were done in fields up to 14 T down to 0.35 K. The zero-field electrical resistivity, another important characteristics of heavy fermion metals, does not provide a straightforward support for the presence of heavy electrons. Maple et al. [91] found that the resistivity, between 8 and 40 K, follows a fermi-liquid temperature dependence (p=po+AT2). Our resistivity data between 8 and 16 K follows the above mentioned dependence (Fig. 6-11) with A =0.009 cm/K2 (in agreement with A found by Maple et al. [91]). As inferred from Kadowaki-Woods (KW) relation (A/7l y x110-5 Rcm(mol K/mJ)2) [60] this value of A implies a small electronic specific-heat coefficient 7 ,30 mJ/K2 mol, comparable to the one measured for LaOs4Sb12. So, evidently there is an upper temperature limit (less than 8 K) for the heavy fermion behavior. Figure 6-12 shows the resistivity of PrOs4Sb12 in H=3, 10, 15, 16, 17 and 18 T in a temperature range of 20 mK to 0.9 K. The resistivity below 200 mK saturates for all fields. This temperature dependence at the lowest temperature was also observed by other groups [91, 92]. Therefore, the resistivity for all other intermediate fields was measured to 350 inK. Figure 6-13 shows the resistivity between 350 mK and about 1.3 K for several relevant fields (3.5, 5.5, 7, 10, and 13 T). Maple et al. [91] proposed the following temperature dependence for fixed magnetic field: p=po+aT", with n>2. In their study (transversal magnetoresistivity) n was ~3 for 3 T and 2.6 for 8 T. In our longitudinal case these exponents are slightly larger (e.g., 3.9 for 3 T). The exponents depend on the temperature range of the fit, i.e., n becomes smaller when the upper temperature limit of the fit decreases. The residual resistivity po values resulted from the fit on different temperature ranges (included in the 350 mK and 0.9 K interval) were close to p at 20 mK. The residual resistivity po attains a maximum near H=10 T, field corresponding to the crossing between the two lowest-energy CEF levels of Pr (Fig. 6-14, lower panel). In this region (around 9-10 T) the lowest two singlets form a quasidoublet possessing quadrupolar degree of freedom. These electric quadrupoles order at sufficiently low temperatures [56] with the ordering temperature having maximum in the crossing field [31, 56]. Resistivity is dominated therefore by the CEF effects or the quadrupolar ordering. This ordering is completely suppressed by fields higher than 15 T. As it can be noticed from Figs. 6-12 and 6-14 the residual resistivity po does not change substantially in fields higher than 15 T. In fact, it can be seen than the residual resistivity po versus H field can map the boundary of the AFQ phase, i.e. a sharp increase of po indeed coincides with the AFQ boundary, indicated by arrows in Fig. 6-14, lower panel. The same conclusion can be drawn from resistivity measurements for high magnetic fields perpendicular to the current [55]. The rate of the increase of the resistivity with temperature is still changing above 15 T (Fig. 6-12). It can be concluded that the reduction of the temperature rate correlates with an increase of the energy between the lowest CEF levels. A precise accounting of these changes is difficult since neither of the functions checked out describe accurately the variation p(T) in a fixed field. A linear dependence of p on T2 is accounted by resistivity (p po+AT2) in different temperature ranges (above 0.4 and 0.5 K), as seen in Figs. 6-13. Using the KW ratio [60] (however, there is no experimental or theoretical studies on A/72 for Pr-based systems) the electronic specific heat coefficients for H=3, 10, and 18 T are about 200, 400, and 200 mJ/K2 respectively. The A coefficient deduced from the narrow range of temperature (Fig. 6-15) increases sharply with the magnetic field and reaches a maximum near 6 T. After a plateau between 6 and 12 T, a strong decrease is encountered. This establishes a correlation between A and both AFQ order and CEFs with a strong increase when approaching the AFQ boundary. A characteristic field dependence of the residual resistivity (Fig. 6-14, low panel) was associated both to CEF effects and long range AFQ order. The CEF effect on the resistivity was considered by Frederick and Maple [93] using the following expression: PCEF [Tr(PQM) + Tr(QA)]. (6-2) The first term represents a contribution due to exchange scattering, and the second term is the contribution due to aspherical (or quadrupolar) scattering. The aspherical Coulomb scattering is due to the quadrupolar charge distribution of the Pr3+. Matrices Pij, QM and Q j are defined as follows: E p(E, Ej) P = (63) Z e- r 1- e-(E, -Es)' (6 3) 1 1 QJ J j2 (i j 2 J 2, (6-4) +2 QAi 22 m=-2 In the above relations Ei are the eigenvalue of the CEF eigenstates, the Wi)'s are the CEF eigenstates, 3=1/(kBT), and the yp's are the operator equivalents of the spherical harmonics for L=2 (i.e., quadrupolar terms) [94]. The Qyi-matrices are normalized to each other [95] such that QYM = Q (2J+)J( 1 J+1)J4 180. (6-6) The most intriguing conclusion is a strong enhancement of the A coefficient with the magnetic field between 2 and 6 T. This could imply an enhancement of m* for fields in this range. Low temperature resistivity calculation for F3 and Fl CEF ground state are shown in Fig. 6-16 (upper and lower panel, respectively). Our residual resistivity (at T=20 mK) seems to be in a better agreement with calculations for F3 than for F1 ground state. Thus, CEF cannot account for the magnetoresistance of PrOs4Sb12. We will return to this interesting problem while describing the magnetoresistivity of La-doped ( i--I '1- Furthermore A seems to have a maximum value near the field separating ordered and non-ordered phases. Note that this is not the crossing field for the lowest CEF levels. Thus, these results -'ir-., -I a possibility of m* enhancement due to strong fluctuation of the AFQ order parameter. ...[.. ..s.... . 4 H= 04 E 0".. T "-" ..."" I" I ' 3T l TOP 6T 02 max. in M vs T 8 4T 0 2 2 K 4 6 T i.i 6- TX max. in M vs T Smax. in dCfdH vs H 2 PrOsSb 1 #,oH <100> Superconductivity H 01 0 1 2 3 4 T IK) Figure 6-1. Specific heat C of PrOs4Sbl2 in fields up to 8 T for f//(10 0) (upper panel). The magnetic field phase diagram H-T of PrOs4Sb12 in fields up to 8 T for 1H//(100) (lower panel) (Reprinted with permission from Aoki et al. [56]). 8000 6000 4000- E * | 2000- *** H=10 T "3 H= 10T v 6000 I ,- I ,- I ,- I 0U 4000- S" 0* H=12T 20001 0 1 2 3 4 5 T (K) Figure 6-2. Specific heat C of PrOs4Sbl2 in 10 and 12 T in the vicinity of FIOP transition for //(10 0) (Reprinted with permission from Rotundu et al. [31]). 8000 6000 * 4000 2000 H= 13 T 8000 I I I I i o E S6000 E O 4000 ** m U. 2000 H= 13.5 T 8000 I I I I 6000 4000 - 2000 H= 14 T 0 1 2 3 4 5 T (K) Figure 6-3. Specific heat C of PrOs4Sb12 in 13, 13.5, and 14 T, for H//(100). A shoulder appears at about 1.2-1.3 K at 13 T and the FIOP transition is suppressed at 13.5 T. 4000 3000 - 2000 1000 H= 16 T 4000 I I I I I i .*** . E 3000 . 1000 2000 - 1000 H= 20 T 4000 I I I I I 3000 2000 1000 H= 32 T 0 1 2 3 4 5 T (K) Figure 6-4. Specific heat C of PrOs4Sbl2 in magnetic fields of 16, 20, and 32 T, for H//(1 00). I I II I 30 H/(100) i - 30 20 25 10 . 0 0123456 20 - I 15 10 [ 0 1 2 3 4 T (K) Figure 6-5. Magnetic field phase diagram H-T of PrOs4Sbl2 for H//(1 00) (H>8 T). Filled squares represent the FIOP transition. Open squares correspond to the Schottky anomaly. The inset is the model calculation of the Schottky anomaly assuming the singlet as the ground state[90]. The solid line represents the FIOP boundary; the dashed line corresponds to a maximum in C (Reprinted with permission from Rotundu et al. [31]). H//[100] H//[ 110] H//[111] -60 -60 -60 -70 Z -70 --70 -80 / -80 -- -80 -90 -90 --90 -100 --100 --100 -110 -110 -110 -120 -120 -120 0 10 20 0 10 20 0 10 20 H (T) Figure 6-6. Zeeman effect calculations for PrOs4Sb12 in the F1 CEF ground state scenario. There is crossing of the two lowest levels for H//(1 0 0) or H//( 111) at around 9 T and anti-crossing at the same field for H//(11 0). The figure shows only the two lowest levels, i.e. the singlet F1 and the triplet F5. H// [100 ] -60 -70 -80 -90 -100 -110 -120 H//[ 110] 0 10 20 0 10 H (T) H//[ 111] -60 -70 / -80 -90 -100 110 -120 20 0 10 20 Figure 6-7. Zeeman effect for PrOs4Sb12 in the F3 CEF ground state scenario. The effect in strongly anisotropic. There is no crossing of the two lowest CEF levels for H1//(11 0) or H//7(111). The figure shows only the two lowest levels, i.e. the doublet F3 and the triplet F5. -90 -100 -110 -120 7000 , H /(116) A 8T A A H=11T 600 9T 6000 H=12T 9.5T , e 10T H=13T 5000 v 10.5T '. H=14T 11T ..1 * 5000 . 40004000 : 4000 A 1 1 - E t 3000 S0.6 0.7 0.8 0.9 1.0 1.1 3000 2000 w A. 0 1 2 3 4 5 T (K) Figure 6-8. Specific heat C of PrOs4Sbl2 for H//( 1 0), H=10, 12, 13, and 14 T. The inset shows C versus T neat T, for 8, 9, 9.5, 10, 10.5, and 11 T. 5000 H=12T 5000 -, ** H /(100) T* 4000 3000 x 0 * E S 2000 I I I i | I | i | 3500 H//(110)- 0 as NO=Em 3000 , x 2500 - 0.0 0.5 1.0 1.5 2.0 T (K) Figure 6-9. Specific heat C of PrOs4Sbl2 in H=12 T, for H//10 0) (upper panel), and H//(1 1 0) (lower panel). The arrow indicates the AFQ transition. 1.0 T (K) Figure 6-10. The magnetic field phase diagram H-T of PrOs4Sb12 for H//(l 1 0) (H>8 T). The inset shows the specific heat Cma, of AFQ versus H. For a definition of symbols see Fig. 6.5. 15 I I I I T =1.85K 10 20 1O 5- 15 0 2 4 6 8 10 12 14 16 T (K) 2- 10 P=Po+AT2 A=0.009 pncm/K2 5 0 50 100 150 200 250 T2 (K2) Figure 6-11. Electrical resistivity p versus T2 for PrOs4Sb12. In the inset is p versus T showing the superconducting transition at T-=1.85 K. 0 U E 6000 * E -E 5000 o 8 9 10 11 H (T) H//(110) 4.2 I H =10T 4.0 3.8 3.6 -H =15 o 3.4 H =16T- 3.2 H =17T 3.2 a. H =18T 3.0 H = 3T 2.8 2.6 I I 0.0 0.2 0.4 0.6 0.8 1.0 T(K) Figure 6-12. Resistivity p(T) between 20 mK and about 0.9 K of PrOs4Sb12 in 3, 10, 15, 16, 17, and 18 T (Reprinted with permission from Rotundu and Andraka [96]). 5.5 7T 5.0 ^ 10T 5.05.5T 4.5 E00,0013T S4.0 3.5 3.5 T 3.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 T (K2) Figure 6-13. Resistivity p versus T2 of PrOs4Sbl2 for 3.5, 5.5, 7, 10, and 13 T. 2.5 1 1 1 5 p=Po+ aT 4 2.0 n>2 C 3 . 2 - S1.5 0 4 8 12 16 S. H(T) EU 1.0 - 0.5 - U 0.0 0 2 4 6 8 10 12 14 16 18 3.5 - 0C 3.0 0 2 4 6 8 10 12 14 16 18 H (T) Figure 6-14. Coefficient a (p=po+aT") versus H for PrOs4Sb12 fields up to 18 T is in upper panel. The residual resistivity po(H) is shown in lower panel. I I I I I U- p=po +AT 2 0 2 4 6 8 H(T) 10 12 14 Figure 6-15. Coefficient A (p=po+AT2) versus H. 2.5 I- 0.5 - IIIIIIIIIIIIII 2 1.4 K 0.5(0.5pmaa + 0.5pACs) S 0.35 K r. ground state (a) 0 5 10 15 20 H (T) 41.4 K / PrOs4 Sb12 Q 4.2 K -------- 21 :3- r. mag ACS (b) S0.35 K r ground state 0 1 I I I 0 5 10 15 20 H MT' Figure 6-16. The calculated p(H) of PrOs4Sb12, for both F3 and Fi scenarios. The vertical line indicates the field crossing of the two lowest CEF levels. Note that the crossing field for the F1 ground state was assumed at 3 T (lower panel) (Reprinted with permission from Frederick and Maple [93]). CHAPTER 7 Pr1_xLaxOs4Sbl2 In this C'!I pter the La-alloying study of PrOs4Sb12 by dc and ac susceptibility, specific heat and resistivity is presented. One of main objectives of this work was to differentiate between different proposed models of the conduction electron mass enhancement in PrOs4Sbl2. Mechanisms that have been considered range from single-ion models, such as the quadrupolar Kondo effect [7, 16] or virtual CEF excitations [32, 36], to cooperative models invoking proximity to a long-range order (proximity to the low temperature state of AFQ order) [56]. While investigating the applicability of these models, close attention was paid to whether the single-ion parameters such as the CEF spectrum and hybridization parameters vary with the alloying. 7.1 Lattice Constant The room temperature X-ray diffraction patterns for several La concentration are given in Fig. 7-1. The results of the X-ray diffraction analysis were consistent with single phase materials. A very small and monotonic increase of the lattice constant, a, with the La content is detected (Fig. 7-2). The lattice constant is calculated from the high index line (8 2 2) of the X-ray diffraction pattern. Calculations from smaller angle lines result in the same dependence of a on x, but had a much larger scatter. These small changes (about 0.0 :'.) between the end compounds is in agreement with previously reported [57] and almost non-existent lanthanide contraction in ternary skutterudites containing Sb, of a general form LnT4Sbl2, where T and Ln are transition element and light lanthanide, respectively. To present this change in a proper perspective we recall that the change of the lattice constant across Pr(Osl-xRux)4Sbl2 [97] is 10 times larger. This is despite the fact that the atomic radii of Os and Ru are almost identical (1.35 and 1.35 A for Os and Ru respectively), while La is much larger than Pr (1.88 versus 1.82 A). In Pr(Osl-xRu1)4Sbl2 the CEF parameters increase monotonically with x. Very small changes in lattice constant in PrlxLa1Os4Sbl2 sil--. -i small, if any, changes in the CEF parameters and hybridization. These parameters depend on the position of ligand atoms with respect to Pr. These small changes of the lattice constant in Prl_-LaOs4Sb12 provide a unique opportunity for the alloying study of superconductivity and other phenomena that are strongly influenced by microscopic inhomogeneities associated with the lattice constants' mismatch. 7.2 DC Magnetic Susceptibility Susceptibility measurements were done on single-crystal samples of masses ranging between a few mg to 50 mg belonging to different batches. Specific heat measurements were performed as well on samples characterized by the susceptibility. Figure 7-3 shows the susceptibilities only in the range 1.85 to 10 K. All data are normalized to a Pr mole. Due to the very small size of the samples used in the initial measurements, the measured moment of most of the samples and the background (the magnetic moment of the sample holder consisting of a piece of a plastic drinking straw) were comparable at 10 K. At room temperature the magnetic moment of the samples was even smaller than the background, especially for dilute concentrations. In order to avoid this background contribution, the magnetic susceptibilities were remeasured (for x=0, 0.05, 0.3, 0.8 and 0.95) using several < iv--I 1- and holding them between two long concentric straws. No background subtraction was needed this time. The Curie-Weiss temperature was found above 150 K and the effective magnetic moment in the range 3.2-3.6PB/Pr atom. The values are in the range of moments reported for pure PrOs4Sb12. Some discrepancy between these values and that expected for Pr3+, 3.58pB [52], can be due to an error in mass determination. Because of the very fragile nature of these < i --i i- some of them broke off during the measurement and small fractions moved in between the two tubes. A further check of the magnetic moment was performed on one large crystal for x=0.67 (~20 mg each). Figure 7-4 shows the susceptibility and the inverse of susceptibility for x=0.67. From the Curie-Weiss fit the high temperature effective moment is found to be 3.62MB/Pr mol, close to the value expected for Pr3. All low temperature susceptibility data (Fig. 7-3) exhibit a broad maximum at 3-5 K due to excitations between the lowest CEF states. Our high magnetic field specific heat study [31], and neutron [32, 84] measurements established F1 singlet as the CEF ground state separated by about 8 K from the first excited F5 triplet. Very small changes in the position of these maxima in the susceptibility are the first indication that CEF are essentially unaltered by the doping as expected from the measurement of the lattice constant. Another interesting aspect of the susceptibility is a strong initial reduction of the low-temperature values of X (normalized to a mole of Pr) by La. The reduction of the maximum susceptibility from approximately 100 for x=0 to about 50 memu/Pr mol for x=0.4 is clearly outside the error bar. The aforementioned measurements on assemblies of crystals for x=0.8 and 0.95 also resulted in a 4 K value of about 505 memu/Pr mol for both compositions. Some broadening and decrease in magnitude of the CEF susceptibility are expected in mixed alloys due to increased atomic disorder. However, the very large initial drop in the susceptibility and lack of variation above x=0.4 might indicate that some characteristic electronic energy (analogous to a Kondo temperature) increases sharply upon substituting La for Pr. A similar suppression of the corresponding maximum is observed in the specific heat data discussed in the next section. 7.3 Zero Field Specific Heat 7.3.1 Specific Heat of PrOs4Sb12: Sample Dependence The main evidence for heavy fermion behavior in PrOs4Sb12 comes from a large discontinuity of the specific heat at T,. Specific heat provides evidence for unconventional superconductivity. The evidence includes a power low dependence of C below Te, and the presence of two distinct superconducting transition. PrOs4Sb12 was initially reported to have a single superconducting transition at T-=1.85 K [7]. More recent specific heat measurements revealed two superconducting transitions (Vollmer et al. [63], Maple et al. [53], Oeschler et al. [64], Cichorek et al. [65]). Before discussing zero field specific heat measurements of Prl-La0Os4Sbl2 we need to comment on the sample dependence of the specific heat of PrOs4Sb12. This is in order to distinguish La-induced changes from variation related to sample quality. In Fig. 7-5 we present the specific heat near T, for three representative samples from three different batches. All three samples were obtained in an identical way. All three samples have pronounced lower temperature transitions. The upper temperature transition is less distinct and sample dependent. Our convention of defining Tji and Tc2 (by local maxima or shoulders in C/T versus T) is illustrated in Fig. 7-5. The upper transition temperature T,~ is identical for all three samples. There seems to be some sample dependence for the lower transition temperature Tc2. However, the variation is very small considering that each C/T at a given T in Fig. 7-5 was obtained by integrating the specific heat over 0.04T interval. The width of the transition defined, for comparison reason, by T3-Tc (Fig. 7-5) is large and approximately equal for all the samples. Finally, A(C/T) defined as the difference between C/T at Tc2 and T3 is about 800 mJ/K2 mol. Our observations are consistent with other, particularly more recent, reports. Almost all recent investigations find two superconducting anomalies, more pronounced at Tc2 and less defined at Tci. An exception to this rule are unpublished data by Aoki et al. [98] that show a sharp peak at T1I, and only a change of slope in C/T at Tc2. The width of the transition, ~0.2 K, defined above, is quite similar for all published data. There is a large distribution of reported A(C/T) at Tc, from 500 to 1000 mJ/K2 mol. A usual determination of A(C/T) by an equal area (conservation of entropy) construction cannot be applied due to the presence of two superconducting transitions. Applying our method, C/T(Ta)-C/T(T3), results in an average A(C/T) of 800 mJ/K2 mol for the most recent results. 7.3.2 Zero Field Specific Heat of Pr1_-LaOs4Sb12 In order to account for normal-electron and phonon contributions to the specific heat of Pr1_-La1Os4Sb12 alloys, the specific heat of LaOs4Sb12 was measured. The normal state specific heat between 1 and 10 K is shown in Fig. 7-6 in the format of C/T versus T2. The results can be expressed by the following equation C/T = 56 + 1.003T2 + 0.081T4 4.260 x 10-4T6, (71) where C/T is expressed in mJ/(K2 mol) and T in K (Fig. 7-6). A significant nonlinearity in C/T versus T2 is probably due to the rattling motion of loosely bound La atoms [99]. Values of 7 and / for LaOs4Sb12 reported by other research groups are: 7 of 36 [54], 55 mJ/K2 mol [100], 56 [44] and j=0.98 mJ/K3 mol [101]. Figures 7-7 and 7-8 present the f-electron specific heat of Pr1_-La0Os4Sb12 alloys, i.e. the specific heat of LaOs4Sb12 and, normalizing to a mole of Pr. Note that the phonon specific heat of pure PrOs4Sb12 in C'! Ilpter 6 was taken from Vollmer et al. [63], which was derived by fitting the total specific heat C to a function representing phonon, conduction electrons, and Schottky contributions. However, using the LaOs4Sb12 specific heat seems to be more justifiable for moderately and strongly La-doped alloys and therefore this way of accounting for phonons is consistently used in this chapter on La alloying. 7.3.2.1 Evolution of T, with the La Doping Figure 7-7 shows the specific heat for x=0, 0.05, 0.1, and 0.2 near the superconducting transition temperature. As already discussed, the pure compound has two superconducting anomalies in the specific heat. The specific heat data for x=0.05 exhibit a shoulder which seems to correspond to the anomaly at T,1 for x=0. The width of the transition (T3-Tc2) for x=0 is about 0.2 K. This width becomes slightly smaller for x=0.05. The specific heat for x>0.3 alloys (Fig. 7-8) exhibits one superconducting transition only. The width of the transition for this group of materials is about a half of the width of the pure material. This reduction cannot be accounted for by the reduction of T, itself. A similar conclusion about a drastic reduction of the width of the transition can be derived from graphs in which T is replaced by a reduced temperature T/T,. The reduction is probably related to a disappearance of one superconducting transition (at TI,). Open symbols in Fig. 7-9 denote the lower temperature superconducting transition, and the filled squares symbolize the higher temperature superconducting transition Tc2. Lanthanum doping has a surprisingly weak impact on Tc. This weak dependence (approximately linear) of T, on x in Pr1_-LaxOs4Sb12 is unusual for heavy fermion alloys. For instance, heavy fermion superconductivity in UBe13 is completely suppressed by only . La [102] substituted for U. Furthermore, since PrOs4Sb12 is clearly an unconventional superconductor (e.g., time reversal symmetry breaking) while LaxOs4Sb12 is presumably a conventional superconductor we would expect, while varying x, a suppression of one type of superconductivity before the other type emerges. Figure 7-10 shows that there is smooth evolution of T, (and superconductivity) between the end-compounds. A somewhat stronger suppression is observed in the case of Ru replacing Os [71, 97]. But even in this case, the T, reduction rate is small if compared with the in i ii i iy of Ce- and U-based heavy fermions and considering the fact that Ru alloying drastically affects CEF energies and hybridization parameters. 7.3.2.2 The Discontinuity in C/T at Tc Existence of two distinct superconducting transitions in PrOs4Sb12 makes the determination of the discontinuity in C/T somewhat arbitrary. Furthermore, it also complicates the interpretation of this discontinuity. Despite a substantial recent improvement in sample quality, the question whether the two transitions correspond to different regions of the sample becoming superconducting at different temperatures or whether the lower transition corresponds to the change of the symmetry of the superconducting order parameter in a homogeneous medium is not completely settled. The comparable magnitude (equal as argued by Vollmer et al. [63]) of the anomalies at T,~ and Th2 precludes a popular speculation that one of these transitions is associated with surface superconductivity. As it was already stressed, this A(C/T) is currently the main evidence for the presence of heavy electrons. The presence of a modified Schottky anomaly near T, makes a direct determination of the electronic specific heat coefficient unreliable. In a BCS-type superconductor, AC/T, is related to the electronic specific heat coefficient 7. In general, AC/T, and 7 are affected by the coupling strength of the Cooper pairs and can vary by a factor of order of 2-3. Nevertheless, for lack of any other measure, we use this quantity as an indication of heaviness of electrons in Pr1_,LaOs4Sbl2. We recall that LaOs4Sb12 is also a superconductor, therefore for dilute Pr concentrations the normalization of C/T to Pr mole (used in Fig.s 7-7 and 7-8) has no meaning. Therefore, Fig. 7-10 shows the total A(C/T) per formula unit. There is a drastic decrease of A(C/T) with x for 0 800 for x=0 to 280 for x=0.2 and further to about 160 mJ/(K2 Pr mol) for x=0.3 (C/T is seven fold reduced with x, for x between 0 and 0.3). A(C/T) -1 i,-; approximately constant with x for 0.3 LaOs4Sbl2. Therefore, these results -i,--.- -1 that the heavy fermion character disappears above x=0.3. Therefore, there is a lack of strong correlation between the heavy-fermion character as measured by A(C/T) at T, or AC/T, and the average T,. Thus, the results argue also for different mechanisms responsible for the heavy fermion state and enhanced value of T, in PrOs4Sb12. 7.3.2.3 The Schottky Anomaly The evolution of Schottky-like anomaly in pure PrOs4Sb12 near 3 K with magnetic fields provided important information on the CEF configuration of Pr. The discrepancy between the theoretically predicted temperature dependence and the observed one has been explained by mixing of f- and conduction-electrons degrees of freedom, reducing the entropy associated with the pure Schottky anomaly. The results on the discontinuity of C/T at T,, presented in the previous section, -Ii--.- -1 that the heavy fermion character is strongly suppressed by the La alloying. Thus, one would expect the mixing to be reduced for strongly La-doped samples and the Schottky anomaly to approach the theoretically predicted temperature dependence. Recall that the maximum value of the specific heat in PrOs4Sb12 is about 7100 mJ/K mol, smaller than the theoretical value of 8500 for singlet to triplet excitations. However, as it can be inferred from Figs. 8.7 and 8.8 the specific heat at the Schottky-like maximum decreases noticeably with the alloying. Some reduction and broadening of such a maximum due to CEF excitations can be explained by disorder effects induced by the alloying. However, this effect should be small. La does not change the local symmetry of Pr nor the distances between Pr and its nearest neighbors. Furthermore, the main reduction takes place between x=0 and x=0.2 and, within our error bars, there is no appreciable change beyond x=0.3. This reduction, by a factor of 2, is about the same as the decrease of the low temperature magnetic susceptibility, discussed in Section 7.2. In order to study further this specific heat reduction, a large crystal of Pr0.33La0.670S4Sb12, for which addenda heat capacity is negligible below 6 K, was investigated. First of all, the effective magnetic moment measured at room temperature for this particular crystal is very close to the expected value for Pr3+. Thus, the reduced specific heat cannot be explained by incorrect Pr stoichiometry, nor by some Pr ions being in a mixed-valent state. The results, after subtracting the specific heat of LaOs4Sb12 and dividing by 0.33, are shown in Fig. 7-11 in the form of C/T. This graph shows also a fit to the function describing a Schottky specific heat for a singlet-triplet excitations, scaled by a factor a=0.44. A similar scaling was used by Frederick et al. [71] to account for the specific heat data of Pr(Osl-xRux)4Sbl2 in terms of the singlet CEF ground state. A necessity to use such a small scaling factor for this model of CEF (of about 0.5) was used by Frederick et al. [71] to argue for a doublet CEF ground state. However, as can be seen from Fig. 7-12, a reasonable fit to the doublet CEF model also requires a scaling factor, although somewhat larger (a=0.73). Finally, a fit to the singlet-to-singlet scattering requires no scaling at all. The fit shown in Fig. 7-13 obtained with a as an adjustable parameter, resulted in a 1.009 (within 1 .). |

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Iamdedicatingthisworktomydearesthumanbeings,myparentsElenaandConstantinRotundu.Icannotndanadequatewaytoexpressmyloveandgratitudetothem.Ithankthemfortheirinnitesupportinordertocompletemyeducation.IamgratefultomymotherandmybrotherRomulusNeculaiRotunduforbelievinginme.Iowemuchtomyadviser,Dr.BohdanAndraka.Hehasbeenanincrediblesourceofguidanceandinspiration.Hewasagreatadviserwithendlesspatience.Myeducationwouldnothavebeenpossiblewithouthisnancialhelp(throughDOEandNSF).Iamdeeplyindebtedtohim.SpecialthanksgotoProf.YasumasaTakanoforteachingmesomanyexperimentaltricks;andfordiscussions,support,andgreatcollaboration,especiallyattheNationalHighMagneticFieldLaboratory(NHMFL).Hewasanendlesssourceofenergy.IwouldliketothankProf.GregoryR.Stewartforlettingmeusehislaboratory.Ithankmyothersupervisorycommitteemembers(Profs.BohdanAndraka,GregoryR.Stewart,YasumasaTakano,PradeepKumarandIonGhiviriga)forreadingthisworkandfortheiradvices.IreceivedhelpwithmanyexperimentsatNHMFLandourlabfromDr.HiroyukiTsujii.IthankDrs.JungsooKimandDanielJ.MixsonIIfortheirtechnicaladvice.OtherpeopleintheeldIwouldliketoacknowledgeareProf.PeterHirschfeld,thenestprofessorIeverhad,whogavemeinsightonthetheoryofcondensedmatterphysics;andDrs.EricC.PalmandTimP.Murphyfortheirhelpandsupportovermorethan4yearsattheSCM1/NHMFL.IthankCenterofCondensedMatterSciencesforthenancialsupportthroughtheSeniorGraduateStudentFellowship.Lastandnotleast,Iwouldliketothankmyhighschoolphysicsteacher(DumitruTatarcan),whoencouragedandguidedmyrststepsinphysics. 4 PAGE 5 page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 14 2THEORETICALBACKGROUND ......................... 17 2.1TheCrystallineElectricField(CEF)forCubicGroup ............ 17 2.2ConductionElectronMassEnhancement(m)MechanisminPrOs4Sb12 22 2.2.1QuadrupolarKondoEect ....................... 22 2.2.1.1ThermodynamicPropertiesoftheQuadrupolarKondoModel ............................. 24 2.2.1.2RelevancefortheCaseofPr3+IoninPrOs4Sb12 25 2.2.2Fulde-JensenModelformEnhancementinPrMetal ........ 26 2.2.3FluctuationsoftheQuadrupolarOrderParameter .......... 26 3PROPERTIESREVIEWOFTHEPrOs4Sb12 30 3.1CrystallineStructure .............................. 30 3.1.1RattlingofPraseodymiumAtom .................... 30 3.1.2Valence .................................. 31 3.1.3CrystallineElectricFields ....................... 31 3.2Normal-StateZero-FieldProperties ...................... 33 3.2.1SpecicHeat ............................... 33 3.2.2deHaasvanAlphenMeasurements .................. 33 3.2.3Resistivity ................................ 34 3.2.4DCMagneticSusceptibility ....................... 35 3.3TheLong-RangeOrderinMagneticFields .................. 35 3.4Superconductivity ................................ 35 3.4.1UnconventionalSuperconductivity ................... 36 3.4.1.1TheDoubleTransition .................... 36 3.4.1.2TemperatureDependenceofSpecicHeatBelowTc 37 3.4.1.3NuclearMagneticResonance(SbNQR) .......... 39 3.4.1.4MuonSpinRotation(SR) ................. 39 3.4.2ConventionalSuperconductivity .................... 40 3.4.2.1NuclearMagneticResonance(SR) ............. 40 3.4.2.2PenetrationDepthMeasurements()bySR ....... 40 3.4.2.3Low-TemperatureTunnelingMicroscopy .......... 40 5 PAGE 6 ........................... 49 4.1TheSamples:SynthesisandCharacterization ................ 49 4.1.1Synthesis ................................. 49 4.1.2X-RaysDiractionCharacterization .................. 51 4.2SpecicHeatMeasurements .......................... 52 4.2.1Equipment ................................ 52 4.2.1.1Cryogenics ........................... 52 4.2.1.2SamplePlatform ....................... 55 4.2.2ThermalRelaxationMethod ...................... 55 4.3MagneticMeasurements ............................ 57 4.3.1DCSusceptibility ............................ 57 4.3.2ACSusceptibility ............................ 58 4.4Resistivity .................................... 59 5MATERIALSCHARACTERIZATION ....................... 65 6PrOs4Sb12 67 6.1InvestigationofCEFCongurationbySpecicHeatinHighMagneticFields 67 6.2MagnetoresistanceofPrOs4Sb12 73 7Pr1XLaXOs4Sb12 93 7.1LatticeConstant ................................ 93 7.2DCMagneticSusceptibility .......................... 94 7.3ZeroFieldSpecicHeat ............................ 95 7.3.1SpecicHeatofPrOs4Sb12:SampleDependence ........... 95 7.3.2ZeroFieldSpecicHeatofPr1xLaxOs4Sb12 96 7.3.2.1EvolutionofTcwiththeLaDoping ............. 97 7.3.2.2TheDiscontinuityinC=TatTc 98 7.3.2.3TheSchottkyAnomaly .................... 99 7.4SpecicHeatinLargeFields .......................... 101 7.5MagnetoresistanceofPr1xLaxOs4Sb12 103 7.6UpperCriticalFieldHc2 107 7.6.1ACSusceptibility ............................ 107 7.6.2DeterminationofHc2(T)bySpecicHeatMeasurementsinSmallMagneticFields ............................. 108 8CONCLUSION .................................... 136 REFERENCES ....................................... 139 BIOGRAPHICALSKETCH ................................ 145 6 PAGE 7 Table page 2-1TherelevantstatesforthequadrupolarKondoeect. ............... 23 3-1Thevaluesreportedbydierentgroups,extractedfromtsofspecicheatbelowTc. ....................................... 38 3-2Thevaluesreportedbydierentgroupsfromothermeasurementsthanspecicheat. .......................................... 38 7 PAGE 8 Figure page 2-1CubicpointgroupsymmetryTh. .......................... 27 2-2Lea,Leask,andWol0srepresentationofCEFforJ=4(Leaetal.,1962). .... 28 2-3RepresentationoftheU4+ionsincubicsymmetryundergoingquadrupolarKondoeect. ......................................... 28 2-4MappingofthequadrupolarKondoHamiltonianontothetwo-channelKondomodel. ......................................... 29 2-5S,C,C=T,andversusT=TKofthequadrupolarKondomodel(SacramentoandSchlotmann,1991). ............................... 29 3-1CrystalstructureofPrOs4Sb12. ........................... 41 3-2Fitsof(T)toeither3or1CEFgroundstate,andCttedbyatwo-levelSchottkyanomaly(Baueretal.,2002). ....................... 42 3-3Fitsof(T)toeither3or1CEFgroundstatemodel,calculatedS(T)inboth3and1CEFgroundstatemodels(Tayamaetal.,2003),andthemeasuredS(T)(Aokietal.,2002). ............................... 43 3-4(T),(T)andC(T)ofPrOs4Sb12(Baueretal.,2002). ............. 44 3-5FermisurfaceofPrOs4Sb12(Sugawaraetal.,2002) ................ 45 3-6H-TphasediagramofPrOs4Sb12bydM(T)=dTanddM(H)=dHmeasurements(Tayamaetal.,2003) ........................ 46 3-7C(T)ofPrOs4Sb12(Vollmeretal.,2003;Meassonetal.,2004)andtherealpartoftheacsusceptibility(Meassonetal.,2004)presentingdoubleSCtransition. 46 3-8ThetwosuperconductingphasesofPrOs4Sb12:phaseAandphaseB(Izawaetal.,2003).TheplotoftheSCgapfunctionwithnodesforbothphases(Makietal.,2003). ...................................... 47 3-9TwoSCtransitionsin(T)ofPrOs4Sb12(Oeschleretal.,2003). ......... 47 3-10Tdependenceoftherate1/T1atthe2Qtransitionof123SbforPrOs4Sb12andLaOs4Sb12(Kotegawaetal.,2003). ......................... 48 3-11TunnelingconductancebetweenPrOs4Sb12andanAutip(Suderowetal.,2004).Thegapiswelldevelopedwithnolow-energyexcitations,signofnonodesintheFermisurfacegap. ................................ 48 4-1PictureofPrOs4Sb12largecrystal(about50mg). ................. 60 8 PAGE 9 .................................. 60 4-3Schematicviewofthe3HecryostatusedinthemeasurementsperformedatUniversityofFlorida. ................................. 61 4-4SchematicviewofthecalorimeterusedintheSuperconductingMagnet1(SCM1),NationalHighMagneticFieldLaboratory. ................... 62 4-5Thesample-platform/Cu-ringassembly. ....................... 63 4-6SpecicheatCmeasurementprocessusingtherelaxationtimemethod. ..... 64 5-1(T)ofPrOs4Sb12.Thehightemperatureeectivemomentis3.65B,veryclosetotheonecorrespondingtofreePr3+,whichis3.58B. .............. 66 5-2(T)ofthenon-fequivalentLaOs4Sb12. ...................... 66 6-1CofPrOs4Sb12ineldsupto8Tfor~H//(100)(upperpanel),andH-Tphasediagramineldsupto8Tfor~H//(100)(lowerpanel)(Aokietal.,2002). ... 78 6-2CofPrOs4Sb12in10and12TinthevicinityofFIOPtransitionfor~H//(100). 79 6-3CofPrOs4Sb12inmagneticelds13,13.5,and14T,for~H//(100). ...... 80 6-4CofPrOs4Sb12in16,20,and32T,for~H//(100). ................ 81 6-5H-TphasediagramofPrOs4Sb12for~H//(100)(H>8T). ............ 82 6-6ZeemaneectcalculationsforPrOs4Sb12inthe1CEFgroundstatescenario. 83 6-7ZeemaneectforPrOs4Sb12inthe3CEFgroundstatescenario. ........ 84 6-8CofPrOs4Sb12for~H//(110). ............................ 85 6-9CofPrOs4Sb12inH=12T,for~H//(100)(upperpanel),and~H//(110)(lowerpanel). ......................................... 86 6-10H-TphasediagramofPrOs4Sb12for~H//(110)(H>8T). ............ 87 6-11versusT2,andversusTforPrOs4Sb12. ..................... 87 6-12(T)ofPrOs4Sb12in3,10,15,16,17,and18T,between20mKandabout0.9K. 88 6-13versusT2ofPrOs4Sb12for3.5,5.5,7,10,and13T. ............... 89 6-14a(=0+aTn)versusHforPrOs4Sb12eldsupto18T(upperpanel).Theresidualresistivity0(H)(lowerpanel). ........................... 90 6-15A(=0+AT2)versusH. .............................. 91 9 PAGE 10 ...................................... 92 7-1X-raydiractionpatternsofPr1xLaxOs4Sb12versusLacontentxforx=0,0.1,0.2,0.4,and1. .................................... 113 7-2LatticeconstantaofPr1xLaxOs4Sb12versusLacontentx. ........... 114 7-3(T)ofPr1xLaxOs4Sb12normalizedtoPrmolebetween1.8and10K,measuredin0.5T. ........................................ 114 7-4(T)ofPr0:33La0:67OsSb12versusT.TheCurie-Weisstathightemperature(T>150K)giveseff=3.62B/Prmole. ...................... 115 7-5C=TversusTforthreedierentPrOs4Sb12samplesfromdierentbatches. ... 116 7-6C=TversusT2aboveTcofLaOs4Sb12. ....................... 117 7-7C=TversusTnearTcforPr1xLaxOs4Sb12forx=0,0.05,0.1,and0.2. ..... 118 7-8C=TversusTnearTcofPr1xLaxOs4Sb12forx0.3 ............... 119 7-9TcversusxofPr1xLaxOs4Sb12. ........................... 120 7-10Total(C=T)atTcand0versusxofPr1xLaxOs4Sb12for0x1.0isat1.8KfromFig.7-3. ................................. 120 7-11C=TversusTofPr0:33La0:67Os4Sb12ttedby15Schottky. ......... 121 7-12C=TversusTofPr0:33La0:67Os4Sb12ttedby35Schottky. ......... 121 7-13C=TversusTofPr0:33La0:67Os4Sb12ttedbysinglet-singletSchottky. ...... 122 7-14Cforx=0(10T)and0.02(8and9.5T). ..................... 122 7-15f-electronspecicheatofPr0:9La0:1Os4Sb12inmagneticelds. .......... 123 7-16f-electronspecicheatofPr0:8La0:2Os4Sb12inmagneticelds. .......... 123 7-17f-electronspecicheatofPr0:4La0:6Os4Sb12inmagneticelds. .......... 124 7-18H-TphasediagramfromCmeasurementsforx=0,0.02,0.1,and0.2. ...... 124 7-19(H)ofPr0:95La0:05Os4Sb12atT=20mKfor~H//Iand~H?I(I//(001)). .... 125 7-20(H)ofPr0:95La0:05Os4Sb12for~H?I//(001)atT=20and300mK. ....... 125 7-21(H)ofPr0:95La0:05Os4Sb12for~H?I//(001)atT=20,310,and660mK. .... 126 7-22versusT2ofPr0:7La0:3Os4Sb12in0and0.5T. .................. 127 7-23versusT2ofPr0:7La0:3Os4Sb12in9and13T. .................. 128 10 PAGE 11 ..................................... 129 7-25(H)ofPr0:7La0:3Os4Sb12when~H//I//(001)at20,310,660,and1100mK. .. 129 7-26(H)ofPr0:33La0:67Os4Sb12at0.35K. ....................... 130 7-27ACsusceptibilityversusT=TcofPr1xLaxOs4Sb12,forx=0,0.05,0.4,0.8,and1. 131 7-28C=TversusTnearTcfortwoPrOs4Sb12samplesfromdierentbatchesinlowmagneticelds. .................................... 132 7-29C=TversusT,nearTc,ofPr0:95La0:05Os4Sb12inlowmagneticelds. ....... 133 7-30C=TversusT,nearTc,ofPr0:9La0:1Os4Sb12insmallmagneticelds. ...... 133 7-31C=TversusT,nearTc,ofPr0:7La0:3Os4Sb12inmagneticelds. .......... 134 7-32dHc2=dTversusx. ................................. 134 7-33p ............................... 135 11 PAGE 12 PrOs4Sb12istherstdiscoveredPr-basedheavyfermionmetalandsuperconductor.Ourhighmagneticeldspecicheatmeasurementsprovidedclearevidenceforthenon-magneticsingletcrystallineelectriceld(CEF)groundstate.ThisCEFgroundstateprecludestheconventionalKondoeectastheoriginoftheheavyfermionbehavior.ThesuperconductivityinPrOs4Sb12isunconventional,asinferredfromthedoublesuperconductingtransitioninthespecicheat.Pr1xLaxOs4Sb12(0x1)crystalsweresynthesizedandinvestigatedinordertoprovideadditionalevidencesforapostulatedCEFconguration,todiscriminatebetweendierentconductionelectronmassenhancement(m)mechanismsproposed,andtoprovideinsightintothenatureofthesuperconductivity.Lanthanumdopinginducesanomalouslysmallincreaseofthelatticeconstant.ThespecicheatresultsinhighmagneticeldsindicatedthatCEFschemeisunalteredbetweenx=0andatleast0.2,followedbyanabrupt(butsmall)changesomewherebetweenx=0.2and0.4.MagnetoresistancemeasurementsonLa-dopedsampleswereconsistentwithasingletCEFgroundstateofPr.InvestigationofthespecicheatdiscontinuityatTcandoftheuppercriticaleldslopeatTcindicatedthattheelectroniceectivemass,m,isstronglyreducedwithx,betweenx=0andxcr0.2{0.3,followedbyaweakdependenceonxforx>xcr.Therefore,wehavepostulatedthatsingle-impuritytypemodelscannotaccountfortheheavyfermionbehaviorofPrOs4Sb12.Investigationofthemagneticphasediagramandmagnetoresistanceprovidedstrongcorrelationsbetween 12 PAGE 13 13 PAGE 14 Intherareearth(Ce,Yb)-andactinide(U,Np)-basedalloystheelectronicstateshaveanenergyordersofmagnitudesmallerthaninordinarymetals,andsince(k)=~2k2=2m,theeectivemassmisordersofmagnitudelargerthanthefree-electronvalue,hencethetermheavyfermion.Thereareseveralexcellentexperimentalandtheoreticalreviews[ 1 { 5 ]onheavyfermions.OnehallmarkoftheheavyfermioncharacteristhelargeSommerfeldcoecientofthespecicheat.ThespecicheatofmetalsinthenormalstateatlowtemperatureisapproximatedbyC=T+T3,whereTistheelectronicspecicheatandT3isthelattice(Debye)contribution.Foranormalmetalisoforderof1mJ/K2mol,andforheavyfermionisfromseveralhundredtoseveralthousandmJ/K2mol.ThemagneticsusceptibilityathightemperaturesfollowstheCurie-Weissform=C/(T+CW),whereCisaconstant,andCWistheCurie-Weisstemperature.Atlowtemperatures(0)rangesfrom10to100memu/mol.Inthemajorityofheavyfermionmetals,theelectricalresistivityatverylowtemperatureshasaT2dependence:=0+AT2,where0istheresidualresistivityandAisontheorderoftensofcm/K2,muchlargerthanthatofnormalmetals. Thereareabout20heavyfermionsystemsthataresuperconductorsandalmostallofthemareCe-orU-based(thereisonePu-basedheavyfermionsuperconductor:PuCoGa5[ 6 ]). ThelledskutteruditePrOs4Sb12istherstdiscoveredPr-basedheavyfermioncompoundthatisasuperconductor[ 7 ]. Intheconventionalheavyfermions,theonlymicroscopictheoriessomewhatsuccessfulinaccountingfortheeectivemassenhancement(m)asmeasuredbythespecicheataretheS=1/2andS=3/2Kondomodels.ThesemodelswereinitiallyproposedforCe-basedsystems,whoseeectivedegeneraciesoff-electronsincrystallineelectriceldsareeither2or4.TheKondoeectinthesesystemsisanomalousbecauseofstrongspin-orbitcoupling.ThereisonefelectronperCeatomandaccordingtoHund'srules 14 PAGE 15 8 9 ]aremoreconsistentwiththef2conguration,allowingforasimilarCEFschemeasthatforPr.Therefore,theinvestigationofPrOs4Sb12withPrhaving2f-electronsmightberelevantandhelptotheunderstandingofthelargeclassofU-basedheavyfermions,sinceCEFcongurationsareusuallyknownforPr.Thenon-magneticcrystallineelectriceldgroundstate(thoughtaseithersingletordoublet[ 7 ])excludestheconventionalKondoeectastheoriginoftheheavy-fermionbehaviorinPrOs4Sb12,whichisconsideredtobethesourceofheavyfermionbehaviorinCe-andU-basedmetals.ThesuperconductivityinPrOs4Sb12isunconventional,butdierentfromthatinCe-andU-basedmaterials.Marksoftheunconventionalityofsuperconductivitycanbeinferredfromthedoublesuperconductingtransitionandpowerlowdependenceofthespecicheatbelowthetransition. Themaingoalsofthisworkare:tosettlethecrystallineelectriceldgroundstateinPrOs4Sb12, 15 PAGE 16 16 PAGE 17 whereqjisthechargeatthejthneighboringion,at~Rj.Ifthemagneticionhaschargeqiat~ri,thenthecrystallineelectriceldHamiltonianHCEFis ThesumPiistakenoverelectronsinunlledshells[ 10 ]. TheCEFpotentialcanbeevaluatedintermsofCartesiancoordinatesorintermsofsphericalharmonics.Hutchings[ 10 ]evaluatedthepotential(2{1)forthesimplest3arrangementsofchargesgivingacubiccrystallineelectriceld.Thethreecasesanalyzedwerewhenthechargesareplacedatthecornersofanoctahedron(sixfoldcoordination),atthecornersofacube(eightfoldcoordination),andatthecornersofatetrahedron(fourfoldcoordination).InCartesiancoordinatesthepotential(2{1)canbewrittenas[ 10 ] 5r4]+D6[(x6+y6+z6)+15 4(x2y4+x2z4+y2x4+y2z4+z2x4+z2y4)15 14r6]; wheredisthedistanceofthepointchargeqfromtheoriginineach3cases.C4andD6are70q/(9d5)and224q/(9d7)fortheeightfoldcoordination,+35q/(4d5)and 17 PAGE 18 Inthesphericalcoordinatesthesamepotentialiswritten[ 10 ]as 14[Y44(;)+Y44(;)]g+D06fY06(;)r 2[Y46(;)+Y46(;)]g; whereD04andD06are56qp Thereare2generalrulesthatcantellusthenumberofnonzerotermsintheCFpotential.Ifthereisacenterofinversionattheionsitetherewillbenoodd-nterms.Secondly,ifthezaxisisnotanm-foldaxissymmetry,thepotentialwillcontainVmn[ 10 ]. However,calculatingthepotentialtermsinCartesiancoordinatesandeveninsphericalcoordinatesistedious.AmoreconvenientmethodisthesocalledoperatorequivalentorStevens0operatortechnique[ 11 12 ].TheHamiltonian(2{2)isofformHCEF=PijejV(xi;yi;zi).Iff(x;y;z)isaCartesianfunction,inordertondtheequivalentoperatortosuchtermsasPif(xi,yi,zi),thecoordinatesx,y,andzarereplacedbyangularmomentumoperatorsJx,Jy,andJzrespectively,takingintoaccountthenon-commutativityofJ0is.Thisisdonebyreplacingproductsofx,y,andzbycombinationsofJ0isdividedbythetotalnumberofcombinations.Asanexamplewecanconsider whereJ=JxiJy. 18 PAGE 19 5r4]+D6[(x6+y6+z6)+15 4(x2y4+x2z4+y2x4+y2z4+z2x4+z2y4)15 14r6]: Usingtheequivalentoperatorrepresentation,theHamiltonianwillbe[ 10 ] or whereB04andB06are+7jejqJhr4i/(18d5)andjejqJhr6i/(9d7)foreightfoldcoordination,7jejqJhr4i/(16d5)and3jejqJhr6i/(64d7)forsixfoldcoordination,and+7jejqJhr4i/(30d5)andjejqJhr6i/(18d7)forfourfoldcoordinationrespectively.Also,hr4iandhr6iarethemeanfourthandsixthpoweroftheradiiofthemagneticelectrons,andthemultiplicityfactorsJ,J,andJareforPr3+(f2)2213/(325112),22/(325112),and2217/(345711213)respectively[ 10 ].Also, 19 PAGE 20 13 ].TheHamiltonianiswrittenas InordertocoverallpossiblevalueoftheratiobetweenthefourthandsixthdegreetermsareintroducedthescalefactorWandtheparameterx,proportionaltotheratioofthetwoterms where1 PAGE 21 15 ] 1:p 12(j+4i+j4i)+p 6j0i(2{19) 23:r 24(j+4i+j4i)r 24j0ir 2(j+2i+j2i) (2{20) (1)4:a1j4ia2j2i+a2j+2i+a1j+4ib1j3i+b2j1i+b3j1i+b4j3i (2)4:a2j4ia1j2i+a1j+2ia2j+4ib2j3ib1j1ib4j1i+b3j3i Ify=0,theeigenstatesarethoseforOhsymmetry[ 13 ] 1:p 12(j+4i+j4i)+p 6j0i(2{23) 3:r 24(j+4i+j4i)r 24j0ip 2(j+2i+j2i) (2{24) 4:r 8j3ir 8j1ir 2(j+4ij4i) (2{25) 5:r 8j3ir 8j1ir 2(j+2ij2i) (2{26) 21 PAGE 22 Theeigenfunctionsandeigenvaluesof1(Th)and23(Th)arethesameasthoseof1(Oh)and3(Oh),thereforearenotaectedbyOt6fromtheHamiltonian.Wheny=0,(1)4(Th)hasthesameeigenfunctionsandeigenvaluesas4(Oh),and(2)4asthosefor5(Oh).Wheny6=0,4and5mixresultingintwo(1;2)(Th)[ 15 ].Therefore,theeigenfunctionsandeigenvaluesofCEFforThandOharedierent.TheOt6terminHamiltonianaectsomeeigenfunctionsandeigenvalues,resultinginachangeofthetransitionprobabilitiesofneutronscatteringinPr3+. 16 17 ].Barnes[ 18 ]foundthatCu2+ionsinthecupratesuperconductorscouldleadtosuchaKondoeectaswell.Later,newevidencebelievedtobehallmarksofaquadrupolarKondoeecthasbeenfoundinthealloysY1xUxPd3[ 19 { 22 ]forx=0.1and0.2. InUBe13,thetotalangularmomentumofU4+(5f2conguration)isJ=4.Thisleads[ 13 ]toa3CEFgroundstateforabouthalfthecrystaleldparameterrange(Fig.2-2).Thef2congurationisexpectedalsoforPr3+inPrOs4Sb12,and,accordingtoCEFcalculationsofLea,Leask,andWolf[ 13 ],a3doubletCEFgroundstateisveryprobable(Fig.2-2).Therefore,theheavyfermionbehaviorinPrOs4Sb12couldbeinprincipleduetoaQKeect.Also,thephysicalpropertiesofUBe13(andU1xThxBe13)arehighly 22 PAGE 23 TherelevantstatesforthequadrupolarKondoeectforU4+.Thelasttwocolumnsaretheprojectionsofthemagneticandquadrupolarmomentsrespectively(ReprintedwithpermissionfromCoxandZavadowski[ 23 ]). Cong. StateJEigenstatehJzih3J2zJ(J+1)i 24[j4ij4i]q 12j0i0+8 2[j2i+j2i]08 2q 6j5 2iq 6j3 2i+5 60 2q 6j5 2iq 6j3 2i5 60 2q 6j5 2i+q 6j3 2i+11 6+8 2q 6j5 2i+q 6j3 2i11 6+8 2j1 2i+1 28 2j1 2i1 28 reminiscentofthoseofPrOs4Sb12.Thus,sincethediscoveryoftheHFstateinPrOs4Sb12itsnormalpropertieshavebeenassociatedwiththeQKeect. ThestatesinvolvedinthequadrupolarKondoeectforU4+aregiveninTable2-1.Thedoublydegenerategroundstatecanbetreatedasatwo-levelsystem(amanifoldwithapseudo-spinof1 2).Theprojectedvalueoftheelectricquadrupolemomentontothe3basisisjQzzj=j3J2zJ(J+1)j=8andtheprojectedvalueofthemagneticdipolemomentiszero,i.e.jJzj=0(Table2-1).Therefore,thecouplingisbetweentheelectricquadrupolemomentandtheconductionelectrons. TheAndersonmodelfortherelevantstatesofthequadrupolarKondoeect(sinceitconsidersonly3,7,and8)iscalledthe3-7-8model.Figure2-3showsaschematicrepresentationoftheAndersonmodelrelevantforU4+ionsinthecubicsymmetry.Thegroundstate3(J=4,4f2)andrstexcited7(J=5/2,5f1)mixonlyviatheconductionpartialwaves8(J=5/2,c1).Thetransitionf1!f2isdonebyremovingaconductionelectronandthetransitionf2!f1isdonebyemittingaconductionelectron.Itcan 23 PAGE 24 23 ](or,inthegrouptheoryframework,37=8). Applyingacanonicaltransformation(SchrieerandWol[ 24 ])tothe3-7-8model,thehybridizationtermcanbeeliminated.Also,thetransformationyieldstoaneectiveexchangeinteractionbetweenpseudospin-1 2andelectricquadrupolemomentsoftheform where3isapseudospin-1 2matrixforthe3quadrupole,8(8)arethepseudospinsformedfromthe8+2,8+1(82,81)partialwaves(Table2-1).TheexchangeintegralJexchangeisproportionalto=fN(0)andisnegative. TheHamiltonianhasatwo-channelKondoform;twodegeneratespeciesofconductionelectronscouplewithidenticalexchangeintegralsJexchangetothelocal3=1 2object.Thechannelindicesarethemagneticindicesofthelocalconductionpartialwavestates.Figure2-4showsschematicallythemappingofthequadrupolarKondotothetwo-channelKondomodel.Thetwo-channelquadrupolarformoftheHamiltoniantellusthattheconductionelectronorbitalmotioncanscreentheU4+quadrupolemomentequallywellformagneticspin-upandmagneticspin-downelectrons. 25 { 27 ].Thesusceptibility(Fig.2-5,lowerpanel)divergeslogarithmicalatT=0,=(eTH)1ln(H=TH),whereTH=(=e)TK[ 28 29 ].Here,eisthebaseofln,i.e.2.71...InthequadrupolarKondomodelthiscorrespondstoadivergentquadrupolarsusceptibility.ForT!0,thefreeenergyinzeroeldisF=1 2Tln2.Therefore,thezero-temperaturezero-eldentropyisequalto1 2ln2[ 25 ].Thenon-zeroentropyatT!0isconsistentwiththedivergenceinsusceptibilityandarguesinfavorofanon-singletgroundstate(asingletisthegroundstateforthestandardKondomodel).Asexpected,theentropyincreasesmonotonicallywithtemperatureandreachesasymptoticallytheln2 24 PAGE 25 2spin)athighT.Also,theT=0entropyincreaseswiththeeld.SincetheS(T=0)decreaseswithHthespecicheatincreaseswithHatintermediateTresultinginlargevaluesof,commonforheavyfermionsystems.AthighTthepseudo-spinisfree,thereforeS=ln2.TheentropychangeS(H)=S(T=1;H)S(T=0;H)increaseswithHfrom1 2ln2toln2forlargeH.IntheC=TplotstheKondopeakscanbeseen. Theinitialmeasurementsofspecicheat[ 16 ]werenotconclusiveforaquadrupolarKondoeectinUBe13.Also,morerecentmeasurementsofnonlinearsusceptibility[ 30 ]areinconsistentwiththequadrupolar(5f2)groundstateoftheuraniumion,indicatingthatthelow-lyingmagneticexcitationsofUBe13arepredominantlydipolarincharacter. 7 ].Laterexperiments[ 31 { 33 ]establishedthecrystallineelectriceld(CEF)groundstateofthePr3+ioninthecubicsymmetryenvironmentofPrOs4Sb12(Thpointgroupsymmetry)asthenon-magneticsinglet1.TheconsequenceofthisisthattheoriginalformulationofthequadrupolarKondoeectcannotbeappliedtotheconductionelectronmassenhancementinPrOs4Sb12. 1isnearlydegeneratewiththe(2)4triplet.Though1itselfdoesn0tcarryanydegreesoffreedom,thepseudo-quadrupletconstitutedby1and(2)4isspeculatedtohavemagneticandquadrupolardegreesoffreedom[ 34 ],andthereforeamagneticorquadrupolarKondoeectisinvokedtoexplaintheenhancementoftheeectivemassofthequasi-particles. Ontheotherhand,themodeldoesnotseemtoberelevantsincethepredictedpropertiesofthequadrupolarKondoeectareindisagreementwiththemeasurements.Butthisisasingle-ionmodel.Possibly,intersiteeectsareresponsibleforthedisagreements.Thereisnolatticequadrupolarmodel. 25 PAGE 26 35 ]. Goremychkinetal.[ 32 ]proposedthatthemassenhancementinPrOs4Sb12canbeexplainedbyabalancebetweentwotypesofinteractions,magneticdipolarandquadrupolarbetweenconductionandthefelectronsofPr.ThetheoryofFuldeandJensen[ 36 ]ofconductionelectronmassenhancementascribesthistotheinelasticscatteringbycrystaleldtransitionsinasingletground-statesystem.Themassenhancementoftheconductionelectronsareduetotheirinteractionwiththemagneticexcitations. TherelevantHamiltoniandescribingtheinteractionbetweentheconductionelectronsandtherare-earthlocalizedmomentsis[ 36 ] whereIsfistheexchangeintegral,gListheLandefactor,~Jnisthetotalangularmomentumofarare-earthionatsite~Rn,and~arePaulimatrices. Themassenhancementduetotheinelastictransitionatenergybetweentwolevels,jiiandjji,is wheregJistheLandefactor,Isfistheexchangeintegralcouplingtheconductingelectronstothef-electrons,N(0)istheconductionelectrondensityofstatesattheFermilevel,andhijJjjiisthemagneticdipolematrixelementcalculatedusingthederivedcrystaleldparameters.Thisformulashowsthatforasmallexcitationenergyleadstoalargeenhancementinm. 37 ],Millis[ 38 ], 26 PAGE 27 39 ](acompletereviewisgivenbyStewart[ 40 41 ]).Allthesemodelsexhibitdivergenceofthelowtemperaturespecicheat. Byanalogy,inPrOs4Sb12,thequadrupoleuctuationsofPrionsarebelievedtoplayanimportantroleintheHF-SCproperties.Therefore,anothermodelproposed(acollective-typemodel)forthemassenhancementmechanismareduetotheuctuationsoftheantiferroquadrupolarorderparameterduetotheproximitytotheAFQorderedphase. PrOs4Sb12exhibitsanantiferroquadrupolarorderedphaseineldsbetweenabout4.5and14T.Forelds5{13TthetwolowestCEFlevelsaresucientlyclosetoformapseudo-doubletwithquadrupolarandmagneticdegreesoffreedom,resultinginalongrangeorder. Thereisnotheory(tothismoment)thatdescribesthemassenhancementduetotheuctuationsofthequadrupolarorderparameter.OurmagnetoresistivitydataofPrOs4Sb12andLaalloyspresentedinChapters6and7seemtosupportthismassenhancementmechanism. Figure2-1. RotationalsymmetryTh.Intheleft(a),thesmallboldbluesegmentisassimilatewiththedistancebetweentwoantimonyatomsbelongingtothesameicosahedra.Arotationwithrespectto(100)anda2 42 ]). 27 PAGE 28 Lea,Leask,andWol0srepresentationofCEFforJ=4(RedrawnwithpermissionfromLeaetal.[ 13 ]). Figure2-3. RepresentationoftheU4+ionsincubicsymmetryundergoingquadrupolarKondoeect.Themodelinvolvesadoubletgroundstateineachofthetwoelectroniclowest-lyingcongurations:f2havingthequadrupolarornon-Kramers3doublet,andf1congurationhavingthemagneticorKramers7doublet.Theconductionelectronsmixthetwocongurationsthroughahybridizationprocess.The8conductionstatecouplesthesetwodoublets(RedrawnwithpermissionfromCoxandZavadowski[ 23 ]). 28 PAGE 29 MappingofthequadrupolarKondoHamiltonianontothetwo-channelKondomodel.a)Thestandardtwo-channelKondomodelinspinspace:twoconductionelectronssc+andsccoupleantiparalleltotheimpurityspinSI.b)InthequadrupolarKondocase,thespinisduebythequadrupolarororbitaldeformations.Thetwochannelscomefromtherealmagneticspinoftheconductionelectrons.TheorbitalmotionoftheelectronsproducesthescreeningoftheU4+orbitaluctuations(RedrawnwithpermissionfromCoxandZavadowski[ 23 ]). Figure2-5. 27 ]). 29 PAGE 30 43 ](a=9.30311A[ 44 ]aftermorerecentmeasurements),spacegroupIm 45 ].LaOs4Sb12isthenon-fequivalentofPrOs4Sb12withasimilarcrystalstructure.AlltheexoticphenomenaofPrOs4Sb12arethoughttobeassociatedwithitsuniquecrystalstructure.Inparticular,thelargecoordinationnumberofPrionssurroundedby12Sband8Osionsleadstostronghybridizationbetweenthe4fandconductionelectrons[ 46 ].Thisstronghybridizationresultsinarichvarietyofstronglycorrelatedelectrongroundstatesandphenomena. 47 ].Theconsequenceisareductionofthethermalconductivity.Thelledskutteruditeswiththecagearefavorableforathermoelectricdevicepossessingahighcoecientofmerit[ 48 ]. TheamplitudeofthisvibrationofthePrioninPrOs4Sb12isabout8timesbiggerthantheamplitudeofOs.EXAFSdata[ 49 ]supportstheideaofarattlingPrllerion(basedonthelowEinsteintemperatureE75K)withinafairlysticageinthismaterial.Besidesthedynamicmovement,astaticdisplacementwasdetectedinwhich 30 PAGE 31 49 ].Gotoetal.[ 50 ],basedonatheoryofCoxetal.[ 23 ],suggestedthatthetunneleectbetweenthetwopositionsofthePrionscouldbelinkedtotheappearanceofthesuperconductivity. 7 ],oreff=3.5BasreportedbyTayamaetal.[ 51 ],andaCurie-WeisstemperatureCW=16K[ 7 ].TheeectivemomentfoundissomewhatlowerthanthemomentofafreeionPr3+whichhaseff=3.58B[ 52 ]. X-ray-absorptionne-structure(XAFS)measurements[ 49 ]carriedoutatthePrLIIIandOsLIIIedgesonPrOs4Sb12suggestthatthePrvalenceisverycloseto3+.EachPrionhastwoelectronsonthefshell(4f2electronicstructure). 13 ]intermsoftheratioofthefourthandsixthordertermsofangularmomentumoperatoroftheCEFpotential,x,andanoverallenergyscalefactorW.Formorethantwodecades,thesymmetrywasthoughtasOh,insteadofTh. Baueretal.[ 7 ]ttedthemagneticsusceptibilitydata(seeFig.3-2)byaCEFmodelinwhichthegroundstatewaschosentobeeitherthenon-magnetic1singlet(W>0)orthenon-magnetic3doublet(W<0).Thepeakpresentinthe(T)datawasthoughttobeproducedwhentherstexcitedstateisatriplet5withaenergy<100Kabove 31 PAGE 32 53 ]andBaueretal.[ 54 ] TheseauthorsusetheconventionalcubiccrystaleldmodelwhichisapplicabletotheO,TdandOhsymmetries.IntheThsymmetry,thenon-Kramersdoublet3correspondstothedegenerate2and3singletstates(denotedas23)and4and5statescoincidewith(1)4and(2)4tripletstates,respectively,whenlasttermiszerointhecrystaleldHamiltonian(2-17).Thesingletstate1isthesameforbothcases. Thelasttermofequation(2-17)isuniquetotheThsymmetryofthismaterialcomingfromtheatomiccongurationofSbionsinthecrystal[ 15 ]andisabsentintheconventionalcubiccrystaleldHamiltonianthatMapleetal.[ 53 ]andBaueretal.[ 54 ]used.TheomittingofthelasttermintheHamiltonian(seeequation2-17)hasimplicationsintheinterpretationoftheinelasticneutronscatteringdata. Theabovementionedtreproducestheoverallshapeofthelowtemperaturepeak,andalsothevalueofthevanVleckparamagneticsusceptibilitywithaneectivemomentcloseto,butsomewhatlowerthanthat,ofthefreePr3+ion.Baueretal.[ 7 ]ttedCassumingadegeneratespectrum Specicheatdatawastted[ 7 ]byasystemwithtwolevelsofequaldegeneracysplitbyanenergy=6.6K(ithasbeenassumedthatthedegeneracyofanylevelisliftedbyCEFwhenthelocalsitesymmetryofthePr3+ionsisnotcubicasaresultofsomekindoflocaldistortion). Theentropyinthe3-5casewasfoundtobeS35=Rln27.6J/(molK)[ 7 ].ThetotalentropyofthebroadpeakjustabovethetransitionisS=R(C(T)=T)dT10.3J/(molK).TheclosenessinvaluesmadeBaueretal.[ 7 ]tofavorthe3groundstatescenario. 32 PAGE 33 53 ]consideringOhsymmetrysuggestthat3istheCEFgroundstateinPrOs4Sb12.Theresistivitydatameasurementswerealsointerpretedintheframeworkofa3CEFgroundstate[ 55 ]. Incontrast,Tayamaetal.[ 51 ]obtainedasomewhatbettertofthemagneticsusceptibility(T)databya1CEFgroundstatemodel(Fig.3-3(a)).Also,thetheoreticalcurvesofS(T)basedon1groundstatemodelshowincreaseofentropywithelds(lowerpanelofFig.3-3(b)).Thistrendisconrmedbymagneticeldspecicheatmeasurements(Fig.3-3(c))byAokietal.[ 56 ]. Therefore,zeroorsmallmagneticeldsdataarecontradictory,moreexperimentsaretobedoneinordertoestablishthetrueCEFgroundstateinPrOs4Sb12. 3.2.1SpecicHeat 43 ],andthenbyBraunetal.[ 57 ].Itwasin2002whenBaueretal.[ 7 ]discoveredsuperconductivityinPrOs4Sb12.Sincethediscontinuityinspecicheatisoftheorderof,thislargevaluediscontinuity((C=T)jTc=1:85K500mJ/K2mol[ 7 ])impliesthepresenceofheavyfermionsbothinthenormalandsuperconductingstates. Thereisnoconsensusregardingtheprecisevalueof,butallthereportedvaluesimplyheavyfermionbehavior.Actually,thisisperhapsthestrongestevidenceforHFstatesinPrOs4Sb12.Consideringtherelation(C=Tc)=1.43theSommerfeldcoecientisfoundtobe350mJ/K2mol.Thephonon(lattice)contributiontothespecicheatCdatacanbedescribedbyT3thatisidentiedwithspecicheatofLaOs4Sb12withD=304K.isrelatedtoDby=(1944103)n=3D,wherenisthenumberofatomsintheformulaunit(e.g.,n=17inLaOs4Sb12). 44 ]comparativewiththebandsstructure(LDA+Umethod[ 58 ])arepresentedin 33 PAGE 34 44 ](whichleaks4felectrons).Thisindicatesthatthe4f2electronsinPrOs4Sb12arewelllocalized.ThesimilartopologyoftheFSforthetwocompoundsissupportedalsobysimilarangulardependenceofthedHvA.ThreeFermisurfacesheets,includingtwoclosed(practicallysphericalshaped)andonemulti-connected,wereidentiedinagreementwiththecalculations. TheeectivemassesmeasuredbydHvAarebetween2.4and7.6m0(m0isthefreeelectronmass).Thesevaluesarewellbelowtheonesreportedfromthespecicheatmeasurements.Theselowvalueshavebeenexplained[ 59 ]intheframeworkofthetwo-bandsuperconductivitymodelinwhichband2correspondstothelightbanddetectedbydHvAmeasurements.Band1isaheavybandhavingmostofthedensityofstates.Theheaviestquasiparticlesareseeninthermodynamicmeasurements(CorHc2)only.However,theapplicabilityofthetwo-bandmodeltoPrOs4Sb12isnotestablished.Furthermore,ourresultspresentedinsection7.6shedssomedoubtsintheinterpretation. 7 ].Thisimplies0116A,vF=1.65106cm/s,andm50m0.ThiscalculationassumesasphericalFermisurface. Theresistivitydatabetween8and40KrevealedaT2dependence0+AT2,withA=0.009cm/K2[ 7 ].TheAcoecientisabouttwoordersofmagnitudesmallerthanthevalueexpectedforaheavyfermioncompound.ConsideringtheKadowaki-Woodsuniversalrelation[ 60 ]betweenAand,A/2=1105cmmol2K2mJ2.Thevalue 34 PAGE 35 7 ],andthisisatypicalvaluefornormalmetalsandismuchsmallerthanofLaOs4Sb12. 7 ]asT!0.Thisisthehallmarkofanonmagneticgroundstate.Above150K,(T)ofPrOs4Sb12canbedescribedbyaCurie-Weisslaw.Thereisalargediscrepancybetweenthehightemperatureeectivemomentreportedbyvariousresearchgroups.TheeectivemomentaccordingtoBaueretal.[ 7 ]iseff=2.97B,andeff=3.5BisthevaluereportedbyTayamaetal.[ 51 ]ThefreeionPr3+hasahightemperatureeectivemomentof3.58B[ 52 ].TheCurie-WeisstemperatureisCW=15K[ 51 ]. Fromthediamagneticonset(inset(ii),Fig.3-4(a))itisfoundthatthetemperatureofthesuperconductingtransitionTcisequaltothevaluefoundfromthespecicheatmeasurements. 56 ]ineldsupto8Tandresistivity[ 55 ]inmagneticeldsuptoabout10Trevealedtheexistenceofaeldinducedorderedphase(FIOP)above4.5T.InthisChapteradiscussionofthenatureoftheFIOPwillbepresentedalongwiththespecicheatdatathatcompletesthemagneticphasediagram.Asimilarphasediagramhasbeenobtainedlaterbymagnetization[ 51 61 ](seeFig.3-6)andbythermalexpansionandmagnetostrictionmeasurements[ 62 ]. 35 PAGE 36 3.4.1.1TheDoubleTransition 7 ]showedasinglesuperconductingtransitionatTcof1.85K.Higherqualitymaterialsrevealedactuallytwosuperconductingtransitions(Vollmeretal.[ 63 ],Mapleetal.[ 53 ],Oeschleretal.[ 64 ]).InFigure3-7panels(a)and(b)areshownspecicheatofPrOs4Sb12presentingtwosuperconductingtransitions,Tc2=1.75KandTc1=1.85KbyVollmeretal.[ 63 ],andTc2=1.716KandTc1=1.887KbyMeassonetal.[ 59 ],respectively.TwosuperconductingtransitionsatthesametemperatureshavebeenreportedbyCichoreketal.[ 65 ]alongwithaspeculationforathirdsuperconductingtransitionat0.6KinferredfromHc1measurements.ItisbelievedthatinclusionsofthefreeOsinthesinglecrystalcannotberesponsiblefortheenhancementofHc1,thoughTcofpureOsis0.66K[ 66 ]basedonsensitiveX-rayandelectronmicroprobestudies[ 65 ]. Therearetwoclassesofexplanationsofthenature(intrinsicornot)ofthedoubletransition.Onearguesinfavoroftwodierentpartsofthesamplewithtwodierentsuperconductingphases,andthereforewithdierentTc's.Thus,thequalityofthesamplesiscrucial.Forinstanceithasbeenconsidered[ 59 ]thatdespitethesharpspecicheattransitions,thesamplesstillpresentspatialinhomogeneities.OnepossibilitywouldbeaninhomogeneouscoexistenceoftwoelectroniccongurationsofPr,4f1and4f2.Thehightemperaturemagneticsusceptibilitymeasurementsareinfavorof4f2,sincetheyhavefound[ 51 ]aneectivemomenteff=3.6B/Pr(theexpectedvaluefor4f1is2.54Bandfor4f2is3.58B). Anotherpossiblescenariothatispresentedinthisdissertationistheexistenceofinhomogeneitiesduetotheclosenessofthesystemtoalongrangeantiferro-quadrupolarorder:clusterswithashort-rangeorderwouldhavedierentsuperconductingparametersthantheremainingpartofthesample. 36 PAGE 37 59 ].Thenatureofthetwotransitionsisnotyetestablished.Thewidthofthetransitionasmeasuredbyspecicheatandac-susceptibilityisthesame,about0.2K. ThesuperconductinggapstructureinvestigatedusingthermaltransportmeasurementsinmagneticeldrotatedrelativetothecrystalaxesbyIzawaetal.[ 67 ]providesanotherevidencefortheunconventionalcharacterofsuperconductivityinPrOs4Sb12.Thechangeinthesymmetryofthesuperconductinggapfunctionthatoccursdeepinsidethesuperconductingstategivesaclearindicationofthepresenceoftwodistinctsuperconductingphaseswithtwofoldandfourfoldsymmetries(Fig.3-8).Thegapfunctionsinbothphaseshaveapointnodesingularitywhichisincontrasttothelinenodesingularityobservedinalmostallunconventionalsuperconductors.Thetwo-bandsuperconductivity(similartothatobservedinMgB2)isobservedinnewerthermalconductivitymeasurements[ 68 ]. Adoubletransitioncanbeseeninthethermalexpansion[ 64 ]experiment(Fig.3-9).Thetwotransitionsareatthesametemperaturesatwhichthespecicheatdiscontinuitiesoccur.UsingtheEhrenfestequation@Tc=@P=VmTc=C,whereVmisthemolarvolume,calculationsshowthatthesuperconductingtransitionsTc2isdecreasedtwotimesfasterunderpressurethanTc1.Thisisinfavorofintrinsicnatureofthetwosuperconductingtransitions. 37 PAGE 38 Thevaluesreportedbydierentgroups,extractedfromtsofspecicheatbelowTc(ReprintedwithpermissionfromGrubeetal.[ 70 ]). Specicheatdata Grubeetal.[ 70 ] 3.70.2 Vollmeretal.[ 63 ] 2.60:2(3) Fredericketal.[ 71 ] 3.10:2(1) Fredericketal.[ 71 ] 3.60:2(3) Table3-2. Thevaluesreportedbydierentgroupsfrommeasurementsotherthanspecicheat(ReprintedwithpermissionfromGrubeetal.[ 70 ]). Experiment GapFunction Tunnelingspectroscopy[ 72 ] 1.7 Nearlyisotropic 73 ] 2.1 Nearlyisotropic 74 ] 2.6 Pointnodes SbNQR[ 75 ] 2.7 Isotropic Inallreporteddatathespecic-heatmeasurementsexhibitarapiddecreaseofCbelowthesuperconductingtransition.Thispointstopronouncedstrong-couplingsuperconductivity. Thesocalled-model[ 69 ]assumesthatthesuperconductivepropertieswhicharemainlyinuencedbythesizeofthegapandthequasiparticle-stateoccupancycouldbeapproximatedbysimplyusingthetemperaturedependenceoftheweak-couplingBCSgap.ThesizeofthegapintheFermisurfaceisafreelyadjustableparameter=(0)/kBTc,where(0)isFermi-surfaceaveragedgapatT=0.Table3-1presentscomparativevaluesobtainedbydierentgroups.Ananalysisusingthe-modelresultsinanextremelylargegapratioof=(0)/kBTc=3.7andahugespecicheatjumpofC/(c)5[ 70 ]. AsummaryofthepublishedsuperconductivegapratiosandgapanisotropyofPrOs4Sb12fromothermeasurementsthanspecicheatarepresentedinTable3-2. Fredericketal.[ 71 ]succeededinmakingabettertforthespecicheatdataofPrOs4Sb12usingapower-lawfunctionbelowthesuperconductingtemperature.Thets, 38 PAGE 39 75 ]inzeroeldshowsaheavyfermionbehaviorandcontroversialconclusionsregardingthenatureofthesuperconductivityinPrOs4Sb12.IntheSCstate,1/T1showsneitheracoherencepeakjustbelowTcnoraT3-likepower-lawbehaviorobservedforanisotropicHFsuperconductorswiththeline-nodegap.Theabsenceofthecoherencepeakin1/T1supportstheideaofunconventionalsuperconductivityinPrOs4Sb12(Fig.3-10).Theisotropicenergygapwithitssize/kB=4.8KseemstoopenupacrossTcbelowT=2.3K.Theverylargeandisotropicenergygap2/kBTc5.2indicatesanewtypeofunconventionalstrong-couplingregime. 76 ]revealaspontaneousappearanceofstaticinternalmagneticeldsbelowthesuperconductingtransitiontemperature,providingunambiguousevidenceforthebreakingoftime-reversalsymmetryinthesuperconductingstate.Thiswillfavorthemultiplesuperconductingphasetransitionsobservedbyspecicheatandthermalconductivitystudiesandsupportthereforetheunconventionalityofsuperconductivity. MagneticpenetrationdepthdatainsinglecrystalsofPrOs4Sb12downto0.1K,withtheaceldappliedalongthea,b,andcdirectionswassuccessfullytted[ 74 ]bythe3HeA-phase-likegapwithmultidomains,eachhavingtwopointnodesalongacubeaxis,andparameter(0)/kBTc=2.6,suggestingthatPrOs4Sb12isastrong-couplingsuperconductorwithtwopointnodesontheFermisurface.Thesemeasurementsconrmedthetwosuperconductingtransitionsat1.75and1.85Kseeninothermeasurements. 39 PAGE 40 3.4.2.1NuclearMagneticResonance(SR) 75 ](Fig.3-10,fullsymbol),whichisthesignatureofaconventionaltypeofsuperconductivity. Figure3-10(opensymbols)alsoplotsthedatafortheconventionalsuperconductorLaOs4Sb12.Forans-wavecasethatisactuallyseenintheTdependenceof1/T1forLaOs4Sb12withTc0.75K,intheSCstate,1/T1showsthelargecoherencepeakjustbelowTc,followedbyanexponentialdependencewiththegapsizeof2/kBTc3:2atlowT.ThisisaclearevidencethatLaOs4Sb12istheconventionalweak-couplingBCSs-wavesuperconductor. 73 ]anexponentialtemperaturedependenceofthemagneticpenetrationdepth,indicativeofanisotropicornearlyisotropicenergygap,indicatingaconventionalsuperconductivitymechanism. Thisisnotseen,todate,inanyotherHFsuperconductorandisasignatureofisotropicpairingsymmetry(eithers-orp-wave,indistinguishablebythermodynamicorelectrodynamicmeasurements),possiblyrelatedtoanovelnonmagneticquadrupolarKondoHFmechanisminPrOs4Sb12.Also,theestimatedmagneticpenetrationdepth=3440(20)A[ 73 ]wasconsiderablyshorterthaninotherHFsuperconductors. 72 ]intheheavy-fermionsuperconductorPrOs4Sb12demonstratesthatthesuperconductinggapiswelldevelopedoveralargepartoftheFermisurface.Theconductancehasbeensuccessfullyttedbyas-wavesuperconductivity 40 PAGE 41 Figure3-1. CrystalstructureofPrOs4Sb12(ReprintedwithpermissionfromAokietal.[ 77 ]) 41 PAGE 42 (a)Fitsofthemagneticsusceptibility(T)ofPrOs4Sb12toCEFmodelwitheither3(solidline)or1(dashedline)groundstate.Thesamesymbolsareusedintheinset,whichshows(T)bellow30K.Intheinset,thesolidlinetsaturatesjustabove=0.06cm3/mol.(b)Cttedbyatwo-levelSchottkyanomaly(ReprintedwithpermissionfromBaueretal.[ 7 ]). 42 PAGE 43 (a)Fitsofthemagneticsusceptibility(T)ofPrOs4Sb12toCEFmodelwitheither3(dashedline)or1(dotedline)groundstate.Thesolidlanerepresentstheexperimentaldata(takenfromTayamaetal.[ 51 ]).(b)ThecalculatedentropyS(T)for~H//(100)inboth3and1CEFgroundstatemodels(takenfromTayamaetal.[ 51 ]).(c)ThemeasuredentropyS(T)for~H//(100)(ReprintedwithpermissionfromAokietal.[ 56 ]). 43 PAGE 44 (a)Resistivity(T)andsusceptibility(T)ofPrOs4Sb12(b)SpecicheatC(T)upto20K[ 7 ](ReprintedwithpermissionfromBaueretal.[ 7 ]). 44 PAGE 45 FermisurfaceofPrOs4Sb12(ReprintedwithpermissionfromSugawaraetal.[ 44 ]). 45 PAGE 46 51 ]).OpenandclosedsymbolsweredeterminedbythedM(T)=dTanddM(H)=dHdata,respectively.Rightpanel,thePrchargedistributionsinducedintheantiferroquadrupolarorderedphaseinmagneticeld(ReprintedwithpermissionfromMeasson[ 45 ]). Figure3-7. (a)C(T)ofPrOs4Sb12presentingdoublesuperconductingtransition(ReprintedwithpermissionfromVollmeretal.[ 63 ])(b)C(T)ofPrOs4Sb12presentingtwosuperconductingtransitions(ReprintedwithpermissionfromMeassonetal.[ 59 ])(c)TherealpartoftheacsusceptibilityofPrOs4Sb12presentingtwodistinctsuperconductingtransitions(ReprintedwithpermissionfromMeassonetal.[ 59 ]). 46 PAGE 47 ThetwosuperconductingphasesforPrOs4Sb12(ReprintedwithpermissionfromIzawaetal.[ 67 ]).ThegapfunctionhasafourfoldsymmetryinAphaseandtwofoldsymmetryinBphase.Right:TheplotofthegapfunctionwithnodesforAphaseandB-phase(ReprintedwithpermissionfromMakietal.[ 78 ]). Figure3-9. TwosuperconductingtransitionsinthethermalexpansioncoecientofPrOs4Sb12.Thetwotransitionsarevisibleforthesametemperaturesofthetwotransitionsinspecicheat(ReprintedwithpermissionfromOeschleretal.[ 64 ]). 47 PAGE 48 Temperaturedependenceoftherate1/T1atthe2Qtransitionof123SbforPrOs4Sb12(closedcircles)andLaOs4Sb12(opencircles)(ReprintedwithpermissionfromKotegawaetal.[ 75 ]). Figure3-11. TunnelingconductancebetweenPrOs4Sb12andanAutip.Thegapiswelldevelopedwithnolow-energyexcitations.ThelineingureisthepredictionfromconventionalisotropicBCSs-wavetheoryusing=270eVandT=0.19K(ReprintedwithpermissionfromSuderowetal.[ 72 ]). 48 PAGE 49 Thischapterdescribesthesamplesynthesis,characterizationandtheexperimentalproceduresused:dcandacsusceptibilities,resistivity,andspecicheatmeasurements.Abriefdescriptionoftheperformedmeasurementsisgiven. 4.1.1Synthesis 54 57 ].Sincetheuxisoneoftheconstituentelementsofthecompounds(i.e.Sb)themethodiscalledself-uxgrowth.High-puritystartingelements(PrandLafromAMESLaboratory,99.99%puritypowderOsfromColonial,Inc.,and99.999%puritySbingotfromAlfaAESAR)areusedintheproportionR:Os:Sb=1:4:20,whentherare-earthelementRisPrandLainvariousproportions.TheRalloysusedascomponentsintheuxgrowthweresynthesizedeventuallybymeltingitsconstituentelementsinanEdmund-BuhlerArcMelterunderahighpurityargonatmosphere.First,smallchunksofSbwereplacedinsideofaquartztube.AbovethatwereplacedtheOsandtheRcomponentsthatwerepre-meltedseparatelytoeliminateanytraceofoxidefromthesurfaceoftheelements.TheOspowderwaspressedinsmallpelletsandthenmelted.ThequartztubewassealedunderlowpressureAratmosphere(20mTorr)afterthetubeispumpedandushed3to5times.ThetubewiththemixturewasplacedinaLindberg51333programablefurnace(digitalcontrolled,Tmax=1200C)usingthefollowingheattreatmentsequence:temperaturerampingto980Cwitharateof200C/hfollowedbyT=980Cfor24h,thencoolingatarateof3C/hdownto650C.Thelaststepwasafastcoolinginthefurnacetoroomtemperatureata200C/hrate.Thesinglecrystalswerethenremovedfromtheantimonyuxexcessbyetchinginaquaregia(HCl:HNO3=1:1).Thecrystalswerecubicorrectangularupto50mginweight(upto3mminsize)dependingontheamountofthestartingelementsandthecoolingrate.Forinstance,using1gofOsandacooling 49 PAGE 50 Thepoly-crystallineRalloys(usedasoneofthestartingcomponentsinthesynthesisofthesinglecrystals)werepreparedbymeltingitsconstituentelementsinanEdmund-BuhlerArcMelterAMunderahalfatmospherehighpurityAr.Theapparatusconsistsofastainlesssteelvacuumchamberwhichsitstightonawatercooledgroovecruciblesinacopperbaseplateandwithanelectrodeatthetop.Thetungstenelectrodeismotordrivenwhichcanbemovedfreelyabovethecrucible.Themeltingprocesscanbeobservedthroughadarkglasswindow.Allimportantcontrolfunctionsareintegratedintheheadoftheelectrodeandensuresafeandconvenientoperation.Whenfedatthemaximumcurrentthetemperatureoftheelectricarcinthemeltercangoashighas4000Candmelts500gofmetals.Thearcmelterhasaipper,amanipulatorforturningthesamplesinsitu.Thisgivesthepossibilitytoipandagainmeltthesample,ensuringitshomogeneity,withoutopeningthechamber.Beforeoperating,thecopperbaseplatewasthoroughlycleanedwithacetonetoavoidanycontaminationofthesamplewithimpurities.Rightatthebeginning,eachoftheconstituentelementswerewellcleanedtoeliminatetheoxidelayeronthesurface.Theprecisioninmassmeasurementswas0.02mg.Startingwiththeradioactiveorthehardestelementwecanadjusttherelativemassesoftheothercomponentstogainthewantedstoichiometricratio.Thetotalmasswasfromteensofmilligramto1g,thesizeofthesamplebeadwasupto1cm.Rightbeforetheelementsweremeltedtogether,azirconiumbuttonwhichwasalsousedforignitionofthearc,wasmeltedjusttoensureaevenhigherpurityoftheAr,whichwaslteredthroughapurierbeforeenteringintothearcchamber.Zirconiumiswellknownasaoxygenabsorber.Theelementwiththehighestvaporpressurewasthenplacedonthecopperplaterightbelowtheelementswithlowervaporpressures.Theaimofthis 50 PAGE 51 51 PAGE 52 whereishalfofthereectionangle,nisaninteger(n=1fortherstorderspectrum),distheinter-planedistance,andisthewavelengthoftheincidentradiation.ThelatticeconstantsarethencalculatedfromdandtheintersectionpointsofthelatticeplanesfromthedesiredspacegroupnumberisgivenintermsoftheMillerindices(hkl).ForacubicsymmetrythesameBraggequationcanberewrittenas sin2=2 whichisderivedfromd(hkl)=a=p 4.2.1Equipment 4.2.1.1Cryogenics Inhouse(StewartLab.,PhysicsDepartmentatUniversityofFlorida)measurementsofspecicheatwereperformedinthetemperaturerangeof0.3to2Kusually,andinsomecasesupto10K.Ahomemade3Hecryostatwasused.TheschematicdrawingisgiveninFig.4-3.Thisprobewasusedforthemeasurementsofspecicheatinmagneticeldsaswell.AspeciallydesigneddewarfromCryogenicConsultantsLimitedwasalsoused.Thesuperconductingmagnetreached14Tat4.2Kbathtemperature.Fortheac 52 PAGE 53 SpecicheatmeasurementsatlowertemperatureandhighermagneticeldswereperformedattheMillikelvinFacility(SuperconductingMagnet1{SCM1),HighMagneticFieldNationalLaboratory,Tallahassee,Floridausingatoploadingdilutionrefrigeratorwhichispermanentlyinstalledina18/20Tsuperconductingmagnet.Themeasurementtemperaturerangewas20mKto2Kcombinedwithamagneticeldofupto20T.Thesmallhomemadecalorimeter(Fig.4-5)wasconnectedtothegeneralpurposesamplemountprovidedbythefacility.Resistivitymeasurementsperformedatthesamefacilityweredoneinthetemperaturerangeof20mKto0.9K.Anothersampleholder,aso-called16pinamplerotator,wasused.Thisallowsthechangeoforientationofthesampleineldduringtheexperiment.Thisholderhas16pins(16connectionwirestothetopoftheprobe)thatallowsuptoamaximumoffourdierentsamplestobemeasuredwithoutpullingouttheprobefromthedilutionrefrigerator,savingprecioustime.Takesupto6hourstoinserttheprobeintotherefrigeratorandcoolthesampleto20mK.Specicheatmeasurementsinmagneticeldsupto32Twereperformedatthe33T,32mmboreresistivemagnet(Cell9),atthesameNationalLaboratory.Anotherhomemade3Heprobesimilartotheonementionedearlierbutwithslightlydierentdimensionsinordertotintothemagnetandalsotoaccommodatethesampleinthemaximumeldstrengthregionwasused.Rightbeforetheinsertionoftheprobeintothemagnetanelectricalcheckwasdoneonwireconnections.ThequalityofvacuumandsealingwascheckedalsousinganAlcatelASM10LeakDetector.Forboth33Tand45Tmeasurements,aspecialpositioningsystemmadeitpossibletocentertheprobeinsidethemagnetsuchthatitdid 53 PAGE 54 Becauseofthelargeamountofheatthathadtoberemoved,theprobewascooledinliquidnitrogen(LN)downtotheboilingpoint(77.35K).Afterabout2hours,whentheprobewasatthermalequilibriumwiththeliquidnitrogen,itwastransferredquicklyintoadewarinwhichittstight.ThedewarwascooledinadvanceinLNaswell.Thedewar(withtheprobeinside)waslledwithliquid4He(LHe)andafterseveralhours(dependingonthevolumeofthecan)thetemperatureoftheprobereached4.2K.After4.2Kwasattainedfollowingtheproceduredescribedabove,the4HepotwaslledwithLHefromthebathbyopeningtheneedlevalve,and3Hegas(alighterisotopeofHe)wastransferredintothe3Hepot+probelineusingahome-made3Hehandlingsystem.Thehandlingsystemconsistsofatanklledwith3He,apumpwhichhelpstotransfertoandbackfromthe3Hepotline,andpressuregaugestodisplaytheamountof3Heleftinthetankandinthetransferlines.Afterclosingtheneedlevalveandpumpinginthe4Helineatemperaturebetween1and2Kwasobtained.Itwasnecessarytorellthe4Hepotbyopeningtheneedlevalveonceinseveralhours.Inordertoattain0.3Kacompletelycontained3Hecoolingpartusingasorptionpumpwasrequired.Whencooled,gasesgenerallyadsorbtosolidsurfaces.Thesorptionpumpisbasedontheideathatat10Kalmostallofthe3Hegasmoleculesareadsorbed,whereasat35Kallofthemoleculesdesorb.ThesorptionpumpconsistsofaCucylinderthatcontainsactivatedcharcoal,whichhasanenormoussurfacearea(tensofsquaremeterspergram).Thecylinderisattachedtothelowerendofametallicrod.Thewholesystem,rod+cylinderwithcharcoal,wasplacedinsidethe3He-gasenclosure.Asthecharcoalwasloweredtowardthe3Hepot,the3Hewasabsorbedbythecharcoalreducingthevaporpressureandloweringthetemperatureofthe3Hepot.Afterthecharcoalbecamesaturatedwith3He,thecharcoalwaswarmedup(byraisingtherodwiththecharcoal),andthegaswas 54 PAGE 55 79 { 81 ].Thethermalrelaxationmethodconsistsofmeasuringthetimeconstantofthetemperaturedecayofthesampleconnectedtotheheatbathbyasmallthermallink.ApowerPisapplied(Fig.4-6)(thermalpowerbyasmallcurrentoftheorderofA)totheplatform-samplesystem.Thetemperatureofthesample,initially 55 PAGE 56 Thetimeconstant1isproportionaltotheCtotal(sample+platform): whereisthethermalconductanceofthewireslinkingthesample+platformatT=T0+TandtheringatT=T0.Theblocktemperatureisregulatedbyablockheater(abundleofmanganinwire)andmeasuredbyathermometerattachedtotheblock.Thetimeconstantisobtainedbymeasuringthetimedecayoftheo-balancevoltagesignalfromaWheatstonebridgeusingalock-inamplier.TwoarmsoftheWheatstonebridgeareavariableresistanceboxandtheplatformthermometer.Thebridgeisbalancedbyadjustingtheresistanceoftheresistancebox.Thismadeitpossibletondtheresistanceofthethermometer.FromaninitialcalibrationofthethermometerRversusT: 1 itispossibletondthetemperaturecorrespondingtotheplatformthermometerresistance.Thethermalconductanceisgivenby: whereP=IVisthepowerappliedtotheplatformheater.Equation(4{3)isvalidifthethermalcontactbetweensampleandplatformisideal(i.e.,sample1).Ifthecontactispoor(i.e.,sample),then 56 PAGE 57 57 PAGE 58 82 ]andtwosecondarycoilsmadefromcopperwire,woundinbothsidesinoppositedirectionsof2700turns.ThecoilsareattachedtotheCublock(whichisinthermalcontactwiththe3Hepot).Theapparatususesthemutualinductanceprinciple.Thesampleissubjecttoanalternatingmagneticeldof0.1Oeproducedbytheprimarycoil(andalsotheEarth'smagneticeld).Theresultingelectromotiveforce(EMF)inducedinthesecondarycoilisdetected.Thebackgroundsignalisnulledbytheidenticalsecondarycoil,connectedinseriesopposition.Forthesamereasonthetwoscrewsareidenticallybuilt.ThesampleisgluedtoonescrewwithGeneralElectric(GE)varnish7031whichensuresagoodthermalandmechanicalcontactatlowtemperatureandalsocanberemovedeasilyusingacetone.Theacsusceptibilitymeasurementswereperformedattwodierentfrequencies:27Hertz(Hz)and273Hz.Itwasdeliberatelyusedthesefrequencies(notintegermultiplesof60Hz)inordertoavoidthenoisecomingfromthecommonelectricaloutlet.Ingeneral,B=0(H+MV)=0H(1+),withHthemagneticeld,MVthevolumemagnetizationand=MV/Histhemagneticsusceptibility.IftheappliedeldHhasasinusoidalform,thetimedependentmagnetizationMV(t)canbeexpressedasaFourierseriesofthenon-linearcomplexacsusceptibility.ApplyingtheinverseFouriertransformtoMV(t)itcanbefoundthenthharmonicofbothrealandimaginaryacsusceptibility.Thefundamentalrealcomponent 58 PAGE 59 59 PAGE 60 PrOs4Sb12largecrystal,about50mg(right).Intheleft,anOsballwithPrOs4Sb12singlecrystalsattached,waitingtobeetchedout. Figure4-2. PrOs4Sb12samplespreparedfor(leftpanel)resistivityand(rightpanel)specicheatmeasurements. 60 PAGE 61 Schematicviewofthe3HecryostatusedinthemeasurementsperformedatUniversityofFlorida. 61 PAGE 62 SchematicviewofthecalorimeterusedintheSuperconductingMagnet1(SCM1),NationalHighMagneticFieldLaboratory. 62 PAGE 63 (a)Topviewofthesample-platform/Cu-ringassembly.(b)Lateralviewofthesapphireplatformandsample. 63 PAGE 64 SpecicheatCmeasurementprocessusingtherelaxationtimemethod(RedrawnwithpermissionfromMixson[ 83 ]). 64 PAGE 65 Allsamplesweresynthesizedusingtheself-uxgrowthmethod,describedinChapter4.Thesamplesarecubicshapedandofsizesrangingfrom1/2mmto3mmandweightingfrom1mgtoabout50mg.X-raydiractionwasperformedtoverifythedesiredcrystalstructure.Fromthediractionpatternitwasalsopossibletodeterminethelatticeconstants.Inadditiontothis,theX-raysconrmedthatthesamplesweresingle-phasewithinanaccuracyof5%. Thequalityofthesampleisalsogivenbythesharpnessofthetransitioninthespecicheat.AmorequantitativelymeasureofthequalityofthesampleistheresidualresistivityratioRRR=(300K)/(T!0).Thisratiorangesfrom50toabout170(PrOs4Sb12samplesstudiedbyMeassonetal.[ 59 ]haveRRR40.) Duetotheverysmallsizeofthesamplesused,thesusceptibilitiesmeasuredforallconcentrationsandthebackground(susceptibilityofthesampleholderconsistingfromaplasticdrinkingstraw)werecomparableat10K.Atroomtemperaturethesusceptibilitywasevensmallerthanthebackground,especiallyfordiluteconcentrations.Inordertoavoidthisbackgroundcontribution,magneticsusceptibilitywereremeasured(forx=0,0.05,0.3,0.67,0.8and0.95)usingbiggersamples.Also,inthesemeasurements,thematerialwaspressedinbetweentwolongconcentrictubessuchthatnobackgroundsubtractionwasneeded. AlltheseadditionalmeasurementsyieldedtoaCurie-Weisstemperaturedependenceabove150K,correspondingtoaneectivemagneticmomentclosetotheoneexpectedforPr3+(Fig.5-1),muchclosertotheexpectedvalueforPr3+thantheinitiallyreportedeff=2.97B[ 7 ]forPrOs4Sb12.TheeectivemomentofthefreePr3+iseff=3.58B[ 52 ].NewmeasurementsbyTayamaetal.[ 51 ]revealedaneectivemomentclosetothisvalue.ThissupportsthenotionofanessentiallytrivalentstateofPrinallPr1xLaxOs4Sb12alloys. 65 PAGE 66 Figure5-2. 66 PAGE 67 7 ](either3or1CEFgroundstate)forPrOs4Sb12implynon-magneticgroundstatesandexcludeaconventionalKondoeect,believedtobethesourceofHFbehaviorinCe-andsomeU-basedmetals. Thecontroversybetweenthetwoschemeswasbroughtaboutbydierentexperimentsthatseemtofavoreitherconguration. AspresentedinChapter3,therstpublishedresultssuchasthezeroeldspecicheat,magneticsusceptibilitydata[ 7 ],resistivityinsmallmagneticelds[ 55 ],inelasticneutronscatteringdatainterpretedusingOhsymmetry[ 53 ]favoredthe3doubletastheCEFgroundstate. Ontheotherhand,magneticsusceptibilitydataofTayamaetal.[ 51 ]andentropychangesinsmallmagneticeldsmeasuredbyAokietal.[ 56 ]werebetterttedbya1CEFgroundstatemodel. ThezeroeldSchottkyanomalyoccurringat3.1Kcanberelatedtothe3-5model,assumingthesetwolevelsaresplitby6.5K,or1-5modelwiththesplittingof8.4K.Thedicultyininterpretingtheselowtemperature,loweldresultsisrelatedtoastronghybridizationof4fandconductionelectrons,inferredfromthelargeelectronic 67 PAGE 68 Inordertopresentourresultsinaproperperspectivewestartfromrecallingthespecicheatdataforeldssmallerthan8TobtainedbyAokietal.[ 56 ] Figure6-1,upperpanel,showsthelowtemperaturespecicheatto8TobtainedbyAokietal.[ 56 ],thelowerpanelacomprehensivephasediagramknownbeforeourmeasurements.4.5TisthelowesteldatwhichasignatureofFIOPisdetectableasasmallkink(at0.7K).Thiskinkevolvesintoasharppeakat0.98Kin6T.TheC(T)peakgrowsandmovesalsotohighertemperaturesforhigherelds. TheFIOPwasconrmedbyspecicheatofVollmeretal.[ 63 ]andmagnetizationstudyofTayamaetal.[ 51 ]. AnumberofobservationsbroughtforwardtheinterpretationofFIOPintermsofantiferroquadrupolar(AFQ)order.Theseobservationsincludedalargeanomalyinthespecicheat(correspondingtoalargeentropyremovedbythetransition)andtheverysmallvalueoftheordered(antiferromagnetic)moment(about0.025Bat0.25Kin8T[ 84 ])measuredbyneutrondiraction,andalsosimilaritiestosystemsdisplayingquadrupolarorder(e.g.,PrPb3[ 85 ]). Figures6-2,6-3,and6-4showthespecicheatineldsrangingfrom10to32T.Thespecicheatmeasurementsineldsupto14TweredoneusingCryogenicConsultantLimitedsuperconductingmagnetattheUniversityofFlorida.Measurementsineldslargerthan14TwerecarriedoutattheNationalHighMagneticFieldLaboratory,Tallahassee,FloridausingaresistiveBittermagnet.Theeldwasappliedalongthecrystallographic(100)direction. Thespecicheatdatainallthreeguresareaftersubtractingthephononbackground(T3with=(1944103)n/3D[ 1 ])correspondingtoaDebyetemperature(D)of165K,proposedbyVollmeretal.[ 63 ]ThisvalueofDobtainedfromthetemperature 68 PAGE 69 7 ]),320K(Aokietal.[ 56 ]),and259K(Mapleetal.[ 53 ])arebasedonspecicheatmeasurementsofLaOs4Sb12. Thelowesttemperatureoftheheatcapacitymeasurements,actualvalue,ischosenrelativelyhighinordertoavoidcomplicationsassociatedwithanuclearcontributionofPr.Thiscontributionisstronglyenhancedbycouplingwithorbitalmomentsoffelectrons[ 86 87 ].Itisdiculttomeasurespecicheatbyaconventionalrelaxationmethodattemperatureswherenucleardegreesoffreedomdominatebecauseofadditionalthetimescaleenteringtheexperiment,nuclearspin-latticerelaxationtimeT1[ 88 ].Stronglynon-exponentialtemperaturedecaysatthelowesttemperatures(e.g.,below0.5Kintheeldof10Tandbellow1.5Kintheeldof32T)indicatetheimportanceofnucleardegreesoffreedomandcannotbeanalyzedusingtheso-called2correction.Therefore,theselowesttemperaturepointscarrylargeuncertainty.Whenthemagneticeldappliedalongthe(100)crystallographicdirectionis10T,thetemperatureofthesharpFIOPpeakappearsat1K(Fig.6-2).Whenincreasingtheeldfrom10Teldto12and13T(Figs.6-2and6-3)theorderingtemperatureTxdecreasesonlyslightlybutC(Tx)issuppressedinastrongmanner. Theresultspresentedhere[ 31 ]combinedwiththoseofAokietal.[ 56 ]andVollmeretal.[ 63 ]showthatTx(peakpositioninC)reachesamaximumvaluearound9T.Also,CatTxismaximumsomewherebetween8and10T. In13TashoulderappearsonthehightemperaturesideoftheFIOPanomaly.Thespecicheatvalueatthisshoulderisabout3400mJ/Kmol.ThisshoulderevolvesintoabroadmaximumforH=13.5T.Above13.5TtheFIOPcannotbeobservedanymoreinthespecicheat.Thus,theseresultsstronglyimplythedisappearanceofFIOPbeforeTxreaches0. Thebroadmaximumthatappearsin13Texistsatalleldsstudieduptoatleast32T.Thetemperatureofthemaximumincreaseswiththestrengthoftheeld(Fig.6-4). 69 PAGE 70 89 ].Theuncertaintyofthespecicheatmeasurementsintheseelds(andattemperatureswherenuclearcontributionissmall)isabout10%.IncreasingDfrom165K,usedinthesubtractionofthephononterm,totheotherextremalvalueproposed,320KwouldraisetheestimateoftheelectronicpartofCbyabout290mJ/Kmolat3.5K.Thus,theextractedvaluesatthemaximumarewellwithintherealisticerrorbarofthetheoretical3650mJ/Kmolforthetwo-levelSchottkyanomaly.Thehighesteldusedof32Tislargeenoughtosplitanydegeneratelevels,thereforetheobservedSchottkyanomalyisduetotheexcitationsbetweentwosinglets.TmisrelatedtotheenergyseparationofthetwolevelsbyTm=0.417[ 89 ].AnextrapolationofTmtoT=0(Fig.6-5)determinestheeldatwhichthetwolevelscross,whichissomewherebetween8and9T. TheseresultcanbeusedtoinfernewinformationregardingtheplausiblecrystaleldcongurationofPr.Prcanbemodeledbythefollowingsingle-sitemean-eldHamiltonian[ 84 ]: whereHCEF,JandOirepresenttheCEFHamiltonianforthecubicThsymmetry,thetotalangularmomentum,andthei-thquadrupolemomentofPrinasublattice,respectively,wheretherearevetypesofquadrupolarmomentoperators:O02,O22,Oxy,Oyz,andOzx.JandQiaretheinter-sublatticemoleculareldcouplingconstantsofspin(exchange)andquadrupolarinteractions,respectively.ThethermalaveragesoftheangularmomentumandquadrupolemomentofthePrinthecounterpartsublatticearehJ0iandhO0ii. UsingtheCEFparametersproposedbyKohgi[ 84 ]forthe1-5CEFconguration,Tm(withQi=0)andtheOyz-typequadrupolarorderingtemperatureTxwerecalculatedfor(100)directionbyAokietal.[ 90 ]AsitisdemonstratedinFig.6-5,themeasured 70 PAGE 71 However,thelevelcrossingforeld(100)directionisalsoexpectedfor3-4model,althoughatsomewhatdierenteld,asdemonstratedbyVollmeretal.[ 63 ]. MoreconclusiveargumentsregardingtheCEFcongurationcanbeobtainedfromthestudyoftheanisotropyoftheZeemaneect.ResultsofourcalculationsfortheZeemaneectfor~H==(100),~H==(110),and~H==(111)areshowninFig.6-6for1CEFgroundstate.TheplotsshowonlythefourlowestCEFlevels.Thehigherlevelsareatabove100Kand200Kfromthegroundstate,andthereforeplaynoroleinthelowtemperatureproperties.ThecalculationsweredoneneglectingexchangeandquadrupolarinteractionsandconsideringtheThsymmetry.Neglectingorretainingthelasttwotermsin(6{1)forthe(100)directionleadtoalmostidenticalresultsforeigenvalues(Aokietal.[ 90 ]andourresults). Thereisacrossingbetween1andthelowest5level(splitbymagneticeld)atabout9Twhen~H==(100)or~H==(111)andanti-crossingwhen~H==(110)aroundthesameeld.Therefore,thecrossingeld,extrapolatedfromthetemperatureoftheSchottkyanomalyathigheldsshouldbeindependentoftheelddirection. Figure6-7showsthesamecalculationsforthe3CEFgroundstatemodel.For~H==(100)thereisacrossingbetweenthetwolowestCEFlevels,althoughataeldsomewhatlargerthantheoneexpectedforthe1CEFgroundstate.However,thereisnocrossingexpectedinvolvingthelowestCEFlevelswhentheeldisappliedalongthe(110)or(111)directioninthe3-5model(Fig.6-7).Therefore,measurementsofspecicheatwhenmagneticeldisappliedinanydirectiondierentthan(100)dierentiatebetweenthetwoscenarios.Measurementsofthespecicheatineldsto14 71 PAGE 72 TheH-TphasediagramispresentedinFigure6-10.For~H//(110)directionweobserveadecreaseofTxvalueswithrespecttothe(100)directionforthecorrespondingelds,consistentwiththepreviousmagnetizationmeasurements[ 51 ](Fig.3-6).Ontheotherhand,withintheuncertaintyofthemeasurement,thereisnochangeinthepositionoftheSchottkyanomalyat13and14T,asexpectedforthe1CEFgroundstateandinconsistentwiththe3scenario.Moreover,forthe(110)orientationtheSchottkyanomalycanbeclearlyseenalreadyat12T.ThislowereldlimitfortheSchottkymaximumisprobablyduetocompetitionbetweenthetwotypesofanomaliesandlowervaluesofTxforthe(110)direction(Fig.6-9). AstraightlinetforthethreeTmpointsresultsinthecrossingeldvalueof91T.Thisvalueagrees,withintheerrorbar,withtheestimateforthe(100)direction.Theexistenceofthecrossingeldforthe(110)directionprovidesanunambiguousevidenceforthe1-5model.Asmallmisalignmentofthesamplewithrespecttotheeldineitherofthemeasurementscannotexplainessentiallyidenticalcrossingeldsforbothdirections.Infact,themeasureddierenceinTxvaluesfor(100)and(110)directionsprovidesanadditionalcheckofthealignment.Similartothe(100)direction,thereseemstobeaclosecorrelationbetweenthecrossingeldandtheeldcorrespondingtoTxmaximum. Figures6-5and6-10implyastrongcompetitionbetweentheeld-inducedorderandtheSchottkypeak.TheFIOPtransitioninthespecicheatabruptlydisappearsbefore 72 PAGE 73 44 51 ],ontheotherhand,wereabletomapTxasafunctionofthemagneticeldallthewaytoTx0.ThisapparentcontradictioncanbeexplainedbyaverysmallentropyavailablefortheFIOPtransitionabove13and12Tforeldsparalleltothe(100)and(110)directions,respectively.Specicheat,beingabulkmeasurement,canbelesssensitivethanmagnetizationtechniquesinthissituation.Astrongcompetitionistobeexpectedinthe1-5scenario.Thegroundstatepseudo-doubletformedatthelevelcrossingcarriesbothmagneticandquadrupolarmoments.Sinceaquadrupolarmomentoperatordoesnotcommutewithadipolarone,thequadrupolarinteractionsleadingtoFIOPcompetewiththemagneticZeemaneect. Therefore,thehighmagneticeldsmeasurementsofspecicheat[ 31 ]providedtherstunambiguousevidenceforthesingletCEFgroundstateofPrinPrOs4Sb12.Thisresultwasconrmedbyrecentinelasticneutronscatteringexperiments[ 32 ]analyzedintheThsymmetry,andourmagnetoresistivityresultsdescribedinSections6.2and7.2. ThemainindicationofheavyelectronsinPrOs4Sb12isthelargediscontinuityinC=TatTc.Themassenhancementinferredfromspecicheatmeasurementsisoftheorderof50[ 7 ].Thisvalueisanestimateandthereisnoconsensusonaprecisevalue.Anuncertaintyexistsinevaluationoftheeectivemassdirectlyfromthelowtemperaturezero-eldspecicheat,becausethereisnostraightforwardmethodofaccountingfortheCEFspecicheat.ThecorrespondingSchottkyanomalyisstronglymodiedbecauseofthehybridizationbetweenthefandconductionelectrons.Thezero-eldspecicheatjustaboveTcisdominatedbyCEFeects. 73 PAGE 74 32 ]suggestedmenhancementtobeabout20.However,theirestimatewasbasedontheFulde-Jensenmodel,whichwedonotbelieveisrelevanttoPrOs4Sb12.Thisenhancementis3{7,accordingtothedeHaas-vanAlphenmeasurements[ 44 ].However,dHvAeectwasanalyzedoverawiderangeofelds3{17TanddidnottakeintoaccountmbeingdependentofH[ 44 ]. TheresidualresistivityratioRRR=(300K)/(T!0)oftheinvestigatedsamplewasabout150.Thisvalueisamongthehighestreported,implyinghighqualityofoursample.Boththecurrentandthemagneticeldwereparalleltothe(100)direction(longitudinalmagnetoresistance).Themeasurementsweredoneusingthe18T/20TsuperconductingmagnetattheMillikelvinFacility,NationalHighMagneticFieldLaboratory,Tallahassee,Florida.Thetemperaturerangewas20mKto0.9K,themaximumeldused20T.MeasurementsattheUniversityofFloridaweredoneineldsupto14Tdownto0.35K. Thezero-eldelectricalresistivity,anotherimportantcharacteristicsofheavyfermionmetals,doesnotprovideastraightforwardsupportforthepresenceofheavyelectrons.Mapleetal.[ 91 ]foundthattheresistivity,between8and40K,followsafermi-liquidtemperaturedependence(=0+AT2).Ourresistivitydatabetween8and16Kfollowstheabovementioneddependence(Fig.6-11)withA0.009cm/K2(inagreementwithAfoundbyMapleetal.[ 91 ]).AsinferredfromKadowaki-Woods(KW)relation(A/21105cm(molK/mJ)2)[ 60 ]thisvalueofAimpliesasmallelectronicspecic-heatcoecient30mJ/K2mol,comparabletotheonemeasuredforLaOs4Sb12.So,evidentlythereisanuppertemperaturelimit(lessthan8K)fortheheavyfermionbehavior. Figure6-12showstheresistivityofPrOs4Sb12inH=3,10,15,16,17and18Tinatemperaturerangeof20mKto0.9K.Theresistivitybelow200mKsaturatesforallelds.Thistemperaturedependenceatthelowesttemperaturewasalsoobservedbyothergroups[ 91 92 ].Therefore,theresistivityforallotherintermediateeldswasmeasuredto 74 PAGE 75 Mapleetal.[ 91 ]proposedthefollowingtemperaturedependenceforxedmagneticeld:=0+aTn,withn>2.Intheirstudy(transversalmagnetoresistivity)nwas3for3Tand2.6for8T.Inourlongitudinalcasetheseexponentsareslightlylarger(e.g.,3.9for3T).Theexponentsdependonthetemperaturerangeofthet,i.e.,nbecomessmallerwhentheuppertemperaturelimitofthetdecreases.Theresidualresistivity0valuesresultedfromthetondierenttemperatureranges(includedinthe350mKand0.9Kinterval)wereclosetoat20mK.Theresidualresistivity0attainsamaximumnearH=10T,eldcorrespondingtothecrossingbetweenthetwolowest-energyCEFlevelsofPr(Fig.6-14,lowerpanel).Inthisregion(around9{10T)thelowesttwosingletsformaquasidoubletpossessingquadrupolardegreeoffreedom.Theseelectricquadrupolesorderatsucientlylowtemperatures[ 56 ]withtheorderingtemperaturehavingmaximuminthecrossingeld[ 31 56 ].ResistivityisdominatedthereforebytheCEFeectsorthequadrupolarordering.Thisorderingiscompletelysuppressedbyeldshigherthan15T.AsitcanbenoticedfromFigs.6-12and6-14theresidualresistivity0doesnotchangesubstantiallyineldshigherthan15T.Infact,itcanbeseenthantheresidualresistivity0versusHeldcanmaptheboundaryoftheAFQphase,i.e.asharpincreaseof0indeedcoincideswiththeAFQboundary,indicatedbyarrowsinFig.6-14,lowerpanel. Thesameconclusioncanbedrawnfromresistivitymeasurementsforhighmagneticeldsperpendiculartothecurrent[ 55 ].Therateoftheincreaseoftheresistivitywithtemperatureisstillchangingabove15T(Fig.6-12).ItcanbeconcludedthatthereductionofthetemperatureratecorrelateswithanincreaseoftheenergybetweenthelowestCEFlevels.Apreciseaccountingofthesechangesisdicultsinceneitherofthefunctionscheckedoutdescribeaccuratelythevariation(T)inaxedeld. AlineardependenceofonT2isaccountedbyresistivity(=0+AT2)indierenttemperatureranges(above0.4and0.5K),asseeninFigs.6-13.UsingtheKWratio[ 60 ] 75 PAGE 76 Acharacteristicelddependenceoftheresidualresistivity(Fig.6-14,lowpanel)wasassociatedbothtoCEFeectsandlongrangeAFQorder.TheCEFeectontheresistivitywasconsideredbyFrederickandMaple[ 93 ]usingthefollowingexpression: Thersttermrepresentsacontributionduetoexchangescattering,andthesecondtermisthecontributionduetoaspherical(orquadrupolar)scattering.TheasphericalCoulombscatteringisduetothequadrupolarchargedistributionofthePr3+.MatricesPij,QMijandQAijaredenedasfollows: 1e(EiEj);(6{3) 2jhijJ+jjij2+1 2jhijJjjij2;(6{4) IntheaboverelationsEiaretheeigenvalueoftheCEFeigenstates,thejii0saretheCEFeigenstates,=1/(kBT),andtheym20saretheoperatorequivalentsofthesphericalharmonicsforL=2(i.e.,quadrupolarterms)[ 94 ].TheQij-matricesarenormalizedtoeachother[ 95 ]suchthat 76 PAGE 77 FurthermoreAseemstohaveamaximumvalueneartheeldseparatingorderedandnon-orderedphases.NotethatthisisnotthecrossingeldforthelowestCEFlevels.Thus,theseresultssuggestapossibilityofmenhancementduetostronguctuationoftheAFQorderparameter. 77 PAGE 78 SpecicheatCofPrOs4Sb12ineldsupto8Tfor~H//(100)(upperpanel).ThemagneticeldphasediagramH-TofPrOs4Sb12ineldsupto8Tfor~H//(100)(lowerpanel)(ReprintedwithpermissionfromAokietal.[ 56 ]). 78 PAGE 79 SpecicheatCofPrOs4Sb12in10and12TinthevicinityofFIOPtransitionfor~H//(100)(ReprintedwithpermissionfromRotunduetal.[ 31 ]). 79 PAGE 80 SpecicheatCofPrOs4Sb12in13,13.5,and14T,for~H//(100).Ashoulderappearsatabout1.2{1.3Kat13TandtheFIOPtransitionissuppressedat13.5T. 80 PAGE 81 SpecicheatCofPrOs4Sb12inmagneticeldsof16,20,and32T,for~H//(100). 81 PAGE 82 MagneticeldphasediagramH-TofPrOs4Sb12for~H//(100)(H>8T).FilledsquaresrepresenttheFIOPtransition.OpensquarescorrespondtotheSchottkyanomaly.TheinsetisthemodelcalculationoftheSchottkyanomalyassumingthesingletasthegroundstate[ 90 ].ThesolidlinerepresentstheFIOPboundary;thedashedlinecorrespondstoamaximuminC(ReprintedwithpermissionfromRotunduetal.[ 31 ]). 82 PAGE 83 ZeemaneectcalculationsforPrOs4Sb12inthe1CEFgroundstatescenario.Thereiscrossingofthetwolowestlevelsfor~H//(100)or~H//(111)ataround9Tandanti-crossingatthesameeldfor~H//(110).Thegureshowsonlythetwolowestlevels,i.e.thesinglet1andthetriplet5. 83 PAGE 84 ZeemaneectforPrOs4Sb12inthe3CEFgroundstatescenario.Theeectinstronglyanisotropic.ThereisnocrossingofthetwolowestCEFlevelsfor~H//(110)or~H//(111).Thegureshowsonlythetwolowestlevels,i.e.thedoublet3andthetriplet5. 84 PAGE 85 SpecicheatCofPrOs4Sb12for~H//(110),H=10,12,13,and14T.TheinsetshowsCversusTneatTxfor8,9,9.5,10,10.5,and11T. 85 PAGE 86 SpecicheatCofPrOs4Sb12inH=12T,for~H//(100)(upperpanel),and~H//(110)(lowerpanel).ThearrowindicatestheAFQtransition. 86 PAGE 87 ThemagneticeldphasediagramH-TofPrOs4Sb12for~H//(110)(H>8T).TheinsetshowsthespecicheatCmaxofAFQversusH.ForadenitionofsymbolsseeFig.6.5. Figure6-11. ElectricalresistivityversusT2forPrOs4Sb12.IntheinsetisversusTshowingthesuperconductingtransitionatTc=1.85K. 87 PAGE 88 Resistivity(T)between20mKandabout0.9KofPrOs4Sb12in3,10,15,16,17,and18T(ReprintedwithpermissionfromRotunduandAndraka[ 96 ]). 88 PAGE 89 ResistivityversusT2ofPrOs4Sb12for3.5,5.5,7,10,and13T. 89 PAGE 90 Coecienta(=0+aTn)versusHforPrOs4Sb12eldsupto18Tisinupperpanel.Theresidualresistivity0(H)isshowninlowerpanel. 90 PAGE 91 CoecientA(=0+AT2)versusH. 91 PAGE 92 Thecalculated(H)ofPrOs4Sb12,forboth3and1scenarios.TheverticallineindicatestheeldcrossingofthetwolowestCEFlevels.Notethatthecrossingeldforthe1groundstatewasassumedat3T(lowerpanel)(ReprintedwithpermissionfromFrederickandMaple[ 93 ]). 92 PAGE 93 7 16 ]orvirtualCEFexcitations[ 32 36 ],tocooperativemodelsinvokingproximitytoalong-rangeorder(proximitytothelowtemperaturestateofAFQorder)[ 56 ].Whileinvestigatingtheapplicabilityofthesemodels,closeattentionwaspaidtowhetherthesingle-ionparameterssuchastheCEFspectrumandhybridizationparametersvarywiththealloying. 57 ]andalmostnon-existentlanthanidecontractioninternaryskutteruditescontainingSb,ofageneralformLnT4Sb12,whereTandLnaretransitionelementandlightlanthanide,respectively. TopresentthischangeinaproperperspectivewerecallthatthechangeofthelatticeconstantacrossPr(Os1xRux)4Sb12[ 97 ]is10timeslarger.ThisisdespitethefactthattheatomicradiiofOsandRuarealmostidentical(1.35and1.35AforOsandRurespectively),whileLaismuchlargerthanPr(1.88versus1.82A).InPr(Os1xRux)4Sb12theCEFparametersincreasemonotonicallywithx.VerysmallchangesinlatticeconstantinPr1xLaxOs4Sb12suggestsmall,ifany,changesintheCEFparameters 93 PAGE 94 Figure7-3showsthesusceptibilitiesonlyintherange1.85to10K.AlldataarenormalizedtoaPrmole.Duetotheverysmallsizeofthesamplesusedintheinitialmeasurements,themeasuredmomentofmostofthesamplesandthebackground(themagneticmomentofthesampleholderconsistingofapieceofaplasticdrinkingstraw)werecomparableat10K.Atroomtemperaturethemagneticmomentofthesampleswasevensmallerthanthebackground,especiallyfordiluteconcentrations.Inordertoavoidthisbackgroundcontribution,themagneticsusceptibilitieswereremeasured(forx=0,0.05,0.3,0.8and0.95)usingseveralcrystalsandholdingthembetweentwolongconcentricstraws.Nobackgroundsubtractionwasneededthistime.TheCurie-Weisstemperaturewasfoundabove150Kandtheeectivemagneticmomentintherange3.2{3.6B/Pratom.ThevaluesareintherangeofmomentsreportedforpurePrOs4Sb12.SomediscrepancybetweenthesevaluesandthatexpectedforPr3+,3.58B[ 52 ],canbeduetoanerrorinmassdetermination.Becauseoftheveryfragilenatureofthesecrystals,someofthembrokeoduringthemeasurementandsmallfractionsmovedinbetweenthetwotubes.Afurthercheckofthemagneticmomentwasperformedononelargecrystalforx=0.67(20mgeach).Figure7-4showsthesusceptibilityandtheinverseofsusceptibilityforx=0.67.FromtheCurie-Weisstthehightemperatureeectivemomentisfoundtobe3.62B/Prmol,closetothevalueexpectedforPr3+. 94 PAGE 95 31 ],andneutron[ 32 84 ]measurementsestablished1singletastheCEFgroundstateseparatedbyabout8Kfromtherstexcited5triplet.VerysmallchangesinthepositionofthesemaximainthesusceptibilityaretherstindicationthatCEFareessentiallyunalteredbythedopingasexpectedfromthemeasurementofthelatticeconstant. Anotherinterestingaspectofthesusceptibilityisastronginitialreductionofthelow-temperaturevaluesof(normalizedtoamoleofPr)byLa.Thereductionofthemaximumsusceptibilityfromapproximately100forx=0toabout50memu/Prmolforx=0.4isclearlyoutsidetheerrorbar.Theaforementionedmeasurementsonassembliesofcrystalsforx=0.8and0.95alsoresultedina4Kvalueofabout505memu/Prmolforbothcompositions.SomebroadeninganddecreaseinmagnitudeoftheCEFsusceptibilityareexpectedinmixedalloysduetoincreasedatomicdisorder.However,theverylargeinitialdropinthesusceptibilityandlackofvariationabovex=0.4mightindicatethatsomecharacteristicelectronicenergy(analogoustoaKondotemperature)increasessharplyuponsubstitutingLaforPr.Asimilarsuppressionofthecorrespondingmaximumisobservedinthespecicheatdatadiscussedinthenextsection. 7.3.1SpecicHeatofPrOs4Sb12:SampleDependence 7 ].Morerecentspecicheatmeasurementsrevealedtwosuperconductingtransitions(Vollmeretal.[ 63 ],Mapleetal.[ 53 ],Oeschleretal.[ 64 ],Cichoreketal.[ 65 ]). 95 PAGE 96 Ourobservationsareconsistentwithother,particularlymorerecent,reports.Almostallrecentinvestigationsndtwosuperconductinganomalies,morepronouncedatTc2andlessdenedatTc1.AnexceptiontothisruleareunpublisheddatabyAokietal.[ 98 ]thatshowasharppeakatTc1,andonlyachangeofslopeinC=TatTc2.Thewidthofthetransition,0.2K,denedabove,isquitesimilarforallpublisheddata.Thereisalargedistributionofreported(C=T)atTc,from500to1000mJ/K2mol.Ausualdeterminationof(C=T)byanequalarea(conservationofentropy)constructioncannotbeappliedduetothepresenceoftwosuperconductingtransitions.Applyingourmethod,C=T(Tc2)C=T(T3),resultsinanaverage(C=T)of800mJ/K2molforthemostrecentresults. 96 PAGE 97 whereC=TisexpressedinmJ/(K2mol)andTinK(Fig.7-6).AsignicantnonlinearityinC=TversusT2isprobablyduetotherattlingmotionoflooselyboundLaatoms[ 99 ].ValuesofandforLaOs4Sb12reportedbyotherresearchgroupsare:of36[ 54 ],55mJ/K2mol[ 100 ],56[ 44 ]and=0.98mJ/K3mol[ 101 ]. Figures7-7and7-8presentthef-electronspecicheatofPr1xLaxOs4Sb12alloys,i.e.thespecicheatofLaOs4Sb12and,normalizingtoamoleofPr.NotethatthephononspecicheatofpurePrOs4Sb12inChapter6wastakenfromVollmeretal.[ 63 ],whichwasderivedbyttingthetotalspecicheatCtoafunctionrepresentingphonon,conductionelectrons,andSchottkycontributions.However,usingtheLaOs4Sb12specicheatseemstobemorejustiableformoderatelyandstronglyLa-dopedalloysandthereforethiswayofaccountingforphononsisconsistentlyusedinthischapteronLaalloying. 97 PAGE 98 102 ]substitutedforU.Furthermore,sincePrOs4Sb12isclearlyanunconventionalsuperconductor(e.g.,timereversalsymmetrybreaking)whileLaxOs4Sb12ispresumablyaconventionalsuperconductorwewouldexpect,whilevaryingx,asuppressionofonetypeofsuperconductivitybeforetheothertypeemerges.Figure7-10showsthatthereissmoothevolutionofTc(andsuperconductivity)betweentheend-compounds.AsomewhatstrongersuppressionisobservedinthecaseofRureplacingOs[ 71 97 ].Buteveninthiscase,theTcreductionrateissmallifcomparedwiththemajorityofCe-andU-basedheavyfermionsandconsideringthefactthatRualloyingdrasticallyaectsCEFenergiesandhybridizationparameters. 63 ])oftheanomaliesatTc1andTc2precludesapopularspeculationthatoneofthesetransitionsisassociatedwithsurfacesuperconductivity. Asitwasalreadystressed,this(C=T)iscurrentlythemainevidenceforthepresenceofheavyelectrons.ThepresenceofamodiedSchottkyanomalynearTcmakes 98 PAGE 99 WerecallthatLaOs4Sb12isalsoasuperconductor,thereforefordilutePrconcentrationsthenormalizationofC=TtoPrmole(usedinFig.s7-7and7-8)hasnomeaning.Therefore,Fig.7-10showsthetotal(C=T)performulaunit.Thereisadrasticdecreaseof(C=T)withxfor0x0.2(Fig.7-7).ThediscontinuityofC=Tissuppressedfrom800forx=0to280forx=0.2andfurthertoabout160mJ/(K2Prmol)forx=0.3(C=Tissevenfoldreducedwithx,forxbetween0and0.3).(C=T)staysapproximatelyconstantwithxfor0.3 PAGE 100 Inordertostudyfurtherthisspecicheatreduction,alargecrystalofPr0:33La0:67Os4Sb12,forwhichaddendaheatcapacityisnegligiblebelow6K,wasinvestigated.Firstofall,theeectivemagneticmomentmeasuredatroomtemperatureforthisparticularcrystalisveryclosetotheexpectedvalueforPr3+.Thus,thereducedspecicheatcannotbeexplainedbyincorrectPrstoichiometry,norbysomePrionsbeinginamixed-valentstate.Theresults,aftersubtractingthespecicheatofLaOs4Sb12anddividingby0.33,areshowninFig.7-11intheformofC=T.ThisgraphshowsalsoattothefunctiondescribingaSchottkyspecicheatforasinglet-tripletexcitations,scaledbyafactora=0.44.AsimilarscalingwasusedbyFredericketal.[ 71 ]toaccountforthespecicheatdataofPr(Os1xRux)4Sb12intermsofthesingletCEFgroundstate.AnecessitytousesuchasmallscalingfactorforthismodelofCEF(ofabout0.5)wasusedbyFredericketal.[ 71 ]toargueforadoubletCEFgroundstate.However,ascanbeseenfromFig.7-12,areasonablettothedoubletCEFmodelalsorequiresascalingfactor,althoughsomewhatlarger(a=0.73).Finally,attothesinglet-to-singletscatteringrequiresnoscalingatall.ThetshowninFig.7-13obtainedwithaasanadjustableparameter,resultedina=1.009(within1%). 100 PAGE 101 103 ]suggestpresenceoftheeldinducedAFQorderformuchhigherLa-concentrations,ashigh0.3. 101 PAGE 102 Thus,theseresults,togetherwithpreviouslydiscussedevolutionofzeroeldproperties,particularlyweaksensitivityofthetemperatureofthemagneticsusceptibilitymaximum,providestrongargumentsforCEFenergiesandeigenstatesbeingunaectedbyLaalloying,atleasttox=0.2.Ontheotherhand,thetemperaturesoftheSchottkyanomalyin13and14Tforx=0.6areabout0.3Klowerthanthoseforx0.2,suggestingapossibilitythattheCEFenergiesandthecrossingeldforx=0.4increasebyapproximately20%.Veryrecentresultsofthehigheldspecicheatstudyofx=0.67byAndraka[ 116 ]alsoimplytheCEFenergiestobe20{25%largerthanintheundopedmaterial.Thus,thereisapossibilityofanabrupt(butsmall)changeofCEFenergiessomewherebetweenx=0.2and0.67.However,towithinourexperimentaluncertainty,weclaimCEFenergies(andeigenstates)tobeidenticalbetweenx=0and0.2,thusintheconcentrationrangewheredramaticchangesofmareanticipatedbasedonmeasurementsof(C=T). Thesehigheldresultsallowustocommentonwhethertheagreementbetweenthemagnitudeofthezeroeldanomalyinmoderatelyandstrongly(x=0.67)dilutedalloysandtheSchottkyspecicheatcorrespondingtosinglet-singletexcitationsisaccidentalormeaningful.Thespecicheatmaximumforx=0.4in14Tisapproximately1500mJ/Kmol(200mJ/Kmol).ThisvalueissignicantlysmallerthanthetheoreticalvaluefortheSchottkymaximum(3650mJ/Kmol).SimilarlythespecicheatofPr0:33La0:67Os4Sb12inmagneticeldsashighas18Tshowsamaximumwhosevalue 102 PAGE 103 103 ]. TheresistivitiesofPr1xLaxOs4Sb12forx=0.05,0.3and0.7ineldsupto18Tweremeasuredintemperaturedownto20mK.Theuncertaintyinthedeterminationoftheabsolutevalueoftheresistivitywasupto30%.Theroomtemperatureresistivitywasapproximatelyequalforallthreecrystals.Therefore,weassumedthattheresistivityatroomtemperatureis300cmforallcrystals,consistentwiththepublishedvalueforbothendcompounds,PrOs4Sb12andLaOs4Sb12[ 92 ].TheratiooftheroomtemperatureresistancetotheresistanceextrapolatedtoT=0(RRR)was50,180,and170forx=0.05,0.3,and0.7,respectively.Withtheexceptionwiththeresultforx=0.05,thesevaluesbelongtothehighesteverreportedforpureanddopedPrOs4Sb12,suggestinggoodqualitiesofoursamples.Thex=0.05crystalwasfromthesamebatchwhoseresultsofspecicheatandsusceptibilityweredescribedinpreviousSections. 103 PAGE 104 Longitudinalmagnetoresistanceforx=0.05attwotemperatures,20and300mK,isshowninFig.7-20.Thetransversemagnetoresistanceforthesamealloyatfortemperatures20,310,and660mKisshowninFig.7-21.ThetwocurvesinFig.7-20showexcellentoverlap,implyinganabsenceoftemperaturedependenceoftheresistivitybelow300mKforanyeldbetween2and18T.Thisoverlapisconsistentwithourresistivitymeasurementsofthepurecompound(Fig.6-12)forwhichtheresistivitywasatbelow300mKforalleldsbetween3and18T. Thus,the20mKresistivityabovethecriticaleldisessentiallyidenticaltotheresidualresistivity,0.0forx=0.05increasesbyafactorof2between2and10T(Fig.7-21).Thisincreaseissignicantlylargerthanthecorrespondingincreaseforx=0whichwasabout25%.Thislargerincreasein0forx=0.05coincideswithsomesuppressionoftheAFQorderwithLa,asdemonstratedbyspecicheatmeasurementinmagneticelds.Alarger0at10Tforx=0.05isduetoasmallerdegreeoftheAFQorder.Thedropin0above10T,ontheotherhand,islesspronouncedforx=0.05thanx=0.ThesetrendscontinuewithfurtherLadoping,x=0.3. Theelectricalresistivityforx=0.3inzeroeld,justaboveTc,isproportionaltothesquareoftemperature,withA=0.16cm/K2(Fig.7-22,upperpanel).ByapplyingtheKadowaki-Woodsformula,A/2=105(withAincm/K2andinmJ/K2mol),wearriveatoforder100mJ/K2mol.However,theapplicationofjust0.5T(approximatelyHc2forthisconcentration)againrevealsthesaturationofresistivityatthelowesttemperatureswhichwasseeninthepurecompound(Fig.6-12).Inhighermagneticelds,thelinearvariationinT2isgraduallyrestrainedtonarrowertemperatureintervals 104 PAGE 105 Figure7-24showstheresistivityat20mKforx=0.3upto18Tforeldsparallelto(001),(011),and(010)andI//(001).Allthreeisothermsexhibitastepcenterednear9{10Tsuperimposedonalinearbackground.Intheinvestigatedeldrangewedonotndthedomestructurecharacteristicofx=0or0.05.Notetheapproximatelyequalslopesofthecurvesbelow3Tandabove14T.Interestingly,Sugawaraetal.[ 92 ]foundthatthemagnetoresistanceofLaOs4Sb12at0.36KisapproximatelylinearinmagneticeldandhasasimilarorientationasthatshowninFig.7-24.ThelargermagnetoresistanceofLaOs4Sb12for(011)thanforthe(001)directioncorrelateswiththelargermagnetoresistanceofx=0.3for(011)thanfor(001)direction(Fig.7-24).Therefore,wecanassumethattheapproximatelylinearinHmagnetoresistanceofPr0:7La0:3Os4Sb12below3Tandabove14Tisduetonormal(non-f)electrons.Subtractingsuchlinearcontributionsresultsinidenticalcurves,almostatbelow6Tandabove13T(insettoFig.7-24).Furthermore,theresultingcurvesareidenticalforallthreedirections,arguingforveryisotropicf-electronmagnetoresistance.ThisisotropicbehaviorisconsistentwiththesingletandinconsistentwiththedoubletCEFgroundstateofPr. TheeldvariationoftheresistivityshownintheinsettoFig.7-24isconsistentwithmodelcalculationsoftheresistivityforthe1-5modelbyFrederickandMaple[ 93 ].Accordingtothesecalculationsthattakeintoaccountmagneticandquadrupolardegreesoffreedom,theresistivityshouldexhibitasharpjumpatthecrossingeld.Theindependenceofthecrossingeldonthecrystallographicdirectionisalsoconsistentwiththe1-5CEFmodel(Fig.6-6).RecallthatCEFlevelcrossinginthe3-5CEFmodeloccursforthe(100)directiononly(Fig.6-7). Figure7-25showsrawdataforthesamesamplewhen~H//I//(100)forseveraltemperatures,20,310,660,and1100mK.Nobackgroundsubtractionhasbeendone. 105 PAGE 106 93 ]. Themorediluteconcentration,x=0.67wasinvestigatedtoabout0.35Kintemperatureandineldsupto14T.Itsresistivityatthelowesttemperaturesexhibitsasimilarmagneticelddependencetothatforx=0.3(Fig.7-26). ThepresentedresistivitydatashowthatthereisnosignofFIOPorderedphaseinthex=0.3material.OurpreviousspecicheatmeasurementsindicatedthattheFIOPphasedisappearssomewherenearx=0.2.However,themaineectoftheLadopingontheAFQanomalyisthesuppressionofitssize,withasomewhatsmallereectonthetransitiontemperatureitself.Thus,itispossiblethatthiseld-inducedAFQorderpersiststoconcentrationslargerthanx=0.2,butitssignaturesinthespecicheatareundetectableduetosmallentropiesinvolved.Thecalculationsoftheresistivitypredictingthestepinthemagnetoresistancewereperformedinasingleimpuritylimit,i.e.,assumingindependentscatteringfromeachPrion.ForPrOs4Sb12thescatteringinbothsmallandlargeeldsshouldbecoherent;i.e.,onemightexpectsmallcontributionfromfionsawayfromthecrossingeldofabout9T.Atthecrossingeld,thePrlatticelosesitscoherencesincesomeoftheionswillbeintheexcitedstate;i.e.,thetranslationalperiodicityislost.Webelieve,thiscoherencemechanismisresponsibleforthedomeshapeof(H)inPrOs4Sb12andthedierencebetweenpureandLa-dopedalloys. Theunchangedeldvalueforthestepintheresistivitybetweenx=0.3and0.67suggeststhatCEFenergiesarenotsignicantlyalteredbytheLadoping.ThisisinagreementwiththealmostunchangedtemperaturepositionofthemaximuminthemagneticsusceptibilityandspecicheatbelievedtobedueexcitationsbetweenthelowestCEFlevels. 106 PAGE 107 59 65 104 ].Thetemperatureseparationbetweenthetwostepsinacsusceptibilityforx=0isabout0.14Kandisapproximatelyequaltothatbetweenthepeaksinthespecicheat.TheeldcooleddcmagnetizationofsamplesperformedbyMeassonetal.[ 59 ]showedaMeissnereectof50%,indicating(likespecicheat)bulksuperconductivity.Therefore,thereisapossibilitythatthetwotransitionsareduetoinhomogeneouscoexistenceoftwosuperconductingphasesinPrOs4Sb12.Forx=0.05bothtransitionsarevisibleandtheoverallwidthofthetransitionisabout0.13K.Theonsettemperatureisapproximatelythesameasforx=0.Fromx=0.2to0.8theacsusceptibilitydataaresimilar.Thereisanincreaseofthetransitionwidth(inthereduced 107 PAGE 108 Themostprobableoriginofthewidesuperconductingtransitionsisinhomogeneities,whoseoriginisnotclear.PrOs4Sb12andLaOs4Sb12areisostructural,butLaOs4Sb12exhibitsaverysharptransition.Therefore,theinhomogeneitiesseemtobeassociatedratherwith4felectronsofPr.OneplausiblescenarioisamixtureoftwoelectroniccongurationsofPr,4f1and4f2.However,hightemperaturemagneticsusceptibilitydatawereingeneralconsistentwith4f2congurationofPr.Also,theLIIIabsorption[ 49 ]andinelasticneutronscattering[ 32 105 ]resultsagreewithavalenceofPrcloseto+3.Anotherscenariofortheexistenceofinhomogeneitiesistheclosenessofthesystemtoalongrangeantiferroquadrupolarorder[ 56 ].Thismeansthatclusterswithashort-rangeorderwouldhavedierentsuperconductingparametersthantheremainingpartofthesample. 108 PAGE 109 71 ].IfthesetwopeaksinC=Tarerelatedtotwodierentsuperconductingphases,thencrystalNo.1hasarelativelylargefractioncorrespondingtothehigherTcphase.Theevolutionofthesespecicheatpeaksinmagneticeldsindicatesthatthereisacorrelationbetweenthesetwophases. ThetwoinsetstoFig.7-28showthecriticaleldversusTdeterminedbyspecicheatforbothsamples.ForsampleNo.1bothsuperconductingtransitionsarevisibleineldsupto0.5T.OpensymbolsmarkthehighertransitiontemperatureTc1andtheclosedcirclesmarkTc2.Thelinesrepresentingthetwotransitionsremainapproximatelyparallel,withatemperatureseparationofabout0.12K.ThemostinterestingfeatureisapositivecurvatureinHc2versusTforH<2000Oe.TheinitialslopeofHc2versusTisdHc2=dT=1T/K.However,forH>2000OetheHc2islinearinT(insettoupperpanelofFig.7-28)andtheslopeisabouttwiceaslarge,i.e.dHc2=dT=2.1T/K.ApositivecurvatureinHc2versusTnearTc,wasalsodetectedinmeasurementsofelectricalresistivity,magneticsusceptibility,andspecicheat[ 7 55 63 67 ].Thisconsistencywasusedtoargueforintrinsicproperty,andnotduetosomeartifactoftransportmeasurementsorcomingfrominhomogeneitiesinthesamples.Meassonetal.[ 59 ]consideredthispositivecurvaturetobeahallmarkofthetwo-bandsuperconductivity.Thetwo-bandsuperconductivitydescription,inwhichtwodierentbandscorrespond 109 PAGE 110 106 ]inmagneticelds. InsampleNo.2wewereunabletodeterminetheevolutionofTc1inelds,thereforeonlyonelineispresentedintheinsettolowerpanel.Withintheresolutionofourmeasurements,thereisnocurvatureinHc2versusTnearTc2,andtheslopeisfoundequaltothatofsampleNo.1foreldshigherthan2000Oe,i.e.2.1T/K.Toourknowledge,thisistheonlymeasurementofHc2(T)nearTcthatdoesnotndthispositivecurvature. Forallconcentrationsotherthanx=0onlyasinglesuperconductinganomalyinthespecicheatmeasurementscouldbeclearlydetected(mostprobablycorrespondingtoTc2inx=0).Ourx=0.02and0.05alloysexhibitasmallbutdetectablecurvature(insettoFig.7-29forx=0.05).Forx=0.05,theinitialslopedeterminedforeldssmallerthan1000Oeisabout0.9T/K.Foreldshigherthan1000Oetheslopeis1.6T/K.QuitepossiblythereisasmallcurvatureinHc(T)forx=0.1,butitcannotbeclearlyresolved(insettoFig7-30).Forx=0.3(Fig.7-31)andalloyswithx>0.3wehaveaconventionalvariationofHc2(T),i.e.,withnopositivecurvaturenearTc.Forx=0.3dHc2=dT=0.5T/K. TheevolutionofdHc2=dTisofgreatinterestsincetheslopeofHc2versusTatTc2isrelatedtotheeectivemassofCooperpairs.Figure7-32showsthedHc2=dTversusLaconcentration.Forconcentrationsx=0,0.02,and0.05twosetsofdHc2=dTvalueshavebeendetermined:theinitialslope(opencircles)andthatforsucientlylargeelds,forwhichHc2isclearlylinearinT(H>2000Oeforx=0andH>1000Oeforx=0.02and0.05)(lledcircles).ThisdHc2=dT,markedbyclosedcircles,isabout2.1T/Kforx=0anddecreasesrapidlywithx,withmostofthereductiontakingplaceforsmallvaluesofx.ThisimpliesthattheeectivemassofcarriersisrapidlyreducedbysmallamountofLa. Inthecleanlimitofsuperconductivity(l),whichisthecaseforPrOs4Sb12andLaalloys,theeectivemassdependsonp 110 PAGE 111 Bothp Usingthetwobandsuperconductivityframework,thetwoslopesinHc2(T)(Fig.7-32)forx PAGE 112 59 108 ].However,insuchacaseonlytheuppertransitionshouldshowupinthespecicheat.Anotherpossibility,notyetconsideredinliterature,isthattheuppersuperconductingtransitionisrelatedtothephasetransitionwithanorderhigherthan2[ 109 { 111 ].Atwo-steptransitionisexpectedinthiscase.Ahigherorderphasetransition(suchasthethirdorder)wouldbeverysusceptibletoimpuritiesandimperfectionsleadingtoasecondordertransitioninsucientlyimperfectcrystals.TheexpulsionofmagneticuxwouldbeweakwhenloweringTfromTc1toTc2,followedbyamorerapidexpulsionbelowTc2.Thisscenariomyaccountforthedierentmagneticeldresponseinmagneticeldsofthetwosamples.Thepossibilityofthethirdorderphasetransitionremainsspeculativesincefewmaterialswerereportedtoexhibitphasetransitionswithanorderhigherthan2[ 109 { 111 ]. 112 PAGE 113 X-raydiractionpatternsofPr1xLaxOs4Sb12versusLacontentxforx=0,0.1,0.2,0.4,and1.Theintensitiesarenormalizedtothehighestpeak. 113 PAGE 114 LatticeconstantaofPr1xLaxOs4Sb12versusLacontentx.Thesolidlineisalinearleast-squaresttoaversusx. Figure7-3. Magneticsusceptibility(T)ofPr1xLaxOs4Sb12normalizedtoPrmolebetween1.8and10K,measuredintheeldof0.5T.Themeasurementswereperformedonindividualcrystalswithmassesrangingbetween1and5mg.Largeuncertaintyareduetolargebackground(ReprintedwithpermissionfromRotunduetal.[ 112 ]). 114 PAGE 115 Magneticsusceptibility(T)ofPr0:33La0:67OsSb12versustemperatureT.IntheinsetistheCurie-Weisstofthehightemperaturedata(T>150K)fromwhichaneectivemomentof3.62B/Prmolehasbeencalculated. 115 PAGE 116 116 PAGE 117 117 PAGE 118 112 ]). 118 PAGE 119 112 ]). 119 PAGE 120 SuperconductingtransitiontemperatureTcversusxofPr1xLaxOs4Sb12(ReprintedwithpermissionfromRotunduetal.[ 112 ]). Figure7-10. TotalC=TdiscontinuityatTcand0versusxofPr1xLaxOs4Sb12for0x1(ReprintedwithpermissionfromRotunduetal.[ 112 ]). 120 PAGE 121 Figure7-12. 121 PAGE 122 Figure7-14. 122 PAGE 123 113 ]). Figure7-16. 113 ]). 123 PAGE 124 Figure7-18. 124 PAGE 125 Figure7-20. 125 PAGE 126 114 ]). 126 PAGE 127 114 ]). 127 PAGE 128 128 PAGE 129 Figure7-25. 129 PAGE 130 114 ]). 130 PAGE 131 ACsusceptibilityversusT=TcofPr1xLaxOs4Sb12,forx=0,0.05,0.4,0.8,and1.SampleNo.1exhibitonesuperconductingtransitionwhilesampleNo.2havetwo(ReprintedwithpermissionfromRotunduetal.[ 115 ]). 131 PAGE 132 115 ]). 132 PAGE 133 115 ]). Figure7-30. 115 ]). 133 PAGE 134 115 ]). Figure7-32. 115 ]). 134 PAGE 135 115 ]). 135 PAGE 136 ThischaptersummarizestheresultsofthermodynamicmeasurementsonPr1xLaxOs4Sb12(0x1)asafunctionoflanthanumconcentration(x),temperature,andmagneticeld.Specicheatmeasurementsineldsbetween8and32TofPrOs4Sb12extendedthepreviouslymeasuredH-Tphasediagramupto8T[ 56 ].TheSchottkyanomaly,duetoexcitationsbetweentwolowestcrystallineelectriceld(CEF)levels,wasfoundforboth~H//(100)and~H//(110)abovetheeldwheretheeld-inducedorderedphase(FIOP)(identiedwithanantiferroquadrupolarorderedphase[ 56 ])iscompletelysuppressed.TheH-TphasediagramshowsweakmagneticanisotropyandimpliesacrossingofthetwoCEFlevelsatabout9{10Tforbothelddirections.CalculationsoftheZeemaneectinthe1CEFgroundstatescenariopredictacrossingbetween1andthelowest5energylevel,betweenbetween9and10T,whichisalmostindependentontheelddirectionSimilarcalculationsforthe3CEFgroundstatemodelpredictbothstronganisotropyofthephasediagramandnocrossingforthe(110)direction.Thus,thisworkhasestablishedthenon-magnetic1singletbeingtheCEFgroundstate.Furthermore,ourinvestigationoftheeld-inducedorderedphase(FIOP)hasprovidedevidencesforthe(near)levelcrossingasthedrivingmechanismofFIOP.Thenon-magneticsinglet1CEFgroundstatecontradictstheideaofaquadrupolarKondoeect,atleastinthepresentformulation,astheoriginoftheheavyfermionbehaviorinPrOs4Sb12. TheLa-alloystudywasperformedtoprovideinsightontheoriginoftheelectronicmassenhancement.ZeroeldspecicheatofPr1xLaxOs4Sb12showedthatthetotal(C=T)atTcisreducedmorethansevenfoldfromabout800mJ/K2molbetweenx=0and0.3andstaysapproximatelyconstantandaboutequaltothatofLaOs4Sb12,whichisaconventionalsuperconductor,forx>0.3.Similarly,measurementsoftheuppercriticaleldsuggestedtheexistenceofacrossoverconcentration,xcr0.2{0.3.TheuppercriticaleldslopenearTcdecreasesrapidlywithxforx PAGE 137 Inordertoverifythatsingle-ionparameters,suchashybridizationandCEFspectrum(consideredbysingleimpuritymodels),arenotseverelyaectedthealloying,measurementsofthelatticeconstantandhigheldspecicheatwereperformed.X-raypowderdiractionofPr1xLaxOs4Sb12revealedananomalouslysmallincreaseoflatticeconstantwithx(0.04%betweentheendcompounds).Thelow-temperaturemagneticsusceptibilityshowedalmostnonexistentconcentrationdependenceofthelowtemperaturemaximum,believedtobeduetoexcitationsbetweenthelowestCEFlevels.Specicheatinmagneticeldsupto14Tforx=0,0.02,0.1,and0.2showedthatthetemperatureoftheSchottkyanomalyhasasimilarelddependenceforalltheseconcentrations.Therefore,CEFenergiesandeigenstatesofPrareunchangedbetweenx=0andatleast0.2,i.e.,inthealloyparameterrangewherealargechangeoftheelectroneectivemassisobserved.Theseresultsreinforceourconclusionofanon-singleimpurityoriginoftheheavyfermionbehaviorofPrOs4Sb12.Inparticular,theyareinconsistentwiththecurrentlyprevailingFulde-Jensenmodel. 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