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DOPING EXPERIMENTS ON MAGNETIC HEAVY FERMION SUPERCONDUCTORS By WEONWOO KIM 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 ACKNOWLEDGMENTS I wish to express my sincere appreciation to Professor Gregory R. Stewart not only for his stimulating suggestions and valuable advice throughout the entire research of this work, but also for his patient encouragement when I had a hard time in my course work and research. In addition, I am very thankful not only to Professor P Kumar for his delightful clear discussions comments in physics, to Professors Takano, Sullivan, Ingersent, and T.J. Anderson for their valuable advice and suggestions. Special thanks go to Dr. Dr. J. B. Andraka for his Kim for his help in experiments, R. Pietri, a valuable discussions and cooperation, nd S. Thomas for his proofreading of the manuscript as well as pleasant relationship. The faculty, postdoctoral fellows, administrative, and technical support from the machine shop, cryogenics, and electronic shop have contributed a great deal to this work. Finally, I dedicate my warm love and gratitude to my family including my wife HeaKyung for her consistent trust and encouragement, two sons DongHwa and Dong In, and relatives in Korea for their support and prayer. HAPPY THE PERSON WHO REMAINS FAITHFUL UNDER TRIALS, BECA USE WHEN HE SUCCEEDS IN PASSING SUCH A TEST, HE WILL RECEIVE AS HIS REWARD THE LIFE WHICH GOD HAS PROMISED TO THOSE WHO LOVE HIM. (James 1:1 TABLE OF CONTENTS pages ACKKNOWLEDGMENTS......... ............................................... LIST OF TABLES ...........................................................................................................v LIST OF FIGURES .............. .......................................................................................... ABSTRACT ...................................................................................................................ix CHAPTERS INTRODUCTION ..................................................................................................... 1 Overview of Heavy Fermion Systems.. 1.1.1 1.1.2 1.1.3 1.1.4 Classification of Heavy Hermion Systems....... M ass Enhancement ............... ................... ....... Superconductivity ............................................ M agnetism ............... .................................... ...... 1 ......2 .....5 ........6 M motivation ................... ................. .......... Experimental Techniques ........................... 1.3.1 1.3.2 Sample Preparation........... Measurement Techniques.. Fermi Liquid Theory......... NonFermi Liquid Theory. 2.2.1 2.2.2 * *. *. *.. .. .... . . ... *. *..o 4 4 .4 *... .. .. Multichannel Kondo Effect...... Phenomenological Theory of a Critical Point at T = O K................. .....38 NONFERMI LIQUID BEHAVIOR OF MAGNETIC HEAVY FERMION 1T TPFPR rfNTT TTTTOR 5 An I(ltl)lllt)((l(llltl11(111(1111111111111 )(11111)1)1)()11(111(111(1)1((1111111111 ...................6 REVIEW OF THEORY. ................... ................... ................... ......... r 3.2.3 NonFermi Liquid Behavior of UxThl.xRu2Si2 for x < 0.07..................49 3.2.4 NonFermi Liquid Behavior of UxThl.xRu2Si2 for 0.07 3.3 Magnetism and Its Connection to NonFermi Liquid Behavior in U i.xM xNi2Al3 {M = Th, Pr, and Y } ............................................ 3.3.1 UNi2Al3 ................ 3.3.2 NonFermi Liquid Behavior in UlxMxNi2Al3 (M = Th, Pr, and Y}... 4.1.1 4.1.2 M magnetic Properties ............................................................................95 4.1.3 Concentration Determ nation ...............................................................96 4.1.4 Isotope Effects........... 4.1.5 PressureComposition Isotherms ................... ................... ................... 97 4.2 Overview of UPd2A3 ........ 4..1 oma tae ......... ........... ........... .S.. 4.3 Experimental Apparatus........... 4.4 Sample Preparation .................. *4.**4* *4 *4**S. *4*** ** '........*4***. *4*. ** *4*44*4 .... *4*. *** *4*. *4** 99 S 4*'**~~* *4**t4~4 *44~ 4~~**~* *4* *4 *******~ .. . .. .. . . 4*4**44 *4~ ~ *4 *4*4**4* 4**~** *4**** *. 4***~*~~ ~4*** 4 e.4 1 c1'ry t lln Saii le. 494**4*4**..... 44 *9444*4**4 4.5* 4 44. *4 *4 4** 49**. 4..... ....... ***11 4.4.3 Pasvto c1 Samxpile. 44...#**.... ... *444*....*..4.. 4444*4**44* .**.. 4.***4..*.. 1 1 J. 4.5 Hydrogen Effects on UPd2A13 ............. 4.5.1 Hydrogen Effects on Uranium..... * *.. .... . .. 4. ..... ... .. . . . 4.5.2 Hydrogen Effects on UPd2A13Hx ..... ...................... .................... 1 22 SUMMARY AND CONCLUSION. .............. .................. ................................ .....148 UxThlxRu2Si2 for x 0. 17........ 5.2 U1xMxNi2Al3 {M = 5.3 UPd2AI3Hx ............. APPENDIX Th, Pr, and Y) PRELIMINARY RESULTS OF UNi2AI3Hx, CeCu2.2Si2Hx, AND UPt3Hx 4* *494.4 15e IR.IFEIR.J.N1I ES.. *4.4.**4_.. *4 .** 4* .*. *4*4*4** 4 *.4.4.*.4.. 9.***444**44*b. .. ... *4*94 4* 44.**4*4**. ** 6 fln~flCZDur Ani Al CVPTC'T 177 ........................................ HYDROGEN DOPING EF~CTS ON UPd2A13 ........................................ 4. 1 Review of Hydrides. ................... ................... ................... ....... Electronic Properties ................... ................... ................... .......... LIST OF TABLES Tables The physical quantities of two channel spin 1/2 and electric quadrupolar (pseudospin 1/2) K ondo effect ............. ....... ............ . ..... . . ....... . . . Kondo temperature TK in UxThi.xRu2Si2 for 0.07 .... 38 x 0.17 .........................5 5 Physical parameters for Uo.9Mo. 1Ni2Al3 (M = Ce, Th, La, Y Physical relations between the U and the doped elements .............................. 73 Parameters for hydrogen doped UPd2A3Hx ................... ............... ......... . 127 Panes and Pr)..............73 LIST OF FIGURES Pages Classification of heavy fermion and intermediate valence compounds..............4 Diagram of heat flow in the system used for thermal relaxation method......... 13 The sample platform and copperring heat reservoir. ..................................... 14 Block diagram of experimental setup for the specific heat measurements....... 16 Specific heat as a function of T/TK in constant magnetic field and n for S 1/2 and S ...........32 Magnetic susceptibility as a function of T/TK in constant magnetic field and n 1 forS ................33 Entropy, specific heat, and specific heat divided by temperature as a function of T/TK in constant magnetic field for S = 1/2 andn = 2................................ 34 Crystal and spin structure of antiferromagnetic superconductor URu2Si2.......46 Magnetic susceptibility of UxThi.xRuzSi2 for x Low temperature resistance normalized to the roomtemperature value R, of Uo. 7Tho.g3Ru2Si2 ............ The anomaly TA in terms of concentration x. ................... ................... ........... 58 Csd/Tas a function of In Tin UxThi.xRu2Si2 for 0.07 < x < 0.17....................59 C/T vs. T2 for Uo.1Tho.9Ru2Si2 in two different magnetic fields.......................60 rlao;fot.;inn nfnlrnnn arrAtam h17 T.71 Q To.nA.. ^ 9 99 9* *9 95 7~ FiguT~ 3/2 3/2 ~ 0.07...................................5 C/T vs. T for UNi2AI3 between 0 lc and 9 K.. ............,.,..............,....... ...*..'.7A0 x(7) between 1.8 K and 300 K for four samples of UNi2Al ..........................75 C/T vs. T2 for four samples of UNi2zA3 and Uo.9Ceoa.Ni2Al3........................... 76 C/T vs. T2 in Uo.9Mo.1Ni2A3 for M 316 = La, Th, Pr, and C5sT vs. In Tfor Uo.9Mo.iNi2A13 (M Cs'/T vs. In T for U.xThxNi2A13 (x C/T vs. T2 for UIxThxNi2Al3 (x = =0.01 0.015 ,0.2, and 0.3) .............. ......... 79 ,0.2, X vs. T72 for Uo.9Mo.0Ni2Al3 (M 320 321 X vs. p as a function of temperature in U0.9Mo.iNi2A13 (M = Pr, Th, and Y)............84 C/T vs. T for Ui.xThxNi2Al3 (x = 0.0, 0.0025, 0.005, and 0.015).................86 323 y(7) at low temperatures in Ul.xThxNi2Al3 (x = 0.0025 and 0.005)...............87 324 41 42 C/Tvs. T for Uo.9Tho.Ni2Al3 in two different magnetic fields........................89 Two pressurecomposition isotherms of a hydride .........................................98 Hexagonal PrNi2AI3 crystal structure observed in UPd2AI3.......................... 101 p(7) and x(7) in UPd2A13 below T =25 K and up to T vs. TofUPd2AI3 H = 0 T and in several magnetic fields....................... 104 Y. ................... .............77 T dependence of Xde as measured for UNi2A13 ................... ................... .........68 p as a fUnction of temperature for UNi2A13................................. Pr, Th, La, and Y) ................... ........._78 and 0.3)...........................80 Pr, Th, and Y) ................... ................... ....82 In T for U0.9Y0.1Ni2A13  83 300 K. ..................1 02   X vs. T of UPd2Al3Hx for 0 Magnetization M vs. magnetic field H for UPd2Al3Hx for = 0 and 1.30.....128 410 Lowtemperature C/T vs. T of UPd2Al3Hx for 0 y( C/T as T 0) vs. X in a InIn plot for a number of heavy systems.......... 132 Schematic model of XPS for UPd2Al3Hx .................................................... 133 411 412 413 414 CurieWeiss X(as T temperature 0 c.w. vs. concentration x for UPd2Al3Hx............136 415 416 C/TasT X vs.  0) vs. temperature T for UPd2Al3Ho.64 in magnetic field ................................ 141 417 Xvs. 4T of UPd2AI3Hx for x 1.09 and I 418 419 AC/Tvs. In T of UPd2A13Hx for Specific heat of UPd2AI3Hx as C/Tvs. Tat the superconducting state.......... 145 C/T vs. T of UNi2Al3Hx (x = 0.0, 0.23, x(7) of UNi2Al3Ho.30 in different magnetic fields.......................................... 157 X(T) of CeCu2. Si2Hx for x M(H) of CeCu2.2Si2Hx at T = 0.0 and 0.07 ........... .... ....... .. ........... .......... 160 = 1.8 K for x X of the polycrystalline sample of UPt3 as a function of temperature............ 162 x(7) of UPt3Hx for x = 0.0 and 0.17 (both systems are pellets).................... 164 C/T vs. T of UPt3Hx for x = 0.0 and 0.17 (both systems are pellets)............165  0) vs. x for UPd2AI3Hx ................... ................... ................... ....... 138 .30. ................... ..........__.__ ...... x<1.3 0. ........._.._._. ................... ..................1 26 x11.30. ................... .....129 x for UPd2AIJ)Ix. ................... ................... .........., 139 1.09 and 1.30. ................... ................... 143 0.3 0, and 0.3 9), ................... ........._155 0.0 and 0.07.............................161 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 DOPING EXPERIMENTS ON MAGNETIC HEAVY FERMION SUPERCONDUCTORS By Weonwoo Kim August, 1995 Chairman: Gregory R. Stewart Major Department: Physics We present experimental results of nonFermi liquid (NFL) behavior in UxTh~x Ru2Si2 and in Uo.9Mo.1Ni2Al3 (M Y, Th, and Pr), and the hydrogen effects on UPd2Al3. The objective of this work is to understand the origin of the NFL behavior and how hydrogen changes the heavy fermion system. In UxThixRu2Si2, we have observed NFL behavior thermodynamic, transport, and magnetic measurements for 0.07 x<0.17. We report a temperature TA in electrical resistivity p measurements which measures the onset of intersite correlations between local magnetic moments. The increase of p with T for 0.3 K 20K, the approximately linear decrease of TA for 0.1 0.14, and the spin glass behavior in dec Ixl may be due to magnetic fluctuations at T = 0 K or collective modes of excitations at low temperatures rather than the singlesite two channel Kondo effect (TCKE). We have found another new system Uo.9Mo.0Ni2AI3 (M = Th, Pr, and Y) which exhibits the NFL behavior. We observe a positive linear p with T forl.0 K < 20 K in Ydoped samples (less wide range in T for Prdoped sample). susceptibility Xdc, T'l (Th and Pr doped) and In T (Y doped) dependence are observed as a sign of NFL behavior. The In T dependence of Cs/T as an NFL characteristic following the destruction of the magnetic order as observed in C measurements, seemed to imply NFL behavior may be due to the proximity of magnetic instability where a Kondocompensated nonmagnetic state and long range magnetic order are nearly degenerate. We have succeeded in doping UPd2Al3 with hydrogen by improving the activation and poisoning of the parent samples. In the normal state, the increase in both X(0) and y(0) with hydrogen uptake can be interpreted as an enhancement of the effective mass supported by the Sommerfeld freeelectron behavior of these physical values On the other hand, magnetic correlations are proposed as a possible origin of the behavior in both X(0) and y(0) based on the observed relations among 0 c.w., critical hydrogen concentration Cc. X(0), and y(0) around the The peak in X for UPd2Al3 seems to be due to the short range magnetic correlations; we observe the shift of the peak to lower T for more hydrogen uptake and at higher magnetic fields for a given hydrogen concentration. In the CHAPTER 1 INTRODUCTION In this chapter, we will summarize heavy fermion systems based on not only a coupling constant between local moment and conduction electron at each lattice site, but also essential topics in this area while keeping in mind the magnetic and superconducting ground states. Our motivation to study nonFermi liquid behavior by alloying experiments and to perform hydrogen doping experiments on these HFS will be given below. describe experimental techniques used to get our data in our laboratory. We also We are going to explain separately experimental techniques apparatus hydrogen doping experiments in chapter 4. Overview of Heavy Fermion Systems Heavy fermion systems (HFS), which are usually intermetallic compounds certain lanthanides or rareearth elements (notably Ce 4f oneelectron and Yb 4f onehole) and actinides (notably U, Pu, and Np), have attracted physicists for more than a decade because of their fascinating physical properties [Stewart, 1984; Lee et al., 1986; Ott and Fisk, 1987; Fulde et al., 1988; Grewe Steglich, 1991]. highly correlated electronic systems of HFS can be regarded as a lattice offions being embedded in a Fermi Although both the Kondo effect [Kondo, 1964] occurnng at each lattice site and the strong lowlying magnetic fluctuations seem to play a central role, there is no accepted microscopic theory either mass renormalization or the pairing mechanism. However, we are briefly going to describe the main topics in HFS so that we may reduce any conceptual gap to understand following chapters. 1.1.1 Classification ofHeaw Fermion Systems Each of these HFS can be characterized by a Kondolattice temperature, T (this temperature corresponds to the Kondo temperature TK for an isolated Kondo impurity). For T , fions carry local felectroninduced magnetic moments which are coupled very weakly to the conduction electrons. , however, this coupling becomes strong enough to give rise to a nonmagnetic low temperature state [Grewe and Steglich, 1991]. ground states HFS lanthanide depend heavily on the competition between the magnetic intersite interaction, i.e., the indirect RudermanKittel KasuyaYoshida (RKKY) exchange interaction, onsite Kondo interaction reducing the local magnetic moments [Doniach, 1977]. If we use the dimensionless coupling constant, g = NF we can express two energies (kI = 1) as T* ~ exp{1/g) TpxxY ~g2 . NF is the conductionband density of states at the Fermi energy EF, and J is the exchange coupling constant between the local fspin and the conductionelectron S_1L rr.f A /lr f nArn ,*nn+ *ninrn ran*A n n4 j. 4k al ,1 1a a..j^ an at ap a1 1 ~nin nnTn r nn Fniini~nn nnnn+nn, the more localized 4f electrons. Thus, two adjacent 5f configurations (5f2 and 5f ) may be nearly degenerate, and one single parameter g may not be sufficient to classify the various ground states of the HFS with 5fions. In Figure 11, we show a schematic classification of HFS and intermediate valence (IV) compounds with 4fions. rareearth magnetism, and for 1, there is a stable moment regime of an ordinary g > 1 there is a charge fluctuation between the fshell and the conduction band exhibiting a nonmagnetic Fermiliquid ground state. HFS exist between above regimes, for 0 < 1. 4f heavyfermion compounds, valence fluctuations are usually ignored so that local spin fluctuations, which have essentially the same origin as in dilute Kondo alloys, exist for the weaker local exchange coupling. Thus, HFS characterize both 'magnetic limit' compounds and the 'concentrated limit' of the Kondo alloys. There also exists a critical value where the onsite and the intersite interactions have the same strength [Steglich et al., 1994]. 1, the strong hybridization between 4f and conduction electrons fully compensates the local fderived moments well below T* resulting in Fermi liquid behavior. However, these Fermi liquids appear to be unstable against AFM and/or superconducting phase transitions at TN/Tc << T* Also there is a very interesting regime around g=gc, i.e., at the magnetic instability, where it seems that nonFermi liquid (NFL) behavior takes place [Kim J.S. et al., 1992; Kim W.W. et al., 1993; Andraka and Stewart, 1993; LOhneysen et al., 1994; Steglich et al., 1994]. g << STABLE MOMENT REGIME <<1) HEAVY FERMION heavy fermior 'local moment I magnet'" (LMM) e.g, Ce Al2! metal LA e.g. I C "band e.g., Ce supercc e.g. ,Ce "NFL EGIME (g<1) INTERMEDIATE 1 metal VALENCE REGIME (g >1) Fermi liquid eCu6 e.g., CePd3 magnet" '(Cu,Ni)2 Ge2 ,nductor Cu2 Si2 "Kondo insulator" e.g.,Ce3 Pt3Bi1 1 g=NFIJI RE 1.1.2 Mass Enhancement The origin of heavy quasiparticles may be due to either a single electron interaction or a many body resonance (we already know that we can not simply apply the theory of a single electron interaction, because we can not understand the development of the large effective mass quasiparticles at low temperatures simply based on the single electron localized theory). T felectrons he single electron light interaction conduction (itinerant) coupling electrons. between This initially well coupling hybridization makes the localized felectrons delocalized, and the felectrons contribute to the Fermi level with a low Fermi velocity. A large part of the mass enhancement is obtained by local magnetic fluctuations reminiscent of the single impurity Kondo effect. This large effective mass is observed by a large coefficient Yo of the electronic specific heat forT > 0 K, a large 'Paulilike' spin susceptibility Xo, and a large coefficient from the Fermi liquid behavior in resistivity, i.e., p = CT2 (C ~ Yo2). Despite these large physical values, the SommerfeldWilson ratio X/'Yo with proper normalization is of the same order as in simple metals. same time, there exists an intersite magnetic coupling which comparable in magnitude to the local Kondo temperature TK, between magnetic moments induced by the partially filled felectron shells. One of the consequences of this interplay is that heavy fermion compounds exist near magnetic instability, i.e., around the transition 1.1.3 Superconductivity possible origin of the superconducting mechanism HFS can be a large magnetic fluctuating medium which furnishes an attractive potential form quasiparticle pairs. Since the quasiparticles exist for < TF/10O, in the Fermi liquid region, such a magnetic fluctuating medium observed in x(q, o) can be the origin of the SC. Here, the parity of the pairing depends on the nature of the magnetic spin fluctuations, e.g., ferromagnetic spin fluctuations (for odd parity) and antiferromagnetic spin fluctuations (for even parity). Thus, we can change the pairing strength by changing the hydrostatic pressure. For instance, the Grainesien parameter U is given by (1.1) where V is a volume. Under pressure, TF increases causing a decrease of the pairing strength which gives rise to a decrease of Tc (C strength of the quasiparticles pairing). '~ , where is related to the Another possibility is to apply a magnetic field to change the pairing potential, e.g., UBel3 [Brison et al. 1988]. 1.1.4 Magnetism About the origin of magnetism in HFS, it seems to be basically similar to a general magnetism which has been constructed on the Heisenberg model [Heisenberg, 1928] and the Stoner model [Stoner, 1938]. The first model describes magnetism in terms of local magnetic moments, and the other one describes magnetism in terms of itinerant electrons "r12 a~ogT, / dlog V 7 There has been a proposed picture [Doniach, 1977; Nozi&res, 1985] for a magnetic state with well localized moments and only moderate enhancement of y, i.e., of order 0.1 J K2mole' as in CeAI2. Starting from band electrons and ionic fstates, there is an end to the quasiparticle renormalization when the spin order of renormalized moments occurs. corresponding RKKY interactions energy scale TpjcYy are mediated renormalized band electrons. From then on, spinflip scattering is frozen out and the band electrons move in the fixed modulated exchange field provided by the magnetic moments. For instance, TRKKY exceeds TM by more than an order of magnitude for CeAI2 [Steglich et al., 1979b] (the long range magnetic ordering temperature, TM, is usually responsible for an antiferromagnetic ordering, but it can be a ferromagnetic ordering one, e.g., CeCu2 [Gratz et al., 1985]). intermediate yvalues above point moderate renormalization for band electrons well above the Kondo lattice temperature T < TM TRKKy. Yo is usually lower due to the reduced density of excitations in the magnetically ordered phase. The ordered moments are correspondingly smaller than the freeion values in the proper CEF ground state. Even though the density of states (DOS) near the Fermi energy EF changes due to an exchangesplit incipient AbrikosovSuhl resonance (ASR) [Abrikosov, Fermi surface 1965; Suhl, 1965], no electronic excitation gap develops anywhere on the In general, the specific heat well below TN follows C = yT + 3T3 with a cubic magnon contribution. the other hand, the quasiparticle band theory can describe heavy fermion moments, have been transferred to the itinerant heavy quasiparticles. Thus, this system shows a strongly reduced magnetic moment and a large yvalue in the electronic specific heat. In particular, a spin density wave (SDW) instability [Doniach, 1987] may open an electronic excitation gap on parts of the Fermi surface, a situation strongly favored by the nesting properties. In that case, the density of quasiparticle excitations is reduced at low temperatures and an exponential decrease of y(T) can take place below for Ut xThxPt3 [Ramirez et al. 1986; Stewart et al. 1986]. However, most of HFS are typically in a crossover regime between the class of local moment systems and of itinerant systems. Also, when both superconducting and antiferromagnetic ground states coexist, there is a larger coupling between magnetic and superconducting order parameters in the local moment system than in the itinerant system because of the its larger ordered moment. Before we close this section, we would like to classify a large variety of ground states in the HFS based on superconductivity (SC) and magnetism. The first group has a nonmagnetic ground state, i.e., paramagnetism without SC, CeCu6 [Stewart et a., 1984b]. 1985]. The second group has a magnetic ground state without SC, The third group has an SC ground state without magnetism, e.g., e.g., UCu5 [Ott et al., UBel3 [Ott et al., 1984]. The fourth group has SC and magnetism in its ground state, and we can divide the fourth group into two subgroups: UPd2Al3 [Geibel et al., One is the system where the two ground states coexist, 1991b], the other is the system where the two ground states Motivation We can change external physical parameters such as temperature and pressure to study a system more thoroughly. Alloying experiments with different elements (which have different electronic configurations, atomic sizes, and local magnetic moments) are also a very interesting and effective method in gaining more understanding and insight into condensed matter physics. In principle, alloying can vary the band structure and the chemical potential causing changes of DOS at EF. understand HFS, It is crucial to vary the DOS at EF to because the DOS at EF in HFS (the DOS is proportional to y from specific measurements) one of the most important topics to be understood. Alloying also changes the lattice constants inducing a variation in hybridization. It causes a change in the interaction mechanism, and sometimes even induces a phase transition. Since NFL behavior studies have been a newly developing field [Seaman et al., 1991; Andraka and Tsvelik, 1991], we have focused on the magnetic HF superconductors (which would have a relatively small ordered moment below antiferromagnetic transition) out of the family of HFS to find new NFL systems, because as we described briefly in section 1.1.1, the magnetic instability can be the origin of NFL behavior. We will discuss this in detail in chapter 3. As we all agree, one of the unavoidable results of alloying experiments is to disturb the occupation of the sublattice. Physicists have an idea to overcome such a disadvantage iitnIrmr. no rnfto+r +bo 1 1 l nii l A~ .a.. ''V tttIIII*It**rI IY;I *ns**uz; **tnl* **f **'.:n E1u\1tt **rnEIfl *lll IIRtI**tI u*IuI** II. r*.*III uIr~nlt (mn 44 4J ~~*1 ft mnCC\nrl I,,~tb nl nm Anlr. 10 system unchanged, even at a relatively high doping concentration, (which is an essential point for a systematic study) because hydrogen atoms can go into the lattice interstitially as a proton (which is in size several orders smaller than the atom) by donating its electron to the Fermi level. Even though hydrogen atoms get extra electrons from the Fermi level in some cases, the effect on the crystalline structure is still smaller than other usual doping elements. However, there are also very difficult problems, which we will describe in detail in section 4.4, that we need to overcome in order to enjoy the advantages mentioned above. After we studied several hydrided systems (usually for hydrogen storage), the hexagonal crystalline structure of LaNi5 (one of most studied systems and a possible candidate for hydrogen storage) stimulated our research on the HFS hydrides. Hydrogen doping experiments can not only give more information about the parent system, but also induce a new ground state in the hydrided system as will be discussed in chapter 4. 1.3 Experimental Techniques We are going to describe in this section briefly how we made our samples and what experimental techniques we used. There is already a very detailed description about this in J.S. Kim' dissertation [1992]. However, we will describe in detail how we made our hydrided samples and what apparatus we used in the hydriding system in sections 4.3 and 4.4. 1.3.1 Sample Preparation synthesized each polycrystalline sample stoichiometric amounts elements within 0.3 % accuracy using a Bihler arc melter. Each sample was melted several times followed by flipping each time to increase the homogeneity of the sample. We especially have kept some airsensitive elements, e.g ., some transition elements and all lanthanides, in sealed Pyrex tubes to have the highest quality samples. a pressure casting technique [Crow and Sweedler, We sometimes used 1973] in the arc melter to make some resistivity samples If necessary, we sealed the samples in evacuated quartz tubes and annealed them in a temperaturecontrolled furnace for a few days or a few weeks. we characterized the samples using a Philips Xray diffractometer to Then, see if there is any second phase and to determine the lattice constants. Measurement Techniaues Measurements of resistivity and specific heat down performed in a to about 4He cryostat by pumping the dewar or an inner temperature measurements for resistivity, specific heat, K have been 4He pot. Even lower ac magnetic susceptibility were done by using 3He cryostat with a movable charcoal rod, which absorbs and releases 3He depending on the temperature and pressure (boiling point of 3He is 3 .2 Kat bar). For the probe used, refer to J.S. dissertation [1992] for a schematic drawing. necessary, magnetic fields T for these experiments were used thermndvnamir meaRuhirementq Kim's 1.3.2.1 Specific heat Specific heat measurements can be classified into two groups, i.e., adiabatic and diabetic methods, depending on the isolation system from its surroundings. Adiabatic methods attempt to isolate the sample system completely from heat sources or heat leaks so that a sample with an internal heat source, e.g., radioactive material is not suitable for this method. On the other hand, diabetic methods in principle accept the fact that no sample can be completely isolated from its environment. The thermal conductance of the main links between the sample system and its surroundings (or heat reservoir) is measured and used to get the specific heat, and this is in contrast to a heat switch used in adiabatic methods in order to stop heat flowing. Thus, we may regard adiabatic methods as special cases of more general diabetic methods, because we can regard the thermal conductance in adiabatic methods as becoming zero. Diabatic methods improve precision of specific heat measurements while they lose accuracy relative to the adiabatic methods. We used a thermal relaxation (or time constant) method [Bachmann et al. Stewart, 1983] which is one of the diabetic methods. small sample calorimetry in HFS. 1972; This method is especially good for In HFS, samples can be as small as 1 mg because of the large specific heat at low temperatures resulting in a much smaller ratio of the addenda contribution to the total specific heat than low yvalue samples. As we show in Figures 12 and 13, after we are in equilibrium temperature To, we 1 +/11 K= AT o C Total C Sample+ C Mdenda Solder uCu W 3/8" Sc Sample Copper Ring Epoxy Pad Evaporated 7% TiCr Heater H31LV Silver Epoxy AuCu Wires Ge Chip 3/8" Sapphire Thermormeter 15 the temperature of sample and platform decays exponentially with a time constant Ti (a measure of the time for the sample and platform to reach thermal equilibrium) which is inversely proportional to the conductance of the wire links. We can express the relation as follow: + ATexp(t/ t, (1.2) The conductance K(T) of the wire links should be measured and calibrated vs. temperature for the later specific heat measurements using a relation K = P/AT. We use Ko, the K value at To + AT/2, to get the specific heat C the addenda contributions. = Ko'i at To + AT/2 followed by subtracting Here, the thermal conductance of sample and platform should be much larger than that of the wire links. Otherwise, there should be some corrections due to the socalled x2 effect [Bachmann et al., 1972] to get a correct value of the specific heat. We would like to describe briefly how to measure specific heat using Figure We set the frequency of an ac excitation in the lockin amplifier (LIA) around fo = 2700 Hz such that it is not the multiple of 60 Hz to avoid any unwanted noise. This frequency setup in the LIA increases the signaltonoise ratio by filtering out frequencies other than fo with a width of frequencies. We then set up the phase to get the maximum sensitivity of the platformthermometer resistance and thereby a best timeconstant value. We achieve this by firstly finding a phase, where the LIA voltmeter shows no response for varying the a C S  n Wheatstone Bridge 1 Platform Platform Block Block thermometer Heater Heater Thermometer The block temperature is controlled by a block heater and measured by a block thermometer at To. We apply power P increasing the temperature of the platformsample assembly to + AT (AT/T, The resistance of the platform thermometer is measure by an ac Wheatstone bridge, and the resistance value is used to get the new platform temperature based on the precalibrated data. After we are at a stable temperature, we turn off the platform heater. We get the time constant tr by digitizing the signal from the Wheatstone bridge using an analogdigital converter and subsequent signal average. With the Kvalue at this temperature, we have the specific heat after correcting addenda contribution. Overall absolute accuracy our measurements approximately within detailed descriptions, see the J. Kim' dissertation [1992]. used Gethermometer block thermometer, a home made evaporated Gealloy was used for ac calorimetry to the platform thermometry (see J. Kim's dissertation [1992] for detailed process). When we measured the specific heat in magnetic fields up to 14 T (we can go up to 16 T if we lower the temperature of magnet below K by pumping the Xplate), we used Speer carbonresistor thermometer for the blocktemperature measurements down to 0.3 K. We calibrated it in zero magnetic field with the block Gethermometer, then did a field correction according to Naughton et al. [1983] We also used a capacitive thermometer above 1 K as a thermometer in magnetic '* 4 4I b r r rr+ r r L .I I II nnri in .t 18 the platform thermometer based on the precalibrated thermometers in magnetic field, and subsequently corrections to the thermal conductance with the help of these two (block and platform) fieldcalibrated thermometers in order to get the right specific heat were made. 1.3.2.2 Magnetic Measurements For the ac magnetic susceptibility, we used a low frequency (around 86 Hz) mutual inductance technique using homemade primary (superconducting wire) and secondary (Cu wire) coils [J.S. Kim, 1992]. After we put a specimen into the primary coil, applied an ac voltage using a lockin amplifier (EG & G Model 124 A) to the coil. signal detected by the secondary coil is amplified by the lockin and gives a voltage difference due to the magnetic flux change in time, d opposite directions there electromagnetic force unintended fluctuations of the applied voltage or field.). This voltage difference is proportional to the magnetization, and therefore the magnetic susceptibility [Signore, 1995]. We used this technique to measure the superconducting transition temperatures of our HFS for 0.3 K For dc magnetic susceptibility, device (SQUID) from Quantum Design for 1.7 K T in two opposite directions. we used a superconducting quantum interference 400 K in magnetic fields up to The basic idea of the dc measurement [Roseinnes and Rhoderick, 1980] is similar to that of the ac method. We move the specimen in the steady magnetic field applied a dc powered solenoid (corresponding to the primary the secondary coil detects the signal with a high sensitivity due to the SQUID. We also measured magnetization vs. magnetic field at a given temperature. 1.3.2.3 Electrical Resistivity used a standard probe technique measure resistance specimen for 0.3 K <70 K. If necessary, we measured the dimension of the specimen to get resistivity. All measurements are automatically performed with computer controlled equipment and data collection [J.S. Kim, 1992]. CHAPTER REVIEW OF THEORY We are going to review briefly the basic concepts and the physical quantities of the Fermi liquid (FL) theory as a function of temperature, because it explains relatively well the phenomena in heavy fermion systems. However, unusual behavior, which can not be understood by the FL theory, has been observed in recent years. There have been very extensive theoretical developments both in a microscopic, e.g., the exact Bethe ansatz and conformal field theory, and in a phenomenological theory to understand the origin of the new behavior. We will concentrate on the physical quantities for which unusual behavior has been observed in, e.g., thermodynamics, transport, and magnetic susceptibility. There is no conclusive theory explaining all of the new phenomena. 2.1 Fermi Liquid Theory The model of noninteracting fermions or a Fermi gas has worked relatively well in a system of Fermi particles, even though the interaction among the fermions is rather strong. Electrons in a metal serve as a classic example. The reason why it does so well is that the scattering rate of electrons is dramatically reduced because of the Pauli exclusion ' : 1, assume a noninteracting quasiparticle model as a good approximation and quasiparticles obey the exclusion principle. He proposed a semiphenomenological method [Landau, 1956]. He assumed firstly the interaction between the particles is adiabatically turned on such that the Fermi gas gradually transits into the excited Fermi gas resulting in a Fermi liquid. Here, there is a onetoone correspondence between the eigenstates of the Fermi liquid and those of the noninteracting Fermi gas. Secondly, he assumed that the temperature is much lower than the Fermi energy, so excitations are confined to with an energy of T (kB = 1) near the Fermi surface, and those excitations have a sufficiently long relaxation to be defined. excitations Fermi are called Like the bare particles, they obey Fermi statistics and have momentum (or crystal momentum) as a good quantum number. be considered as However, since the quasiparticles can particles in a selfconsistent field, they have a renormalized effective mass, and the energy of the whole system is no longer the same as the sum of the energy of each particle in isolation. Instead, the energy is a functional of the distribution function n(k). If the distribution function is changed by an infinitesimal quantity 6n(k), Sn(k) =n(k) (2.1) where no(k) is the distribution function of quasiparticles in the ground state, then the total energy of the system changes by an amount x ek6n(k), (2.2) quasiparticles.  n,(k), ~ f(k, k' )n(k'), (2.3) where jfk,k') is the interaction energy of the quasiparticles and of order 1/V, of the system. From Eqs. the volume (2.2) and (2.3), we have z f(k,k' )n(k )6n(k'). (2.4) k k' If the system is isotropic for k and k' on the Fermi surface, f"a)(k,k') depends only on the angle 0 between directions k and Here f '(k,k') f '(k,k') are the antisymmetric and symmetric parts of the quasiparticle interactions, respectively They can be expanded in a series of Legendre polynomials as follows: 00 f*'(k,k' z f$(&) P,(cosO) where is completely determined by the set of coefficients /" and a. It is convenient to express f in reduced units by setting N(EF) .f "' Fi a) , where N(EF) is the density of states at the Fermi level. After using elementary thermodynamics, we can relate the physical quantities obtained from experiments to the Landau parameters Fi '(9) and F '(0) as follows [Nozi&res, 1963; Pines and Nozieres, 989]: Sx2k, N(EF) 3V N(E,) (2.6) VK Fm t2k2 = k (1+Fo (2.8) n=E; + O~Mlb\l E ,vrr~r~lT 23 where V is the volume of the system, C, is the specific heat, m is the bare mass of a Fermi particle, m* is the effective mass of quasiparticle, S is the sound velocity. electrical resistivity due to the impurity at low temperatures can be derived by applying a Fermi liquid approach to the Anderson model for degenerate orbitals assuming electron hole symmetry near the Fermi level, and it is proportional to T and Zawadowski, 1978]. Landau's theory was constructed for a neurtal Fermi long range interactions. The only neutral Fermi liquid foun [Yoshimori, 1976; Mihily gas in which there are no d in nature is the quantum liquid it becomes degenerate temperatures. If there interaction between the 3He atoms, the Fermi temperature or the degeneracy temperature would be given by Eo k ht2k' 2mk, ~5K. (2.10) with m as a linear x 1024g and, from the particle density c, kF T dependence in specific heat and Such quantum behavior T independent magnetic susceptibility are experimentally found at about 0.05 K. This reduct comes from the interaction between the 3He atoms, ion in TF by two orders of magnitude the enhancement in the effective mass of the quasiparticles. Now let us consider the charged Fermi liquid rather than the neutral Fermi liquid. Let us consider a system of noninteracting electrons which move in the periodic potential was = 3 X2 crln where Uik has the periodicity of the lattice. are good quantum numbers. Both j, the band index, and k, a wave vector, When we take account of the Coulomb interaction between electrons, we can again establish a onetoone correspondence between the eigenstates of the real system and those of the noninteracting system by using the adiabatic switching procedure as discussed in the neutral Fermi liquid. A Bloch wave possesses an energy esO, and a velocity on the Fermi surface is given by vk = V(eoj) which is different from the velocity of a free electron k/m. After we turn on the interaction adiabatically, the quasiparticles have an energy Ekj = dE/Sn(k). Ekj is constant over the Fermi surface, and is equal to the chemical potential p. We can again define at every point of the Fermi surface a velocity vk, Now, which will be different from v k. we know that the difference between vk of interacting electrons and k/m of a free electron is due to firstly, influence of the periodic lattice on each electron secondly, the "many body" effect arising from the Coulomb interaction. Thus, Landau's theory of a Fermi liquid as applied to a spatially homogeneous state is suitable in the case of the Coulomb interaction between electrons [Silin, 1958]. Just as for the neutral Fermi liquid, we can now define an interaction energy between quasiparticles Ak,k') as the second derivative of the energy with respect to n(k). Since the Fermi surface of a metal is generally not isotropic (e.g., due to the anisotropy of the crystalline lattice), Ek depends on the direction of k and fAk,k') depends on both k and Also, the divergence offlk,k') due to, e.g., the inhomogeneity of the system or the mteraction between screened quasiparticles, quasiparticle its associated screening cloud [Pines and Nozieres, 1989]. However, the Fermi surface is almost spherical, as in the alkali metals for example, if the system' behavior is nearly isotropic. Thus, we can introduce a crystalline mass or a band mass me such that where jk is the current carried by a quasiparticle with wave vector k, and me is different from the bare electron mass m because of the periodic potential acting on the electron The real quasiparticle effective mass m* due to the quasiparticle interaction is defined by where vk is the velocity of a quasiparticle on the Fermi surface. The relation between m* and me is given by (2.14) = m(l +F I/3). We can also express the spin susceptibility X, in terms of the paramagnetic susceptibility X, due to the noninteracting free electron gas as follows: m(1 + Fo . (2.1 ) Equation (2.15) is meaningful only when Otherwise, the long wavelength fluctuations of the magnetic moment become unstable giving rise to a ferromagnetic correlation. Phenomena that are not influenced by quasiparticle interactions can be described in terms of a one electron model with renormalized energies. The de HaasVan Alphen effect and the low frequency Hallconstant in a high magnetic field are examples. On the other hand, phenomena affected by the quasiparticle interaction f(k,k') can not be exactly described by the one electron model. Instead , they can be expressed in terms of the Landau parameters above. Some properties of heavy electron systems (HFS), the Paulilike magnetic susceptibility linear temperature dependence specific temperatures can be explained by the Landau FL theory even if there are strong onsite correlations at low temperatures for T An analogy has been drawn between liquid He and the HFS, and it seems that there is relatively good agreement with experiments. The Landau Fermi liquid theory is likely to work well in metals as long as there is no critical behavior such as a phase transition. Before we finish this Section, let us mention that there are some problems in translating the 3He results directly to the HFS. One of those [Lee et al., 1986] is the lack of Galilean invariance in the HFS , so that the effective mass is not simply related to the Landau parameter F, s as shown in Eq.(2. Furthermore, the Fermi liquid theory does not consider the effects of spinorbital coupling, crystal symmetry, and band structure. 27 2.2 NonFermi Liquid Theory We define nonFermi liquid (NFL) behavior as that deviating from Fermi liquid results, e.g., a logarithmic divergence of C/T at low temperatures instead of saturation. One possible theory to explain the NFL behavior is the multichannel Kondo effect. will discuss the multichannel Kondo effect mostly based on the exact Bethe ansatz solution [Schlottmann and Sacramento, 1993]. Subsequently, we will summarize another proposed phenomenological theory, which is based on the scenario of magnetic quantum fluc tuations at 0 K, to explain the NFL behavior. Multichannel Kondo Effect The nchannel Kondo model for an impurity spin S and an integer nnumber of orbital conduction electron channels is given by Nozi&res and Blandin [1980] z Ea + kkmkina S .mak ,o ac'mo',. (2.17) k,k ,m,o,o where S is the spin operator describing the magnetic impurity, J is the antiferromagnetic coupling constant, a are the Pauli matrices, and m labels the orbital channels. The different orbital channels are strongly correlated close to the impurity and form an orbital singlet; i.e., the spins of the conduction electrons at the impurity site are glued together to form a total spin n/2 [Nozieres and Blandin, 1980; Andrei and Destri, 1984; Tsvelik and Wiegmann, 1984]. In general, three qualitatively different situations /%lM Ori ouh twn 4Kif* +10 f0 ontn nl ,lmnllno tn an k*mn .i o ntnn Ifn = the number of conduction electron channels is exactly sufficient to compensate impurity a singlet giving rise to Fermi liquid behavior. Classic examples are FeCu, FeAg, and CrCu. < 2S, the impurity spin is only partially compensated (undercompensated spin), because there are not enough conduction electron channels to yield a singlet ground state. An effective degeneracy of {(2S+1n) remains at low temperatures (and in zero field). This case may describe the integer valent limit of impurities with two magnetic configurations like Tm. >2S, the impurity is said to be overcompensated. A critical behavior is obtained as the temperature and the external field tend to zero. The quadrupolar Kondo effect [Cox, 1987a] is one possible realization of this case. 2.2.1.1 Compensated Impurity Spin The basic model describing the interaction of a transition metal (or magnetic) impurity with a metallic host is the orbitally degenerate Anderson model: where ++ d k kma kma +d mcid, (2.18) SrVk (dockmo + ckd. ) kma cbna (CkAo) creates (annihilates) a conduction electron with momentum k=lkl in the partial wave 1 = 2 with zcomponent m (Iml < 1) and spin , and d m, (dm,,) creates (annihilates) an delectron with quantum numbers m and a, Ek is the conductionstate z king Smnc' m+o' ma~aToa,,o crystalline electric field (CEF) quenches the orbital angular momentum. Third, the rather large hybridization width of the 3d levels smears out the energy splitting of the ionic term. Lastly, the direct orbital exchange, which leads to an orbital singlet, is expected to occur TK [Okada and Yosida, 1973]. This model can also be applied to rare earth (4f) or actinide (5f) impurities. spatial extension of the wave function is reduced in such cases: decreases the magnitude of the hybridization Vk. low temperatures Fermi liquid properties can extracted resulting in the singlet ground state, where the coefficient y of the linear T term in the specific heat can be obtained via the Sommerfeld expansion as follows: (2.19) (n+ 2)TK and the zero magnetic field susceptibility is given by (2.20) The Wilson ratio for the impurity is given by 432) 3 X,= 2(n+ ), (2.21) which is same as (Ax/x)/(AC/C) [Nozieres and Blandin, 1980], where AX and AC are the contribution of impurities in the susceptibility and specific heat, respectively. As we Eq. (2.21) reduces to the Wilson ratio of the ordinary Kondo problem for n = 1. The < 5f 2n:T, liquid approach assuming electronhole symmetry near the Fermi level [Yoshimori, Mihaly and Zawadowski, 1978] as follows: 1976; Pimp Poll 1 5; 8 (n+ itT 2)TK Thus, we see the T independent specific heat divided by temperature and susceptibility along with T 2 dependence of resistivity at low Tin the compensated impurity spin. 2.2.1.2 Undercompensated Impurity Spin Let us compare the compensated case, n = 1 and S 1/2, with the case n 1 and = 3/2 (all cases with S > n/2 are qualitatively the same). The specific heat as a function of temperature is shown in Figure . For spin S = 1/2 and n see Figure 2la, the height of the peak grows with field and asymptotically approaches the value of a free spin Schottky anomaly. For S << TK, e.g., = 0.01 TK in Figure 1 b, the specific heat has two independent peaks; the peak at lower T corresponds to the Zeeman splitting of the ground multiple and the one at higher Tto the Kondo screening. For S = 1/2 and n , the susceptibility has a maximum at low T due to the singlet ground state in H free spin behavior. << TK. For 8 On the other hand, its maximum at about H *1 I and H  T is reminiscent of = 0, the Curie law as a straight line is shown in Figure However there is a smooth change in the Curie constant due to the change of the effective spin from S at high at low The magnetic field lifts the degeneracy at low T and reduce the susceptibility to a finite value resulting from the 2.2.1.3 Overcompensated Impurity Spin Since there are more conduction electron channels to compensate the impurity spin, the remaining conduction electron spin of (n2S) is delocalized giving rise to a critical behavior. The magnetic susceptibility in small field, H <~c 1k, diverges with a power law given by Wiegmann and Tsvelik [1983] and Sacramento and Schlottmann [1991a] 1+2/n) n>2, (2.23) For n and S the susceptibility diverges with a logarithmic dependence on the field [Desgranges, 1985]. Note that the critical exponent in Eq. ) only depends on the number of channels, but not on the spin. The entropy, s, of the impurity in H = 0and = 0 is given by Tsvelik [1985], Desgranges [1985], and Sacramento and Schlottmann [1991b] = 0,H sin[7t(2S+1) / (n + 2)1 (2.24) sinrx t/ + 2)] In the presence of a magnetic field, the ground state is a singlet giving the corresponding entropy equal to zero. At high temperature the impurity spin behaves like a free spin in a magnetic field. At low temperatures, the zero field susceptibility X(7) and the specific heat divided temperature C(T)/T diverge critical behavior given Schlottmann [1991c], Ii' 0.35 0.30 0.25 0.20 0.10 0.jQ 3 ojIT 102 101 100 101 102 / TK 10'2 10I 100 101 0"3 10.2 10 "' 10 10 t 102 T/TK 1500 1000 H 0.15 0.30 0.45 X (T=O) I0~2 UN 1 o10t 10o 101 102 Figure 23 (a) Entropy, (b) specific heat, and (c) specific heat divided by temperature as n fi r,.*;, fa TrIT.. ; rannctant mr nnansth fiGAd fCnr .V = 1/9 and n = 7 The. ntrnnv ij ningiilar For n and S (two channel Kondo model, see Figure 23), the critical exponent vanishes and again a logarithmic dependence on the temperature arises [Tsvelik, 1985; Desgranges, 1985; Sacramento and Schlottmann, 1989b, 1990, 1991a]. In Figure 23a, we see that in zero field the entropy changes smoothly between the value {In }/2 given by Eq. (2.24) for low T and the asymptotic free spin entropy, high T. The results for the entropy for n _ 2 is qualitatively different from the traditional Kondo problem (n = 1), where there is only one energy scale TK, because as n increases Kondo screening is less pronounced another dependent energy scale TK(HITK)' +2 exists. The specific heat as a function of T/TK in constant field for n = 2 and S = 1/2 shown in Figure 23b, and these curves just correspond to the slope, C, = T(asl/T),, of those in Figure 23a. The H = 0 specific heat curve shows the Kondo resonance. small magnetic field and at low T, another peak develops due to the second energy scale and the two peaks merge into one characterized by a Schottky anomaly at intermediate fields H 1k. For H 0 the specific heat at low T is proportional to T and can be characterized by a coefficient y. In Figure 23c, we see that the value y for H = 0 does not saturate as T + 0 K, but it becomes finite for H Also a maximum in C/T arises for larger fields as a consequence of the developing Schottky anomaly. In Figure 23e, we see the logarithmic divergence of X in T for H = 0 and n and X decreases with field but has a maximum which correlates with the low T peak of the specific heat. of freedom of the U 5fshell in the heavy fermion compound UBet3. It can explain the a magnetic dependence electronic specific magnetic susceptibility, but cannot explain the strong negative magnetoresistivity. The main assumptions of the model are the following: configuration of the U ions is 5f Firstly, the stable 5fshell and this leads to a ground multiple with total angular momentum J = 4 according to Hund's Secondly, the CEF splits the J = 4 multiple into a F 1 singlet, 3 doublet, and f4 and F triplets. The 13 nonKramers doublet is assumed to be the ground state. Thirdly, the excited 4 triplet explains a Schottky anomaly in the specific heat, and the coupling to the F3 via the magnetic field gives rise to the van Vleck susceptibility. Kondo effect is incorporated hybridizing electrons conduction states and by considering virtual excitations into the 5f configuration with a = 5/2 ground multiple, cubic symmetry. which is split into a ground state F 7 double and Fs quartet in A nonvanishing matrix element between the F3 states of 5f and the F states of the 5f can be obtained via the J = 5/2 partial wave, in particular with the conduction electron states having Fs symmetry . By projecting the 5f states out obtains the following exchange interaction, '(0)1, (2.26) where J >0, S are the pseudospin operators for S /2 of the electric quadrupole JS 10,(0)+0, magnetic Kondo Hamiltonian corresponds tetragonal (quadrupolar) splitting related to a deformation of the cubic symmetry. The magnetization is the electric quadrupolar moment, thus the magnetic susceptibility represents actually the electric quadrupolar susceptibility [Cox, 1987a, 1988a, and 1988b]. Also we should keep in mind that even if the concentration of U ions is very small, they are going to interfere with each other at low behavior. T due to a divergent correlation length associated with a critical This interference competes with the local lattice distortion induced by a single quadrupolar Kondo ion. At low T(T TK), the resistance is given by Ludwig and Affleck [1991] R(T) /R(O) (T/ TKT)"2 (2.27) The specific heat divided by temperature, C/T, and the electric quadrupolar susceptibility Xq are proportional to In T. However, the proportionality coefficient of Xq is usually very small so that we do not see the effect. Instead, we do see the van Vleck magnetic susceptibility which is not due to the ground state, but due to the coupling between the ground state and the excited CEF level via the magnetic field. =Xvv It is given by (2.28) where Xvv is the Tindependent van Vleck magnetic susceptibility. The coefficient r is inversely proportional to the CEF splitting. now summanze basic physical quantities in Table 2.1 as a function of temperature between the two channel spin 1/2 (or magnetic dipole) Kondo effect and the two channel electric quadrupole (pseudospin 1  rl(Tl T,)"2] Possible examples of quadrupolar Kondo effect systems are UxY .xPd3 for = 0.2 [Seaman et al. 1991], UxThlxRu2Si2 for x < 0.07 [Amitsuka et al. 1993 Amitsuka and Sakakibara, 1994], and Ceo.lLao.CCu2.2Si2 [Andraka, 1994c]. Table . The physical quantities as a function of temperature of the two channel spin 1/2 Kondo effect and the two channel electric quadrupole (pseudospin 1/2) Kondo effect. Two Channel Spin Two Channel Electric 1/2 Kondo Effect Quadrupole Kondo Effect p TT T C/T In T In T Xq a ln T ,a << 1 Xn, In T 1 T 2n 11 ~ Xvv ; for nonmagnetic ground state. In T ; for a magnetic ground state. 2.2.2 Phenomenological Theory of a Critical Point at T = 0 K with a Scaling Analysis Even though there are several experimental results [Andraka and Stewart, Lohneysen et al. 1994] which reveal a magnetic nature in the vicinity of a critical point at = 0 K as an origin of the NFL behavior, there is no microscopic theory explaining the NFL behavior. However, recently there was a report [Tsvelik and Reizer, 1993] of a possible explanation for the NFL behavior not based on a microscopic theory but on a phenomenological theory. Tsvelik Reizer proposed phenomenological theory to explain NFL behavior, which exhibits strong deviations from the Fermi liquid theory, observed in the heavy fermion metallic alloys U0.2Yo.sPd3. Their theoretical development is based on the the resistivity are observed. Secondly there is a scaling in the magnetization M in the following form, H f(H = f( ) ~TP '" (2.29) where y  0.3 ,I3+. ~ 1.3 , and f(x) is a nonsingular function. Thirdly same scaling of H holds for the specific heat, C,(H, T) S(o, T) H (2.30) Lastly, the behavior of the thermodynamic quantities becomes even more singular UCu4Pd, which is closer to the antiferromagnetic (AFM) part of the phase diagram. these properties hold below T 10K. The materials behave like Kondo alloys well above temperature T dependence p [Seaman et al ., 1991] originating from the scattering of conduction electrons off U ions. The Fermi liquid (FL) theory is based on the assumption that the low energy excitations are dominated by fermionic excitations. This assumption via the Pauli principle implies a natural energy scale exists in the theory, chemical potential EF and thermodynamic properties depend on T/sF and H/SF. The dependence of C and M to the magnetic field and temperature behave as described in Eqs. (2.29) and (2.30). Their scaling clearly demonstrates the absence of a natural energy scale and thus the irrelevance of NFL behavior with fermionic excitations. Therefore, Tsvelik and Reizer suggested that the low T thermodynamics in the transition occurs at a finite critical temperature T7, the notation T in Eqs. (2.29) and (2.30) should be changed T  T1. Although the nature of the T = 0 K instability remains obscure, glass fluctuations are suggested as an essential origin based on the observed experimental results in the alloys. The free energy is given by T  Tf ( T ^J K (2.31) S' where h, are external fields are dimensions. fields are relevant if their dimensions are positive 6, In the scaling region, one must consider only the leading singularity of the free energy F and then perform the limit excitation spectrum o at the critical point is given by w q, . Suppose that the then the dimensional analysis gives F ~ Tl+d/z ~ 1 In the limit TK oo and h, , the indices of the singular X are given +I+d (2.3 'TT If ,(0) are related to the thermodynamic correlation functions of the operators c,(t,x) as a function of time and real space and the concept that the correlation must decay at t oo is applied at finite temperature to restrict y's (see the paper by Tsvelik and Reizer [1993]), the following relation is obtained: The comparison of the specific heat result, C  T n T, with the specific heat from the free T2S, xi (O) do not control logarithms), and the excitation spectrum at the critical point scales as This result along with Eq. (2.33) means that 6, < 2 and this simple inequality is very important, because in both a spin glass transition and a quadrupolar fluctuation a magnetic field couples to a relevant field quadratically, i.e., 2. The data from the alloys show that the magnetic field scales as T with 6 = 1.3 +0.1 which means that H2 has the scaling dimension 6(H 2) requirement 6, = 26 so that it rules out not .6 +0 . This estimate contradicts the only the scenario related to a spin glass transition but also the one of an impurity quadrupolar fluctuations. Recently, there was a study of two channel Kondo effect with two impurities to consider a more realistic system where intersite and onsite interactions compete all the time [Ingersent et al., 1992]. CHAPTER 3 NONFERMI LIQUID BEHAVIOR OF MAGNETIC HEAVY FERMION SUPERCONDUCTORS There has been a lot of interest in a nonFermi liquid (NFL) behavior in felectron (Ce and U) heavy fermion materials for the past few years. The physical properties of the NFL usually exhibit a weak power law or logarithmic divergence at low temperatures. critical point at T = 0 K has been suggested as an origin. The possible origins of a 0 K critical point are an unconventional single effect fluctuations order parameter near 0 K resulting in a second order phase transition. Our attention will be given to the NFL systems whose parent systems have the antiferromagnetic (AFM) state and superconductivity (SC) at low UxMixNi2Al3 (M . A more thorough emphasis will be given to the Th, Pr, and Y) and UxTh.xRu2Si2 systems with experimental data to suggest a possible ground state of these systems. 3.1 Overview of NonFermi Liquid Systems Almost a dozen NFL systems have been discovered in recent years (see the review papers, e.g., Maple et al. [199 Andraka [1994], and Steglich et al. [1994a]). These systems are Ce or U intermetallics doped a nonmagnetic element with a few conduction electrons resulting in magnetic order, the nonmagnetic quadrupolar moments interact with the charges of the conduction electrons participating in a quadrupole ordering at low The NFL materials exhibit several unusual physical properties in, e.g., specific heat C, electrical resistivity p, and magnetic susceptibility Xm in contrast to the local Fermi liquid theory with which we can describe some of the heavy fermion systems. For instance, C/T is expressed in terms of In T dependence rather than constant in temperature, p is linear in T"' or T or In T rather than in TP , and Xm is linear in In T or T'n rather than constant at low temperatures. Thus, the existence of a critical point at T =OK has been suggested. One of the possible origins comes from an unconventional moment screening process such as multichannel Kondo effect [Schlottmann and Sacramento, 1993]. Another possibility is a = 0 K secondorder phase transition such as a long range magnetic order [Andraka Tsvelik, Tsvelick Reizer, 1993] (another possibility is due to dilute magnetic impurities in a disordered metal where a singularity comes from the probability distribution for Kondo temperatures [Dobrosavljevic et al., 1992]). There have been extensive experimental studies to confirm above theoretical suggestions NFL behavior, The parent heavy fermion materials, of which doped ones exhibit vary from the nonmagnetic (CeCu6) to the magnetic system (UCus), from the superconductor (UBe13) to Kondo insulator (CeRhSb), and even systems in which and antiferromagnetic states coexist (URu2Si2). We can roughly divide these systems into 44 1994], CeCus.7Auo.3 at pressure [Bogenberger and LOhneysen, 1995], UCu3.,Pd.s5 [Andraka and Stewart, 1993], CexThi.xRhSb (may be disorder) [Andraka, 1994b], and CePtSio.9Geo.i and UCus.6Al.4 [Steglich et al., 1994]. The second group members, being described to exhibit single ion two channel Kondo effect (TCKE), are Uo.9Tho.1Be13 [Aliev al.,1995] and maybe UxThi.xPd2Al3 [Maple et al., 1995]. The last group members may be described by either the first collective modes or the second single particle excitation, e.g., Ceo.1Lao.9Cu2Si2 [Andraka, 1994c], UxMi.xPd3 (M and Tsvelick, Y, Sc) [Seaman et al., 1991; Andraka 1991; Gajewiski et al., 1994], PrxYi.xCuzSi2 [Sampathkumaran, 1993], and UxTht~xRu2Si2 [Amitsuka and Sakakibara, 1994]. So we present here some experimental results for UxThi.xRu2Si2. It seems to us that possible collective modes at low temperatures dominate over the single ion excitations in the system. Furthermore, we will present another new NFL materials Uo.9Mo.INi2A13 (M = Th, Pr, and Y) and in the end we hope to convince that our new systems are more related toa T = 0 K long range phase transition than other possible origins. Investigation of the Ground State in UxThl.xRu2Si2 for x < 0.20 There has been a very interesting report [Amitsuka et al., 1993] such that they have found nonFermi liquid behavior in UxThi.xRu2Si2 for x < 0.07 The experimental results of C/T, p, and X due to Uion exhibit a In T dependence for one or two decades of temperatures. They have attempted to explain the results of x and C/T in terms of the 45 predict [Tsvelick, 1990; Ludwig and Affleck, 1991] that the resistivity subtracted from the residual one, p(7)  p(O), increases as Ta with a positive a for T >0. We firstly summarize briefly the parent system URu2Si2 and its doping experiments on the Usite with Th, , La, and Ce. We then describe the main points of UxThtxRu2Si2 experiments for x 0.07 by Amitsuka et al., [1993] with their recent new interpretation [Amitsuka and Sakakibara, 1994]. Our emphasis will be given to the possibility of a magnetic transition at low T with the resistance, specific heat, and magnetic susceptibility measurements in UxThixRu2Si2 for 0.07 x <0.17 3.2.1 URu2Si, The physical properties of the superconductivity (Tc = netism (TN 1.5 K) and antiferromag 17.5 K) in URu2Si2 [Palstra et al., 1985; Schlabitz et al., 1986; Maple et al., 1986] have attracted condensed matter physicists. structure is bodycentered tetragonal ThCr2Si2 with lattice parameters a = 4.219 A and C = 9.575 A as shown in Figure 31. The spins on the Usites are aligned antiferromagnetically to the cdirection with alternate ferromagnetic sheets in a/2 apart. It is a semiheavy electron system with y, the coefficient of electronic specific heat, being 64 mJ KI mole"' [Palstra et al., 1985]. also shows a large slope of the upper critical magnetic field Hc2 at the superconducting critical temperature (dHcJdT)Tc, which is about [Rauchschwalbe, 1987]. ThRu2Si2 has the same crystalline structure as URu2Si2, and y is about 10 mJ K'2mole"' Ou Ru oS For the AFM ordering there are several possibilities. Firstly, it may be due to the development of a charge density wave (CDW) or spin density wave (SDW) with a gap of meV from specific heat measurements [Maple et al., 1986] or a gap of 7 meV produced by the hybridization of U5f states with conductionelectron states from the inelastic neutron scattering [Walter et al., 1986]. In this case, the nonmagnetic singlet ground state is responsible for the transition. However, there is a maximum around 30 K in the electronic specific heat vs. temperature in addition to the anomaly at TN. The total entropy connected both peaks is close to R implying magnetic temperature state of URu2Si2 is formed by localized 5f electrons in a CEF doublet ground state [Schlabitz et al. , 1986]. Secondly, it may be due to the long range RudermanKittel KasuyaYoshida (RKKY) interaction of the localized U5f magnetic moments supported Xray resonance magnetic scattenng [Isaacs et al , 1990] magnetic neutron scattering [Mason et al., 1990]. But, there is no indication of a Schottky anomaly in specific heat up to 350 K even if most of the CEF levels should be populated at 200 K. The entropy released below TN is about 0. 17R ln2 although the ordered moment is as small as 0.03 Pa [Broholm et al., 1987]. In the magnetic susceptibility ym, there is a broad peak around 55K, which is interpreted as being due to a short range magnetic correlation [Schlabitz et al. the coherence effect in the Kondo lattice [Mydosh, 1987], 1986], or or AFM fluctuations of 5f state [Amitsuka et ., 1992]. High magnetization experiments reveal several sharp [Palstra et aL 1985; Schlabitz et al., 1986; Broholm et al. 1987]. This invokes the possibility of elucidating the interplay between the two ground states. Doing Effects of URuSi, There have been several doping experiments of UtxMxRu2Si2 (M , La, and Ce) into Usite to understand the fascinating properties of the parent system mentioned above [Torre et al., 992; Park, 1994; Park et al., 994; Amitsuka et al., 1992 Mihalik et , 1993]. In U,xLaxRu2Si2 for 0 x < 0.3 [Amitsuka et al., 1992], TN decreases for 0.07 and starts to increase for 0.07 x < 0.3 as the lattice parameters gradually expand. Their argument is that 5f character continuously changes from an itinerant to a localized nature with doping based on YX experiments. There is a linear correlation between Tmn, where Xm shows a maximum peak, and TN for 0.07 x<0.3 , indicating the same nature of both TN and Tm,. So they concluded the maximum peak in Xm around 5K is related to the AFM fluctuations of the 5f state. In the Urich case of UixThxRu2Si2 for 0.0 x < 0.05 [Torre et al., 1992], small doping of Th suppresses both TN and Tc, and it can be explained in terms of the decreased coupling constant J between conduction electrons. other words, expanding of the lattice parameters causes a suppression of hybridization as shown in Eq. ) assuming the energy difference between the Fermi level and 5f level is unchanged based on tetravalent character of U and Th. Thus, both decrease of J and increase of Ri  [JN(E,)12 I where (3.2) VFf is the hybridization matrix, Er the Fermi level, Ef the energy of the 5flevel, N(EF) the density of states at the Fermi level, and R0 the distance between lattice sites i andj. Torre et al. [1992] use the concept of "Kondo hole" to explain the minimum in the resistivity. However, it might be ruled out as shown by Ydoping experiments [Park, 1994], specifically, the minimum in p at low temperatures exists up to 50 % doping. the other hand, there have been a lot of interesting phenomena in the Udilute case, namely "nonFermi liquid" behavior which we will describe in more detail in the following section. NonFermi Liquid Behavior of UxTht.xRuSi, for x < 0.07 A Japanese group has reported [Amitsuka et al., 1993; Amitsuka and Sakakibara, 1994] that they have found nonFermi liquid behavior in a single crystal of UxThixRuzSi2 for x 0.07, i.e., Udilute case, such that C/T, p, and X due to the Uions show a In T dependence. They have attempted to explain the experimental results in terms of the two channel Kondo effect (TCKE) [Nozieres and Blandin, 1985; Sacramento and Schlottmann, 1980; 1991a; Seaman et al., Tsvelick and 1991] with a single impurity based on the tetragonal 5f 2 configuration even though the resistivity data is totally in disagreement with the model, which predicts the T'" dependence of electrical resistivity. They have claimed that the T independence of X'm, magnetic susceptibility along the basal plane or aaxis, originates from van Vleck paramagnetism due to 3.2.3 TRKKY Wiegmann, state was a magnetic doublet Fs5 (3H4, U4 , 5f2) rather than a nonmagnetic doublet. Thus, TCKE in this Udilute system UxThlxRu2Si2 x < 0.07 is caused dipolar fluctuations in the ground state. This ruled out the possibility of the TCKE caused by quadrupolar fluctuation in the ground state of 3 nonmagnetic doublet for the cubic and tetragonal Ucompounds [Cox, 1988b]. For instance, there is no such magnetic ground state phenomena in the cubic symmetric systems such as U0.9Th0oBe13 [Aliev et al., 1993; Aliev et al., 1995] [Seaman et al., 1991] which are also believed to exhibit nonFermi liquid behavior (there has been polarized inelastic neutron scattering in which they observed the critical fluctuations around the antiferromagnetic position in the U0.2Yo.,Pd3 similar to that of the Uo.45Y0.55Pd3 whose ground state is a magnetic F s triplet). In cubic symmetry, e.g. Uo.2Yo.8Pd3, the selection rules used [Cox, 1993] restrict the lowest CEF state to the F doublet that couples to the F excited doublet via the F 8 symmetry conduction states. this case the degeneracy of 13 comes only from the quadrupolar degrees of freedom. the other hand, in the tetragonal symmetry, e.g. UxThixRu2Si2 for x < 0.2, the TCKE is expected only when the Fst (5f ) lowest doublet couples to the frt (5f ') excited doublet via the hybridization effects with the F6t+ F7t symmetry conduction states at the Usite. the In T dependence of Xem of UxThixRu2Si2 for x < 0.07 is the very case not in the electric quadrupolar field but in the magnetic field [Amitsuka and Sakakibara, 1994]. Thi nharht rclu ll ;e trnl anl, uhian the rulMhlt r.. U0.2Y0.8Pd3 51 Xq, to show In T divergence at low temperatures because Fst is doubly degenerate with the quadrupolar moment cplane. investigate possibility ultrasound measurements, from which the quadrupolar (or strain) susceptibility can be extracted, are desired. 3.2.4 NonFermi Liquid Behavior of UxTh..xRu2Si2 for 0.07 x 0.17 A purely logarithmic temperature dependence of Xm (Xc'" in Figure 32) in the cdirection breaks down at low temperatures (T 100 inK) for x > 0.03 resulting in a peak anomaly at temperature Tm of the order of 0.1 K as shown in Figure anomaly may come from an intersite magnetic correlation between Uions, e.g., spin glass ordering. While we were doing the doping experiments in UxThi.xRu2Si2 for 0.07 we found a kink below 2 K in the resistance data. This fact allowed us to speculate T dependence of Xm about decades of temperature observed Uo.olTho.99Ru2Si2 might correspond to magnetic fluctuations near = 0 critical point [Andraka and Tsvelik, 1991], proposed already for other Ubased alloys displaying similar low temperature properties. Since the temperature of the anomaly increases with concentration x, we have focused our investigation [Kim W.W. et al., 1994] on the less dilute systems corresponding to laboratory, but we limited x > 0.07 to operate at more accessible temperatures in our x < 0.17 to avoid the problem of a miscibility gap for 0.2 0.9 [Torre et al., 1992]. 1 1 10 100 1000 Johnson Matthey Inc. (JMI), and 99.999 3 in detail. Al from JMI as we described in section 1.3.1 Some resistivity samples were cast into a particular shape in a watercooled crucible using a pressure casting technique inside an arc furnace [Crow Sweedler , 1973]. After we took the sample out of the arc furnace, we subsequently annealed it at 900 C for one week. All samples showed single phase according to Xray powder diffraction data taken on a Philips diffractometer with a scan of 6 o per minute in 20. Resistance. These measurements were performed via a standard dc fourprobe method to temperature as low as 0.3 K. If we look at the resistance data shown in Figure 33 in terms of T. we see a kink below 2 K for all our concentrations. The position of the kink anomaly decreases as we dilute Uconcentrations for x < 0.14 as shown in Figure 34, and decreases abruptly for x> 0.14. crystalline sample [Amitsuka et al., 1993 We think the peak at 250 mK in Xm from single Amitsuka and Sakakibara, 1994] corresponds to an anomaly TA in the resistance for =0.07 Although the TA for = 0.07 was not clear in our polycrystalline sample, it is interesting to note that the TA approaches zero for x 0if we assume the linear decrease of TA for x < 0.1we can see such a possibility for 0.1 <0.14. Thus, it is desirable to check this behavior with single crystalline samples for . This possibility allows us to speculate that the In T dependence of X'm observed in Uo.oiTho.99Ru2Si2 may correspond to magnetic fluctuations near T = 0 critical point. [Amitsuka Sakakibara, 1994]) in contradiction to the TCKE, which predicts decrease in p with temperature caused by Kondo scattering. Thus, it seems that the behavior of p is closely related to the onset of coherence and correlated scattering. other words, the NFL behavior of the resistivity in the UxThl.xRu2Si2 for Udilute region (x < 0.14) seems to be related to the intersite interaction of U (short range correlation or even long range one as T Specific heat. Sp > 0) rather than the single site two channel Kondo effect. ,ecific heat measurements did not reveal any signature consistent with a phase transition at TA. Only the specific heat for the 'most magnetic' alloy of 0.14 showed a change of slope or a peak in C/Tat about 0.45 K shown in Figure 35; this temperature is roughly three times smaller than the corresponding TA. NFL behavior in p measurements at low temperatures, Since we saw the we can expect an unusual Fermi liquid behavior in specific heat rather than Fermi liquid behavior, observe a saturation of C/T at low temperatures. in which we usually We show specific heat due to the U5f contribution divided by temperature CsfT vs. In Tfor 0.07 x 0.17 in Figure 35. CwjT for a given concentration is fairly linear in In T again NFL behavior (the In T dependence of Cs of Tdeviating for T scale over more than one decade implying = 0.07 holds for only one decade 3 K and this is also observed by Amitsuka et al. [1994]). When we applied a magnetic field, CIT monotonically decreased as we increased the field up to 14 T . For instance, C/T at 1 K is decreased by about 42 % at H = 14 T compared to the value in the H = 0 for x = 0.1, and it follows surprisingly well Fermi [Steglich et al. 1994] as we apply magnetic fields, because a singlesite interaction is induced by the magnetic fields which destroy the weak long range correlations between local magnetic moments. A similar behavior of the magnetic field dependence has been reported [LBhneysen et al., antiferromagnetic transition 1994] in system CeCus~.Auo.i, = 0 Kin H which is believed to have an = 0, and comes back to the Fermi liquid behavior magnetic According to TCKE [Schlottmann Sacramento, 1993] as we described in Figure 23c, however, CIT should increase from the zero field value for the magnetic field roughly comparable to TK. For instance, C/T for x 0.1 should increase at least by a factor of five at H = 10 T and T Kfor TK 10K, T/TK = 0.1 and H/TK = 1 (see Table 31). TK listed in Table 3 1 has been obtained from the slope of the linear fit of Figure 3 using the formula (3.3) from TCKE model and assuming TK is independent of magnetic field: C5f/T 1R T In(.41 0.41TK )+A, (3.3) where A is a constant and R is a gas constant 8.314 J K' mole . But, we did not see such an increasing behavior of C/T implying the ineligibility of the TCKE in this system. Table 3 Summary of Kondo temperature TK using Eq. (3.3) for all our concentrations x<0. 17. concentration (x) TK (K) 0.07 6 0.10 10 0.125 11 As we dilute the Uconcentration from 0.17 to 0.07 the lattice constants increase due to the larger atomic size of Th than U giving rise to a decrease of both TK and TRKKY assume that the TA corresponds to a magnetic phase transition, . If we we see in Figure 37 that the long range correlations (TRKKY) are dominant over the onsite interaction (TK) in our system because decreases as the coupling constant between local magnetic moments and the conduction electrons decreases. Amitsuka et al. [1994] also suggested a possibility of "channel reduction" that there is a crossover from the TCKE with a magnetic ground state F such to the usual onechannel Kondo effect in high magnetic fields and at low temperatures. But it is not probable, because it always generates the Kramers doublet as the lowest state so that a Curielaw susceptibility is expected even along the hard axis (in this case aaxis) and this contradicts almost T independent behavior Ca lm Thus, such a magnetic dependence of the specific heat can not be explained by the simple TCKE as shown by both the decrease of C/T with fields and the inadequate argument of the channel reduction to explain the FL behavior in magnetic field. There is another NFL system of Uo.2Yo.sPd3, where the physical properties such as specific heat and magnetic susceptibility were suggested to obey TCKE [Seaman et al., 1991; Maple et al. , 1995]. Also, as the theory [Tsvelik, 1985; Sacramento Schlottmann, 1989b] predicts, there seems to be a remnant entropy {R In2) =OK [Seaman et at., 1991]. Thus, C/T should increase at low temperatures, i ST/TK <0.1 o0.4 0.36 I I 5 * a S . * S.. * S* . S S S S  S S S S a S S a a U S S  a S S. S a S S S S S S S a ..5**' 0.405 "Se 4. 'Pr  7 0.395 S a S S a S S S. S a a S S I S SS a S S... U.JL13 a SS a a S a a a SS SS SS a 1 S 0.375 0.5 1 1.5 2 2.5 a S I a I A A I I A (K) I' ci: 0.7 0 O r S p  0.2 400  200 E U C1 .5 0 Log o 0.5 (K) C Q.nX la ot nf T C:,,,,, 1 350 250 150 100 TRKKY Magnetic 4fmetal Magnetic CKS Nonmagnetic CKS 2/w 62 observed by Andraka and Tsvelik [1991] at 0.35 K and 14 T (there should be about order of one increase in C/T for T/TK < 0.01 and H/TK  0.3 from the Figure 23c) indicating the ineligibility of the TCKE theory to Uo.2 Susceptibility. Y0.5Pd3. Dc magnetic susceptibility measurements were performed from room temperature down to 1.8 K in a commercial SQUID magnetometer. No apparent signature consistent with a magnetic anomaly was found in our samples in the investigated range of temperaturesthis is not surprising if we accept the possible relation between TA and T,, e.,TA ":(3  4) Tm so that Tm is expected below However there was a distinct difference between the zeromagneticfieldcooled (ZFC) and fieldcooled (FC) susceptibilities concentrations investigated. Moreover, ZFC some concentrations flattened out at the lowest temperature investigated indicating that there might be a possibility of an anomaly at temperatures lower than 1.8 K. depicted in Figure 38 for This tendency is = 0.14. It has been reported [Amitsuka and Sakakibara, 1994] that there was an excellent scaling behavior in the xm for x < 0.0 implying a single site Kondo effect and a fairly good fitting of Xcm data was possible by the TCKE as shown in the inset of Figure 3 However, we suggest interesting to look there is a onetoone correspondence between TA and Tm by measuring xgm (even TA for better resolution) with single crystalline samples for 0.07 x 0.14. Because, if such a correspondence exists, it O E I 3  16 E G) U E >, +'> F~C FC p I 1 i iL  (K) __ 64 In summary, firstly we observed TA in p measurements, below which the intersite interactions between local magnetic moments occur, for 0. x < 0.17 at low . This anomaly at TA seems to be related to the onset of coherence, and can not be explained by singlesite TCKE interpretation. Furthermore, increase resistivity temperature for 0.3 K coherence 20K contradicts the Thus, the NFL behavior in UxThlxRu2Si2 for 0.1 TCKE, instead implying the onset of x < 0.17 seems to be due to magnetic fluctuations at T rather than the singlesite TCKE. in the Udilute regime exhibits th = 0 K or collective modes of excitations at low temperatures Secondly, we know the specific heat of UxThixRu2Si2 NFL behavior. Also, thermodynamic measurements in magnetic fields reveal that these systems transforms to the Fermi liquid regime from the magnetic instability or magnetic intersite correlation regime caused by the destruction of the intersite interactions between local magnetic moments. Thirdly, we observed a spin glass anomaly for concentrations 0.0 x <_ 0.17 in the FC and ZFC Xm measurements using the polycrystalline samples. Also , the confirmation of the relation between Tm and TA using single crystalline samples is desirable to understand the origins of Tm and the logarithmic behavior of Uo.o1Tho.99Ru2Si2 in Xm. Although we have focused on the anomaly of TA and Tm in the resistivity and magnetic susceptibility, it is desirable to check if our system obeys the scaling behavior in the specific heat and magnetization measurements at several magnetic fields as a fiurther work 65 3.3 Magnetism and its Connection to NonFermi Liquid Behavior in UI.xMxNNiAls (M = Th. Pr. and Y1 Since there was a report of the discovery of a new heavy fermion system UNi2AI3 [Geibel et al., 1991c], there have been several studies to understand the system. While we were conducting doping experiments on the Usite UNi2AI3, we found unusual temperature dependence of the physical properties inconsistent with Fermi liquid behavior indicating a possible NFL system. Thus, we are going to investigate the doped system more thoroughly and give a possible explanation for the origin of our newly discovered [Kim W.W 3.3.1 .et a., 1993] NFL system. UNi2A1 took several years physicists discover new heavyfermion superconductor, UNi2Al3, as CeCu2Si2 [Steglich et al. URu2Si2 [Schlabitz et al., [Geibel et al., 1991c] after discovering the previous ones such 1979a], UBei3 [Ott et al., 1983], UPt3 [Stewart et al. 1986]. 1984a] Superconductivity takes place at UNi2At3, AFM transition at TN = 4.6 K reflects a weak delocalization of 5f electrons. It has a hexagonal PrNi2AI3 crystalline structure, shown in Figure 39, with the lattice parameters 5.207 A and c = 4.018 As we see in Figure 39, the UU distance is already larger than the Hill limit (3.4 A) [Hill, 1970] to exhibit heavy fermion Tho .amnnl i annnsalsd tn rmnT/P cEAfnnnralrv nhbsec (tcmullv T TAI1 which vYict hbpah; inr Te If we look at X vs. temperature as shown in Figure 310, there is a cusp at TN and a broad maximum around = 100 There is a flat region for Tm in electrical resistivity p vs. T, which is shown in Figure 311. Also, Geibel et al. [1991c] found a corresponding maximum in p around T,m after they subtracted the contribution from the electronphonon part, and suggested a transition from localized to weakly delocalized 5f behavior in the spirit of the Kondo lattice concepts. AFM transition not clear in p, is exhibited a meanfieldtype relatively sharp transition at TN and a broad curve below vs. Tas shown in Figure 312. as high as H TN down to the onset of Tc in The TN is weakly depressed by an external magnetic = 8 T as shown in the inset of Figure 312, resembling the case of, e.g., the heavy fermion antiferromagnet CeCu2Ge2 [de Boer et al., 1987]. Magnetic entropy released at TN gives a value of S = 0.13 R ln2 indicating a rather small ordered Umoment. This has been confirmed by neutron scattering measurements [Schroder et al., 1994] which give ordered moments rd= (0.24 + 0.10) pW/Uatom in the basal plane, incommensurate with the nuclear lattice. which is However, positivemuon spin rotation (p'SR) measurements [Amato et al., 1992b] were reported such that the o = 0.1 pB/Uatom along cdirection (perpendicular to basal plane) commensurate AFM ordering. The superconductivity is exhibited by a sharp transition in p, shown in the inset of Figure 311, and by a broadened jump in the specific heat around at Tc = K, shown in Figure 312. The electronic specific heat coefficient 120 mJ K'2mole1' is deduced, S~~ . I ,.di TV .f I, m '7' ,r Tr fi n 0 At Pd 100 200 300 1000 750 500 250 0 100 200 300 200 150 ~100 I .*. . 200 170 10O 71 measurements of Tc with magnetic fields, the initial slope of the upper critical mag data between Tc and 7 K shows a linear behavior, shown in the inset of Figure 311, usually preceding the asymptotic low T behavior, p , of Fermi liquid. From the netic field, c2 =  ( jc2//)rTc, is also reported to be 1.4 T/K. From Yo, Tc, po, H2, and assuming an isotropic one band model, they estimate [Geibel et al., 1991c] the effective carrier mass = 70 mo, where mo is the mass of the bare electron. 3.3.2 NonFermi Liquid Behavior in Ui.xMxNbAl_{M Thk Pr. and Y} Since our group found a possible separation single effects from correlation effects in doping experiments [Kim J.S. et al., 1991; Jee et al., 1990], doped the parent system UNi2A3 on the Usite with Ce, La, Y, Th, and Pr in order to vary observed behavior. While we were doing our experiments, we found unusual temperature dependence of the physical properties in, e.g., C/T, x, and p inconsistent with the FL behavior. Thus, we performed systematic studies to understand the origin of the NFL behavior. Samples were prepared via arc melting together the starting elements, followed by (with exception of the sample #1 in UNi2Al3 discussed below) annealing for one week at samples maintained PrNi2Ah3 crystalline structure with lattice parameters shown in doped elements in Table Table 33 We also compare the physical properties of U and the to help the understanding of the doping effects in these The dc magnetic susceptibility Xdc was measured samples. The value obtained at 1.8 K for our UNi2AI3 sample # was about 8 % less than that found in the discovery work by Geibel et al. [1991 c]. In addition, they saw a small (about 6 %) peak Table 3 Parameters for Uo.9Mo.iNi2A13. M = a (A) c (A) X (1.8 K) _memu/formula unit] U (parent) 5.184 4.023 4.55, 3.30, 3.95, 3.50 a La 5.200 4.023 4.20 Y 5.184 4.024 5.35 Ce 5.188 4.030 4.20 Th 5.182 4.032 5.15 Pr 5.184 4.027 8.35 a) The values correspond to the sample #1 through #4 from the left. Table 33. Comparison of U and the doped elements. at 4.8 K in Xdc (see the inset in Figure 3 10), whereas our data for sample #1 are, within %, featureless in this region as shown in Figure 3 As will be discussed below more thoroughly, the sample dependence of the magnetic transition at TN is an important point. Thus, weprepared three additional samples of UNi2Al3, whose Xdc (1.8 K 300 K) and (1K 10 K) were measured (see Figures 3 13 and 3 We see a significant atomic size electronic configuration effective magnetic moments U~ Y tetravalent (4+): Th U > Pr, U > Ce U < La, Pr, Th, Ce trivalent (3+): La, Y, Pr, Ce Y, La, Th = 0 was annealed at 1000 C for six days, while the subsequent samples were annealed in a different furnace at 900 0C for seven days. Unless the furnace used for sample #1 defective, the absence of the AFM transition shown in the C/T Recently, there was a report [Schank et al. UNi2AI3. and Xdc, seems unusual. 1994] of the influence of offstoichiometry on They did see the absence of AFM transition and superconductivity down to 40 mK in UNi2.05A12.95, while they found about a 10 % suppression of TN, and Tc"~ with a much broader transition width in UNi1.95A13.05. 0.66 K From these results we see that the offstoichiometry of the Ni (the excess) and Al (the deficiency) is very crucial to the magnetic order and the superconductivity in UNi2A13. show C/T vs. T2for Uo.9Ceo.0Ni2Al3 in Figure 3 14 and for Uo.9Mo.1Ni2Al3 (M , Th, La, and Pr) in Figure 3 . The 10 % doping destroys the AFM order completely and the C/T at low temperatures seems to diverge (except for Ce doping). For the small peak around K in the 10 o Cedoped sample, shown in C/T vs. T we do not see a corresponding anomaly in Xdc down to 1.8 K. We checked the temperature dependence of the 5f electronic contribution of specific heat over temperature Csf/T of 10 % Y Th, La and Pr doped systems. As shown in Figure 3 16, a in T dependence for more than one decade of temperatures is observed in the 10 % Th and Prdoped systems. we did not see a broad in T dependence of Csf /T for the Even though 0 % Ydoped systems for T we saw a In T dependence down to about 0.3 K leaving about a decade of NFL temperature behavior, shown in Figure 3 16 (all NFL systems mentioned in section 31 show the logarithmic temperature dependence of C/T due to magnetic ions). The 10 % we estimate Kondo temperature using the formula (3.3), we have 77 K, 69 K, and for Pr, Th, and Y 10 % doped systems, respectively. The question arises: how does this unusual behavior change upon further doping For instance, the shrinking of the T range, where In T dependence of C/T is observed, is seen upon further Udoping in Uo.2Yo.sPd3 [Seaman et al., 991]. In UCu3.sPdi.s [Andraka and Stewart, 993], varying the Pd to UCu4Pd totally destroys the In T dependence; instead giving rise to C/T cT 0.32 from 1 to 10 K. In our systems, the temperature range over which this In T behavior is observable, on the other hand, shrinks upon increasing Th (or decreasing U) content, up to 3 .2 K for 20 % Th and 2 K for 30 % Th as shown in Figure 3 other words, unusual behavior is too limited T to significant credence. As seen in Figure 3 18, using Th doping as a representative example, the temperature at which the upturn in C/T begins is depressed with more doping and the negative slope at low temperatures is steeper for more Th doping resulting in the deviation from the In T dependence in C/T. The low Tvalues of Xdc vary as detailed in Table 3 The Xdc results for 1.8 K < 9K of = 0.1 Th and Pr obey T in dependence, shown in Figure 3 9, as predicted by TCKE for the nonmagnetic ground state [Cox, 1987a and 1987b]. The theory says that we can express magnetic susceptibility Xm as Xm  nT l' (refer to Eq. 28) where nr is proportional to the T independent van Vleck susceptibility, Xvv, and inversely proportional to the CEF splitting. The coupling via conduction electrons between the nonmagnetic 100 200 300 (K) P .mira2...12 IIrLA~ .r*.J MaGnetic suscentibilitv as a function of T between 8 K and 100 K fnr fnwir I C' 100 100 E150 140 100 100 .5 0 Log10 0.5 (K) Figure electronic contribution specific over temperature, moaaorpAd fr TL ,,Pr^ .iN'LAl. T L ^Th. .Ni.Al, T ^V .NLAl. an T ^T a .NMiAI. ik nirnttArl 0.5 1 Loglo (K) 160 140 120 100 we have seen the sample dependence in the parent system, we prepared a second sample of Uo.,Tho. Ni2AI3. The data also show Xdc dependence. On the other hand, found that Xde of the 10 % Ydoped system exhibits a more singular behavior in the T 12 scale than the Th and Prdoped systems as shown in Figure 319. Thus, the doped system follows logarithmic temperature dependence rather dependence as shown Figure 320. Since quadrupolar susceptibility, where aspherical charge distribution of the 5f ions play the role of a pseudospin, is usually too small to see a response, Xde for Y 0 % doped sample may be described by the effect of magnetic ground state as also suggested [Amitsuka and Sakakibara, 1994] in the Udilute UxThi.xRu2Si2. However, as we will discuss thoroughly in the specificheat measurements by a small amount of doping, it seems to be unlikely that we could ignore the intersite correlations in the Uconcentrated (90 %) systems. As we showed in Table 3 there are differences for Pr, and Non magnetic Th and Y doped systems exhibit a smaller Xdc values at low temperatures than the magnetic Pr doped one. Also, the different T dependence in Xdc seems to be more related to the atomic size of doped elements, i.e., Th and Pr are larger than Y (which is almost same in size with U) as shown in Tables and 33 rather than the electronic configuration and the magnetic moment of each element. Furthermore, we measured the electrical resistivities of 10 % Th, Pr, and Y doped T 'n   8.7 O E E .3 . 5.1 (1)7 2.2 3 1 (K tI;mlri )rrn nra ro arl nn n fi 1rl1 ** *3t, Eti WVI EtCtI*ItI iluttJ*ll t.ia *.~tI.. *I TT,.Th, ,N;, dl, IIIIIIIIIV II /"1' 4.9 0.2 0.7 1.2 Loglo (K) (K) for about 8.0 K < 20 K saturating at lower temperatures as shown in Figure 321. decrease resistivity at lower temperatures contradicts TCKE, where opposite behavior is expected because of Kondo scattering. Thus, the observed positive 6 seems to be related to the onset of coherence and the correlated scattering rather than a singleion scattering. [Lbhneysen et at, 1 This kind of behavior having positive 6 is observed in CeCus.9Auo 994] out of the several NFL systems, and also observed in highTc superconducting oxides over a wide temperature range leading to proposal of a marginal Fermi liquid [Varma et al., 1989]. The resistivity p for the above three systems of 10 % doped with Th, Pr, and Y varies linearly in T with a smaller positive 6 value for 0.3 K 1.0 K approaching residual resistivity (the data of p for 0.3 K 1.0 K were indistinguishable in the T and T2 scales due to the scatterings of data). To understand the origin of the NFL behavior in our systems, we doped small amount of Th in U.xThxNi2Al3 (x = 0.0025 0.005 and 0.01 ) as shown in Figure 3 The magnetic transition temperature, as may be measured by the position of the maximum in the specific heat anomaly, decreases with increasing doping at the rate of about K/%Th, while the size of the anomaly, AC, decreases by 40 % for = 0.005. Magnetic susceptibility, Xdc, for these series of samples shows a maximum at about 4.2 K (3.7 K) for x = 0.0025(0.005) as shown in Figure 323, undoped UNi2Al3. while there is a peak at about If the decreasing rate of TN is linear for Th doping, estimated K for the about 180 " 150 120 0 50 100 Figure 322 C/Tvs. T2 for UL.xThxNi2Al3, x = 0.0 (sample #2), 0.0025, 0.005, and 0.01 normalized per Umole. The suppression of magnetism is clearly seen to be achieved with extremely small doping levels. Note the divergence in C/T as T >0 is already present in yr Al 0 .I' an I *ff i h )\ f ^i nn y / i ^ ^^ rnn.aa TO^Y r ^ w a n arnrl n a i f rt rI I n an at ,+ n r I I I .1 i' \ I  0.005 0 00 0 0 o 0 v v 0.0025 v V V V , Y I, (K) In T dependence below about 3 K resulting in a more saturation than the = 0.01 which are shown in Figure 3 We also performed specific heat C measurements on Uo.9TholNi2A3 in magnetic We found about a 2 % decrease of C at H = 14 T at 1 K and almost no change for 4 10 K as shown in Figure 324. This small magnetic field dependence of C in the doped system is not surprising we remember there is almost no magnetic dependence of the AFM anomaly in the parent system, shown in Figure 312. If this system follows the TCKE, where we estimated TK ~ 70 K using Eq. (3.3), there should be an increase of C/T of order of one at 14 T and 1 K as can be seen in Figure 23c (T/TK 0.014 and H/TK ~ 0.2, where we assumed there is no change of TK in magnetic field) which was not observed. However, we can not explain our system in terms of the scaling theory suggested by Tsvelick and Reizer [1993], either. For instance, we do not see a power law dependence of Xdc as a function temperature which is necessary to obey the scaling properties. It is also interesting to note that the upturn in C/T observed at low temperatures in UixMxNi2Al3 with several doping elements, is already present above TN in the parent compound, as seen in Figures 314 and 322. The broad upturn above TN is apparently not due simply a broadened AFM transition with entropy spread out to higher temperatures due to sample inhomogeneity. Instead, we see in Figure that the maoncftirf trnroitinmn rcac p E t rlnfi1 hrylnd n it c cunnrsed tn 1wr ftmneratiuree 125 115 105 5 10 (K) concentrations. Thus, this fact indicates that the disorder interpretation [Dobrosavljevic et al., 1992] seems not to be correct, because for weak disorderas one would expect for parent compoundthe theory predicts unusual temperature dependence ultralow temperatures. Thus, we see that dilution causes a suppression to lower temperatures and results disappearance of TN for more doping. 90 % concentration, where we found the NFL behavior of C/T, p, and X at low T, seems to be still close to the magnetic instability region. Therefore, it seems to us that there may be a relatively broad crossover region between (weakly) magnetic and nonordering in which we are finding the NFL behavior in C, x, and p in the UixMxNi2AI3 (M = Th, Pr, and Y) systems. In summary, we found the new system U0.9Mo.1Ni2A3 (M = Th, Pr, and Y) which exhibits the NFL behavior in thermodynamic, transport, and magnetic properties. the resistivity being linear in temperature at least for 1.0 K Firstly, < 20 K (the Prdoped one exhibited the same behavior in a less wide range) contradicts the Fermiliquid theory. Also, the monotonic increase of resistivity with temperature for 0.3 K 20 K can not be explained by the singleion TCKE, but seems to be related to the onset of coherence and the correlated scattering. Secondly, the T mn (Th and Pr doped) and In T (Y doped) dependence in Xm were observed as a sign of the NFL behavior. The difference between  0 