SSuperconducting phases of f-electron compounds
Christian Pfleiderer ∗ Physik Department E21, Technische Universit¨at M¨unchen, D-85748 Garching, Germany
Intermetallic compounds containing f-electron elements display a wealth of superconductingphases, that are prime candidates for unconventional pairing with complex order parameter sym-metries. For instance, superconductivity has been found at the border of magnetic order as wellas deep within ferro- and antiferromagnetically ordered states, suggesting that magnetism maypromote rather than destroy superconductivity. Superconductivity near valence transitions, or inthe vicinity of magneto-polar order are candidates for new superconductive pairing interactionssuch as fluctuations of the conduction electron density or the crystal electric field, respectively.The experimental status of the study of the superconducting phases of f-electron compounds isreviewed.
Contents
I. INTRODUCTION
II. INTERPLAY OF ANTIFERROMAGNETISMAND SUPERCONDUCTIVITY X
62. The series Ce n M m In n +2 m III. INTERPLAY OF FERROMAGNETISM ANDSUPERCONDUCTIVITY IV. EMERGENT CLASSES OFSUPERCONDUCTORS Si 402. CeMX V. MULTIPLE PHASES VI. PERSPECTIVES Acknowledgments References ∗ Electronic address: christian.pfl[email protected]
I. INTRODUCTION
Superconductivity was discovered almost a centuryago. Yet, unexpected and fascinating new variants ofthis same old theme are being found at an increasingpace. This is due to great technical advances in ma-terials preparation and an increasingly more system-atic screening of new compounds. Prior to the late1970s all known superconductors could be accounted forin terms of a condensate of Cooper pairs, where theCooper pairs form due to electron-phonon interactions.With the discovery of the superfluid phases of He thisunderstanding began to change in two ways (Osheroff et al. , 1972; Vollhardt and W¨olfle, 1990). First, Heprovided an example of non-electron-phonon mediatedpairing. Second, it provided an example of a super-fluid condensate that breaks additional symmetries. Thediscovery of heavy-fermion superconductivity as a primecandidate for complex order parameter symmetries andnon-electron-phonon mediated pairing in f-electron com-pounds nearly three decades ago was long recognized asan important turning point in the history of supercon-ductivity. However, progress in heavy fermion supercon-ductivity until not long ago seemed to have been slow.In recent years especially the superconductivity in thecuprates, ruthenates, cobaltates, pyrochlores and iron-pnictides received great attention. However, a spectac-ular series of discoveries and developments in f-electronsuperconductors took place at the same time. While inthe first twelve years following the discovery of heavy-fermion superconductivity in CeCu Si only five moreheavy fermion superconductors could be identified, overtwenty five additional systems have been found in thepast fifteen years (see Fig. 1). By now over thirty sys-tems are known, about half of which were discovered inthe past five years alone. This illustrates the speed ofdevelopment the field of f-electron superconductivity haspicked up despite its long tradition.As a result there is growing appreciation that super-conducting phases of f-electron compounds frequently ex-ist at the border of competing and coexisting forms ofelectronic order. For the majority of systems, includ-ing the original heavy-fermion superconductors, an in- a r X i v : . [ c ond - m a t . s up r- c on ] M a y FIG. 1 Evolution of the total number of f-electron heavy-fermion superconductors. Systems included in this plot andcovered in this review: 1979 CeCu Si ; 1984 UBe , UPt ;1986 URu Si ; 1991 UPd Al , UNi Al ; 1993 CeCu Ge ;1996 CePd Si , CeNi Ge ; 1997 CeIn ; 2000 CeRhIn , UGe ;2001 CeIrIn , CeCoIn , URhGe; 2002 PuCoGa , PrOs Sb ;2003 Ce RhIn , PuRhGa ; 2004 CeNiGe , Ce Ni Ge , UIr,PrRu P , CePt Si, CeIrSi , CeRhSi ; 2005 PrRu Sb ; 2007UCoGe, NpPd Al , CeCoGe , CePd Al . terplay with antiferromagnetism is observed. However,there are also several examples of superconductivity thatcoexists with ferromagnetism. Further examples includesuperconductivity at the border of polar order and nearelectron localization transitions. Finally, several heavy-fermion superconductors have even been discovered withnon-centrosymmetric crystal structures and coexistentantiferromagnetic order. The large variety of systemsfound so far establishes unconventional f-electron super-conductivity as a rather general phenomenon. It alsosuggests the existence of further unimagined forms of su-perconductivity.The objective of this review is it to give a status re-port of the experimental properties of the candidatesfor unconventional f-electron superconductivity. For along time the search for a unified microscopic theoryof f-electron superconductivity has been hampered bythe large differences of the small number of known sys-tems. Even though the increasing number of systemshas allowed great progress in understanding, a criticaldiscussion of the theoretical scenarios is well beyond thelength constraints of the present review. For reviewsof selected compounds and theoretical scenarios we re-fer to (Flouquet, 2006; Flouquet et al. , 2006; Grewe andSteglich, 1991; Joynt and Taillefer, 2002; Maple et al. ,2008; Mineev and Samokhin, 1999; Sauls, 1994; Sigrist,2005; Sigrist and Ueda, 1991; Thalmeier and Zwicknagl,2005; Thalmeier et al. , 2005).The outline of this paper is as follows. The intro-duction is continued in section I, with a short accountof conventional superconductivity and its interplay withmagnetism, Fermi liquid quasiparticle interactions andadvances in materials preparation. In section II we ad-dress the interplay of antiferromagnetism and supercon-ductivity. Section III is concerned with ferromagnetism and superconductivity, while we review the properties ofemergent classes of new superconductors, discovered veryrecently, in section IV. Finally, in section V, we sum-marize evidence for multiple superconducting phases inUPt and tentative indications for such behavior in othersystems as well as for the formation of textures. The pa-per closes with a short section on the general perspectivesof this field. A. Superconductivity versus Magnetism
Superconductors derive their name from being perfectelectrical conductors. However, in contrast to ideal con-ductors superconductors display, as their second defin-ing property, perfect diamagnetism, i.e., in the supercon-ducting state sufficiently low applied magnetic fields arespontaneously expelled. Flux expulsion identifies super-conductivity as a thermodynamic phase.Following the discovery of the two defining proper-ties of superconductors, notably perfect conductivityand perfect diamagnetism by Onnes in 1911 (Onnes,1911a,b,c) and Meissner and Ochsenfeld in 1933 (Meiss-ner and Ochsenfeld, 1933), respectively, it took until 1957when Bardeen, Cooper and Schrieffer (BCS) proposeda remarkably successful theoretical framework (Bardeen et al. , 1957). There is a large number of excellent in-troductory and advanced level textbooks and review pa-pers, e.g., (de Gennes, 1989; Parks, 1969; Sigrist, 2005;Tinkham, 1969; Waldram, 1996). BCS theory identifiessuperconductivity as the quantum-statistical condensa-tion of so-called Cooper pairs, which are bound pairs ofquasiparticle excitations in a Fermi liquid. For a simpleHamiltonian describing attractively interacting quasipar-ticles in a conduction band, it is possible to show the for-mation of an excitation gap ∆ in the quasiparticle spec-trum at the Fermi level E F .A superconducting transition exists for quasiparticlesystems with both attractive and repulsive componentsof the quasiparticle interactions (Morel and Anderson,1962). For instance, in the presence of electron-phononinteractions the Coulomb repulsion of conduction elec-trons is screened and exhibits a retarded attractive inter-action component below the Debye frequency. Physicallyspeaking, the electrons avoid the bare Coulomb repulsionand attract each other in terms of a polarization tracethat decays slowly as compared with the speed of travelof the electrons. The mathematical form of the T s is es-sentially the same as for purely attractive interactions,but the Coulomb interaction enters in a renormalizedform. The same is also true when keeping track of the fullretarded solution in the Eliashberg strong-coupling for-malism (Eliashberg, 1960), which leads to the MacMillanform of T s (Allen and Dynes, 1975; MacMillan, 1968).We return to more complex quasiparticle interactions ofstrongly correlated electron systems in section I.B.The experimental characteristics of conventional su-perconductors derive from the formation of an isotropicgap at the Fermi surface. This implies, that bulk prop-erties such as the specific heat show an exponential tem-perature dependence below T s and, in the weak couplinglimit, an anomaly ∆ C/γT s = 1 .
43. At the heart of theunderstanding of the superconducting state is the for-mation of quantum mechanical phase coherence as seenin several microscopic probes. For instance, the NMRspin-lattice relaxation rate shows coherence effects likethe Hebel-Slichter peak and an exponential freezing outbelow T s (for a pedagogical discussion with examples see(Tinkham, 1969; Waldram, 1996)). The rigidity of thesuperconductivity against external perturbations is ex-pressed by the phase stiffness of superconducting conden-sate, as measured by the coherence length ξ . The lengthscale of the variations of the superconducting order pa-rameter is expressed by the Pippard or Ginzburg-Landaucoherence length.Taking into account electron-phonon coupling the re-sulting screened, retarded quasiparticle interactions areshort-ranged, representing essentially contact interac-tions. For the corresponding Cooper pair wave-function,which is composed of the product of an orbital and spin-contribution, this implies that the orbital contributionhas to be in the l = 0 channel (no angular momen-tum) and the spin part has to have spin-singlet character( s = 0, opposing spin directions). Otherwise the range ofthe attractive interaction component is shorter than theaverage distance of the electrons.Characteristic length scales that determine the way ap-plied magnetic fields suppress superconductivity are thecoherence length ξ , on the one hand, and the penetra-tion depth λ , on the other hand. If the ratio κ = λ/ξ exceeds 1 / √ = h/ e (Abrikosov, 1952). Theflux lines are organized in a lattice with a geometry thatminimizes the ground state energy. All the compoundsaddressed in this review are strong type II superconduc-tors and the morphology of the flux line lattice yieldskey information on the nature of the superconductivity(for recent work in Nb see (Laver et al. , 2006; M¨uhlbauer et al. , 2009)).Microscopically, applied magnetic fields suppress su-perconductivity by interacting either with the orbital orspin momentum. For pure orbital limiting the uppercritical field, H orbc ( T →
0) = Φ / (2 πξ ) is connectedwith the initial slope of H orbc near T s as H orbc ( T →
0) = − . dH c /dT | T s (Saint-James et al. , 1969). This is con-trasted by pure Pauli limiting of the upper critical field,which is related to T s as H P aulic ( T →
0) = 1 . T s ,where H is in T and T s in K (Chandrasekhar, 1962;Clogston, 1962). The ratio of orbital to Pauli limitingis expressed by the Maki parameter α = √ H P aulic /H orbc (Saint-James et al. , 1969). It was also noticed that thetransition at H c for pure Pauli limiting becomes firstorder below T † = 0 . T s (Ketterson and Song, 1999; Saint-James et al. , 1969).Since the early days of research in superconductivity,the effect of internal magnetic fields (cf. exchange fields)on the superconductivity was of great interest. Theo-retical work suggested that static or dynamic internalmagnetic fields would prevent superconductivity (Berkand Schrieffer, 1966; Ginzburg, 1957). In the limit of ex-treme purity and pure Pauli limiting, i.e., large values of α , a novel state was predicted to be possible, that con-sists in real-space modulations of superconductivity witha weakly spin-polarized normal state (Fulde and Ferrell,1964; Larkin and Ovchinnikov, 1965). We return to theexperimental status of this so-called FFLO phase in f-electron systems in section V.B.1.Experimentally the question for internal magneticfields in superconducting materials was at first fol-lowed up in studies of binary and pseudo-binary sys-tems with rare earth impurities (R) such as La − x R x and(Y − x R − x Os ) (Matthias et al. , 1958b). Early studiessuffered from metallurgical complexities due to cluster-ing and glassy types of magnetic order and were some-what inconclusive. They motivated, however, more de-tailed studies which led to a fairly advanced understand-ing of paramagnetic impurities in superconductors. Re-views have been given in, e.g., (Maple, 1976, 1995, 2005;White and Geballe, 1979). Overall it was accepted thatmagnetic impurities are detrimental to superconductiv-ity, while it was also appreciated that conventional super-conductivity is fairly insensitive to nonmagnetic defects(Anderson, 1959).The upshot of these studies has been, that the rate ofsuppression of T s is the highest in the middle of the rare-earth series (Maple, 1970; Matthias et al. , 1958b), consis-tent with the strongest pair breaking due to magnetic ex-change interactions (Abrikosov and Gor’kov, 1961; Her-ring, 1958; Suhl and Matthias, 1959). An exception isCe, which causes an anomalously large depression of T s due to the strong hybridization of the f-electrons withthe conduction electrons. A more detailed understand-ing of the effect of magnetic impurities on superconduc-tors requires an understanding of the properties of mag-netic moments dissolved in a non-magnetic host. In theKondo effect, the conduction electrons hybridize with themagnetic moment, eventually forming a screening cloudbelow a characteristic temperature T K , the Kondo tem-perature (for an introduction see e.g. (Hewson, 1993)).Alternatively, the moment may by quenched by low lyingcrystal fields. While the former leads to strong Cooperpair breaking, the latter reduces the effects of pair break-ing.Pioneering studies of Ce − x La x Al revealed the pres-ence of Kondo screening with a Kondo temperature T K ∼ . − x La x Al displays reentrantsuperconductivity (Maple et al. , 1972; Riblet and Winzer,1971), i.e., the superconducting transition at T s is fol-lowed by a second characteristic temperature T s < T s below which superconductivity vanishes again. The reen-trance may be understood as resulting from an increasingstrength of the pair breaking of the paramagnetic im-purities with decreasing temperature, because T K < T s (M¨uller-Hartmann and Zittartz, 1971). The strength ofpair breaking due to Kondo screening was also studied inhigh pressure experiments on La − x Ce x alloys, where thesuperconductivity vanishes in a finite pressure intervalfor x = 0 .
02 as the Kondo temperature increases underpressure (Maple et al. , 1972). As a side-effect of detailedstudies in Ce − x La x Al it was finally also recognized,that even pure CeAl displays a Kondo-effect, thus qual-ifying as the perhaps first example of a Kondo-lattice(Buschow and van Daal, 1970; van Daal and Buschow,1969; Maple, 1969). The effect of crystal electric fields(CEF) in removing the magnetic moment was studied,e.g., in the series La − x Pr x Tl , where superconductivityvanishes only slowly, because the crystal fields reduce thepair breaking strength with increasing x (Bucher et al. ,1972).In contrast to a purely competitive form of supercon-ductivity and magnetism doping studies in the seriesCe − x Gd x Ru also suggested the possibility of a coex-istence of superconductivity and magnetism in small pa-rameter regimes (Hein et al. , 1959; Matthias et al. , 1958a;Phillips and Matthias, 1961). By the late 1970s two seriesof compounds had been discovered, which display such anextremely delicate balance of superconductivity and mag-netism intrinsically, notably the series RRh B where Ris a rare earth and the Chevrel phases such as DyMo S (Bulaevskii et al. , 1985; Fertig et al. , 1977; Ishikawa andFischer, 1977; Moncton et al. , 1977). These compoundsare frequently referred to as magnetic superconductors.As a key feature T s in these systems is always larger thanthe magnetic ordering temperature.The interplay of magnetism and superconductivity isexemplified by the series Er − x Ho x Rh B , in which theonset of ferromagnetism destroys superconductivity. Ina tiny temperature interval for small x magnetic ordersucceeds in coexisting with superconductivity by forminga modulated state. This firmly suggests that supercon-ductivity and magnetism are antagonistic forms of order.However, for selected antiferromagnetic members of thisseries even a constructive interplay of magnetism and su-perconductivity could be inferred from an increase of the H c below T N . In contrast to the systems reviewed heresuperconductivity and magnetism may be viewed as re-siding in separate microscopic subsystems. Comprehen-sive reviews of this field may be found in (Fischer, 1990;Fischer and Maple, 1982; Maple and Fischer, 1982).As a remark on the side, these compounds also pro-vided first hints of the Jaccarino-Peter effect (Jaccarinoand Peter, 1962), notably an enhancement of the super-conductivity when an applied field cancels any internalmagnetic fields. In recent years further compounds havebeen discovered with a coexistence in separate subsys-tems, notably the Ruthenocuprates (Frazer et al. , 2001;Klamut et al. , 2001; Otzschi et al. , 1999) and the Boro-carbides RNi B C (R=Gd-Lu, Y) (Budko and Canfield, 2006; Mazumdar and Nagarajan, 2005).The possibility of unconventional superconducting or-der parameter symmetries had been anticipated theoreti-cally, when the superfluid phases of He were discovered;excellent reviews may be found in (Leggett, 1975; Voll-hardt and W¨olfle, 1990; Wheatly, 1975). In particular He provided a first example of a constructive interplayof superconductivity and the magnetic properties of thesystem. Theoretically it had been suggested that ferro-magnetic fluctuations may mediate superconductive pair-ing (Fay and Appel, 1980b; Layzer and Fay, 1971) andthat superconductivity may even exist in itinerant ferro-magnets (Fay and Appel, 1980a). However, for a longtime there was no evidence supporting this suggestion inreal materials.During the 1970s great advances were also made in theunderstanding of intermediate valence compounds, seee.g. (Buschow, 1979; White and Geballe, 1979). As a keyfeature nonmagnetic members of this group of materi-als exhibit enhanced Fermi liquid coefficients such as thelinear specific heat γ = C/T or quadratic temperaturedependence of the resistivity A = ∆ ρ/T . A number ofcompounds even displayed particularly strong renormal-ization effects of the Fermi liquid coefficients, like CeAl (Andres et al. , 1975). They are known as heavy-fermionsystems. Amongst the heavy fermion systems supercon-ductivity was for the first time observed in, CeCu Si (Steglich et al. , 1979). Due to the large specific heatanomaly of CeCu Si at the superconducting transitionit was immediately appreciated, that the strongly renor-malized quasiparticle excitations take part in the pair-ing. Moreover, under tiny changes of stoichiometry theground state of CeCu Si was found to become magneti-cally ordered. This vicinity to magnetic order suggestedan important role of magnetic correlations in the super-conductive pairing.The discovery of heavy fermion superconductivity cre-ated intense experimental and theoretical efforts. Forearly reviews we refer to (Grewe and Steglich, 1991; Stew-art et al. , 1984). However, in the first twelve years fol-lowing the discovery of superconductivity in CeCu Si (Steglich et al. , 1979) only 5 more heavy fermion su-perconductors were discovered (UBe (Bucher et al. ,1975; Ott et al. , 1983), UPt (Stewart et al. , 1984),URu Si (Maple et al. , 1986; Palstra et al. , 1985; Schlab-itz et al. , 1984, 1986), UPd Al and UNi Al (Geibel et al. , 1991a,b)). Because the microscopic details of thesesystems proved to be remarkably different, a unified theo-retical understanding turned out to be a great challenge. B. Roadmap to superconducting phases
In recent years several ingredients have come to lightthat prove to be almost universally important in thesearch for further examples of superconducting phasesof f-electron compounds. First, an improved appreci-ation of the quasiparticle interactions in Fermi liquids.Second, the experimental ability to tune these interac-tions in pure metallic systems in a controlled mannerby means of a non-thermal control parameter such aspressure, stress or magnetic field. Third, and most im-portant, great advances in materials preparation. In thefollowing we briefly discuss these developments.A simple plausibility argument shows, that the super-conductive pairing in heavy fermion systems is proba-bly not driven by electron-phonon interactions and thatthe order parameter is most likely unconventional, i.e.,the order parameter breaks additional symmetries. Go-ing back to the importance of retardation for electron-phonon mediated pairing and the local character of theinteraction, it is helpful to keep in mind that the speedof travel of a quasiparticle excitation in heavy fermionsystems typically is reduced by nearly three orders ofmagnitude. In turn the effects of repulsive quasiparticleinteraction components for a conventional pairing sym-metry ( l = 0 and s = 0) can no longer be avoided. How-ever, the repulsive components of the interactions maybe avoided in higher angular momentum and spin statesof the Cooper pairs.A systematic search for novel forms of superconduc-tive pairing interactions and pairing symmetries hencerequires a systematic quantitative determination of thequasiparticle interactions in the presence of strong elec-tronic correlations. Second, it requires very clean sam-ples, since unconventional pairing tends to be extremelysensitive to non-magnetic defects. As a rule of thumb, thecharge carrier mean free path needs to be substantiallylarger than the coherence length for superconductivity tooccur.Quite generally the quasiparticle interactions may beexpressed in terms of the generalized, dynamical responsefunction of the system. For instance, in systems at theborder of magnetic order this is expressed in terms of thewave vector and frequency dependent magnetic suscepti-bility χ ( (cid:126)q, ω ); for a pedagogical introduction see (Lon-zarich, 1997). Experimentally quasiparticle excitationspectra and the related interaction potentials may beexplored in quantum oscillatory studies. Careful com-parison of the experimentally observed quasiparticle en-hancements with the response function determined in,e.g., neutron scattering allows the development of a sim-ple description of the generalized quasiparticle interac-tions.A program of this kind was first systematically carriedout in the 1980s for weakly and nearly magnetic tran-sition metal compounds and selected f-electron systems.For reviews of this work we refer to (Lonzarich, 1980,1987, 1988). More recent reviews of quantum oscilla-tory studies may be found in (Onuki, 1993; Onuki andHasegawa, 1995; Settai et al. , 2007b). As an importantaspect of the early work, it became at the same time pos-sible to calculate quantitatively the magnetic orderingtemperature of weakly magnetic itinerant-electron sys-tems (Lonzarich and Taillefer, 1985; Moriya, 1985). Thispaved the way for a quantitative analysis of supercon- ducting pairing interactions in weakly ferromagnetic andantiferromagnetic compounds, see e.g. (Dungate, 1990),and eventually allowed an educated guess of which sys-tems to study (see also (Monthoux et al. , 2007)).The quasiparticle interactions were finally tuned bymeans of high hydrostatic pressures in pure samples.The experiments served to clarify two questions. First,to identify possible examples of magnetically mediatedsuperconductivity (for early attempts see e.g. (Cordes et al. , 1981)). Second, to investigate the nature of themetallic state in the vicinity of a quantum critical point.Here we briefly note that quantum phase transitions,quite generally, are defined as phase transitions that aredriven by quantum fluctuations. In practice this meansthat quantum phase transitions are zero temperature sec-ond order phase transitions. In recent years this defini-tion has been relaxed somewhat and zero temperaturephase transition in general are referred to as quantumphase transitions (Pfleiderer, 2005). It transpires thatquantum phase transitions represent an extremely richfield of condensed matter physics. For reviews we referto (Belitz et al. , 2005; Hertz, 1976; v. L¨ohneysen et al. ,2007; Monthoux et al. , 2007; Sachdev, 1999; Stewart,2001, 2006; Vojta, 2003).Besides the advances in understanding the metallicstate in the presence of strong electronic correlations,great advances have also been achieved in the experi-mental techniques (for a recent review on the 5f states inactinides see (Moore and van der Laan, 2009)). Studiesunder extreme conditions such as very low temperatures,high pressures and high magnetic fields are now routinelyavailable in numerous laboratories.Probably most important are, however, major im-provements in materials preparation. For instance, ma-jor improvements have been achieved by means of thepurification of the starting elements. Electro-transportof, e.g. uranium, under ultra-high vacuum was foundto be extremely efficient in removing impurities such asFe and Cu (Fort, 1987; Haga et al. , 1998). Electro-transport in combination with annealing under ultra-highvacuum has also been used to promote the formation oflarge single-crystal grains and improve the sample qual-ity (Haga et al. , 2007; Matsuda et al. , 2008; Schmidt andCarlson, 1976). For the growth of high vapor pressurecompounds a closed crucible annealing technique was de-veloped (Assmus et al. , 1984). In many materials thecombination of ultra-high vacuum with an inert gas at-mosphere is sufficient to obtain large high-quality singlecrystals (McDonough and Huxley, 1996). This cannotbe underestimated given that both the rare earth andactinide elements readily react, especially with oxygen,hydrogen and nitrogen. Advances in the understandingof the phase diagrams of binary and ternary compoundshas motivated the improvement and extensive use of tech-niques like traveling-solvent float-zoning or the controlleduse of flux methods, e.g., in the skutterudites or the seriesof Ce n M m In n +2 m compounds. Finally in recent yearsan increasing number of groups explores the use of opti-cal floating-zone furnaces for the growth of intermetalliccompounds, see e.g. (Souptel et al. , 2007). For examplelarge single crystals have been grown of UNi Al (Mi-halik et al. , 1997) and of URu Si (see e.g. (Pfleiderer et al. , 2006)). It is expected that this technique will playa very important role in the future.A frequent objection in materials preparation concernsthe relative importance of the various aspects. For in-stance, it is believed that the accuracy at which per-fect stoichiometry can be achieved generally outweighsany efforts put into the purification of the starting ele-ments. Empirically this is contrasted by the impressivelist of unusual phenomena such as unconventional su-perconductivity that have been discovered in ultra-purecompounds. The perhaps most important challenge inmaterials preparation is the lack of methods for charac-terization. Generally speaking, high sample quality isproven by the combination of standard characterization(x-ray diffraction, microprobe, etc.) plus the physicalproperties themselves. This shows that despite all of thetechnical achievements the growth of high quality sin-gle crystals continues to require great physical intuition,systematic work and a fair bit of luck. II. INTERPLAY OF ANTIFERROMAGNETISM ANDSUPERCONDUCTIVITY
In this section we review the interplay of antiferromag-netism and f-electron superconductivity. Section [II.A],reviews systems where superconductivity emerges at theborder of itinerant antiferromagnetism. In particularproperties of the series CeM X and Ce n M m In n +2 m areaddressed. Section [II.B] is concerned with supercon-ductivity in antiferromagnetic compounds. This includeslarge moment systems like UPd Al and UNi Al as wellas small moment systems like UPt and URu Si . A. Border of antiferromagnetism
1. The series CeM X The discovery of heavy fermion superconductivityin CeCu Si (Steglich et al. , 1979) marked the start-ing point of unconventional superconductivity (Greweand Steglich, 1991; Sparn et al. , 2006; Stewart et al. ,1984; Thalmeier et al. , 2005). CeCu Si crystallizes inthe tetragonal ThCr Si crystal structure as summa-rized in table I. The heavy-fermion superconductivity inCeCu Si generated great interest in the isostructuralseries of CeM X compounds, where M is a transitionmetal (M=Cu, Au, Rh, Pd, Ni) and X=Si or Ge. Be-cause most members of this series exhibit antiferromag-netic order (Grier et al. , 1984; Thompson et al. , 1986), itrepresented a major break-through for the entire field,when superconductivity was discovered in CeCu Ge ,CeRh Si , and in particular CePd Si , as well as incipi-ent superconductivity in CeNi Ge . For a summary see FIG. 2 Variations of the tetragonal BaAl crystal structure.The ThCr Si structure is frequently found amongst CeM X compounds reviewed in section II.A.1. The BaNiSn crys-tal structure, which lacks inversion symmetry, is typical ofthe CeMT compounds reviewed in section IV.A. Amongstf-electron systems with the non-centrosymmetric CaBe Ge structure no compounds are known that exhibit supercon-ductivity. Plot taken from (Kimura et al. , 2007b). also table I. Being a derivative of the BaAl parent struc-ture, the ThCr Si structure is intimately related to theBaNiSn and CaBe Ge types of structures as illustratedin Fig. 2. A surprise in recent years was the discovery ofsuperconductivity in several Ce-based compounds withthe non-centrosymmetric BaNiSn structure, because itwas believed that triplet superconductivity cannot existin crystal structures lacking inversion symmetry. For anaccount of this work we refer to section IV.A.2. Interest-ingly no superconductivity has so far been found amongstthe CaBe Ge relatives of the ThCr Si series, which isalso non-centrosymmetric. a. CeCu Si & CeCu Ge The ground state properties ofCeCu Si are extremely sensitive to the precise Cu con-tent, which may be controlled by an annealing procedureunder Cu vapor (Assmus et al. , 1984). Samples withheavy-fermion superconductivity, antiferromagnetism ora combination thereof may be obtained, which are re-ferred to as (S), (A) or (AS), respectively. For Cu de-ficient samples the SDW order is stabilized and super-conductivity destroyed, while the SDW is destabilizedand the superconductivity stabilized for Cu excess. Onthe level of changes of less than a few % of composi-tion achieved under Cu annealing, it is believed that thechanges of properties originate mostly in changes of unitcell volume (Trovarelli et al. , 1997). This may be inferredalso from doping with Ge which, being larger than Si, sta-bilizes the SDW, while hydrostatic pressure destabilizesthe SDW and stabilizes the superconductivity (Krimmeland Loidl, 1997; Trovarelli et al. , 1997). In the followingit proves to be convenient to address S-type samples first.Quite generally the normal state of CeCu Si is char-acteristic of a heavy Fermi liquid with C/T = γ =1 J / mol K and an equally enhanced Pauli susceptibility.The heavy fermion state develops in a crystal electric fieldground state Kramers doublet and a first and second ex- TABLE I Key properties of superconductors in the series CeM X (M: Cu, Pd, Rh, Ni; X: Si, Ge) and various miscellaneousCe-based systems. Missing table entries may reflect more complex behavior discussed in the text. References are given inthe text. Critical field values represent extrapolated T = 0 values. (AF: antiferromagnet, SC: superconductor, ISC: incipientsuperconductor) CeCu Si CeCu Ge CePd Si CeRh Si CeNi Ge CeNiGe Ce Ni Ge CePd Al structure tetragonal tetragonal tetragonal tetragonal tetragonal orthorh. orthorh. tetragonaltype ThCr Si ThCr Si ThCr Si ThCr Si ThCr Si SmNiGe U Co Si ZrNi Al space group I4/mmm I4/mmm I4/mmm I4/mmm I4/mmm Cmmm Ibam I4/mmm a (˚A) 4.102 4.186 4.223 4.092 4.150 21.808 9.814 4.156 b (˚A) 9.930 10.299 9.897 10.181 9.842 4.135 11.844 4.156 c (˚A) 9.930 10.299 9.897 10.181 9.842 4.168 5.963 14.883 c/a T N (K) 0.8 4.15 10 36, 25 - 5.5 5.1, 4.5 3.9, 2.9 (cid:126)Q (0.22, 0.22, 0.53) (0.28, 0.28, 0.53) (0.5, 0.5, 0) (0.5, 0.5, 0) - - - -- - - (0.5, 0.5, 0.5) - - - - µ ord ( µ B ) 0.1 1 0.62 1.42, 1.34 - 0.8 0.4 - γ (J / mol K ) 1 0.062 0.027 - - 0.09 0.056 T max s (K) 0.7, 2.5 0.64 0.4 0.42 0.3, 0.4 0.45 0.26 0.57 p max s (kbar) 0, 30 70 28 10 0, 18 70 36 108∆ C/γ n T s H abc (T) 0.45 2 0.7 - - 1.55 0.7 0.25 ddT H abc (T/K) - -11 -12.7 - - -10.8 - -1.04 H cc (T) - - 1.3 0.28 - - - - ddT H cc (T/K) - - -16 -1 - - - - ξ ab (˚A) - 90 300 - - 100 210 - ξ c (˚A) - - 230 340 - - - -year of disc. 1979 1993 1996 1996 1996 2004 2005 2007 cited doublet at 12.5 meV and 31 meV (Horn et al. , 1981).S-type samples of CeCu Si display T s = 0 . C/γT s = 1 . H c ≈ .
45 T (Rauchschwalbe et al. , 1982).The leading order temperature dependence of the spe-cific heat C ( T ), thermal expansion α (Lang et al. , 1991)and penetration depth λ (Gross et al. , 1988) vary as T characteristic of line nodes. NMR and NQR show theabsence of a Hebel-Slichter peak and a power law depen-dence of the spin-lattice relaxation rate, also suggestingline nodes (Ishida et al. , 1999; Kawasaki et al. , 2004).The magnetic phase diagram of CeCu Si for magneticfield applied along the a -axis in the basal plane is fairlycomplex (Bruls et al. , 1990, 1994; Steglich et al. , 2001).Ultrasound and thermal expansion measurements earlysuggested the presence of two spin density wave phases,referred to as A- and B-phase, respectively. Magneticfield suppresses at first the superconductivity above H c ,where the A-phase is restored. The B - T boundary ofthe A-phase is reminiscent of H c ( T ), as if the A-phase encompasses the superconductivity. Above a critical field H c ≈ . Si wasmissing for nearly 25 years. Progress was made onlyrecently by tracking systematically the incommensuratespin-density wave order of CeCu Ge as a function ofincreasing Si-content. It was found that the orderingwave vector changes little as function Si content, yieldinga value of (cid:126)Q = (0 . , . , . / (r . l . u) in AS sam-ples of CeCu Si (Stockert et al. , 2004). The neutronscattering studies identified an incommensurate spin-density wave in the A-phase with a small ordered moment µ ord ≈ . µ B per Ce site in CeCu Si , which evolvesfrom the antiferromagnetic order in CeCu Ge continu-ously with increasing Si-content. The ordering wave vec-tor agrees thereby with the nesting wave vector found inFermi surface calculations (Zwicknagl and Pulst, 1993).For the (AS) samples antiferromagnetism and supercon-ductivity are mutually exclusive and separated by a firstorder phase transition (Sparn et al. , 2006).As a function of pressure T s in S-type CeCu Si in-creases around 2 GPa and enters a plateau of 2.25 Kabove 2.5 GPa, followed by a moderate decreases witha small shoulder around 7 GPa as shown in Fig. 3 (Bel-larbi et al. , 1984; Thomas et al. , 1993). The unusualpressure dependence early on suggested that T s ( p ) maybe explained in terms of two or more pairing interac-tions. The discovery of superconductivity in CeCu Ge (Jaccard et al. , 1992) and CeRh Si (Movshovich et al. ,1996), but in particular in CePd Si (Mathur et al. ,1998), underscored the idea that superconductivity inCeCu Si is somehow related to the magnetic proper-ties. At the border of the A-phase NFL properties ofthe normal metallic state were observed characteristic ofquantum critical spin fluctuations, where ∆ ρ ( T ) ∝ T / and C/T = γ = γ − α √ T (Gegenwart et al. , 1998).Under moderate doping with Ge, which introducespair breaking defects, the superconducting phase dis-integrates into two domes as shown in Fig. 3 (Holmes et al. , 2004; Yuan et al. , 2004). Based on these studiestwo pairing interactions were proposed: antiferromagnet-ically mediated pairing in the vicinity of the SDW andpairing by charge density fluctuations in the vicinity of avalence transition at high pressures. The latter consistsin fluctuations that originate in a Ce to Ce +4 change ofvalence, where the 4f electron is delocalized in the highpressure Ce state. We also address the question ofcharge fluctuation mediated pairing in section IV.B.1.The superconducting pairing symmetry has been re-visited with the knowledge of the incommensurate SDWand band structure calculations based on the renormal-ized LDA (Thalmeier et al. , 2005). A superconductingd-wave singlet state, d x − y , and the SDW order arehere treated as two competing ordering phenomena. Themodel accounts for the change from the incommensurateSDW order in the A-phase to superconductivity, whereboth order parameters have the Γ symmetry imposedby the crystal electric fields.Analogies of the thermopower in superconductingsamples of CeCu Si with CeCu Ge finally motivatedhigh pressure experiments in CeCu Ge (Jaccard et al. ,1992). At ambient pressure CeCu Ge orders anti-ferromagnetically below T N = 4 .
15 K into an incom-mensurate sinusoidally modulated structure with (cid:126)Q =(0 . , . , . ± .
001 and an ordered moment µ ord = 0 . µ B (Knopp et al. , 1989; Krimmel et al. , 1997).The low temperature properties develop in a crystal elec-tric field environment of a ground state doublet and afirst excited quartet at 19.1 meV. The metallic state ofCeCu Ge is moderately enhanced.Under pressure the N´eel temperature of CeCu Ge de-creases and vanishes at p N ≈
70 kbar (Jaccard et al. ,1992). Superconductivity appears above the critical pres-sure, where T s ≈ .
64 K is only weakly pressure depen-dent and extends over a wide range, where H c ≈ dH c /dT = −
11 T / K. This sug-gests a coherence length of order ξ = 90 ˚A. The struc-tural similarity and lack of pressure dependence of T s FIG. 3 Temperature versus pressure phase diagram ofCeCu Si − x Ge x . The pressure dependence of the supercon-ducting transition temperature, here denoted T c , in S-typeCeCu Si exhibits a plateau between 20 and 70 kbar (blackdots). Weak impurity scattering in moderately Ge dopedCeCu Si decomposes the superconductivity into two domes,one (red) at the border of antiferromagnetism and the other(green) at a presumed valence transition. Plot taken from(Thalmeier et al. , 2005), representing a compilation of severalstudies as described in the text. suggested an intimate similarity with CeCu Si . In factevidence for a valence transition at ∼
150 kbar, where T s is largest, has been inferred from x-ray diffraction (On-odera et al. , 2002). b. CePd Si & CeNi Ge The intense studies of thequasiparticle interactions in weakly ferromagnetic tran-sition metal compounds and selected f-electron systemsmentioned above (Lonzarich, 1980, 1987, 1988) resultedin quantitative estimates of magnetically mediated super-conductivity at the border of weak ferromagnetism. Thismotivated detailed high pressure studies in MnSi (Pflei-derer et al. , 1993, 2001a, 1997b, 2004) and related com-pounds. It inspired also studies of the suppression of an-tiferromagnetism under pressure of the isostructural andisoelectronic siblings CePd Si and CeNi Ge reviewedin the following.At ambient pressure CePd Si may be described asintermediate valence system. At high temperatures theresistivity varies weakly with temperature, followed bya rapid decrease below ∼
50 K and a sharp drop at theonset of antiferromagnetic order at T N ≈
10 K. The an-tiferromagnetic order in CePd Si consists of alternat-ing ferromagnetic sheets with (cid:126)Q = (1 / , / ,
0) wherethe moments are oriented along [110], i.e., they residein the tetragonal basal plane (Grier et al. , 1984). Themagnetism has been interpreted as local moment like,with a reduced ordered moment µ ord = 0 . µ B in thecrystal field environment (van Dijk et al. , 2000). TheCEF level scheme of the localized Ce electronshave been determined from the susceptibility and inelas-tic neutron scattering as a sequence of three Kramersdoublets: Γ (1)7 (0), Γ (19 meV), Γ (2)7 (24 meV). The metal-lic state may be described as a Fermi liquid with a mod-erately enhanced value of γ = 0 .
062 J / mol K (Steeman et al. , 1988).Under pressure T N in CePd Si decreases and vanisheslinearly at p c ≈
28 kbar (Grosche et al. , 1996; Mathur et al. , 1998; Thompson et al. , 1986). Superconductivityhas been observed in the immediate vicinity of p c with amaximum of T s ≈ . et al. , 2001). The gradualdecrease of T N with pressure suggested that the antifer-romagnetic order vanishes continuously at p c . This hasbeen confirmed more recently in neutron scattering ex-periments of the staggered magnetization (Kernavanois et al. , 2005). The expected abundance of quantum crit-ical spin fluctuations near p c is consistent with the tem-perature dependence of the electrical resistivity, whichdisplays a power law dependence ∆ ρ ∼ T . over a widertemperature range (Grosche et al. , 1996; Mathur et al. ,1998). In the context of these fluctuations it has beensuggested that the fluctuations in CePd Si exhibit a re-duced dimensionality.Based on its vicinity to a quantum critical point the su-perconducting pairing interaction was attributed to theexchange of antiferromagnetic spin fluctuations (Grosche et al. , 1996; Mathur et al. , 1998). Several pieces of ev-idence suggest unconventional pairing, where a d-wavestate appears to be the most promising candidate. Theupper critical field and its initial variation are largeand anisotropic, where H cc = 1 . dH cc /dT = −
16 T / K and H abc = 0 . dH abc /dT = − . / K(Sheikin et al. , 2001). These values suggest anisotropicPauli limiting, where weak or strong coupling behaviorcannot be distinguished unambiguously. The correspond-ing coherence lengths are quite short with ξ ab = 300 ˚Aand ξ c = 230 ˚A.The search for a compound in the CeM X series withlattice parameters and electronic structure at ambientpressure that are akin to CePd Si near p c motivated fur-ther detailed studies of CeNi Ge (publication of thesestudies was delayed for a long time and have been re-viewed in (Grosche et al. , 2000)). The temperature ver-sus pressure phase diagram of single-crystal CeNi Ge asdetermined in resistivity measurements is shown on theright hand side of Fig. 4. The phase diagram is dominatedby a non-Fermi liquid form of the resistivity at ambientpressure and indications of incipient superconductivitybelow T s ∼ . Ge at ambient pressure displays a genuinenon-Fermi liquid ground state (Gegenwart et al. , 1999).In particular the specific heat shows a logarithmic diver-gence C/T ∼ ln T /T and the susceptibility a square rootdivergence χ ∼ √ T for T → FIG. 4 Combined temperature versus pressure phase diagramof the isostructural, isoelectronic pair of systems CePd Si and CeNi Ge , where superconductivity is observed at theborder of antiferromagnetism and at low and high pressures,respectively. Plot taken from (Grosche et al. , 2000). Neutron scattering in CeNi Ge established high-energy spin fluctuations with a characteristic energyof 4 meV at an incommensurate wave-vector (cid:126)q =(0 . , . , .
5) (F˚ak et al. , 2000). The wave-vector isin remarkable agreement with the ordering wave-vectorof the spin density wave in CeCu Si and CeCu Ge .The spin fluctuations are quasi-two dimensional, charac-teristic of a sine-modulated structure with the magneticmoments in the [110] plane. With decreasing temper-ature no critical slowing down of the high-energy spinfluctuations in CeNi Ge is observed.Another surprise in CeNi Ge was the observation ofadditional hints of superconductivity at high pressure(Grosche et al. , 2000, 1997b). The origin of this super-conductivity could not be related to a particular instabil-ity in the spirit of a quantum phase transition, where ananomaly of unknown origin in the normal state resistivitywas denoted T x . One possibility is a valence transitionlike that considered in CeCu Si , but this has not beenexplored further. The evidence for superconductivity inCeNi Ge is purely based on the resistivity, while no ev-idence for bulk superconductivity has been found in thesamples studied to date. c. CeRh Si A pressure induced transition from an an-tiferromagnetic ground state to superconductivity existsalso in CeRh Si (Movshovich et al. , 1996). The obser-vation of superconductivity in this system is remarkable,because it occurs at a fairly pronounced first order quan-tum phase transition that may be related to the delocal-ization of the 4f electron. The properties of CeRh Si have been reviewed in (Settai et al. , 2007b).At ambient pressure CeRh Si orders antiferromag-netically below T N ≈
36K (Thompson et al. , 1986).Neutron scattering establishes an ordering wave vec-tor (cid:126)Q = (1 / , / ,
0) with the moments aligned along[1 , ,
0] (Kawarazaki et al. , 2000). The single- (cid:126)Q structure0changes into a four- (cid:126)Q structure below T N ≈
24 K de-scribed by two ordering wave vectors, (cid:126)Q = (1 / , / , (cid:126)Q = (1 / , / , / . µ B at the corner site of the tetragonal structure and1 . µ B at the body-centered Ce site. The size of theordered moment is consistent with CEF-split localized4f state of the Ce-atom, which when taken togetherwith the entropy released at T N , ∆ S ( T N ) ≈ R ln 2 sug-gests a CEF Kramers doublet. As a function of magneticfield the antiferromagnetism is suppressed above ∼
26 T(Settai et al. , 1997). The metallic state is describedby a weakly enhanced linear term in the specific heat γ = 0 .
027 J / mol K (Graf et al. , 1997) and a quadratictemperature dependence of the resistivity (Araki et al. ,2002a; Grosche et al. , 1997a; Ohashi et al. , 2002).As a function of pressure both T N and T N decreaseand vanish at p N = 10 kbar and p N = 6 kbar, respec-tively (Kawarazaki et al. , 2000). A narrow dome of su-perconductivity emerges precisely at p N , where T max s ≈ .
42 K (Movshovich et al. , 1996) with H c = 0 .
28 T and dH c /dT = − / K for the c-axis, corresponding to acoherence length of the order ξ c ≈ T N , whereboth drop fairly abruptly at p N . The specific heat coef-ficient γ increases to 0 .
08 J / mol K at p N and graduallydecreases at higher pressures (Graf et al. , 1997), whilethe resistivity exhibits a T resistivity at all pressures,where the T coefficient tracks the pressure dependenceof γ consistent with the Kadowaki-Woods ratio (Araki et al. , 2002a; Grosche et al. , 1997a; Ohashi et al. , 2002).These features early on suggested a first order transi-tion at p N . Unambiguous evidence for a first ordersuppression of antiferromagnetism was obtained in quan-tum oscillatory studies as a function of pressure (Araki et al. , 2001, 2002b). More specifically, from the Fermisurface sheets observed it was concluded that the 4f elec-tron changes discontinuously from a local to an itinerantstate at p N . This scenario has received further supportin recent studies of the thermal expansion under pressure(Villaume et al. , 2007).
2. The series Ce n M m In n +2 m The series Ce n M m In n +2 m with M=Co, Ir, Rh displaysheavy fermion superconductivity with very high transi-tion temperatures, as compared to other Ce-based sys-tems. The systems of interest are summarized in tableII. This suggests that a reduction from 3 to 2 dimensionsis favorable to superconducting pairing. The supercon-ductivity in these systems appears to be tied to the anti-ferromagnetic order, where similarities with the cuprateshave been pointed out. The interplay of magnetismwith superconductivity includes thereby several tenta-tive quantum critical points under pressure and magneticfield, which are all of general interest. There is finallystrong evidence for the formation of a FFLO state in FIG. 5 Depiction of the structural series Ce n M m In n +2 m .The infinite layer system CeIn is shown on the left, the sin-gle layer systems are shown in the middle, the double layersystems, which are intermediate to the infinite and the sin-gle layer systems are shown on the right. Also indicated aretypical muon stopping sites in the double layer system. Plottaken from (Morris et al. , 2004) CeCoIn . Status reports on the series of Ce n M m In n +2 m compounds have been given in (Sarrao and Thompson,2007; Settai et al. , 2007b).For a more detailed review it is helpful to begin withthe crystal structure of the series Ce n M m In n +2 m . CeIn crystallizes in the cubic Cu Au structure, space groupPm3m, with a lattice constant a = 4 .
690 ˚A. The tetrago-nal crystal structure of the series Ce n M m In n +2 m maybe derived from the cubic parent compound CeIn interms of n -fold layers of CeIn separated by m -fold lay-ers of MIn . For the single-layer compounds n = m = 1(CeMIn ) one layer of MIn is added while in the double-layer compounds n = 2, m = 1 (Ce MIn ) a single layerof MIn is added for every two layers of CeIn . Withinthis general scheme CeIn may therefore be referred toas ∞ -layer system ( n = ∞ , m = 0).The low temperature properties of Ce n M m In n +2 m de-velop in a crystal electric field scheme that is intimatelyrelated for all members of the series. For CeIn the CEFssplit the J = 5 / ground state doubletand a Γ quartet at around 12 ˙meV (Benoit et al. , 1980;Christianson et al. , 2004; Groß et al. , 1980; Lawrenceand Shapiro, 1980; Murani et al. , 1993). For the seriesCeMIn the quartet is further split into two Γ and Γ Kramers doublets, where values of the first and secondenergy levels are given in table II (Christianson et al. ,2004).The series Ce n M m In n +2 m exhibits metallic behaviorwith a fairly weak temperature dependence of the resis-tivity at high temperatures. With decreasing tempera-ture the resistivity decreases monotonically with a shoul-der around 50 to 100 K before decreasing drastically toa very low residual value of a few µ Ωcm. The normalstate resistivity and magnetic anisotropy for the single-and double-layer series are weakly anisotropic by a factorof two. The susceptibility displays a strong Curie-Weiss1
TABLE II Key properties of the series Ce n M m In n +2 m and Pu- and Np-based heavy-fermion superconductors. Missing tableentries may reflect more complex behavior discussed in the text. References are given in the text. Values of H c are extrapolatedfor T → n M m In n +2 m CeIn CeCoIn CeRhIn CeIrIn Ce RhIn PuCoGa PuRhGa NpPd Al structure cubic tetragonal tetragonal tetragonal tetragonal tetragonal tetragonal tetragonalspace group Pm 3m P4/mmm P4/mmm P4/mmm P4/mmm P4/mmm P4/mmm I4/mmm a (˚A) 4.690 4.614 4.652 4.668 4.665(1) 4.2354 4.3012 4.148 c (˚A) - 7.552 7.542 7.515 12.244(5) 6.7939 6.8569 14.716 c/a B (GPa) 67 . ± . . ± . . ± . . ± . . ± . dB /dp . ± . . ± .
41 5 . ± .
62 5 . ± .
58 3 . ± .
31 - - - κ a (10 − GPa − ) 4 . ± .
13 4 . ± .
08 3 . ± .
08 3 . ± .
06 4 . ± .
04 - - - κ c (10 − GPa − ) 4 . ± .
13 3 . ± .
16 4 . ± . . ± .
08 4 . ± .
11 - - -CEF scheme (Γ , Γ ) (Γ , Γ , Γ ) (Γ , Γ , Γ ) (Γ , Γ , Γ ) - - - -∆ , ∆ (meV) 12 8.6, 25 6.7, 29 6.9, 24 - - - -state AF, SC SC AF, SC SC AF, SC SC SC SC T N (K) 10.2 - 3.8 - 2.8, 1.65 - - - (cid:126)Q ( , , ) - ( , , , ,
0) - - - µ ord ( µ B ) 0.48 - 0.37 - 0.55 - - - µ aeff ( µ B ) - - - - - 0.75 0.8 3.22Θ aCW (K) - - - - - - - -42 µ ceff ( µ B ) - - - - - 0.75 0.8 3.06Θ aCW (K) - - - - - - - -139 γ (J / molK ) 0.14 - 0.4 0.72 0.4 0.077 0.07 0.2 p N (kbar) 25 - 17 - ∼
25 - - - T s (K) 0.19 ( p N ) 2.3 2.12 ( p N ) 0.4 1.1 18.5 8.7 4.9∆ C/γ n T s - 4.5 0.36 0.76 - 1.4 0.5 2.33 H abc (T) 0.45 11.6-11.9 - 1.0 5.4 - 27 3.7 ddT H abc / (T/K) -3.2 -24 - -4.8 -9.2 -10 -3.5 -6.4 H cc (T) 0.45 4.95 10.2 0.49 - - 15 14.3 ddT H cc /T (T/K) -2.5 -8.2 -15 -2.54 - -8 -2 -3.1 dT s /dt (K / month) - - - - - -0.24 -0.39 - ξ ab (˚A) 300 82 57 260 - - 35 - ξ c (˚A) - 53 - 180 77 - 45 - κ GL,a (˚A) - 108 - - - - - - κ GL,c (˚A) - 50 - - - - - - κ GL (˚A) - - - - - 32 - 28discovery of SC 1997 2001 2000 2001 2003 2002 2003 2007 temperature dependence, where the effective fluctuatingmoment for the easy axis corresponds to the free Ce ion. The specific heat is characteristic of strong elec-tronic correlations with a strongly enhanced electroniccontribution. However, closer inspection shows that thetemperature dependence of these electronic contributionsare more complex and typical of non-Fermi liquid behav-ior, as discussed below. a. CeIn We begin with the cubic system CeIn , whichdisplays a strikingly simple temperature versus pressurephase diagram shown in Fig. 6. Here the superconduc-tivity forms a well-defined dome around an antiferro-magnetic QCP. This makes CeIn an important pointof reference for those systems in the series that are moretwo-dimensional.The properties of CeIn are typical of a valence fluc-tuating compound, i.e., by comparison to traditionalheavy fermion systems they are moderately enhancedwith γ = 0 .
14 J / mol K . The characteristic spin fluc-2tuation temperature is fairly high T SF = 50 −
100 K(Lawrence, 1979; Morin et al. , 1988). At ambient pres-sure CeIn orders antiferromagnetically below a N´eeltemperature T N = 10 . (cid:126)Q = (1 / , / , / et al. , 1980; Lawrence andShapiro, 1980). The zero temperature ordered moment µ ord ≈ . µ B is reduced as compared to the value of ∼ . µ B , expected in the CEF ground state given by aΓ doublet. It is also reduced as compared to the Curie-Weiss moment. This is typical of weak itinerant mag-netism, where inelastic neutron scattering shows antifer-romagnetic magnons as well as quasielastic and crystalfield excitations (Knafo et al. , 2003).Under hydrostatic pressure the N´eel temperature inCeIn decreases and vanishes continuously at p N ≈
25 kbar (Morin et al. , 1988) consistent with a QCP(Fig. 6). The temperature dependence of the electricalresistivity changes from a quadratic temperature depen-dence at ambient pressure to ∆ ρ ∼ T . in a narrow in-terval near p N (Knebel et al. , 2001; Walker et al. , 1997).This suggests scattering of the charge carriers by an-tiferromagnetic quantum critical fluctuations. In fact,CeIn is one of the very few systems for which the pres-sure and magnetic field dependence of the resistivity isin excellent agreement with the predictions of an anti-ferromagnetic QCP (Hertz, 1976; Millis, 1993). The ex-istence of a QCP is contrasted by In-NQR measure-ments, which show that the spin lattice relaxation rate1 /T T ∼ constant near p N as expected of a Fermi liq-uid (Kawasaki et al. , 2001). Quantum oscillatory stud-ies through p N further establish a reconstruction of thetopology of the Fermi surface, interpreted as localizedto delocalized transition of the 4f-electrons (Settai et al. ,2005). As p c is approached the cyclotron effective massbecomes strongly enhanced for at least one major Fermisurface sheet reaching m ∗ = 60 m .In pure samples of CeIn with residual resistivities be-low 1 µ Ωcm the QCP is surrounded by a narrow domeof superconductivity, which exhibits a maximum T s ≈ .
22 K (Walker et al. , 1997). A detailed study up to100 kbar with a different set of samples and pressurecells showed that the phase diagram is rather robustand highly reproducible (Knebel et al. , 2001). Undermagnetic field T s initially decreases with dH c /dT = − . / K, where H c ( T →
0) = 0 .
45 T, both character-istic of heavy fermion superconductivity (Knebel et al. ,2001; Onuki et al. , 2004). The upper critical field maybe accounted for in a strong-coupling framework in theclean limit, where the coherence length ξ = 300 ˚A andthe charge carrier mean free path l = 2000 ˚A.The location of the superconducting dome at the bor-der of antiferromagnetic order, the evidence for quantumcritical fluctuations in the resistivity and the sensitivityof the superconductivity to sample purity (Knebel et al. ,2001; Walker et al. , 1997) provide circumstantial evidenceof unconventional pairing. Microscopically this question FIG. 6 Temperature versus pressure phase diagram of CeIn . T M denotes the coherence maximum in the resistivity, T N theN´eel temperature, T c the superconducting transition temper-ature and T the upper boundary of the regime with Fermiliquid resistivity. Plot taken from (Knebel et al. , 2001). has been explored in NMR and NQR studies. The spin-lattice relaxation rate 1 /T lacks a Hebel-Slichter peakbut the low value of T s did not permit to determinethe temperature dependence below T s (Kawasaki et al. ,2002). From a theoretical point of view it has been ar-gued that the antiferromagnetic quantum critical spinfluctuations may provide a pairing interaction consistentwith the size of T s (Mathur et al. , 1998). A more de-tailed theoretical analysis suggests that the gap symme-try due to pairing by antiferromagnetic fluctuations near (cid:126)Q = (111) is either d x − y or d z − r (Fukazawa andYamada, 2003). b. Introduction to CeMIn We next turn to the single-layer systems in the series of Ce n M m In n +2 m ( n = m =1). Key properties are summarized in table II, where ref-erences to the original publications may be found in thetext. Much of the appeal about this series is based onthe sequence Co → Rh → Ir representing isovalent substitu-tions. In this order the unit cell volume increases, whilethe c/a ratio of the lattice constants decreases.Electronic structure calculations show that the Fermisurface in all three systems is highly two-dimensionalwith several cylindrical sheets, even though the electricalresistivity and magnetic susceptibility are not stronglyanisotropic (see e.g. (Settai et al. , 2001)). Band structurecalculations suggest, as an important aspect for under-standing the evolution of the physical properties withinthis series, that the transition metal element affects theelectronic properties only indirectly (Sarrao and Thomp-3son, 2007). This may be related to the Ce atoms andthe transition metal atoms residing in different crystal-lographic planes, which may also explain why substitu-tional doping provides a comparatively controlled ap-proach to tuning the ground state properties withoutmetallurgical segregation and excessive effects of disor-der (Pagliuso et al. , 2002a,b,c, 2001; Zapf et al. , 2001).The presentation proceeds as follows. We begin withthe general phase diagrams of CeCoIn , CeRhIn andCeIrIn and discuss the tentative evidence for QCPs.This is followed by a discussion of the interplay of anti-ferromagnetism and superconductivity and the evidencefor unconventional superconductivity. The section con-cludes with a brief discussion of possible analogies withthe cuprates. c. CeCoIn CeCoIn is a superconductor with a recordhigh value T s = 2 . et al. , 2001b). For the c-axis H cc = 4 .
95 Tand for the ab-plane H abc = 11 . H c may be accounted for by the effective mass model(Ikeda et al. , 2001; Petrovic et al. , 2001b). Before re-viewing the superconducting state of CeCoIn it is help-ful to consider the normal state properties, which arein many ways anomalous. The electrical resistivity ofCeCoIn varies as ρ ( T ) = ρ + a (cid:48) T (Sidorov et al. , 2002)up to ∼ T s . Taking into account CEF contri-butions, the normal state electronic specific heat variesas C/T ∝ − ln T , and the c-axis susceptibility divergesas χ ∝ T − . , while the basal-plane susceptibility is es-sentially constant, χ − ∝ a + bT . (Kim et al. , 2001a;Petrovic et al. , 2001b). These normal state non-Fermiliquid temperature dependences differ distinctly from aheavy Fermi liquid state and suggest the vicinity to anantiferromagnetic quantum critical point.In applied magnetic fields the normal state retains cer-tain NFL characteristics regardless of field direction, be-fore Fermi liquid behavior is recovered well beyond H c (Bianchi et al. , 2003b; Malinowski et al. , 2005; Paglione et al. , 2003; Ronning et al. , 2005). This is surprisingsince the NFL characteristics due to a QCP are normallyrapidly suppressed in a magnetic field. For instance, at H cc the specific heat C/T diverges logarithmically reach-ing
C/T = 1 . / mol K at the lowest temperatures stud-ied (Petrovic et al. , 2001b), while Fermi liquid behavioris only observed above 8 T. Likewise the d.c. suscepti-bility at H c diverges as χ ( T ) = χ + C/ ( T α + a ) with α = 0 . − et al. , 2002).The electronic structure of CeCoIn has been stud-ied microscopically by angle-resolved photoemission(ARPES). The dispersion and the peak width of theprominent quasi-two-dimensional Fermi surface sheet dis-plays an anomalous broadening near the Fermi level(Koitzsch et al. , 2008). Using resonant ARPES a flatf-band is observed with a distinct temperature depen-dence. These observations are consistent with a two-levelmixing model. Direct microscopic evidence of a NFL normal state issupported by de Haas–van Alphen oscillations for mag-netic field along the c-axis. Here strongly spin depen-dent mass enhancements are observed in the immediatevicinity of H cc , that are inconsistent with the Lifshitz-Kosevich expression and thus Fermi liquid theory (Mc-Collam et al. , 2005). This is supplemented by the spin-lattice relaxation rate T in In nuclear quadrupoleresonance, which displays a temperature dependence1 /T characteristic of antiferromagnetic spin fluctuations(Kawasaki et al. , 2003; Kohori et al. , 2001).As discussed below, the electronic correlations at theheart of the NFL behavior are likely to be responsible forthe superconductivity in CeCoIn . This raises the ques-tion for their origin and the possible nature and locationof the QCP. The T coefficient of the resistivity for H along the c-axis diverges at an extrapolated field valuebelow H c suggesting a QCP within the superconduct-ing regime, but the precise location has not been settled(Bianchi et al. , 2003b; Malinowski et al. , 2005; Paglione et al. , 2003). More recently even a dimensional cross-overfrom three-dimensional to two-dimensional quantum crit-icality near H c was inferred from the thermal expansion(Donath et al. , 2008). Entirely unexplained is the obser-vation of a giant Nernst effect in the normal state (Bel et al. , 2004; Izawa et al. , 2007). In fact, one scenario of-fered to explain the giant Nernst effect and scaling of thenormal state resistivity as a function of field direction inthe basal plane is the formation of a d-density wave (Hu et al. , 2006).Further support of unconventional superconductivitywith a d-wave gap has been observed in inelastic neu-tron scattering studies (Stock et al. , 2008). In the nor-mal state slow commensurate fluctuations ( (cid:126) Γ = 0 . ± .
15 meV at (cid:126)Q = (1 / , / , / (cid:126) ω = 0 . ± .
03 meVdevelops with (cid:126) Γ < .
07 meV. The spin resonance is in-dicative of strong coupling between f-electron magnetismand superconductivity. The similarity of this spin reso-nance with the properties of UPd Al and the cupratessuggest that it may be understood in a common frame-work.The specific heat anomaly of the superconducting tran-sition is exceptionally large, ∆ C/γT s = 4 . γ at T s . This would suggest an ex-treme case of strong coupling superconductivity. How-ever, when considering ∆ C/γ normally the extrapolatedzero temperature value of γ is used, which due to theNFL behavior here is ill-defined. The initial varia-tion of H c near T s is large and characteristic of heavyfermion superconductivity, dH cc /dT = −
11 T / K and dH abc /dT = −
24 T / K (Ikeda et al. , 2001). The shortcoherence length ξ a = 82 ˚A and ξ c = 35 ˚A and large pene-tration depth as inferred from microwave measurements, λ ( T →
0) = 1900 ˚A (Ormeno et al. , 2002), along withthe low Fermi energy and large charge carrier mean freepaths of several 1000 ˚A identify CeCoIn as a type II su-4perconductor ( κ a = 108 and κ c = 50) in the super-cleanlimit (Kasahara et al. , 2005).A large number of properties suggest an unconven-tional form of superconductivity in CeCoIn . For in-stance the depression of T s with rare earth substitutioncorrelates with the mean free path (Paglione et al. , 2007).The following experimental evidence suggests line nodesof a d x − y state, notably: (i) the power law temperaturedependence of the specific heat, C ∝ T (Movshovich et al. , 2001), (ii) the variation of the specific heat withfourfold symmetry for magnetic field in the basal plane(Aoki et al. , 2004) (maxima along [110]), (ii) the power-law dependence of the thermal conductivity, κ ∝ T (Movshovich et al. , 2001), (iii) the variation of the ther-mal conductivity with a fourfold symmetry for magneticfield in the basal plane (maxima along [110]) (Izawa et al. ,2001), (vi) the variation of the H c of 1.2% with a four-fold symmetry in the basal plane (maxima along [100])(Weickert et al. , 2006), and (vii) the differential conduc-tance spectra as interpreted in the extended Blonder-Tinkham-Klapwijk model (Park et al. , 2008).In contrast, a d xy pairing symmetry has been inferredfrom the symmetry and the field and temperature depen-dence of the in-plane torque magnetization (Xiao et al. ,2008). Moreover, the magnetic field and temperature de-pendence of the thermal conductivity, was found to be in-consistent with unpaired electrons (Seyfarth et al. , 2008).The latter study points at multi-band superconductivityand a related complex multi-gap state.Microscopic information on the pairing symmetry maybe inferred from the Knight shift, which decreases forboth field directions. This shows that the spin sus-ceptibility decreases for all directions, consistent witheven parity superconductivity (Kohori et al. , 2001). TheNMR/NQR spin-lattice relaxation rate shows no Hebel-Slichter peak and a power-law temperature dependence1 /T ∝ T (Kohori et al. , 2001) further suggesting a nons-wave state.Small angle neutron scattering (SANS) shows a six-fold symmetry of the flux line lattice at low fields andlow temperatures. As a function of magnetic field theflux lattice symmetry undergoes a sequence of transi-tions from hexagonal to orthorhombic, to square, backto orthorhombic and finally hexagonal symmetry near H c (Bianchi et al. , 2008; DeBeer-Schmitt et al. , 2006;Eskildsen et al. , 2003). Most remarkably, the form factorof the FLL as traced all the way to H c increases withincreasing field, in stark contrast with the predictionsof Abrikosov-Ginzburg-Landau theory (Bianchi et al. ,2008). This behavior has been attributed to a combi-nation of Pauli paramagnetic effects around the vortexcores and the vicinity of the system to a quantum criticalpoint. The temperature dependence of the penetrationdepth may be described as λ ⊥ ∝ T . . This has beenexplained in terms of a temperature dependent coher-ence length related to the vicinity to quantum criticality( ¨Ozcan et al. , 2003). Alternatively, the penetration depthhas been described as λ ⊥ ∝ aT + bT and λ (cid:107) ∝ T , where FIG. 7 Temperature versus pressure phase diagram ofCeCoIn . From the resistivity a ’pseudo-gap’ at T pg is in-ferred that merges with the maximum in the onset of thesuperconductivity (SC). At high pressures the superconduc-tivity condenses out of a Fermi liquid temperature dependenceof the resistivity below T FL . Plot taken from (Sidorov et al. ,2002). a cross-over from weak to strong coupling superconduc-tivity was proposed (Chia et al. , 2003).The properties of CeCoIn respond sensitively to hy-drostatic pressure as shown in Fig. 7 (Knebel et al. , 2004;Miclea et al. , 2006; Nicklas et al. , 2001; Shishido et al. ,2003; Singh et al. , 2007; Sparn et al. , 2002; Tayama et al. , 2005; Yashima et al. , 2004). Up to 30 kbar T s traces out part of a dome; an initial increase is fol-lowed by a decrease for p >
16 kbar. The specific heatanomaly ∆
C/γT s decreases under pressure monotoni-cally by nearly 80% up to 30 kbar (Knebel et al. , 2004;Sparn et al. , 2002). H c increases for the ab-plane whileit decreases for the c-axis from 4.95 to 2 T at 30 kbar(Shishido et al. , 2003), so that the anisotropy of H c in-creases from 2.34 at p = 0 to 3.78 at 30 kbar (Tayama et al. , 2005).Despite these rather drastic effects, In NQR showsthat the spin-lattice relaxation rate T below T s remainsqualitatively unchanged T ∝ T − up to 20 kbar. Thissuggests that the nature of the superconductivity remainsunchanged (Yashima et al. , 2004). An increase of the spinfluctuation temperature T SF may be consistently inferredfrom (i) the decrease of the normal state value of γ , (ii)an increase of the coherence maximum in the resistivityfrom 50 to nearly 100 K at 15 kbar and (iii) a change ofthe normal state spin-lattice relaxation rate. All of theseproperties suggest, that pressure moves CeCoIn awayfrom quantum criticality.We finally mention that CeCoIn combines a uniqueset of properties: it shows strong Pauli limiting, the elec-tronic structure is quasi-two dimensional and samplesmay be grown at ultra-high purity. These are the pre-conditions for the formation of a FFLO phase. Indeed,striking evidence exists that CeCoIn stabilizes the firstexample of such a state as discussed in section V.B.1.5 d. CeRhIn In comparison to CeCoIn , the unit cell vol-ume of CeRhIn is larger and the c/a ratio smaller asshown in table II. Taking into account the anisotropiccompressibility for the a- and c-axis the properties ofCeRhIn at high pressure may be expected to resem-ble those of CeCoIn . Considering the bulk modulusof CeIn , the CeIn units may be viewed as experienc-ing an effective pressure of 14 kbar (Hegger et al. , 2000).The electronic properties of CeRhIn emerge in a CEFΓ ground state and Γ and Γ first and second excitedstate at 6 and 29.1 meV, respectively (Christianson et al. ,2004).At ambient pressure CeRhIn orders antiferromagneti-cally below T N = 3 . et al. , 2000), with a tem-perature independent antiferromagnetic ordering wavevector (cid:126)Q = (1 / , / , . µ ord = 0 . µ B is strongly reduced as com-pared to the moment expected in the CEFs of 0 . µ B (Bao et al. , 2000, 2003). It spirals transversely alongthe c-axis, while the nearest-neighbor moments on thetetragonal basal plane are aligned antiferromagnetically.Based on µ -SR it has been suggested that a small or-dered moment also exists at the Rh site (Schenck et al. ,2002). The antiferromagnetic transition as seen in neu-tron scattering and the bulk properties is second order,where the specific heat is characteristic of an anisotropicspin-density wave that gaps nearly 90% of the Fermi sur-face (Cornelius et al. , 2001). The entropy released at T N corresponds to the small ordered moment (Hegger et al. ,2000).The normal state specific heat of CeRhIn is charac-teristic of a heavy fermion state with γ = 0 .
42 J / mol K (Cornelius et al. , 2000). In contrast, the thermal ex-pansion shows a non-Fermi liquid divergence of α/T for[001] above T N while the basal plane is well behavedwith α/T ≈ constant (Takeuchi et al. , 2001). More-over, while the susceptibility displays the Curie-Weissbehavior of nearly free Ce moments at high temper-atures, χ keeps increasing even at low temperatures be-low a shoulder around 30 K, where χ − ab ∝ a + bT . and χ − c ∝ a + T . (Kim et al. , 2001a). Similar anomalousbehavior is also seen in the temperature dependence ofthe normal state electrical resistivity (Hegger et al. , 2000;Muramatsu et al. , 2001).Microscopic evidence of an abundance of critical anti-ferromagnetic fluctuations up to 3 T N has been seen ininelastic neutron scattering (Bao et al. , 2002a) and thetemperature dependence of the In NQR spin-latticerelaxation (Mito et al. , 2001). The magnetic phase di-agram of CeRhIn as a function of an applied magneticfield has been studied up to 50 T for the [110] direction.A spin-flop transition is observed at 2 T and a metam-agnetic transition (spin flip) around 45 T (for 3 K) (Cor-nelius et al. , 2001; Settai et al. , 2007b; Takeuchi et al. ,2001). The c-axis is the easy magnetic axis.Under pressure the N´eel temperature decreases. Su-perconductivity was first observed in CeRhIn above15 kbar, where an abrupt, first order change from anti- ferromagnetism to superconductivity was reported (Heg-ger et al. , 2000). Recent studies suggest, that highquality single crystals display superconductivity even inthe antiferromagnetic state at ambient pressure below T s ≈ . − .
11 K (Chen et al. , 2006; Paglione et al. ,2008). The bulk properties of the superconductivity atambient pressure by comparison with other systems arecharacteristic of being far from quantum criticality.As function of pressure T s increases, while T N de-creases until T N = T s ≈ . p ∼ . p (Knebel et al. , 2006). Neutron scattering showsthat the ordering wave vector and ordered moment ini-tially change weakly (Llobet et al. , 2004; Majumdar et al. , 2002) and a second magnetic modulation emerges(Christianson et al. , 2005). For high pressures of 15and 17 kbar the incommensurate propagation vector is (cid:126)Q hp = (1 / , / , . (cid:126)Q = (1 / , / , . et al. , 2008a). A competitive coexistence of AFand superconductivity up to p is supported by In-NQR (Mito et al. , 2003). Homogenous volume supercon-ductivity is observed above p with a maximum value of T s ≈ . et al. , 2006; Knebel et al. , 2006). Resistivity measurements in CeRhIn ex-tending up to 85 kbar initially indicated the presence of asecond superconducting dome as shown in Fig. ?? (Mura-matsu et al. , 2001). This finding could not be confirmedlater as reviewed by (Knebel et al. , 2008).Electronic structure calculations suggest that the 4felectron is localized in CeRhIn (Elgazzar et al. , 2004).The mass enhancement seen in the specific heat hastherefore been attributed to spin fluctuations abovefrozen magnetic states, which become itinerant and addto the spectrum of fluctuations when going to CeCoIn .De Haas–van Alphen studies show that the electronicstructure of CeRhIn is highly two-dimensional (Cor-nelius et al. , 2000; Hall et al. , 2001). Under hydrostaticpressure a new branch emerges around 24 kbar, the ex-trapolated pressure where T N vanishes. The similaritywith CeCoIn indeed suggests a delocalization of the 4felectron at this pressure (Shishido et al. , 2005).The NFL normal state properties and immediate vicin-ity of the superconductivity to antiferromagnetism inCeRhIn are circumstantial evidence suggesting uncon-ventional pairing. The superconductivity is, nevertheless,rather unexplored. The most direct evidence for uncon-ventional pairing may be the spin-lattice relaxation rateof In-NQR in the superconducting state which doesnot show a Hebel-Slichter peak and varies as 1 /T ∝ T (Mito et al. , 2001).The structural similarity of CeRhIn with CeCoIn raises the question for an analogy of the superconductingphase diagram. In CeCoIn the normal state propertieshint at a quantum critical point that is masked by the su-perconducting dome. Under pressure H c for B ⊥ [001]initially tracks the increase of T s and displays a maxi-6 FIG. 8 Temperature versus magnetic field and pressure phasediagram of CeRhIn as reported by (Park et al. , 2006). De-tailed studies of the specific heat suggests a well defined lineof quantum criticality, separating a regime where supercon-ductivity and antiferromagnetism coexist with a regime ofhomogenous bulk superconductivity. Plot from (Park et al. ,2006) as shown in (Sarrao and Thompson, 2007). mum just above p . Specific heat measurements underpressure and magnetic field in CeRhIn reveal a phaseboundary separating homogenous volume superconduc-tivity and a phase coexistence of antiferromagnetic orderand superconductivity as reported by (Park et al. , 2006)and shown in Fig. 8. In the magnetic field versus pressureplane the phase separation line increases from zero at p and reaches H c at p . The normal state properties inthe B versus p plane are consistent with at a quantumcritical point for B → p . Taken together withthe de Haas–van Alphen studies this provides evidenceof a quantum critical point at p that may be related toa delocalization transition of the 4f electrons. e. CeIrIn The heavy-fermion superconductor CeIrIn ,finally, has the largest unit cell volume and smallestc/a ratio as shown in table II. At ambient pressure thenormal state properties are characteristic of strong elec-tronic correlations that develop in crystal electric fieldsrelated to those of CeIn (Christianson et al. , 2004).The specific heat exhibits a large enhancement with γ = 0 .
72 J / mol K (Petrovic et al. , 2001a). The suscep-tibility exhibits a broad shoulder around 7 K (Takeuchi et al. , 2001), but continues to diverge slowly (Kim et al. ,2001a). This and the resistivity, which varies as ∆ ρ ∝ T n with n ≈ . et al. , 2001a).The bulk properties are consistent with the spin-latticerelaxation rate inferred from In-NQR measurements,which suggests that CeIrIn is an anisotropic, incipientantiferromagnet (Kohori et al. , 2001; Zheng et al. , 2001).More detailed information on the normal state has beeninferred from the Hall effect and the magnetoresistance,which also show non-Fermi liquid behavior. Notably,there is a breakdown of Kohler’s rule and the Hall angle varies as cotΘ H ∝ T (Nair et al. , 2008; Nakajima et al. ,2008). When taken together, in the T versus B phase dia-gram the magneto-transport properties suggest a precur-sor regime of the normal metallic state that shares somesimilarities with the pseudo-gap regime in the cuprates(Nair et al. , 2008).Various properties of CeIrIn suggest two supercon-ducting transitions. At T s = 0 .
75 K the resistivity van-ishes and there are strong indications of an intrinsic formof filamentary superconductivity. At T s = 0 . et al. , 2001a) and in In-NQR (Kawasaki et al. , 2005).Specific heat and In-NQR under pressure show that T s increases to 0.8 K at a pressure of 16 kbar (Borth et al. ,2002; Kawasaki et al. , 2005). The increase of T s is con-sistent with the observed decrease of γ , which may beinterpreted as an increase of the characteristic spin fluc-tuation temperature. H c is anisotropic where the resis-tivity and susceptibility show H ac = 6 . H cc = 3 . H ac = 1 . H cc = 0 .
49 T for S1 and S2, respec-tively (Petrovic et al. , 2001a). The temperature depen-dence and anisotropy of H c of the incipient supercon-ducting state and the bulk superconducting state trackeach other qualitatively (Petrovic et al. , 2001a), wherethe anisotropy may be accounted for by the anisotropicmass model (Haga et al. , 2001).The specific heat, which varies as C ∝ T (Movshovich et al. , 2001) and the thermal conductivity for heat cur-rent along the a-axis, which varies as κ ∝ T with a finiteresidual T = 0 value of κ/T = 0 .
46 W / K m are consis-tent with an unconventional superconducting state andline nodes. This is supported microscopically by NMRand NQR, which shows (i) no Hebel-Slichter peak, (ii)a temperature dependence of the spin-lattice relaxationrate 1 /T ∝ T at pressures up to 21 kbar, and (iii) adecrease of the Knight shift in the superconducting statewith decreasing temperature for all field directions (Ko-hori et al. , 2001). However, the thermal conductivitywith heat current along the c-axis does not chow a resid-ual term at low temperatures (Shakeripour et al. , 2007),ruling out line nodes running along the c-axis. Insteadthe formation of a hybrid gap structure with E g symme-try has been proposed. f. Ce RhIn The properties of the double layer com-pound Ce RhIn are intermediate between CeIn and thesingle-layer compound CeRhIn as may be expected fromthe larger fraction of CeIn building blocks in the crystalstructure. At ambient pressure Ce RhIn develops anti-ferromagnetic order below a second order phase transi-tion at T N = 2 . (cid:126)Q =(1 / , / ,
0) and an ordered moment µ ord ≈ . µ B / Ce(Bao et al. , 2001). The magnetic structure is more akinto that of CeIn , where the specific heat shows that only8 % of the Fermi surface are gapped in comparison toover 90 % in CeRhIn (Cornelius et al. , 2001). A secondantiferromagnetic transition is observed in the resistivity7at T N = 1 .
65 K, which does not appear to be accompa-nied by an anomaly in the specific heat (Nicklas et al. ,2003). The magnetic phase diagram at ambient pressureis reminiscent of that of CeRhIn (Cornelius et al. , 2001).Hydrostatic pressure suppresses both T N and T N ,where T N vanishes below 1 kbar and T N extrapolatesto zero around p N ≈ (Nicklas et al. , 2003). Specificheat under pressure shows a broadening of the antiferro-magnetic transition and an decrease of γ consistent withan increase of the spin fluctuation temperature (Lengyel et al. , 2004). A superconducting dome surrounds p N with a maximum T s = 2 . et al. , 2003). At16.3 kbar H c = 5 .
36 T and the initial temperature de-pendence dH c /dT = − .
18 T / K are large and compara-ble with other compounds in this series. g. Substitutional doping in Ce n M m In n +2 m Particularlyappealing in the series CeMIn is the relative metallur-gical ease with which substitutional doping studies maybe carried out. Three different aspects have been at thecenter of interest: (i) the sensitivity to doping of the f-electron element, (ii) the stability of the ground stateunder replacement of the transition metal element, and(iii) the sensitivity to disorder on the In site.Substitutional doping of Ce in CeMIn has been car-ried out with La, U, Pu and Nd. In CeCoIn La-dopingresults in a two-fluid state, notably a combination ofsingle-impurity Kondo and dense Kondo lattice behav-ior (Nakajima et al. , 2004; Nakatsuji et al. , 2002). It issurprising, that La doping does not yield additional com-plexities. Further, unconventional superconductivity inprinciple is very sensitive to disorder. However the super-conductivity is remarkably insensitive to La-doping andvanishes only for x ≥ .
15. In the superconducting statethe residual electronic thermal conductivity decreaseswhile the residual electronic specific heat increases with x , i.e., the thermal conductivity does not track the elec-tronic degrees of freedom that become available underdoping. This has been taken as evidence of extrememulti-band superconductivity in CeCoIn (Tanatar et al. ,2005). Finally, Nd doping allows to study the evolutionfrom local moment magnetism to heavy-fermion super-conductivity (Hu et al. , 2008).Across the series Ce m Rh n In n +2 ( n = 0 , m =1 ,
2) increasing the La substitution of Ce leads to asuppression of T N . For the tetragonal systems ( n = m = 1 ,
2) the critical concentration is x c ≈ . x c ≈ .
65 (Pagliuso et al. , 2002a). La-dopingof CeRhIn leaves the antiferromagnetic wave vector es-sentially unchanged up to x = 0 . et al. , 2002b).The observation of the same value of x c for the tetrago-nal systems suggests, that antiferromagnetic order is es-sentially controlled by the CeIn building blocks. Thepressure dependence of La and Sn doped CeRhIn , no-tably Ce . La . RhIn and CeRhIn . Sn . , shows thatLa doping shifts the phase diagram to higher pressures, while Sn doping shifts it to lower pressures (Ferreira et al. , 2008). This implies that the strength of the on-site Kondo coupling represents the dominant energy scalecontrolling the phase diagram of CeRhIn .Several studies have explored the evolution of the se-ries CeMIn under the isovalent replacement of Co by Rhand Ir, and of Rh by Ir. In the series CeCo − x Rh x In and CeRh − x Ir x In this allows the study of the evolu-tion between superconductivity and antiferromagnetism,while the series CeCo − x Ir x In allows the study of theevolution between two unconventional superconductorsas summarized in Fig. 9. In the series CeCo − x Rh x In acoexistence of antiferromagnetism and superconductivityis observed for a large range of x (Zapf et al. , 2001). Thetotal entropy released at the two transitions is therebyconstant. This suggests that the two ordering phenomenaare intimately related representing two sides of the samecoin. NQR studies of the normal state show that Rh dop-ing boosts antiferromagnetic spin fluctuations (Kawasaki et al. , 2006). The antiferromagnetic structure of CeRhIn changes from an incommensurate, (cid:126)Q = (1 / , / , . (cid:126)Q = (1 / , / , /
2) and (cid:126)Q = (1 / , / , .
42) at interme-diate concentrations (Yokoyama et al. , 2008). Fluctua-tions with respect to these wave vectors may be relevantfor the superconductivity at intermediate concentrations.Rh doping of CeIrIn initially suppresses the filamen-tary transition at T s so that only the superconduct-ing transition at T s remains. However, for higher Rhconcentrations the bulk T s increases, until antiferro-magnetism emerges for x > . et al. , 2001;Kawasaki et al. , 2006; Pagliuso et al. , 2001). Within theantiferromagnetic state superconductivity in coexistencewith antiferromagnetism survives (Christianson et al. ,2005; Zheng et al. , 2004). Perhaps most importantly,the superconductivity is insensitive to the disorder asso-ciated with the doping. This suggests, that the transitionmetal element affects only indirectly those parts of theFermi surface on which superconductivity is stabilized.Finally, substitutional doping on the In site has pro-vided some remarkable hints concerning the nature ofthe superconductivity. Extensive studies have been car-ried out in CeCoIn − x Sn x , where the superconductivityvanishes rapidly for x ≥ .
15 (Bauer et al. , 2005c). Notethat this represents a much smaller concentration than x = 0 .
15 in La-doping. EXAFS studies have therebyestablished that the Sn atoms preferentially occupy theIn(1) site in the CeIn planes (Daniel et al. , 2005), high-lighting that the superconductivity is particularly sensi-tive to disorder in the CeIn planes. Interestingly thecritical Sn concentration, when referred to the CeIn planes, yields an average distance of the Sn atoms ofthe size of the superconducting coherence length. Thesuppression of superconductivity in CeCoIn − x Sn x maybe compared with the suppression of antiferromagneticorder in CeRhIn − x Sn x at x c ≈ .
35, where a quantumcritical point is generated (Bauer et al. , 2006a). Assum-ing that the Sn atoms as for CeCoIn − x Sn x occupy the8 FIG. 9 Compilation of the evolution of superconductivityand antiferromagnetic order in the series CeMIn , where M=Co, Rh and Ir. Note the continuous evolution of the su-perconducting transiton temperature when going from Ir toCo despite the presence of disorder. This continuous evolu-tion is interrupted by an antiferromagnetic dome in the seriesCo → Rh → Ir. Plot from (Pagliuso et al. , 2002b) as shown in(Sarrao and Thompson, 2007).
In(1) site, this reveals that details of the electronic struc-ture within the CeIn planes control the stability of bothantiferromagnetic order and superconductivity.Rather surprising, substitutional Cd doping of the Insite induces a change from superconductivity to longrange antiferromagnetic order, where the phase diagramscales with the pressure dependent phase diagram ofCeRhIn (Pham et al. , 2006). Electronically Cd doping,in leading order, acts as the removal of electrons, whichin turn compares with the effect of pressure on CeRhIn .However, NMR studies of the series of Cd doped CeCoIn establishes a microscopic coexistence of the two forms oforder, where the ordered moment of 0 . µ B is essentiallyunchanged and the magnetic order may be attributed tothe local environment of the Cd dopant (Urbano et al. ,2007). Thus the magnetic order is not the result of agradual modification of the Fermi surface, but emergesin terms of droplets that coalesce at the onset of long-range antiferromagnetism.Both the superconductivity and the antiferromag-netism respond sensitively to nonmagnetic disorderwithin the CeIn building blocks, while they are rela-tively insensitive to out-of-plane disorder. As a possibleexplanation this behavior may be related to the warpingof the Fermi surface, which is affected by local distortionscreated by the replacement of transition metal elements.However, the detailed mechanisms that control the be-havior in doping studies have not yet been identified. h. Common features and analogies We now discussthe more general features of the entire series ofCe n M m In n +2 m compounds. We begin with materialspecific aspects and conclude this section with a discus-sion of possible analogies with other layered supercon-ductors, notably the cuprates.A major theme across the literature on theCe n M m In n +2 m compounds is the tentative role of aquantum critical point in driving superconductivity. Thisis embodied and was first pointed out with respect toCeIn (Mathur et al. , 1998; Walker et al. , 1997). A natu-ral question concerns, which mechanisms control the T s .For spin fluctuation mediated pairing it has been pointedout that a reduction of the effective dimension, notablymagnetic and/or electronic anisotropy, favour supercon-ductivity (Monthoux and Lonzarich, 2001, 2002). This isconsistent with an empirical observation of T s as a func-tion of the c/a ratio as reported by (Bauer et al. , 2004b)and shown in Fig. 10 .Similarities of the series Ce n M m In n +2 m with thecuprate superconductors have been taken as evidencethat spin fluctuations are responsible for the pairingmechanism in the cuprates (see, e.g., (Mathur et al. ,1998)). It is instructive to summarize these similaritiesin further detail. The consideration begins with the tem-perature versus pressure phase diagram, which shows asuperconducting dome in the vicinity of antiferromag-netic order. At least in CeIn a major qualitative differ-ence is, that a proper antiferromagnetic transition van-ishes at the putative QCP, while the equivalent featurein the cuprates is a pseudogap of ill-defined nature. Herethe phase diagram of CeCoIn is in better analogy withthe cuprates where, however, the temperature ranges ofanomalous behavior in CeCoIn are rather small.Likewise, the analogy with the cuprates may also beseen in the sibling pair CeCoIn and CeRhIn , wherepressure induces superconductivity in CeRhIn as well asin Ce RhIn . The similarity of the phase diagrams is alsoloosely reflected in the doping studies, when keeping inmind that the underlying microscopic processes, notablys-f hybridization in f-electron systems versus pure chargetransfer in the cuprates are radically different. Dopingwith Rh, driving CeCoIn antiferromagnetic, is akin tohole doping in the cuprates. Likewise Cd doping may beunderstood as electron doping, stabilizing antiferromag-netic order. Moreover, the complex pressure, magneticfield and temperature phase diagram in CeRhIn yieldsanother analogy in that magnetic field stabilizes a coexis-tence of superconductivity and antiferromagnetism (Park et al. , 2006). A related effect of magnetic field has alsobeen found in certain cuprates (Lake et al. , 2002).The analogy with the cuprates is not just based onqualitative features of the phase diagram, but also on thebulk properties. As for the cuprates the normal metallicstate exhibits non Fermi liquid behavior. While the re-sistivity, susceptibility and specific heat are not in greatagreement, there is a remarkable similarity of the Hall ef-fect and regarding the breakdown of Kohlers rule in the9 FIG. 10 Evolution of the superconducting transition temper-ature as a function of c/a ratio of the lattice parameter in theseries CeMIn (left-hand and bottom axis) and PuMGa (topand right-hand axis). Plot as shown in (Sarrao and Thomp-son, 2007) magnetoresistance in CeCoIn . In fact, a quadratic tem-perature dependence of the Hall angle and a breakdownof Kohler’s rule have also been observed in CeIrIn , wherethey were interpreted as a precursor phase in the normalmetallic state that share similarities with the pseudo-gapin the cuprates (Nair et al. , 2008). Moreover, just as forthe cuprates a spin resonance at a frequency ω has nowbeen observed in the spectrum of slow antiferromagneticfluctuations in CeCoIn , where the ratio of resonance fre-quency to gap, (cid:126) ω / ≈ .
74, is remarkably similar forCeCoIn , UPd Al and Bi Sr CaCu O δ (Stock et al. ,2008).It is at the same time also important to emphasize thedifferences between Ce n M m In n +2 m and the cuprates.First, CeRhIn appears to exhibit two superconductingdomes. This suggests that at least two pairing mecha-nisms may exist, where possible candidates are a SDWand a valence instability as in CeCu Si . The secondsuperconducting dome makes the analogy with CeCoIn less obvious and it is interesting to ask if there are furthersuperconducting domes in CeCoIn under pressure. Sec-ond, quantum oscillatory studies of the electronic struc-ture show a change of Fermi surface topology through thequantum critical point in CeIn and CeRhIn that ap-pears to be related to a delocalization of the f-electron.A stimulating question concerns, whether, quite gener-ally, an instability of the Fermi surface topology drivesthe superconductivity in the cuprates.Finally, CeIrIn has an even larger unit cell volumethan CeRhIn , but there is no antiferromagnetic ordernearby. The phase diagram shown in Fig. 9 may con-sequently be interpreted differently. Perhaps the entireseries of CeMIn is superconducting where T s increasesin a linear fashion when going from Co to Rh to Ir. How-ever, a slight change of electronic structure of the Rhsystem changes the balance from a superconducting toan antiferromagnetic ground state. This does not ruleout quantum critical fluctuations as a key ingredient of the superconductivity, though the origin of these fluc-tuations may differ from the conventional scenario of asimple quantum critical point.
3. Miscellaneous Ce-systems
We next present several examples of compounds, whereevidence for superconductivity has been observed at theborder of antiferromagnetic order. These compounds arethe first of a given crystal structure, being possibly thefirst member of a new class of f-electron superconduc-tors. The properties of these ’miscellaneous Ce-systemsare summarized in table ?? . The first two examples,CeNiGe and Ce Ni Ge , are members of the ternary Ce-Ni-Ge series. The third system, CePd Al , is isostruc-tural to NpPd Al (cf. section IV.B.2).For completeness it is worthwhile to also mentionbriefly CeCu Au, where antiferromagnetic order is sup-pressed around 40 kbar and a tiny drop of the resistiv-ity is observed that may be related to superconductivity(Wilhelm et al. , 2001) . a. CeNiGe We begin with the discovery of super-conductivity in CeNiGe , which crystallizes in theorthorhombic SmNiGe -type structure (space groupCmmm, see also table I) (Kotegawa et al. , 2006; Naka-jima et al. , 2004). At ambient pressure CeNiGe ordersantiferromagnetically below a N´eel temperature T N =5 . µ eff = 2 . µ B as expected of Ce . The antiferromag-netic transition is accompanied by a distinct anomaly,where the entropy released, ∆ S = 0 .
65 R ln 2, is char-acteristic of localized moments in a 4f crystal field dou-blet ground state. The magnetic structure has been ex-plored by powder neutron diffraction, which revealed twotransitions at T N and T N with ordering wave vectors (cid:126)Q = (1 , ,
0) and (cid:126)Q = (0 , . , / µ ord = 0 . µ B .The electrical resistivity of polycrystalline CeNiGe isdominated by a maximum around 100 K and a sharp dropat T N , where no details have been seen of a second tran-sition. The pressure dependence of the polycrystallinesamples was investigated with two different pressure tech-niques: a diamond anvil cell with NaCl (Nakajima et al. ,2004) and Daphne oil (Kotegawa et al. , 2006) as pres-sure transmitter, respectively. In the following we onlyaddress the results obtained with the latter set-up whichproduces better homogeneity. As a function of pressure T N initially rises to nearly 8 K at 40 kbar followed by afairly rapid decrease. T N vanishes at p c ≈
70 kbar, wherethe T resistivity crosses over to a temperature depen-dence ∼ T . in the range 60 to 70 kbar. The residualresistivity ρ increases and reaches a plateau above p c .0For pressures in the range 20 to 100 kbar hints for su-perconductivity are observed in terms of a zero resistancetransition below 40 kbar and an incomplete resistive tran-sition above 40 kbar, where T s is as high 0.45 K. Thetransition temperature exhibits two broad maxima sep-arated by a shallow minimum near p c . The ac suscep-tibility shows diamagnetic screening. H c increases un-der pressure from 0.015 T to 1.55 T. Correspondingly thecoherence length decreases under pressure from 2000 ˚Ato ∼ H c near T s increasesand reaches dH c /dT = − . / K for the maximum T s ≈ .
45 T. For low pressures orbital limiting is ob-served, while there is Pauli limiting for the highest valuesof T s around 70 kbar. b. Ce Ni Ge Another system in the Ce-Ni-Ge seriesthat attracts increasing interest is Ce Ni Ge (Chevalierand Etourneau, 1999). This compound crystallizes inthe U Co Si -type structure (space group Ibam; see alsotable ?? ). A discussion of structural similarities withCeNi Ge may be found in (Nakashima et al. , 2005).The metallic state of Ce Ni Ge is characteristic of aKondo lattice system with T K ≈ moments is observedat high temperatures and antiferromagnetic order at lowtemperatures (Hossain et al. , 2000). The magnetizationshows two transitions at T N = 5 . T N = 4 . S = 0 .
67R ln 2 at T N is character-istic of reduced moments. Powder neutron diffractionshows collinear antiferromagnetic order below T N withthe magnetic moments aligned along the a-axis and asmall ordered moment of µ ord = 0 . µ B at 1.4 K (Duri-vault et al. , 2002).In comparison with Ce Ni Ge the unit cell volumein the sibling compound Ce Ni Si is 9.6% smaller.Ce Ni Si exhibits a nonmagnetic valence fluctuatingsystem (Mazumdar et al. , 1992). This suggests that hy-drostatic pressure suppresses the antiferromagnetic or-der. Indeed T N in polycrystalline samples decreasesunder pressure and vanishes at p c = 36 kbar, where azero resistance transition is observed at T s = 0 .
26 K(Nakashima et al. , 2005) with H c = 0 . ξ = 210 ˚A. c. CePd Al Another miscellaneous C-based supercon-ductor is CePd Al (Honda et al. , 2008a), which isisostructural to NpPd Al reviewed in section IV.B.2.At ambient pressure CePd Al displays two antiferro-magnetic transitions at T N = 3 . T N = 2 . γ =0 .
056 J / mol K . The resistivity and susceptibility aswell as the magnetization suggest crystal field levels at∆ = 197 K and ∆ = 224 K. Under pressure T N and T N at first increase, where T N can only be tracked ashigh as ∼
30 kbar. T N displays a maximum around 50 kbar and appears to vanish around 90 kbar. The resis-tivity displays a superconducting transition in the pres-sure range 80 to 120 kbar, with a maximum T s = 0 .
57 Kat 108 kbar.
B. Coexistence with antiferromagnetism
In a number of f-electron systems superconductivityemerges deep inside an antiferromagnetically orderedregime, i.e., T s (cid:28) T N . The presentation of these systemsmay be grouped in two parts, large and small momentsystems. We first discuss the large moment antiferromag-nets UPd Al , UNi Al and CePt Si. For these com-pounds the coexistence of antiferromagnetism and su-perconductivity appears to be homogenous. The secondclass are antiferromagnets with tiny ordered moments,notably UPt and URu Si . While the tiny moments inUPt appear to be homogenous, there is growing evidencefor a small volume fraction of large ordered moments inURu Si .
1. Large moment antiferromagnets
Superconductivity in the sibling pair of low tempera-ture antiferromagnets UPd Al and UNi Al was discov-ered in 1991 (Geibel et al. , 1991a,b). Both compoundscrystallize in the hexagonal PrNi Al structure (spacegroup P6/mmm) as summarized in table III. Large sin-gle crystals may be grown of UPd Al , while the metal-lurgy of UNi Al is more complex, i.e., there are fewersingle-crystal studies for UNi Al . In turn the body ofwork on UPd Al is much more complete. In the follow-ing we first review the present understanding of UPd Al before turning to the properties of UNi Al at the endof the section. We address only briefly the coexistenceof superconductivity and antiferromagnetism in CePt Si,which is reviewed extensively in section IV.A.1. a. UPd Al The electrical resistivity of UPd Al de-creases monotonically as a function of temperature be-low a broad maximum around 85 K (Sato et al. , 1992).In single crystals the resistivity is weakly anisotropic bya factor of two with ρ c > ρ ab . The susceptibility exhibitsa broad maximum around 35 K in the basal plane andan anisotropy of ∼ . χ c < χ ab ) (Geibel et al. , 1991b).Above ∼
100 K a Curie-Weiss dependence is observedwith a fluctuating moment µ eff that changes from 3.2 to3 . µ B / U around 300 K (Grauel et al. , 1992). To accountfor the temperature dependence of the susceptibility thefollowing crystal electric field scheme of a tetravalent ura-nium configuration has been proposed: Γ singlet groundstate, Γ singlet first excited state at 33 K, two Γ dou-blets at 102 K, two Γ doublets at 152 K; Γ singlet at562 K; Γ at 1006 K (Grauel et al. , 1992). Crystal fieldexcitations at a temperature around 30 K have also been1 TABLE III Key properties of uranium based heavy-fermion superconductors. Missing table entries may reflect more complexbehavior discussed in the text. H c represents the extrapolated value for zero temperature. References are given in the text.(AF: antiferromagnet; FM,F: ferromagnetism; HO: hidden order; SC: superconductor)U-based UBe UPt URu Si UPd Al UNi Al UGe URhGe UCoGe UIrstructure cubic hexagonal tetragonal hexagonal hexagonal orthorh. orthorh. orthorh. monoclinictype NaZn - ThCr Si PrNi Al PrNi Al - - - -space group O h Fm3c P6 /mmc I4/mmm P6/mmm P6/mmm Cmmm Pnma Pnma P2 a (˚A) 10.248 5.764 4.128 4.189 5.207 3.997 6.875 6.845 5.62 b (˚A) - - - - - 15.039 4.331 4.206 10.59 c (˚A) - 4.899 9.592 5.382 4.018 4.087 7.507 7.222 5.60state SC AF, SC HO, SC AF, SC AF, SC FM, SC FM, SC FM, SC F1, F2, F3, SC γ (J / mol K ) - 0.44 0.07 0.2 0.12 0.032 0.164 0.057 0.049 T N , T C (K) - 5 17.5 14.2 4.6 52 9.5 3 46easy axis - - - - - a b, c c [10¯1]hard axis - - - - - b, c a a,b [010] (cid:126)Q - ( ± / , ,
1) (0,0,1) (0 , , ) (1 / ± δ, , /
2) - - - - δ = 0 . ± . µ ord ( µ B ) - 0.01 0.03 0 . ± .
03 0 . ± .
10 1.48 0.42 0.07 0.5, 0.05, 0.1 µ eff ( µ B ) - - - - - 2.7 1.8 1.7 - T s (K) 0.95 0.530, 0.480 1.53 2.0 1.1 0.8 0.25 (S1) 0.8 0.15 (in F3)0.4 (S2)∆ C/γ n T s H (cid:107) c (T) 14 2.1 3 3.9 0.9 - - - - ddT H (cid:107) c (T/K) -45 − . ± . H ⊥ c (T) - 2.8 14 3.3 0.35 - - - - ddT H ⊥ c (T/K) - − . ± . H c (T) - - - - - - - - 0.0265 ddT H c (T/K) - - - - - - - -10.8 - ξ ⊥ , ξ (cid:107) (˚A) 50 ∼
120 100, 25 85 - - - 150 1100 λ (cid:107) , λ ⊥ (˚A) 4000 4500, 7400 ∼ λ GL (˚A) - - - - - - 9100 (S1) - - κ
80 44 70 52 11 - - - -year of disc. 1984 1984 1986 1991 1991 2000 2001 2007 2004 inferred from a dip in the elastic constants (Modler et al. ,1993).A key characteristic of crystal electric fields in ura-nium compounds is, that they hybridize very stronglywith the conduction electrons. This is also the casein UPd Al , where time-of-flight inelastic neutron scat-tering fails to detect well defined crystal field excita-tions. Instead very broad spectra consisting of quasielas-tic Lorentzians plus additional inelastic scattering are ob-served (Krimmel et al. , 1996). The quasi-elastic scatter-ing thereby limits to an intrinsic width of 5 meV consis-tent with the 50 K energy scale seen in the susceptibilityand resistivity. When subtracting lattice contributionsby means of reference measurements in ThPd Al , theremaining inelastic scattering is consistent with the crys-tal field scheme given above.UPd Al develops strong electronic correlations at low temperatures with an enhanced linear temperature de-pendence of the specific heat in the paramagnetic state γ ∼ .
21 J / mol K (Geibel et al. , 1991a). Antiferro-magnetic order is observed below T N = 14 K (Geibel et al. , 1991b). The magnetic entropy released at T N is a substantial fraction of the local Zeeman entropy, S m = 0 .
65 R ln 2. The resistivity displays a change ofslope at T N (Caspary et al. , 1993). There is no evidencesuggesting the formation of a density-wave, e.g., like thesmall maximum in the resistivity near T in URu Si .In the ordered state the linear temperature dependenceof the specific heat is also enhanced, γ ∼ .
15 J / mol K (Geibel et al. , 1991a).Early single-crystal studies suggested that the mag-netic moments in UPd Al are oriented in the basal-planeof the hexagonal crystal structure, i.e., UPd Al has aneasy magnetic plane (Sato et al. , 1992). In zero magnetic2field neutron scattering shows commensurate antiferro-magnetic order with a wave vector (cid:126)Q = (0 , , /
2) andan ordered moment µ ord = 0 . µ B / U (Krimmel et al. ,1993). This corresponds to ferromagnetic planes stackedantiferromagnetically along the c-axis. The ordered mo-ment in UPd Al displays a mean-field temperature de-pendence that corresponds essentially to the form of a S = 1 / S = 1 / Al maybe seen in a large number of properties. For instance, (i)the thermal expansion, which shows a large sensitivityto uniaxial stress (Link et al. , 1995), (ii) the longitudi-nal and transverse elastic constants (L¨uthi et al. , 1993;Modler et al. , 1993), (iii) a kink in the Al spin-latticerelaxation rate and a gradual increase of the Al NMRline width (Kyogaku et al. , 1993), (iv) the emergence ofa gap in tunneling spectroscopy (Aarts et al. , 1994), and(v) an increase of the thermal conductivity (Hiroi et al. ,1997).The magnetic phase diagram of UPd Al , which yieldskey information of the nature of the magnetic and super-conducting order, has been studied in considerable de-tail. For magnetic fields applied in the basal-plane threetransitions at H = 0 . H = 4 . H m = 18 Tmay be distinguished (Oda et al. , 1994; Sugiyama et al. ,1993, 1994; de Visser et al. , 1992). In contrast, for thec-axis no field induced transition may be observed upto 50 T, the highest field studied. At H the orderedstate changes from commensurate antiferromagnetism toa canted state (Grauel et al. , 1992; Kita et al. , 1994). Themetamagnetic transition at H m has attracted consider-able interest. At H m the magnetization increases from ∼ . µ B / U to ∼ . µ B / U (de Visser et al. , 1992). Be-low 4.2 K the transition becomes hysteretic (Sakon et al. ,2001). The magnetoresistance displays a peak at H m for H and i (cid:107) (cid:104) (cid:105) , while there is a discontinuous step in themagnetoresistance for H (cid:107) (cid:104) (cid:105) and i (cid:107) (cid:104) (cid:105) (de Visser et al. , 1993).The critical field H m increases when tilting the fielddirection towards the c-axis. It exceeds 50 T, the highestfield measured, for an angle larger than 60 ◦ (Oda et al. ,1994; Sugiyama et al. , 1994). The angular dependence isconsistent with XY-type of order. Torque magnetizationmeasurements show that the basal plane anisotropy per-sists up to 60 K (S¨ullow et al. , 1996). The metamagnetictransition field changes only weakly as a function of tem-perature, terminating in a tricritical point around 12 K(Kim et al. , 2001b). For temperatures well above the tri-critical point a cross-over survives at H m , reminiscent ofthe metamagnetic transition (Oda et al. , 1999). Whentaken together, the latter properties suggest that crystalelectric fields and the electronic structure at the Fermilevel play an important role in controlling the metamag-netic transition, possibly related to a change of 5f local-ization.Superconductivity in UPd Al is observed below T s = 2 K. Even though T s is amongst the highest of allheavy fermion systems, it is nearly an order of mag-nitude smaller than T N . This distinguishes UPd Al and UNi Al from the systems reviewed above. Thesuperconducting transition is accompanied by a distinctanomaly in the specific heat, with ∆ C/γT s ≈ .
48. Be-low T s the specific heat varies as C ( T ) = γT + AT , sug-gesting the presence of line nodes (Caspary et al. , 1993).Also consistent with lines nodes is the cubic temperaturedependence of the thermal expansion, α ∝ T (Mod-ler et al. , 1993). The ratio of the thermal conductivitydivided by the temperature, κ/T , shows a finite contri-bution for T → et al. , 1997). Near T s a cross-over isobserved rather than a sharp kink, followed by a depen-dence κ/T ∝ T providing further evidence for line nodes(Hiroi et al. , 1997). In magnetic field κ/T increases,a kink appears at T s ( H ) and the temperature depen-dence changes slightly. Recently angle-resolved magneto-thermal transport measurements showed the absence ofan orientation dependence in the basal plane, while atwo-fold symmetry exists in the plane perpendicular tothe basal plane (Watanabe et al. , 2004a). From this itwas concluded that the gap has a single line node orthog-onal to the c-axis, while the gap is isotropic in the basalplane and may be given as (cid:126) ∆( (cid:126)k ) = ∆ cos( k z c ).The upper critical fields H ac = 3 . H cc =3 . ∂H ac /∂T | T s ≈ − . / Kand ∂H cc /∂T | T s ≈ − .
45 T / K are remarkably isotropic(Ishiguro et al. , 1995; Sato et al. , 1996). They corre-sponds to a coherence length ξ GL ≈
85 ˚A, where the pen-etration depths λ ⊥ (0) = 4800 ±
500 ˚A and λ (cid:107) (0) = 4500 ±
500 ˚A (with respect to the c-axis) inferred from magneti-zation and µ -SR measurements (Feyerherm et al. , 1994;Geibel et al. , 1991a). It establishes UPd Al as strongtype 2 superconductor with κ GL ≈
52. The anisotropyof H c for T → et al. , 1997a). It is instructive tocompare the observed field values with the conventionalweak-coupling orbital and paramagnetic limiting fields,where H p = 3 . H ∗ a = 6 . H ∗ c = 7 . ˚A. UPd Al exhibits hence type 2 superconductivityin the clean limit, that is dominated by paramagnetic lim-iting. This motivated an interpretation of an anomalousdip in the AC susceptibility and magnetization near H c in terms of a FFLO state (Gloos et al. , 1993; Norman,1993). However, further studies suggest that the anoma-lous dip exists at all temperatures below T s in contrastto the finite temperature range predicted theoretically.Taken together the anomalous behavior near H c is morecharacteristic of the peak effect (Haga et al. , 1996). Fora further discussion of FFLO states we refer to sectionV.B.1.3For the further discussion of the interplay of super-conductivity and antiferromagnetism it is instructive toconsider the nature of the 5f electrons. The U-U spac-ing in UPd Al and UNi Al , given by d U − U = 4 .
186 ˚Aand d U − U = 4 .
018 ˚A, respectively, are well above the Hilllimit of 3.4 ˚A (Hill, 1970). This implies that any itineracyof the f-electrons must be related to a hybridization withother electrons. The larger spacing in UPd Al is therebyconsistent with the evidence of stronger localization ofthe 5f electrons. Several properties of UPd Al are char-acteristic of local uranium moments. For instance, thesusceptibility at high temperatures shows a Curie-Weissdependence with a fluctuating moment µ eff = 3 . µ B / U.Polarized neutron scattering of the magnetic form factorsin an applied field of 4.6 T shows the lack of magneticpolarization at the Pd site, i.e., the magnetic polariza-tion is well localized at the uranium site (Paolasini et al. ,1993). It was however not possible to infer unambigu-ously from the magnetic form factor, whether uranium istetravalent. The observed ratio of the orbital to the spinmoment R = µ L /µ S ≈ − .
01 is closer to the value ofU ( R = − .
56) than for U ( R = − . et al. , 1998).The evidence for local moment magnetism is con-trasted by optical conductivity and quantum oscilla-tory studies of the Fermi surface, which clearly showstrongly renormalized quasiparticle conduction bands(Inada et al. , 1999; Terashima et al. , 1997). In the opticalconductivity Drude behavior is observed with ultra-slowrelaxation rates (Scheffler et al. , 2005). At the metam-agnetic transition at H m a reconstruction of the Fermisurface topology is observed without substantial varia-tion of the renormalization. This may be related to amagnetic field induced transition from an antiferromag-netic to a ferromagnetic exchange splitting, but does notappear to be driven by a localization of the f-electrons.Experimentally several properties of UPd Al suggesta dual state, where part of the 5f electrons are local-ized and the other part are itinerant, i.e., a combinationof both characteristics may be seen in the same physi-cal quantity. This was first noticed in measurements ofthe specific heat under pressure up to 10.8 kbar, whereamongst other things the size of the anomaly at the an-tiferromagnetic transition is strongly suppressed, whilethe superconducting transition is not (Caspary et al. ,1993). Also, the magnetic properties are anisotropicas opposed to the superconducting properties which areisotropic (Feyerherm et al. , 1994; Ishiguro et al. , 1995;Sato et al. , 1996). Neutron scattering (Krimmel et al. ,1993) and NMR/NQR studies (Kohori et al. , 1994) fur-ther show that the antiferromagnetic order survives es-sentially unchanged in the superconducting state. Thissuggests that both forms of order may be carried by dif-ferent subsystems. Finally, the spectrum of excitationsexhibits different contributions. Resonant 5d - 5f pho-toemission shows a sharp peak near E F and a broad hump at a binding energy ∼ et al. , 1994; Takahashi et al. , 1995).As a function of temperature photoemission establishesthat the electronic properties change from itinerant tolocalized (Fujimoto, 2007; Sato, 1999). Inelastic neutronscattering shows a weakly dispersive mode at an energyof ∼ T N , consistent with earlystudies (Petersen et al. , 1994), and a quasi-elastic sig-nal at the antiferromagnetic ordering wave vector (Sato et al. , 1997b). UPd Al -Pb tunnel junctions show a su-perconducting gap around 0.235 meV and antiferromag-netic spin wave mode around 1.5 meV, consistent withthe neutron scattering studies (Jourdan et al. , 1999).At first sight the dispersive and the quasi-elastic exci-tations in UPd Al seen in neutron scattering may ap-pear to be disconnected. However, polarized neutronscattering shows that the dispersive mode and the quasi-elastic signal are both transversely polarized. This sug-gests a common origin (Bernhoeft et al. , 1998). As partof this study it was further shown that the spectrum ofantiferromagnetic spin fluctuations in the framework ofconventional paramagnon theory (Lonzarich and Taille-fer, 1985; Moriya, 1985) is quantitatively consistent with T N and the enhancement of the normal state specificheat. As the temperature decreases below T s the quasi-elastic spectrum changes and a steep maximum emergesat very small ω . The maximum is also referred to asresonance mode. When plotting the maximum as a func-tion of temperature, a remarkable agreement with thetemperature dependence of a BCS gap is found, where2∆ = 3 . k B T s (Bernhoeft et al. , 1999; Metoki et al. ,1998). Under magnetic field the resonance vanishes at H c (Blackburn et al. , 2006a). In a spin-echo neutronscattering study the vanishing of spectral weight in thesuperconducting state was investigated at µ eV resolu-tion (Blackburn et al. , 2006b). The experiments establishthat the intensity vanishes completely, placing a strongconstraints on the pairing symmetries.Self-consistent LDA band structure calculations treat-ing the 5f states in UPd Al as being itinerant repro-duce the ordered magnetic moment, magneto-crystallineanisotropy and de Haas-van Alphen spectra (Sandratskii et al. , 1994). These studies also showed that an anti-ferromagnetic and ferromagnetic ground state are nearlydegenerate, consistent with the metamagnetic transitionat H m = 18 T. In these calculations the two largest Fermisurface sheets have markedly different 5f contributions,where one is almost purely 5f and the other yields 30 %5f character, respectively (Kn¨opfle et al. , 1996). Thesedifferences may provide a tentative explanation for thedual behavior.In recent years a controversy has developed concerningthe interplay of antiferromagnetic order and supercon-ductivity in UPd Al . In the traditional view of heavy-fermion systems the f-electron orbitals are screened by asinglet coupling with the conduction electrons and thencondense into a heavy Fermi liquid at low temperatures.4In this scenario the f-electrons are itinerant and the su-perconductivity is due to an abundance of soft magneticfluctuations. The effects of spin-orbit coupling may thenbe treated by a two-component susceptibility (Bernhoeft et al. , 1998; Bernhoeft and Lonzarich, 1995). Here theobservation that the correlation length associated withthe resonance peak matches the superconducting coher-ence length inspired an interpretation of the resonancepeak as a key feature of the Copper pairs themselves.The main objection against the traditional scenario is itslack of material specific aspects.In an alternative scenario it has been proposed thatonly one of the three 5f uranium electrons is itiner-ant, whereas the other two are localized (Sato et al. ,2001; Zwicknagl et al. , 2002). The microscopic under-pinning of this so-called duality-model are strong intra-atomic correlations that are subject to Hund’s rules andweak anisotropic hopping (Efremov et al. , 2004). Inthe duality-model the exchange interaction between theitinerant- and localized-electron subsystems drives thesuperconductivity in terms of a magnetic exciton. Themain objection against the duality-model and a pairingmediated by crystal field excitations is, that the crystalfield levels cannot be distinguished experimentally. Themodel nevertheless proves to be quite powerful. In a firstanalysis an A g order parameter symmetry was predicted(Miyake and Sato, 2001). Further implications havebeen worked out in a strong coupling approach whichwere found to be compatible with experiment (McHale et al. , 2004). The theoretical analysis established thatthe emergence of unconventional superconductivity re-sults in a resonance peak in the spectrum of magneticexcitations, consistent with neutron scattering (Chang et al. , 2007).We conclude this section with a brief review of theproperties of UPd Al at high pressure. The electri-cal resistivity under pressure shows that T N decreasesfrom 14 K to about 8 K at a pressure of 65 kbar, whilethe normal state maximum in the resistivity increases(Link et al. , 1995). At low pressures elastic neutronscattering shows an initial increase of µ ord , followedby a decrease above 5 kbar with a rate dµ ord /dp = − µ B / kbar. This is tracked by T N which decreasesat a rate dT N /dp ≈ − .
05 K / kbar at high pressures(Honma et al. , 1999). Up to 11 kbar the lattice constantsdecrease at a rate c − dc/dp = 7 . × − kbar − and a − da/dp = 4 . × − kbar − .High-pressure x-ray diffraction in UPd Al andUNi Al up to 400 kbar shows that both compoundshave essentially the same bulk modulus B = 159(6) GPa(Krimmel et al. , 2000). In UPd Al these studies furtherrevealed a structural phase transition at p c =250 kbarfrom a high-symmetry hexagonal to low-symmetry or-thorhombic state with space group Pmmm. Up to230 kbar the c/a ratio remains essentially constant. Inthe high-pressure phase the compressibility is a factor oftwo larger. The structure above p c belongs to space groupPmmm, which is a subgroup of Cmmm, which in turn is a non-hexagonal non-isomorphic subgroup of P6/mmm.The shortest metal-metal spacing in UPd Al is the U-Pddistance, which reaches 1.51 ˚A at p c . Interestingly thiscorresponds to the sum of ionic radii of U and Pd ,suggesting a U valence fluctuating state below p c andU to U transition at p c , where the ionic radius of U is reduced by 15%. A combination of resonant inelastic x-ray scattering with first principles structure calculationsis consistent with a delocalization from U +4 − δ to U +4+ δ (Rueff et al. , 2007). Finally, the extrapolated pressure,where the superconductivity in UPd Al vanishes corre-sponds to the critical pressure of the structural transition(Link et al. , 1995). While this may be completely fortu-itous, it might alternatively identify the tetravalent Uconfiguration as a precondition for superconductivity. b. UNi Al In comparison to UPd Al the magnetismand superconductivity in UNi Al are much more typicalof itinerant 5f electrons. The antiferromagnetic order isan incommensurate spin density wave, and the supercon-ductivity is a candidate for spin-triplet pairing. Further,at the antiferromagnetic transition at T N = 4 . T N is small, S m = 0 .
12R ln 2 (Tateiwa et al. , 1998). Likewise the resistivity only shows a faintfeature at T N (Dalichaouch et al. , 1992). As comparedwith UPd Al the smaller U-U distance, d U − U = 4 .
018 ˚Ain UNi Al , is also compatible with the more itinerantcharacter of the 5f electrons. As mentioned above, be-cause the U-U distance in both compounds is above theHill limit (3 . T ∗ ∼
100 K, characteristic of a dominant energyscale, but the coherence temperature may be as high as300 K (Sato et al. , 1996). The normal state properties ofUNi Al at low temperatures show the presence of strongelectronic correlations. This is best seen in the specificheat, which shows an enhanced Sommerfeld coefficient γ = 0 .
12 J / mol K and an enhanced T resistivity (Geibel et al. , 1991a).Selected microscopic probes nevertheless suggest a cer-tain degree of 5f localization. Photoemission exhibitsa combination of a sharp peak near E F , a smaller fea-ture around 0.6 eV and broad hump at 2 eV (Yang et al. ,1996). The features near E F and at 0.6 eV have beenattributed to itinerant and localized 5f electrons, respec-tively, while the hump at 2 eV is related to the Ni 3dstates. The photoemission studies compare with polar-ized neutron scattering and circular dichroism measure-ments, which show a nearly spherical magnetization dis-tribution at the uranium sites of the order 86% in bothUPd Al and UNi Al . In UPd Al the remaining 14%are due to diffuse background, while in UNi Al the re-maining 14% can be attributed to the Ni site (7%) anddiffuse background (7%) (Kernavanois et al. , 2000). The55f orbital contribution observed in circular dichroism isconsistent with that inferred from the polarized neutronscattering study. A local character of the 5f electronshas finally also been inferred from µ -SR measurements(Amato et al. , 2000; Schenck et al. , 2000). A peculiarityof the µ -SR studies in UNi Al are extended muon stop-ping sites, where the muon may tunnel along a ring ofsix m-sites that surrounds the b-site (0 , , / Al is compara-ble to UPd Al and of the order 3 to 5 depending on thetemperature (Sato et al. , 1996; S¨ullow et al. , 1997). Theproposed crystal electric field scheme to account for thesusceptibility is the same for UPd Al , however, withlarger values. Specifically, a Γ singlet ground state isfollowed by a Γ first excited singlet at 100 K, two Γ doublets at 340 K, two Γ doublets at 450 K, one Γ sin-glet at 1300 K and a Γ doublet at 1800 K (S¨ullow et al. ,1997). It is interesting to note, that the ratio of orderedmoment to T N in both compounds is consistent with thecrystal field scheme. As for UPd Al the experimentalevidence hence also supports a tetravalent uranium con-figuration.Neutron scattering experiments in UNi Al at firstfailed to detect the antiferromagnetic order (Krimmel et al. , 1992), while µ SR and NMR showed numerousfeatures hinting at incommensurate antiferromagnetismwith a small ordered moment (Amato et al. , 1992; Kyo-gaku et al. , 1993). Moreover, Al NMR shows an en-hancement of the spin lattice relaxation rate near T N characteristic of an abundance of spin fluctuations (Kyo-gaku et al. , 1993). Single-crystal elastic neutron scat-tering eventually revealed a second order phase transi-tion of incommensurate antiferromagnetic order at T N =4 . (cid:126)Q = (1 / ± δ, , / δ = 0 . ± . ξ m ≈
400 ˚A, are typical of heavy fermion systems. Theordered moment µ ord = (0 . ± . µ B is indeed small(Lussier et al. , 1997; Schr¨oder et al. , 1994) with a crit-ical exponent β = 0 . ± .
03 characteristic of three-dimensional order. In particular the latter feature con-trasts the small moment antiferromagnetism in UPt andURu Si . Spherical neutron polarimetry finally estab-lished that the magnetic structure may indeed be viewedas a spin-density wave, where the moments point in the (cid:126)a ∗ direction and the amplitude is modulated (Hiess et al. ,2001). The antiferromagnetic planes are stacked alongthe c-axis. The magnetic phase diagram of UNi Al as in-ferred from the bulk properties is fairly isotropic (S¨ullow et al. , 1997). An exception is the crystallographic b-axis,where an additional transition has been taken as evidenceof an incommensurate to commensurate phase transition,i.e., magnetic field allows to tune the commensurabil-ity. Taken together, the magnetic order in UNi Al andUPd Al differ considerably.UNi Al superconducts below a temperature T s =1 .
06 K. In polycrystalline samples the specific heatanomaly is distinct but small, with ∆
C/γT s ≈ . H ac ≈ .
002 T and H ac ≈ .
52 T imply type II superconductivity witha Ginzburg-Landau κ ≈
11 (Sato et al. , 1996). Incontrast to UPd Al , which shows a fairly isotropic H c and initial slope near T s and paramagnetic limit-ing UNi Al displays marked anisotropies where H cc ≈ . dH cc /dT = − .
14 T / K and H ac ≈ .
35 T, dH ac /dT = − .
42 T / K, respectively (Sato et al. , 1996).As for UPd Al these values may be compared with theexpected paramagnetic limit H p = 0 .
18 T and orbitallimits H ∗ ac = 0 .
79 T and H ∗ cc = 0 .
29 T. Thus H c ex-hibits orbital limiting H c ≈ H ∗ c and H c < H p in starkcontrast to UPd Al . At first sight this comparison sug-gests pure orbital limiting consistent with triplet pairing(Ishida et al. , 2002). However, it may also be reconciledwith the coexistence of superconductivity and antifer-romagnetic order (Sato et al. , 1996). In any case, thesuperconductivity clearly shows numerous hints for un-conventional pairing. For instance, NMR measurementsshow the absence of a Hebel-Slichter peak at T s (Kyogaku et al. , 1993), where the decrease of 1 /T in the supercon-ducting state is consistent with line nodes (Tou et al. ,1997). The Knight shift remains, moreover, unchangedin the superconducting state characteristic of spin-tripletpairing (Ishida et al. , 2002). This contrasts the behaviorobserved in UPd Al , where the decrease of the Knightshift indicates spin-singlet pairing. Spin triplet pairingin bulk samples of UNi Al is also contrasted by prelim-inary studies of thin epitaxial films of UNi Al . Thesestudies suggest that T s depends on the current direction,where H c implies spin-singlet pairing (Jourdan et al. ,2004).Early µ -SR measurements suggested a genuine coexis-tence of superconductivity and antiferromagnetism (Am-ato et al. , 1992). Elastic neutron scattering shows an ef-fective increase of the ordered magnetic moment in thesuperconducting state (Lussier et al. , 1997). Inelasticneutron scattering shows quasi-elastic scattering around (cid:126)Q = (0 . , , .
5) similar to what is observed in UPd Al ,but with a reduced intensity of about 10%. However,there is neither a build-up of additional intensity nor agap developing, nor a gapped spin wave excitation (Aso et al. , 2000). Further studies established quasi-elasticscattering along ( H, , n/ n is an odd integer,and the width is ∼ et al. , 2002).As for UPd Al only a small pressure dependence of T N and T s is observed in UNi Al , given by dT N /dp ≈− .
12 K / kbar and dT s /dp = − (0 . ± . / kbar(Wassermann and Springford, 1994). In fact, substitu-tional doping of Ni by Pd appears to act dominantlylike pressure. Likewise the bulk modulus determined byx-ray diffraction up to 385 kbar is similar and given by B = 150(5)GPa without evidence for a structural phasetransition up to 385 kbar (Krimmel et al. , 2000). InUNi Al the pressure where an U-Pd spacing is reachedthat is equivalent to UPd Al at p c may be extrapolatedas 725 kbar.6In summary both UPd Al and UNi Al do not seemto be located in the immediate vicinity of a zero temper-ature instability, that may be reached with hydrostaticpressure. This may provide an important hint, that crys-tal electric fields indeed provide a key ingredient for thesuperconductivity to occur in both compounds. c. CePt Si The discovery of heavy-fermion supercon-ductivity in the antiferromagnetic state of CePt Si hasattracted great interest, not so much because it coex-ists with antiferromagnetic order, but because the crys-tal structure of CePt Si lacks inversion symmetry (Bauer et al. , 2004a). The low temperature properties of thiscompound are characterized by the onset of commen-surate antiferromagnetic order at T N = 2 . (cid:126)Q = (0 , , / et al. , 2004), chiral compo-nents or a canting of the magnetic order have so far notbeen observed.The value of T s = 0 .
75 K first reported for CePt Si isfairly high. In contrast, more recent work suggest a lower T s = 0 .
45 K in combination with sharper magnetic andsuperconducting transitions (Takeuchi et al. , 2007). Dueto the lack of inversion symmetry CePt Si may be viewedas the first representative of a new class of heavy-fermionsuperconductors. Further members of this class discov-ered so far are CeRhSi , CeIrSi and CeCoGe . Theproperties of the non-centrosymmetric superconductorsincluding CePt Si is reviewed in section IV.A.
2. Small moment antiferromagnets a. UPt The heavy fermion compound UPt exhibitstwo forms of order at low temperatures. At T N ≈ orders antiferromagnetically. This is followed bya superconducting transition at T s = 0 .
54 K. BecauseUPt so far is the only intermetallic compound, whichunambiguously displays multiple superconducting phaseswith different order parameter symmetries, it has beenstudied in great detail. In the following we briefly reviewkey features of the magnetic order and superconductivityto put them in perspective with the antiferromagneticcompounds addressed so far. The evidence for multiplesuperconducting phases is addressed in section V. For anextensive review of the properties of UPt we refer to(Joynt and Taillefer, 2002).UPt crystallizes in a hexagonal structure, space groupP6 /mmc, point group D h . The lattice parametersare a = 5 .
764 ˚A and ˜ c = 4 .
899 ˚A, where ˜ c is the dis-tance between neighboring planes. It is convenient todefine the b-axis perpendicular to the a-axis (and thusparallel to the a ∗ axis). The molar volume is V m =42 . × − m / mol U and the nearest U-U distance with d U − U = 4 .
132 ˚A quite large. The compressibilities have been inferred from measurements of the sound velocity.They are given by κ a = − a − da/dp = 0 .
164 Mbar − , κ c = − c − dc/dp = 0 .
151 Mbar − and for the volume κ V = 2 κ a + κ c = 0 .
479 Mbar − (de Visser et al. , 1987).Several transmission electron microscopy studies have re-ported a possible incommensurate structural modulation.However, it is now generally believed that this modula-tion results from ion milling and is not present in bulksamples (Ellman et al. , 1997, 1995).The normal state properties of UPt at low temper-atures are well described as a heavy Fermi liquid. Thenormal state specific heat in UPt up to 1.5 K is linearin temperature with C/T ≈ . ± .
02 J / K mol and aweak cubic term T ln( T /T ∗ ) as discussed in (de Visser et al. , 1987). At higher temperatures an additional T contribution emerges consistent with a Debye tempera-ture Θ D ≈
210 K. For
H > H c an unexplained ad-ditional strong upturn in C/T emerges below ∼ . et al. , 1994).As a function of temperature the resistivity of UPt decreases monotonically from a room temperature value ρ ab ≈ µ Ωcm and ρ c ≈ µ Ωcm (Kimura et al. , 1995;de Visser et al. , 1987). At low temperatures a quadratictemperature dependence of the resistivity is observed ρ ( T ) = ρ + AT where A ab ≈ . ± . µ Ω cm K − and A c ≈ . ± . µ Ω cm K − (e.g., (Kimura et al. ,1995; Lussier et al. , 1994; Suderow et al. , 1997). At lowtemperatures the anisotropy of the resistivity is essen-tially temperature independent with ρ b /ρ c ≈ .
6. Theanisotropy is attributed to differences of Fermi veloci-ties. The charge carrier mean free path inferred from theresidual resistivity and quantum oscillatory studies is ofthe order 5000 ˚A. Under pressure the A coefficient of theresistivity decreases at a rate d ln A/dp ≈ −
40 Mbar − (Ponchet et al. , 1986; Willis et al. , 1985). A comparisonof the T resistivity with the linear temperature depen-dence of the specific heat establishes consistency of theratio γ/ √ A with other heavy fermion systems (Kadowakiand Woods, 1986). The observation that UPt forms aslightly anisotropic three-dimensional Fermi liquid withstrong electronic correlations is underscored by tempera-ture dependence observed in thermal conductivity mea-surements (Lussier et al. , 1994; Suderow et al. , 1997).The normal state magnetic properties of UPt arestrongly enhanced. The uniform susceptibility in thebasal plane exhibits a strong Curie-Weiss dependence athigh temperature and a broad maximum around 20 K(Frings et al. , 1983). The susceptibility is anisotropicwith χ c < χ ab . The behavior seen in the uniformsusceptibility is tracked in P t
NMR (Tou et al. ,1996). Inelastic neutron scattering establishes a com-plex spectrum of antiferromagnetic fluctuations (Aeppli et al. , 1988, 1987). At moderate temperatures a fluc-tuation spectrum characteristic of large uranium mo-ments ( ∼ µ B ) is observed with a characteristic energyof 10 meV. Below ∼
20 K antiferromagnetic correlationsdevelop at (cid:126)Q = (0 , ,
1) that peak around 5 meV. Thesefluctuations correspond to correlations between adjacent7nearest-neighbor uranium sites. When decreasing thetemperature well below 20 K additional antiferromag-netic correlations develop around (cid:126)Q = ( ± / , ,
1) witha characteristic energy ∼ . ∼ . µ B . These fluctuations correspond to inter-site correlations within each hexagonal plane. Finally,slow magnetic fluctuations with a dispersive relaxationrate exist at low temperatures (Bernhoeft and Lonzarich,1995). Thus the excitation spectrum yields a duality ofslow and fast excitations somewhat similar to UPd Al .In what way these fluctuations affect the unconventionalsuperconductivity in UPt is an open issue.The magnetic properties of UPt finally include alsoan elastic component of the magnetic correlations at (cid:126)Q = ( ± / , ,
1) with a tiny ordered moment around0.01 to 0 . µ B / U. The antiferromagnetic order was firstnoticed in µ -SR and later confirmed by neutron scatter-ing (Aeppli et al. , 1988). The magnetic order is collinearand commensurate with fairly short correlation lengths ∼
300 ˚A. It appears to be insensitive to sample qual-ity. Perhaps most remarkably, the only experimentalprobes that are sensitive to the antiferromagnetic orderare neutron scattering and µ -SR. Notably, the antiferro-magnetism is not seen in NMR (Tou et al. , 1996), specificheat (Fisher et al. , 1991) and magnetization. It has there-fore been suggested that the magnetic order is essentiallydynamic in nature.Microscopic evidence that UPt forms a heavy-fermionground state par excellence was obtained in quantum os-cillatory studies (Taillefer and Lonzarich, 1988; Taillefer et al. , 1987). The studies revealed a wide range of massenhancements up to 120 times of the free electron mass.Despite these strong mass enhancements the spectra werefound to be in remarkable agreement with density func-tional theory taking the 5f electrons to be itinerant (see(Joynt and Taillefer, 2002) and references therein). Mostof the frequencies, especially those corresponding to largeportions of the Brillouin zone could be identified satis-factorily. In summary the Fermi surface consists of sixsheets of uniformly high effective masses. In fact, theFermi velocities on the observed sheets are extremelyslow (cid:104) v F (cid:105) bc ≈ / s and do not differ by more than15%. In contrast to the topology of the Fermi surface,functional density theory fails to account for these largemass renormalizations.It has been proposed that the mass enhancement inUPt is due to a duality of the 5f electrons in the spiritof that discussed for UPd Al and UNi Al (Zwicknagl et al. , 2002). In this scenario one f electron is itinerantwhile the other two are localized. The mass enhancementin UPt can be accounted for, when assuming a crystalfield level scheme similar to UPd Al with a Γ groundstate and Γ first excited state. A potential weakness ofthis assumption is that the crystal field levels hybridizeso strongly with the conduction electrons, that inelasticneutron scattering fails to detect them. The relation-ship of the duality model as applied to UPt and the ex-perimentally observed tiny ordered moments is thereby also an unresolved issue. The recent thorough analysisof quantum oscillatory studies of the Fermi surface are,finally, in much better agreement with fully itinerant f-electrons (McMullan et al. , 2008).Measurements of the resistivity, specific heat and ACsusceptibility establish UPt as a bulk superconductor(Stewart et al. , 1984). Early studies of the ultrasoundattenuation in magnetic field (M¨uller et al. , 1987; Qian et al. , 1987; Schenstrom et al. , 1989) and of H c (Taille-fer and Lonzarich, 1988) suggested the possibility of twosuperconducting phase transitions. This was eventuallyconfirmed in high resolution specific heat measurements(Fisher et al. , 1989; Hasselbach et al. , 1989). Furtherstudies establish that there are three superconductingphases, denoted A, B and C. The antiferromagnetic or-der can be shown to introduce an additional symmetrybreaking that stabilizes these phases. In summary threepieces of evidence identify UPt as unconventional su-perconductor. First, several transport quantities displaymarked anisotropies, most notably the ultrasound veloc-ity and the thermal conductivity. Second, there is evi-dence for phase transitions within the superconductingstate as seen in the specific heat and ultrasound attenu-ation. Third, several properties show activated tempera-ture dependences instead of the exponential freezing outof excitations. The superconducting phases of UPt willbe described in further detail in section V. b. URu Si The body-centered tetragonal uraniumcompound URu Si , space group I4/mmm, crystallizeswith lattice constants a = 4 .
128 ˚A and c = 9 .
592 ˚A.At low temperatures it undergoes two phase transitions(Schlabitz et al. , 1984): a transition to an hitherto un-known form of order at T ≈ . T s ≈ . et al. , 1986; Palstra et al. , 1985; Schlabitz et al. , 1986). The entropy released at T is given by∆ S ≈ .
2R ln 2. Despite intense experimental and the-oretical efforts the ordering phenomenon accounting forthis entropy reduction has still not been identified. Thephase below T in URu Si has in turn become known as”hidden order” (HO). The hidden order exhibits manycharacteristics of an electronic condensation: (i) the spe-cific heat is consistent with a BCS gap (Maple et al. ,1986), (ii) the resistivity at T is strongly reminiscent ofthe density-wave system chromium (Fawcett, 1988), (iii)slight doping suppresses the resistivity anomaly rapidly(Kim et al. , 2004), (iv) the magnetization at T suggeststhe formation of a spin gap (Park et al. , 1997), whileoptical conductivity indicates a charge gap (Bonn et al. ,1988). Recent thermal conductivity measurements alsopoint towards a gap formation (Sharma et al. , 2005). TheHall effect and magnetoresistance suggest near compen-sation of particle- and hole-carriers and a strong inter-play between the stability of the hidden order under Rh-doping and the degree of polarization of the Fermi liquidand the Fermi surface topology (Jo et al. , 2007; Oh et al. ,82007).Neutron diffraction in URu Si shows antiferromag-netic order below T with a [001] modulation of tiny mo-ments, (0 . ± . µ B / U , and the spins aligned alongthe c-axis (Broholm et al. , 1987). The magnetic order isthree-dimensional with strong Ising-type spin anisotropy.Within a local-moment scenario the antiferromagnetismdoes not account for ∆ S . This contrasts antiferromag-netism with a large moment of 0 . µ B /U and the sameIsing anisotropy, which emerges under large hydrostaticpressure (Amitsuka et al. , 1999). A recent phase dia-gram is shown in Fig. 11 (Amitsuka et al. , 2006). NMR(Matsuda et al. , 2003, 2001) and µ -SR (Amitsuka et al. ,2003) measurements suggest that the tiny-moment an-tiferromagnetism at ambient pressure represents a tinyvolume fraction of large moment antiferromagnetism. Asfunction of pressure T increases, where dT /dp increases p ∗ ≈
14 kbar. In fact, the increase of dT /dp at p ∗ evenpersists under Re-doping (Jeffries et al. , 2007). Thereis currently growing consensus, that the small antiferro-magnetic moment is not an intrinsic property of the hid-den order. However, a spin-density-wave close to perfectnesting may exhibit the combination of a small momentwith a large reduction of entropy (Chandra et al. , 2003;Mineev and Zhitomirsky, 2005).The hidden order in URu Si is bounded by more con-ventional behavior at high excitation energies, high pres-sure and high magnetic fields. Inelastic neutron scatter-ing shows a gap ∆( T → ≈ . et al. ,1991). At low energies and temperatures, dispersivecrystal-field singlet–singlet excitations at the antiferro-magnetic ordering wave vector are observed. These prop-agating excitations merge above 35 meV or for T > T ,respectively, into a continuum of quasi-elastic antifer-romagnetic spin fluctuations, as normally observed inheavy-fermion systems. The excitations exhibit the Isinganisotropy up to the highest energies investigated exper-imentally. A rough integration of the fluctuation spectrasuggests that the size of the fluctuating moments wouldbe consistent with ∆ S , provided that these moments areinvolved in the ordering process (Broholm et al. , 1991;Wiebe et al. , 2007). Under large applied magnetic fieldsparallel to the c-axis the antiferromagnetic moment and T decrease, where T collapses to zero at B m = 38 T(Bourdarot et al. , 2005, 2003; Mason et al. , 1995; Santini et al. , 2000). At B M a cascade of metamagnetic transi-tions is observed, in which a large uniform magnetizationis recovered (Harrison et al. , 2003; Kim et al. , 2003b). Upto B m the entropy reduction at T stays approximatelyconstant (Kim et al. , 2003a), while the gap ∆, as seen inneutron scattering, increases at least up to 17 T (Bour-darot et al. , 2003). For a recent review see, e.g., (Harrison et al. , 2004).The antiferromagnetic order in URu Si is stabilizedunder uniaxial stress along certain crystallographic di-rections and hydrostatic pressure. NMR (Matsuda et al. ,2001), µ SR (Amitsuka et al. , 2003) and neutron scatter-
FIG. 11 Temperature versus pressure phase diagram ofURu Si inferred from various experimental probes. The on-set of the hidden order T is weakly pressure dependent. Thehidden order changes to large moment Ising antiferromag-netism above 7 kbar without pronounced effect on the evolu-tion of T / T N . Superconductivity vanishes with the appear-ance of the antiferromagnetism. HO: hidden order; AF: largemoment antiferromagnet; SC: superconductivity. Plot takenfrom (Amitsuka et al. , 2006). ing (Amitsuka et al. , 1999) measurements suggest, thatthe AF volume fraction increases and reaches 100% above p c ∼
14 kbar. An analogous increase of the AF signal isalso seen in neutron scattering under uniaxial stress of afew kbar along the [100] and [110] directions (Yokoyama et al. , 2005, 2002), but not under uniaxial stress along the c -axis [001]. Inelastic neutron scattering under pressureshows that the dispersive crystal-field singlet excitationsat low energies vanish at high pressures (Amitsuka et al. ,2000), consistent with them being a property of the HOvolume fraction.A major challenge are measurements of the Fermi sur-face. For instance, de Haas–van Alphen (dHvA) studiesunder hydrostatic pressure (Nakashima et al. , 2003) donot resolve abrupt changes of the dHvA frequencies andcyclotron masses at p c . This contrasts naive expectationof a distinct phase separation at p c . In these studies themost important observation is a considerable increase ofthe cyclotron mass with increasing pressure. New in-sights may be achieved with ultra-pure samples, thathave recently become available (Kasahara et al. , 2007;Matsuda et al. , 2008).A large number of microscopic scenarios have been pro-posed to explain the hidden order. These include variousversions of spin- and charge-density wave order (Maki et al. , 2002; Mineev and Zhitomirsky, 2005), forms ofcrystal electric field polar order (Kiss and Fazekas, 2005;Ohkawa and Shimizu, 1999; Santini and Amoretti, 1994),unconventional density waves (Ikeda and Ohashi, 1998)and orbital antiferromagnetism (Chandra et al. , 2002),Pomeranchuk instabilities (Varma and Zhu, 2006) or ne-matic electronic phases (Barzykin and Gorkov, 1993),9combinations of local with itinerant magnetism (Okunoand Miyake, 1998) and dynamical forms of order (Bern-hoeft et al. , 2003; F˚ak et al. , 1999). None of the mod-els was able to satisfactorily explain all of the availableexperimental data; some models are purely phenomeno-logical yet lack material-specific predictions that can bereadily verified by experiment, while others focus onlyon selected microscopic features. This leaves consider-able space for fresh theoretical input.The nature of the superconductivity in the hidden or-der of URu Si is still comparatively little explored. T s depends sensitively on sample quality. It is as high as T s = 1 .
53 K in the purest samples, which have residualresistivities as low as several µ Ωcm and charge carriermean free paths l ∼ et al. , 1995). In the specific heatthe onset of superconductivity is accompanied by a pro-nounced anomaly, where ∆ C/γT s ≈ . et al. , 1994; Fisher et al. , 1990). However, this value isreduced by comparison to the weak coupling BCS valueof 1.43. Between T s and 0 . T s the specific heat variesapproximately as C ∝ T akin that seen in UPt . Thisis consistent with line nodes of either a E u (1 ,
1) or B g state (Hasselbach et al. , 1993). Line nodes and uncon-ventional superconductivity has also been inferred from Si NMR and
Ru NQR, where 1 /T is found to showno coherence peak and decreases as 1 /T ∝ T below T s ,while the Knight shift is unchanged (Kohori et al. , 1996;Matsuda et al. , 1996). However, it has been pointed out,that the specific heat data are equally well explained interms of s-wave pairing in the presence of antiferromag-netism, where nodes are generated by the magnetic order(Brison et al. , 1994).Further information of the possible location and natureof the nodal structure has been inferred from the angularfield dependence of the specific heat, where the absenceof an agular dependence in the tetragonal basal planeand marked anisotropy between a- and c-axis suggeststhat the gap nodes are rather localized near the c-axis(Sakakibara et al. , 2007). An anisotropic gap has alsobeen inferred from point contact spectroscopy, consistentwith d-wave pairing (De Wilde et al. , 1994; Hasselbach et al. , 1992; Naidyuk et al. , 1996). If the experimental ev-idence for nodes is indeed due to the antiferromagnetismas suggested above, this requires, that the small anti-ferromagnetic moments are an intrinsic property of thehidden order, or that the hidden order interacts with thesuperconductivity in the same way antiferromagnetismwould do.A different scenario of the superconductivity has re-cently been proposed based on the electrical and ther-mal transport properties in ultra-pure URu Si (Kasa-hara et al. , 2007; Matsuda et al. , 2008). Here the Halleffect and magnetoresistance suggest multiband super-conductivity in a compensated electronic environment.Most remarkably, in the low temperature limit the ther-mal conductivity divided by temperature, κ/T displaysa rapid increase at low fields followed by a plateau up to some intermediate field H s < . H c . Above H s evolvesdifferently for field parallel and perpendicular to the c-axis, but κ/T drops abruptly just below H c characteris-tic of H c being first order (the first order behavior occursbelow ∼ . K . Based on their observations (Kasahara et al. , 2007) suggest a two-component order parameter,with two distinct gaps: line nodes perpendicular to thec-axis on a spherical light hole band and point nodesalong the c-axis on the elliptical heavy electron band.This scenario, notably the first order behavior and pointnodes are consistent with the magnetic field dependenceof the specific heat in the superconducting state (Yano et al. , 2008). Interestingly, the thermal conductivity inthe same ultra-pure samples also suggest a melting tran-sition of the flux line lattice and the formation of a co-herent quasiparticle Bloch state (Okazaki et al. , 2008).The lower critical field of the superconductivity inURu Si of H c ( T → ≈ . × − T, is essentiallyisotropic and displays a weak temperature dependence(W¨uchner et al. , 1993). H c is in contrast stronglyanisotropic with H ac = 14 T and H cc = 3 T. This im-plies strong type 2 behavior and short coherence lengths ξ a ≈
100 ˚A and ξ c ≈
25 ˚A. The anisotropy of H c maybe accounted for reasonably well by an anisotropic massmodel (Brison et al. , 1994). For the c-axis H c can beexplained by Pauli limiting, while it can be described bya combination of Pauli and orbital limiting for the a-axiswith strongly anisotropic Pauli limiting between the a-and c-axis (Brison et al. , 1995).An additional weak increase of H c for the c-axis at lowtemperatures that exceeds Pauli limiting has been con-sidered as tentative evidence for an FFLO phase. Alsounusual is the temperature dependence of the anisotropy H ac /H cc , which initially increase below T s and becomesconstant below ∼ . T s . In fact, the Ginzburg-Landauparameter inferred from the magnetization exhibits agradual decrease well below T s , somewhat slower thanthe behavior anticipated from H c but consistent withparamagnetic limiting (Tenya et al. , 2000). Finally, asmall positive curvature in the temperature dependenceof H c near T s has been considered as possible evidenceof a multicomponent order parameter that couples to anantiferromagnetic moment (Kwok et al. , 1990; Thalmeierand L¨uthi, 1991). Taken together, it is presently acceptedthat URu Si does not display multiple superconductingphases in terms of real-space or momentum-space mod-ulations (cf sections V.A.2.e and V.B.1).The thermal expansion displays pronounced anoma-lies at T s with ∆ α a = − . × − K − and ∆ α c =0 . × − K − (van Dijk et al. , 1995). Thus, thesuperconductivity varies sensitively with uniaxial pres-sure, notably dT s /dp a = − .
062 K / kbar and dT s /dp c =+0 .
043 K / kbar, consistent with experiment (Bakker et al. , 1991). The qualitative temperature dependenceof H c for uniaxial pressure applied along the a-axis re-mains thereby unchanged (Pfleiderer et al. , 1997a). Forcomprehensive information on the elastic constants werefer to (L¨uthi et al. , 1995).0The interplay of hidden order, small moment antiferro-magnetism and superconductivity in URu Si is largelyunresolved. Early neutron scattering studies suggested,that the small antiferromagnetic moments remain eitherunchanged in the superconducting state (Broholm et al. ,1987; Mason et al. , 1990; Wei et al. , 1992) or may bedecreasing by 1 to 2% (Honma et al. , 1999). This maybe consistent with a microscopic coexistence of hiddenorder and superconductivity. Under hydrostatic pressure T s decreases and vanishes between 5 and 14 kbar (Brison et al. , 1994; Jeffries et al. , 2007; McElfresh et al. , 1987).The magnetization and specific heat thereby shows, thatthe superconducting volume fraction decreases or, alter-natively, that the superconducting gap vanishes (Fisher et al. , 1990; Tenya et al. , 2005; Uemura et al. , 2005).Since the suppression of superconductivity is accompa-nied by an increase of volume fraction of large antifer-romagnetic moments, the large moment antiferromag-netism and superconductivity must represent competingforms of order. In contrast, the HO may even represent aprecondition for the superconductivity in URu Si to oc-cur, which points at an unknown superconducting pairinginteraction. C. The puzzling properties of UBe In the following we briefly review the properties ofUBe . Being the second system in which heavy-fermionsuperconductivity was identified this compound remainsone of the most puzzling materials amongst the systemsknown to date. For a long time UBe seemed to be out-side any of the patterns observed in the other systems.Recent work suggests the possible vicinity to an antifer-romagnetic quantum critical point under magnetic field(Gegenwart et al. , 2004). It is not unlikely, however, thatincipient antiferromagnetism is only part of the story.UBe crystallizes in the cubic NaZn structure, spacegroup O h or Fm3c with lattice constant a=10.248 ˚A(Pearson, 1958). There are 8 formula units per unit cell,with two Be sites; the uranium atoms are surrounded bycages of 24 Be atoms (Goldman et al. , 1985). The U-atoms form a simple cubic sublattice, with a large U-Uspacing d U − U = 5 .
13 ˚A, well above the hill limit of 3.4 ˚A,suggesting that any broadening of the uranium f-statesinto bands is due the hybridization with the conductionbands and not the result of direct overlap of the f-orbitals.By comparison with other heavy-fermion superconduc-tors the properties of UBe are fairly insensitive to sam-ple quality. In the normal metallic state of UBe thespecific heat exhibits a shallow maximum around 2 K,with a large linear term C/T = γ ≈ . / mol K (Ott et al. , 1983, 1984a). The susceptibility displays a strongCurie Weiss dependence with µ eff ≈ µ B and a Curie-Weiss temperature Θ ≈ −
70 K. The electrical resistivityincreases with decreasing temperature and reaches a largevalue of order 240 µ Ω cm before it decreases around 2 Kand reaches value of 130 µ Ωcm at the onset of supercon- ductivity. The extrapolated zero temperature residualresistivity is ρ = 60 µ Ωcm (Maple et al. , 1985).Superconductivity was first observed in the resistivityof UBe in 1975 (Bucher et al. , 1975) - four years priorto the discovery of superconductivity in CeCu Si . How-ever, the zero-resistance transition at T s = 0 . et al. ,1983). The specific heat anomaly is characteristic ofstrong coupling superconductivity with ∆ C/γT s ≈ . H c near T s is exceptionally large dH c /dT = −
45 T / K (Maple et al. , 1985; Thomas et al. ,1995). In the zero temperature limit H c ( T →
0) = 14 T. H c exhibits strong Pauli limiting and as an additionalfeature a change of curvature at T /T s ∼ . H c has been at-tributed to a combination of very strong coupling su-perconductivity and the tendency to form a FFLO state(see also section V.B.1). While the coupling constant λ = 15 in these calculations is suspiciously large and ex-ceeds coupling constants in comparable systems by anorder of magnitude, this scenario finds further supportin the pressure dependence of λ , which tracks the massenhancement inferred from dH c /dT | T s and the specificheat (Gl´emot et al. , 1999).Several properties suggest the presence of zeros of thesuperconducting gap. The power law dependence of thespecific heat C ∼ T (Mayer et al. , 1986; Ott et al. , 1987,1984b) and penetration depth λ ∼ T (Einzel et al. ,1986; Gross et al. , 1986) suggest point nodes, whereasthe NMR spin lattice relaxation rate suggests lines nodes(MacLaughlin et al. , 1987). This identifies UBe as un-conventional superconductor, a conjecture that is sup-ported by the behavior under substitutional Th doping(Lambert et al. , 1986). U − x Th x Be displays a complexphase diagram as show in Fig. 12 with multiple supercon-ducting phases (Ott et al. , 1986). Thermal expansion andspecific heat measurements identify a precursor of this ef-fect in pure UBe (Kromer et al. , 1998, 2000). We referto section V.A.2.f for a brief discussion of the details ofthis phase diagram.The calculated electronic structure of UBe is rel-atively simple for itinerant f-electrons (Norman et al. ,1987; Takegahara and Harima, 2000). The nature of theheavy fermion state in UBe has nevertheless provided amajor puzzle. By comparison to other heavy fermion sys-tems the susceptibility and specific heat vary only weaklyunder magnetic field. This is contrasted by a strongnegative magnetoresistance (Rauchschwalbe et al. , 1985;Remenyi et al. , 1986) providing tentative evidence thatUBe is a low density carrier system (Norman et al. ,1987; Takegahara et al. , 1986). Under hydrostatic pres-sure the normal metallic state assumes the more con-ventional form of a coherent Kondo lattice with a broadmaximum at several 10 K and a decreasing resistivity atlow temperatures (Aronson et al. , 1989; McElfresh et al. ,1 FIG. 12 Temperature versus Th concentration inU − x Th x Be . The upper curve corresponds to the on-set of superconductivity in the resistivity. In the range2% < x <
4% a second transition is observed in the specificheat, which may be related to magnetic order and/or anothersuperconducting phase. Plot taken from (Maple, 1995). T s extrapolates to zero around 40 kbar. Inter-estingly the residual resistivity decreases strongly around40 kbar, suggesting that the scattering mechanism caus-ing the residual resistivity may be involved in the super-conducting pairing UBe .First neutron scattering studies revealed a broad quasi-elastic Lorentzian spectrum of magnetic fluctuations witha half-width of 13.2 meV (Goldman et al. , 1986). Theyfailed to observe evidence for a narrow f-resonance of an-tiferromagnetic correlations (Goldman et al. , 1986; Lan-der et al. , 1992). Recent studies however, reveal short-range antiferromagnetic correlations below ∼
20 K for (cid:126)Q = (1 / , / ,
0) with a characteristic energy width of 1to 2 meV (Coad et al. , 2000; Hiess et al. , 2002).New studies of the normal metallic state as a functionof magnetic field establish non-Fermi liquid behavior with ρ ∼ T / and a related logarithmic divergence of the spe-cific heat (Gegenwart et al. , 2004). For field above H c aregime with T resistivity emerges. This Fermi liquid be-havior has been linked with the suppression of a featurein the thermal expansion that has been interpreted as afreezing of three-dimensional antiferromagnetic fluctua-tions. When taken together this has motivated specula-tions on a field-tuned antiferromagnetic quantum criticalpoint ∼ , at least as a facet of the complexcobination of properties of UBe . III. INTERPLAY OF FERROMAGNETISM ANDSUPERCONDUCTIVITY
Several f-electron ferromagnets have been found in re-cent years, that exhibit superconductivity with T s (cid:28) T C (cf table III). These systems contrast the reentrant su-perconductivity observed in ErRh B and related com-pounds, where ferromagnetic order appears well belowthe superconducting transition temperature and bothforms of order originate in separate microscopic subsys-tems. We begin this section with a review of systemsthat exhibit superconductivity in the ferromagnetic state,notably UGe and URhGe. We next address supercon-ductivity at the border of ferromagnetism in UCoGe andUIr. A. Superconducting Ferromagnets
1. UGe The superconducting ferromagnet UGe crystallizes inthe orthorhombic ThGe crystal structure, space groupCmmm (no. 65), with lattice constants a = 3 . b = 15 . c = 4 . et al. , 1997;Oikawa et al. , 1996). The crystal structure of UGe isdominated by zig-zag chains of the U atoms along thea-axis, where the U-spacing, d U − U = 3 .
85 ˚A. As forUPd Al and UNi Al the U-U distance is above the Hilllimit and without hybridization with other electrons thef-electrons would be localized. The U-chains are stackedwith Ge atoms at interstitial positions to form corrugatedsheets. These sheets are separated by further Ge atomsalong the b-axis, giving the crystal structure a certaintwo-dimensional appearance perpendicular to the b-axis.As discussed below the two-dimensional crystallographicappearance manifests itself in the electronic structure,which is dominated by a large cylindrical Fermi surfacesheet along the b-axis (Shick et al. , 2004; Shick and Pick-ett, 2001).At ambient pressure UGe develops ferromagnetic or-der below T C = 52 K with a zero temperature orderedmoment µ s = 1 . µ B / U aligned along the a-axis. Bycomparison with the a-axis, the b- and c-axis exhibitlarge magnetic anisotropy fields ( ∼
100 T for the c-axis)(Onuki et al. , 1992). The magnetic anisotropy imposesa strong Ising character on the magnetic properies. Inturn the temperature dependence of the ordered momentvaries as M ( T ) ∝ ( T − T C ) β between 0 . T C and T C ,where β = 0 .
33 is close to calculated value β ≈ .
36 ofa 3D Ising ferromagnet (Huxley et al. , 2001; Kernavonis et al. , 2001).Neutron depolarization measurements down to 4.2 Kestablish, that the magnetic moments are strictly alignedalong the a-axis, with a typical domain size in the bc-plane of the order 4 . µ m (Sakarya et al. , 2005). Thiscontrasts earlier reports of macroscopic quantum tunnel-ing of the magnetization below 1 K, where the inferred2domain size was only ∼
40 ˚A (Lhotel et al. , 2003; Nish-ioka et al. , 2002).The susceptibility of the paramagnetic state isanisotropic exhibiting a Curie-Weiss dependence forthe a-axis with a corresponding fluctuating moment of µ CW = 2 . µ B , that exceeds the ordered moment con-siderably. Taken by itself, the reduced ordered momentas compared with the free uranium ion value does notproof itinerant magnetism, but may be reconciled withthe presence of strongly hybridized crystal electric fields.We note that inelastic neutron scattering fails to detectdistinct evidence for crystal electric fields, as commonfor uranium based compounds. However, the reductionof the ordered moment as compared with the Curie-Weissmoment provides clear evidence of 5f itineracy.The degree of delocalization of the 5f electrons hasbeen explored by a variety of experimental techniques.The perhaps most direct probe is a combination of quan-tum oscillatory studies with band structure calculations,showing dominant f-electron contributions at E F (Shick et al. , 2004; Shick and Pickett, 2001; Terashima et al. ,2001). We will discuss these studies in further detailbelow. Polarized neutron scattering shows that the mag-netic order is strictly ferromagnetic without additionalmodulations (Kernavonis et al. , 2001). The magneticform factor of the uranium atoms is equally well ac-counted for by a U or U configuration (Huxley et al. ,2001; Kernavonis et al. , 2001), where a magnetic field of4.6 T does not induce any magnetic polarization at the Gesites. However, the ratio of the orbital to spin moment, R , does not vary substantially as a function of tempera-ture between the paramagnetic and ferromagnetic states.As compared with the free ion value it is systematicallyreduced, suggesting a delocalization of the 5f electrons.Also, the value of R for U is in better agreement withcircular dichroism measurements (Okane et al. , 2006) andLDA+U band structure calculations (Shick and Pickett,2001), which support a trivalent uranium state.Evidence for some delocalization of the f-electrons inUGe may also be seen in the specific heat and the spec-trum of low lying magnetic excitations. At the Curie tem-perature the specific heat displays a pronounced anomaly,where ∆ C/T ≈ . / mol K . This compares with amoderately enhanced Sommerfeld contribution C/T = γ = 0 .
032 J / mol K at low temperatures (Huxley et al. ,2001). The strong uniaxial anisotropy causes a largeanisotropy gap for spin wave excitations. In turn in-elastic neutron scattering near T C only shows stronglyenhanced spin fluctuations, that are characterized by afinite relaxation rate Γ q for q → et al. , 2003a). In a one-band approx-imation the finite relaxation at q = 0 would imply thatthe magnetization is not conserved, which is not true inmulti-band systems. The Ising character of the spin fluc-tuations underscores, that they are intermediate betweenlocal moment and itinerant electron fluctuations.Itinerant ferromagnetism may finally be inferred fromthe fact, that UGe forms a very good metal. High qual- ity single crystals may be grown with residual resistivi-ties well-below 1 µ Ωcm. As a function of decreasing tem-perature the resistivity decreases monotonically with abroad shoulder around 80 K. At the ferromagnetic tran-sition the resistivity shows a pronounced decrease char-acteristic of the freezing out of an important scatteringmechanism. As an additional feature the resistivity dis-plays a down-turn around T x ≈
25 K, that is best seen interms a broad maximum in the derivative dρ/dT (Oomi et al. , 1995). Further evidence for anomalous behaviorat T x has been observed in terms of a minimum in thea-axis thermal expansion (Oomi et al. , 1993), a drasticdecrease of thermal conductivity (Misiorek et al. , 2005),a pronounced minimum in the normal Hall effect (Tran et al. , 2004) and a broad hump in the specific heat (Hux-ley et al. , 2001). Finally, high resolution photoemissionshows the presence of a narrow peak in the density ofstates below E F that suggests Stoner-like itinerant ferro-magnetism (Ito et al. , 2002).As explained below, the behavior at T x yields the keyto an understanding of the superconductivity in UGe .The available experimental evidence suggests that thedensity of states near T x is increased, i.e., thermal fluc-tuations with respect to the Fermi level are sensitive tofine-structure of the density of states such as local max-ima or changes of slope. It is helpful to briefly commenton two specific scenarios that have been proposed to ac-count for the features at T x .The first scenario is inspired by the chain-like arrange-ment of the uranium atoms in UGe . The structural sim-ilarity with α -U, which develops a charge density waveat low temperatures (Lander et al. , 1994), has motivatedconsiderations that the anomaly at T x may be related to acoupled spin-and charge-density wave instability (Watan-abe and Miyake, 2002). Electronic structure calculationspredict a dominant cylindrical Fermi surface sheet withstrong nesting (Shick and Pickett, 2001). However, be-cause the U-U spacing in UGe is larger than for α -U,nesting is less important. Moreover, despite great exper-imental efforts so far no direct microscopic evidence hasbeen observed that would support a density-wave insta-bility (Aso et al. , 2006; Huxley et al. , 2003b, 2001). Infact, detailed inelastic neutron scattering studies of thephonons in UGe show that the hump in the specific heatnear T x does not hint at soft phonons (Raymond et al. ,2006). This contrasts the structural softness expected ofan incipient charge density wave.The second scenario is also based on the electronicstructure calculations in the LDA+U, which accountfor the ordered moment and the magneto-crystallineanisotropy (Shick and Pickett, 2001). In these calcu-lations the ordered moment is identified as the sum oflarge, opposing spin and orbital contributions. Closer in-spection of the results shows the presence of two nearlydegenerate solutions, that differ in terms of the orbitalmoment (Shick et al. , 2004). The upshot of these cal-culations is, that the anomaly at T x may be related tofluctuations between these two orbital states.3 FIG. 13 (a) Pressure versus temperature phase diagram ofUGe . Superconductivity is observed well within the ferro-magnetic state in the vicinity of transition between a largemoment and small moment ferromagnet. (b) Magnetic fieldversus pressure phase diagram of UGe . The ferromagnetictransition at p x and at p c are both first order as seen by firstorder metamagnetic transitions at H x and H m . Plot takenfrom (Pfleiderer and Huxley, 2002). Under modest hydrostatic pressures UGe exhibits arich phase diagram as shown in Fig. 13. The Curie tem-perature is suppressed monotonically and collapses con-tinuously at p c = 16 kbar. AC susceptibility studies es-tablish that the ferromagnetic transition changes fromsecond to first order for pressures above ∼
12 kbar (Hux-ley et al. , 2000). A first order transition at p c is con-firmed by the DC magnetisation, which drops discontin-uously at p c (Pfleiderer and Huxley, 2002). Note that thediscontinuous change of the ordered moment is perfectlyconsistent with the continuous variation of T C . Furtherevidence for a first order transition at p c is provided bya discontinuous change of the spin-lattice relaxation rate1 /T T (Kotegawa et al. , 2005) and quantum oscillatorystudies (Terashima et al. , 2001).The broad anomaly at T x is also suppressed under pres-sure and vanishes at p x = 12 kbar. This was first inferredfrom the derivative of the resistivity (Huxley et al. , 2001;Oomi et al. , 1995), but may also be seen in the ther-mal expansion (Ushida et al. , 2003) and the specific heat(Tateiwa et al. , 2004). In the magnetization a broadhump emerges near T x , which turns into a sharp ferro-magnetic phase transition near p x with increasing pres-sure (Huxley et al. , 2001; Pfleiderer and Huxley, 2002;Tateiwa et al. , 2001a). Below T x the ferromagnetic mo-ment increases. The low temperature, large momentphase is referred to as FM2, while the high temperaturelow-moment phase is referred to as FM1 (cf. Fig. 13).Neutron scattering of the magnetic order is compara- tively straight forward. Due to the cancellation of nu-clear scattering lengths certain Bragg peaks are purelymagnetic. Comparison of selected Bragg peaks stronglysuggests, that both FM1 and FM2 are strictly ferromag-netic (Huxley et al. , 2003b). Moreover, neutron scatter-ing at a pressure just below p x shows that the intensityof the (100) Bragg spot scales with the square of the bulkmagnetization. This shows that the FM2 state does notbreak up just below p x .Finally, within a finite pressure interval ranging from ∼ p c the resistivity and AC susceptibility showa superconducting transition (Huxley et al. , 2001; Sax-ena et al. , 2000). As a function of pressure T s increasesbelow p x and decreases above p x with the possibility ofa small discontinuity exactly at p x (Huxley et al. , 2001;Nakashima et al. , 2005).As a function of pressure the zero temperature fer-romagnetic moment drops discontinuously by ∼
30 % at p x , followed by a discontinuous drop at p c (Pfleiderer andHuxley, 2002). Application of a magnetic field along thea-axis at pressures above p x restores the full ordered mo-ment at a characteristic transition field H x , that emergesat p x and increases rapidly under pressure (Fig. 13 (b)).For pressures above p c the application of a magnetic fieldrestores initially the ordered moment of the FM1 phasewhen crossing the transition field H m that emerges at p c .This is followed by the recovery of the full moment at H x .At low temperatures the transition at H x and H m bothare discontinuous (Pfleiderer and Huxley, 2002). Thelines of first order transitions at T = 0 at H x ( p ) and H m ( p ) are expected to end in a quantum critical pointfor very high fields. Likewise, as a function of increasingtemperature at constant pressure the transition fields H x and H m terminate in critical end-points. The importanceof this finite temperature criticality to the superconduc-tivity is an open issue.The Sommerfeld contribution γ to the specific heatis essentially unchanged at pressures well below p x .Just below p x the value of γ increases and settles in anearly four-fold larger value γ ≈ .
11 J / mol K above p x (Tateiwa et al. , 2004, 2001b). Even though the pressuredepencence of γ is sometimes described as a maximumat p x , real data rather display the shape of a plateaucharacteristic of an increased linear specific heat term inthe FM1 phase. This is supported by the temperaturedependence of the resistivity, which shows a T form ev-erywhere. The T coefficient A increases as a functionof pressure from below to above p x . For magnetic fieldsabove H x it varies as A ∝ / √ H − H x (Terashima et al. ,2006).To explore the nature of the transitions at p x and p c detailed quantum oscillatory studies have been car-ried out for magnetic fields parallel to the b-axis (Set-tai et al. , 2001; Terashima et al. , 2001). This probesthe predicted cylindrical Fermi surface sheets, withoutadding the complexities of the transitions at H x and H m (Shick and Pickett, 2001). In the FM2 phase startingfrom ambient pressure three fundamental frequencies are4observed with F α = 6800 ±
30 T, F β = 7710 ±
10 T and F γ = 9130 ±
30 T. These frequencies exhibit considerablemass enhancements of m ∗ α /m = 23 ± m ∗ β /m = 12 ± m ∗ γ /m = 17 ±
2, that are weakly pressure dependent with dF α /d ln p = 3 . ± . × − kbar − , dF β /d ln p = − . ± . × − kbar − , dF γ /d ln p ≈ ± . × − kbar − (Terashima et al. , 2001). The mass enhancement is con-sistent with the specific heat.Between 11.4 and 15.4 kbar, the regime of the FM1phase, the de Haas–van Alphen spectra change in thefollowing manner: (i) the α and γ branches vanish, (ii)the β branch initially decreases followed by a steep risewith a substantial increase of the mass enhancement to39 . ± δ -branch emerges, which is similar to the β branch,where F δ = 4040 ±
40 T, m ∗ δ /m = 22 ± dF δ /d ln p =15 ± × − kbar − .It is interesting to note that no minority-spin coun-terpart to the β -branch is observed, characteristic of afully spin-polarized state. Under the assumption thatthe Fermi surface volume remains unchanged through p x ,it is not necessary to invoke a complete reconstructionof the Fermi surface to understand the data. When as-signing the β - and α -branches to extremal orbits of themajority-spin Fermi surface and the γ branch to a Fermisurface sheet with hole character, the δ branch may beunderstood as resulting from a shrinking and breaking-up of the γ hole surface. Again the mass enhancement isconsistent with the specific heat.For the paramagnetic state above p c the situation dif-fers. Here the spectra consist of four new branches, thatare not connected in any obvious manner with the spec-tra in the FM1 and FM2 phase. This suggests that theFermi surface completely reconstructs at p c . Because thechange of the frequencies is abrupt, the reconstructionappears to be first order. Preliminary studies have alsobeen carried out for magnetic field along the a-axis (Haga et al. , 2002; Terashima et al. , 2002). For fields above H x in the FM2 phase the spectra and mass enhancementsvary weakly with pressure. In contrast, very little infor-mation could be obtained below H x .The very weak pressure dependence of the ordered mo-ment in the FM1 and FM2, and the fact that the tran-sition between FM1, FM2 and paramagnetism may becontrolled either by pressure and/or magnetic field sug-gests an important role of maxima in the density of states(Huxley et al. , 2001; Pfleiderer and Huxley, 2002; Sande-man et al. , 2003). However, several properties show thatpurely spin-based models or the delocalisation of the 5felectrons would be too simple as an explanation. For in-stance, the derivative of the magnetization χ (cid:107) = dM/dH measures the longitudinal susceptibility, i.e., the sensitiv-ity for changes of amplitude of the ordered moment. Acomparison of the pressure dependence of χ (cid:107) for the a-and c-axis establishes, that the anisotropy of the longi-tudinal susceptibility increases strongly under pressure,i.e., the magnetic response becomes more anisotropic in-stead of less (Huxley et al. , 2003b; Pfleiderer and Huxley, 2002).We further note, that the transition at p x is probablynot controlled by a density wave instability either. Neu-tron scattering of the crystal structure at high pressureshows that U-U spacing at 14 kbar reduces to d U − U ≈ . et al. ,2001). It is conceivable that the requirements for nestingwould be much too sensitive to survive these fairly largestructural changes up to p x . Second, the observation ofquantum oscillations on large Fermi surface sheets seemsinconsistent with a charge-density wave gap in the FM2phase. Moreover, measurements of the uranium mag-netic form factor show, that it may still be accounted forby either a U or U configuration, but the ratio oforbital to spin moment, R = µ L /µ S increases across p x so that R F M /R F M ≈ . ± .
05 (Huxley et al. , 2003b;Kuwahara et al. , 2002). This contrasts a delocalizationof the 5f electrons, since the orbital contribution shouldthen decrease. It is interesting to note, that the increaseof R through p x is consistent with the proposed degener-acy of orbital contributions in the FM1 and FM2 phasesas calculated in the LDA+U (Shick et al. , 2004). Thissuggests that the FM2 to FM1 transition at p x and re-lated properties may be driven by fluctuations betweentwo different orbital moments.Having reviewed the metallic and magnetic state ex-tensively, we finally turn to the superconductivity inthe ferromagnetic state of UGe . The initial experi-ments suggested that the superconductivity in UGe isextremely fragile. The critical current density, of order j c ≈ . / cm , is between one and two orders of mag-nitude smaller than for heavy-fermion systems such asUPt and even three orders of magnitude smaller than forconventional superconductors (Huxley et al. , 2001). Thereduced values of j c may be reconciled with flux flow re-sistance, where the flux lattice forms spontaneously evenat ambient field due to the internal field (the orderedmoment corresponds to 0.19 T). The expected flux linespacing at this field is of the order 600 to 1000 ˚A (Huxley et al. , 2001). Further, the susceptibility depends sensi-tively on the excitation amplitude, consistent with verylow j c , and reaches full diamagnetic screening only forvery small amplitudes (Saxena et al. , 2000). The diamag-netic shielding as seen in the AC susceptibility is largestat p x . Note that this does not show the volume fractionof Meissner flux expulsion. Instead it may be the resultof changes of sensitivity to the AC excitation amplitude.Interestingly, the diamagnetic screening and the pressuredependence of T s do not reflect in a simple manner thedifference of 30% of the ordered moment in the FM1 andFM2 phase.Bulk superconductivity in UGe was at first inferredfrom the magnetic field dependence of the flux flow resis-tance, which displays the characteristic convex increaseup to H c (Huxley et al. , 2001). Less ambiguous infor-mation provided the specific heat, which was found toshow a small, yet distinct, anomaly ∆ C/γT s ≈ . et al. , 2001b). The spin lattice relaxation5rate in Ge NQR shows a change of slope at T s . How-ever, in contrast to the resistivity and susceptibility thespecific heat suggests that bulk superconductivity existsonly in a very narrow interval surrounding p x (Tateiwa et al. , 2004). Such a narrow interval of bulk supercon-ductivity at p x is supported by dH c /dT | T s , which inthe same narrow interval is ten-fold increased, exceeding dH c /dT | T s < −
20 T / K (Nakashima et al. , 2005).The superconductivity in UGe is remarkable, because T s is always at least two orders of magnitude smaller than T C . The superconductivity hence emerges in the presenceof a strong ferromagnetic exchange splitting, estimatedto be of the order 70 meV. This suggests an unconven-tional form of superconductivity pairing. For what isknown about the Fermi surface, odd-parity equal-spintriplet pairing is thereby the most promising candidate.This state is equivalent to the A1 phase of He.As a first experimental hint for an unconventional statethe superconducitivity in UGe is fairly sensitive to thesample purity, i.e., superconductivity vanishes when thecharge carrier mean free path becomes shorter than thecoherence length (Sheikin et al. , 2001). Inferring tripletpairing from the mean free path dependence when dop-ing with selected impurities was previously employed instudies of UPt (Dalichaouch et al. , 1995) and Sr RuO (Mackenzie et al. , 1998). As for UGe the conclusion oftriplet pairing has been questioned on the basis of super-conductivity observed in polycrystalline UGe sampleswith ρ ≈ µ Ωcm (Bauer et al. , 2001). However, thepurity dependence in polycrystals is still within the un-certainty at which the charge carrier mean free path canbe inferred from from ρ . Interestingly the specific heatof the polycrystals only shows a faint supercondutinganomaly and thus bulk superconductivity at 14.7 kbar.This may be caused by the presence of internal strainsbetween the crystal grains (Vollmer et al. , 2002).In single-crystals the maximum specific heat anomaly∆ C/γT s ≈ . γ /γ ( T > T s ) ≈ . T dependence of C/T more specifically sug-gests line nodes.The strongest evidence supporting p-wave supercon-ductivity thus far are comprehensive studies of H c (Sheikin et al. , 2001). Absolute values of H c varystrongly as a function of pressure and crystallographicdirection, where typical values are in the range of a fewT. Below p x the coherence lengths inferred from H c arefairly isotropic and of the order 100 ˚A. In contrast, above p x the coherence lengths display a marked anisotropy,e.g., for 15 kbar ξ a = 210 ˚A, ξ b = 140 ˚A and ξ c = 700 ˚A.It is helpful to address at first two unusual featuresfor the a-axis, that are outside the more general patternof behavior. At small magnetic fields H ac displays neg-ative curvature, that may be attributed to the internalfields associated with the ferromagnetic order. Second,for pressures just above p x , pronounced reentrant behav-ior is observed in H c , when the magnetic field crosses the transition at H x (Huxley et al. , 2001). This reentrant be-havior in H c may also provide a possible explanation forthe pronounced extremum in dH c /dT (Nakashima et al. ,2005). Keeping these two aspects in mind, the more gen-eral features of H c may be summarized as follows: (i) H c exceeds conventional paramagnetic and orbital limit-ing for all field directions, except very close to p c , wherethe a- and b-axis show more conventional limiting, (ii)the anisotropy of H c in the vicinity of T s may be de-scribed by the effective mass model, (iii) the anisotropyseems to relate to the inverse of the magnetic anisotropy,i.e., H c for the c-axis is always the largest.A remarkable feature of the critical field for the c-axisis the presence of positive curvature at temperatures aslow as 0 . T s . The general form of H cc is reminiscent ofthat observed in UBe . It may be accounted for in astrong-coupling scenario, where the coupling parameter λ decreases rapidly with increasing pressure from λ =14, 7 and 1.7 at p = 12, 13.2 and 15 kbar, respectively.We note that for conventional electron-phonon mediatedsuperconductivity these high values of λ would imply anincipient lattice instability.Neutron scattering shows that the ferromagnetic scat-tering intensity at (100) remains unchanged to within lessthan a percent when entering the superconducting state(Aso et al. , 2005; Huxley et al. , 2005, 2001; Pfleiderer et al. , 2005). However, these studies were probably notcarried out sufficiently close to p x to provide informationon the narrow regime, where bulk superconductivity isseen in the specific heat. When taken together the avail-able experimental evidence makes it highly unlikely, thatthe superconductivity is carried by tiny sections of theFermi surface, where the exchange splitting vanishes.The observation that the superconductivity in UGe is confined to the ferromagnetic state has created greattheoretical interest. We conclude this section with avery brief account of some of the theoretical contribu-tions UGe has inspired. The microscopic coexistenceof ferromagnetism and superconductivity has been ad-dressed in a number of contributions, e.g., (Abrikosov,2001; Kirkpatrick and Belitz, 2003; Machida and Ohmi,2001; Sa, 2002; Spalek, 2001; Suhl, 2001). Possible orderparameter symmetries of superconducting ferromagnetsfor given crystal structures and easy magnetizations axishave been classified in (Mineev, 2002a,b, 2004, 2005a,b;Mineev and Champel, 2004; Samokhin, 2002; Samokhinand Walker, 2002). For instance, it has been pointed outthat ferromagnetic superconductors with triplet pairingand strong spin-orbit coupling are at least two-band su-perconductors. Without spin-orbit coupling it is generi-cally expected that separate superconducting transitionstake place, for the majority and minority Fermi surfacesheet (Belitz and Kirkpatrick, 2004; Kirkpatrick and Be-litz, 2004). The upper critical field in these systems isdetermined by a novel type of orbital limiting, and theprecise order parameter symmetry depends on the ori-entation of the ordered magnetic moment. The latterproperty, in principle, allows to switch the superconduct-6ing order parameter through changes of orientation of themagnetization. The precise impact of spin-orbit couplingin this scenario, which of course is strong in f-systems,awaits further clarification.The greatest fascination has generated the absenceof superconductivity above p c , because it even suggestsferromagnetism as a precondition for superconductivity.Experimentally the reconstruction of the Fermi surfacetopology supports a less generic explanation. It is how-ever interesting to note, that a large number of mecha-nisms could be identified that promote superconductiv-ity as confined to the ferromagnetic state. These includehidden quantum criticality, the enhancement of longitu-dinal (pair-forming) spin fluctuations in the ferromag-netic state, special features of the density of states andthe possible coupling of spin- and charge density wave or-der (Karchev, 2003; Kirkpatrick et al. , 2001; Sandeman et al. , 2003; Watanabe and Miyake, 2002). As a newthread several studies have considered the possible inter-play of magnetic textures with the superconductivity andspontaneous flux line lattices. We briefly return to thisquestion in section V.B.2.
2. URhGe
The series UTX, where T is a higher transition metalelement and X=Si or Ge, crystallize in the orthorhombicTiNiSi crystal structure, space group Pnma (Sechovskyand Havella, 1998; Tran et al. , 1998). Even though thecrystal structure of this series differs from that of UGe italso shares certain similarities. In particular, as for UGe the uranium atoms form zig-zag chains. For URhGe theU-U spacing d U − U ≈ .
48 ˚A compares well with the Uspacing in UGe at a pressure of 13 kbar. This has moti-vated detailed studies of high quality crystals, which letto the discovery of superconductivity in the ferromag-netic state of URhGe (Aoki et al. , 2001). Further studieshave revealed a metamagnetic transition within the ferro-magnetic state, surrounded by superconductivity (L´evy et al. , 2005). For clarity we refer in the following to thesuperconductivity at ambient field as S1 and for that themetamagnetic transition at S2.At ambient pressure URhGe displays a paramagneticto ferromagnetic transition with a Curie temperature T C = 9 . µ ord = 0 . µ B / U(Aoki et al. , 2001; Prokes et al. , 2002). Neutron scatter-ing studies show that superconducting samples (S1) arestrictly ferromagnetic. This contrasts earlier studies ofpolycrystalline samples which displayed a non-colinearmagnetic structure (Tran et al. , 1998). Electronicstructure calculations in the LSDA (Shick, 2002) andLAPW+ASA (Divis et al. , 2002) reproduce the orderedmoment and magneto-crystalline anisotropy (LDA+Uappears to be not necessary). These calculations alsoshow the possibility for a canted antiferromagnetic state.In any case, as for UGe the ordered moment is the resultof strongly opposing spin and orbital contributions. In the following we discuss the properties of ferromagneticURhGe only.The ferromagnetic moment in URhGe is aligned withthe crystallographic c-axis (Huxley et al. , 2003b). Incontrast to UGe the magnetic anisotropy field is onlylarge for the a-axis. As discussed in further detail be-low, a magnetic field H R = 11 . .The easy-axis susceptibility in URhGe follows a Curie-Weiss dependence above T C with a fluctuating moment µ eff = 1 . µ B / U (Aoki et al. , 2001), while the b-axissusceptibility varies with temperature as expected of an-tiferromagnetic order at low temperatures (Huxley et al. ,2003b). This strongly suggests itinerant ferromagnetismwith strongly delocalized 5f electrons.The ferromagnetic transition shows a λ -anomaly at T C ,where the magnetic entropy released at T C is small S m =0 . et al. , 2000). At low temperaturesthe specific heat follows a dependence C ∼ γT + bT where γ = 0 .
164 J / mol K . The λ anomaly is rapidlysuppressed for magnetic fields applied parallel to the c-and b-axes, where γ in a field of 15 T decreases by ∼ ∼ T s ≈ .
25 K (S1) (Aoki et al. , 2001). In polycrys-talline samples H c = 0 .
71 T corresponds to a Ginzburg-Landau coherence length ξ GL ≈
180 ˚A. Measurements ofthe magnetization show the onset of weak flux expulsionto be consistent with a penetration length λ l = 9100 ˚A.The specific heat shows a clear anomaly at T s charac-teristic of bulk superconductivity, where ∆ C/γT s ≈ . T s decreases and vanishes for low sample quality, consis-tent with unconventional superconductivity. H c of the S1 state is anisotropic, where the anisotropycompares with the inverse of the magnetic anisotropy,i.e., H c is largest for the a-axis and and smallest forthe c-axis (Hardy and Huxley, 2005). This suggests anintimate connection between superconductivity and fer-romagnetism. For all directions H c ( T →
0) exceedsparamagnetic limiting. As a function of sample qual-ity it is found that H c ( T →
0) varies ∝ T s , showingthe intrinsic nature of the large critical field values. Thecomparatively small anisotropy shows, that large criticalfield values are not due to a reduced g -factor or electronicanisotropies.Because the superconductivity (S1 and S2) occurs inthe ferromagnetic state, it is expected that the pair-ing dominantly occurs on the spin-majority Fermi sur-face akin the odd-parity equal-spin p-wave pairing of theA1 phase of He. This is consistent with the reducedspecific heat anomaly as compared to the BCS value of7∆
C/γT s = 1 .
43 and residual zero temperature specificheat γ ( T →
0) = γ/ T > T s ) (Aoki et al. , 2001).For the crystallographic point group of URhGe, a ferro-magnetic moment parallel to the c-axis and strong spin-orbit coupling only two odd-parity states are possible(Hardy and Huxley, 2005). The temperature dependenceof the ratios of the upper critical fields allows to distin-guish between these two states. The observed combina-tion of 20% increase of H ac /H bc with decreasing temper-ature while H ac /H bc =constant strongly supports an oddparity p-wave state with gap node parallel to to magneticmoments. Finally, the temperature dependence of H c isin excellent agreement with strong coupling calculations,when the initial slope dH c /dT near T s is taken fromexperiment.It is interesting to note that the ratio of the Curietemperature to the maximal superconducting transitionin UGe ( T C /T s ≈ / . .
5) compares well thatin URhGe ( T C /T s ≈ . / .
25 = 38 . α a = 3 . × − K − so that dT aC /dp = 0 . / kbar, ∆ α b =1 . × − K − so that dT bC /dp = 0 . / kbarand ∆ α c = 2 . × − K − so that dT cC /dp =0 . / kbar. This yields a volume thermal expansionand pressure dependence of T C , ∆ V a = 7 . × − K − and dT C /dp = +0 . / kbar, respectively (Sakarya et al. , 2003), i.e., T C increases under pressure. This hasbeen confirmed in experimental studies up to 140 kbar(Hardy et al. , 2005). In these studies T s is found to besuppressed for pressures above ∼
30 kbar. Despite theincrease of T C under pressure the ordered moment de-creases with dµ ord /dp = − . × − µ B / kbar (Hardy et al. , 2004).As depicted in Fig. 14 magnetic field applied paral-lel to the b-axis may be used to tune the ferromag-netic transition (green shading) towards zero. Closeto the field value where T C would vanish the depen-dence of T C versus field bifurcates. Because the tran-sition is continuous throughout, the bifurcation repre-sents a tricritical point (TCP). Application of magneticfield with suitably chosen components along the b-axisand c-axis allows to further reduce the T s , until it van-ishes at a field tuned quantum critical point (QCP) for (cid:126)H = (0 , H b = ±
12 T , H c = ± T C ( H )suggests that the excitation spectrum includes longitudi-nal fluctuations.In the vicinity of the TCP and QCPs of URhGe su-perconductivity (S2) emerges (L´evy et al. , 2005). For amagic angle in the range 30 ◦ to 55 ◦ S2 even stabilizes forfield components along the c-axis. The maximum value
FIG. 14 Temperature versus magnetic field phase diagram ofURhGe, for magnetic fields in the bc-plane. The critical endpoint of the reorientation transition of the magnetic order issurrounded by a dome of superconductivity (S2). Plot takenfrom (L´evy et al. , 2007). of T s = 0 . H c diverges and ex-ceeds 28 T, the highest field studied (L´evy et al. , 2007).The anisotropy of the upper critical field may be ac-counted for in terms of an anisotropic mass model, where H sc = Φ / (2 πξ c ) (cid:113) ξ a cos ( γ ) + ξ b sin ( γ ), with H sc a =Φ / (2 πξ c ξ b ) = 2 .
53 T, H sc b = Φ / (2 πξ c ξ a ) = 2 .
07 Tand H sc c = Φ / (2 πξ b ξ a ) = 0 .
69 T. Further, assumingthat the anisotropy of the critical fields that is observedat zero applied field remains unchanged for the high fieldsuperconductivity, a geometric average of the coherencelength, ξ = √ ξ a ξ b ξ c , can be inferred. Remarkably, thecoherence length ξ as a function of applied magnetic fieldfor the b-axis diverges at H R , where the magnetic fielddependence of ξ of both superconducting phases fall onthe same line. The coherence length thereby decreasesfrom ξ ( H b = 0) = 143 ˚A to ξ ( H R ) <
44 ˚A. The com-mon field dependence of the coherence length suggests,that both superconducting phases have the same origin,notably the quantum critical point at high fields.
B. Border of ferromagnetism
Recently two superconducting ferromagnets have beendiscovered, notably UIr (Akazawa et al. , 2004a,b) andUCoGe (Huy et al. , 2007), in which the ordered moment8of the ferromagnetic state is small as compared with thecompounds introduced so far. The superconductivity inboth compounds is observed at the border of ferromag-netism, rather than deep inside the ferromagnetic state. a. UCoGe
UCoGe is orthorhombic and isostructural toURhGe with lattice constants a = 6 .
645 ˚A, b = 4 .
206 ˚Aand c = 7 .
222 ˚A. It was long thought the UCoGe isparamagnetic, but polycrystalline samples were recentlyfound to exhibit ferromagnetic order with a small orderedmoment µ ord = 0 . µ B / U below T C = 3 K. The orderedmoment is much smaller than the fluctuating momentobserved in the paramagnetic state µ eff = 1 . µ B / U.The specific heat shows a small anomaly at T C , wherethe magnetic entropy released is tiny, S m = 0 .
03 R ln 2,and the normal state specific heat is moderately en-hanced,
C/T = γ = 0 .
057 J / mol K . The thermal ex-pansion shows a volume contraction, where the ideal-ized discontinuity in α is estimated to by ∆ α = − . × − K − . Thus, according to the Ehrenfest relation,the Curie temperature would decrease under pressure ata rate dT C /dp = V m T C ∆ α/ ∆ C = − .
25 K / kbar andis expected to vanish around 12 kbar ( V m = 3 . × − m / mol is the molar volume).Polycrystalline samples of UCoGe display supercon-ductivity with T s ≈ . T s is much smaller than T C . The superconducting transition is seen in the resis-tivity, AC susceptibility, specific heat and thermal expan-sion. In the AC susceptibility the diamagnetic screen-ing is of the order 60 to 70%. In the specific heat theanomaly corresponds to ∆ C/γT s ≈
1, which is smallerthan the weak-coupling BCS value. The thermal ex-pansion displays a positive anomaly, with an idealizedchange of length at T s of the order ∆ L/L ≈ − × − .This implies that T s increases under pressure at a rate dT s /dp ≈ +0 .
048 K / kbar.Experimentally it is found, that T C in polycrystalsrapidly drops under pressure and appears to vanish be-tween 8 and 20 kbar, while T s is essentially unchangedconsistent with the thermal expansion. The width ofthe superconducting transition and additional features ofthe normal state resistivity, such as the residual resistiv-ity and the temperature dependence, suggest a quantumcritical point already at 7 kbar (Hassinger et al. , 2008).In any case, the superconductivity in UCoGe appearsto survive in the non-ferromagnetic state at high pres-sure. However, data available to data do not rule outthat a ferromagnetic moment survives at high pressures.The pressure dependence is supplemented by the vari-ation of the superconductivity and ferromagnetism as afunction of Si substitution in polycrystals, UCoGe − x Si x ,which shows a simultaneous suppression of T C and T s atthe same critical concentration x c ≈ .
12 (de Nijs et al. ,2008).The upper critical field of polycrystalline UCoGe variesnear T s as dH c /dT ≈ − . / K for the sample with thelargest T s . This implies a fairly short coherence length ξ ≈
150 ˚A as compared with the charge carrier meanfree path l = 500 ˚A inferred from the residual resistivity ρ = 12 µ Ωcm. In other words the samples are in theclean limit, a precondition for unconventional supercon-ductivity. An unconventional superconducting state isalso inferred from H c , which exceeds 1.2 T, the highestfield measured, which is thus clearly larger than the Paulilimit.NMR and NQR measurements in polycrystals alsopoint at unconventional pairing (Ohta et al. , 2008). Inthe normal state the spin-lattice relaxation and the Kightshift are characteristic of ferromagnetic quantum criticalfluctuations, where T C ≈ . T C = 2 K and T s = 0 . et al. , 2008). The ferromagnetic moment m s = 0 . µ B is aligned with the c-axis and the a- and b-axis are magnetically hard. Thus UCoGe is an easy-axisferromagnet like UGe , in contrast with the hard-axisferromagnetism in URhGe. H c of single-crystal UCoGeshows a marked anisotropy between the ab-plane and thec-axis, where B c for field parallel to the a- and b-axisexceeds the Pauli limit with B ac (cid:39) B bc ≈ (cid:29) B cc ≈ . dB a,bc /dT ≈ − / K is also large.This suggests an equal-spin pairing state with an axialsymmetry of the gap function and with point nodes alongthe c-axis. Moreover, an upward kink of B ac may indi-cated multiband superconductivity. b. UIr The signatures of the superconductivity in UIrare still rather incomplete as the superconductivity ex-ists at high pressures and very low temperatures. Be-cause the crystal structure of UIr lacks inversion sym-metry, the properties of UIr are presented in more detailin section IV.A.3, which deals with non-centrosymmetricsuperconductors. c. Note on d-electron ferromagnets
It is worthwhile tocomment briefly on two ferromagnetic d-electron systemsin which superconductivity has been reported. First,high-purity samples of iron exhibit superconductivityabove 140 kbar (Jaccard et al. , 2002; Shimizu et al. ,2001). It turns out that the superconductivity occurs inthe hexagonally closed packed (cid:15) -phase of iron, which isbelieved to represent an incipient antiferromagnet (Mazin et al. , 2002; Saxena and Littlewood, 2001). Neverthelessseveral hints, such as great sensitivity to sample purityand a non Fermi liquid temperature dependence of the9resistivity near the highest value of T s , suggest uncon-ventional pairing.The other system is the weak itinerant electron mag-net ZrZn , where an incomplete resistivity transition hasbeen reported (Pfleiderer et al. , 2001b). Here more recentwork suggests that the superconductivity is not intrinsic,but due to the Zn depletion of spark eroded sample sur-faces (Yelland et al. , 2005). IV. EMERGENT CLASSES OF SUPERCONDUCTORS
A growing number of intermetallic compounds exhibitunusual forms of superconductivity that do not fit intothe general category of magnetism and superconductivitycovered in sections II and III. These compounds promiseto be representatives of new classes of superconductors.The following section is dedicated to a review of theseemergent classes of f-electron superconductors. We dis-tinguish non-centrosymmetric systems, materials at theborder to a valence transition and systems at the borderof polar order.
A. Non-centrosymmetric superconductors
In general the strong electronic correlations in heavyfermion systems may be viewed as an abundance ofmagnetic fluctuations, which, being pair-breaking, sup-press conventional s-wave superconductivity. This is con-trasted by spin-triplet pairing, which may occur as longas time reversal symmetry and inversion symmetry aresatisfied (Anderson, 1984). In turn it is was long believedthat pure spin-triplet heavy-fermion superconductivitycannot exist in non-centrosymmetric systems. This iscontrasted by the recent discovery of supercondcuctivityin the antiferromagnets CePt Si (Bauer et al. , 2004a),CeRhSi (Kimura et al. , 2005), CeIrSi (Suginishi andShimahara, 2006) and CeCoGe (Kawai et al. , 2008b;Settai et al. , 2007a). Perhaps most remarkably super-conductivity has even been discovered at the border offerromagnetism in the non-centrosymmetric compoundUIr (Akazawa et al. , 2004a,b). In this section we willreview the current understanding of these compounds.Because their properties may be explained by a mixed s-plus p-wave pairing state they may be representatives ofa new class of superconductors, outside the traditionalscheme of classification.From a theoretical point of view non-centrosymmetricheavy-fermion superconductors are interesting, becausein these materials the Fermi surface exhibits a splittingdue to antisymmetric spin-orbit coupling α ( (cid:126)k ×∇ φ ) · σ . Intwo-dimensional electron gases this splitting is referred toas Rashba- and in bulk compounds as Dresselhaus-effect(Dresselhaus, 1955; Rashba, 1960). As a reminder, spin-orbit coupling is a purely relativistic effect that is due togradients of the electric potential ∇ φ = (cid:126)E transverse tothe motion of the electrons. It can be shown that anti- FIG. 15 Qualitative depiction of chiral exchange splitting ofa spherical Fermi surface by Rashba spin-orbit interactions.Also shown are Cooper pairs that may form under such anexchange splitting, notably a mixed singlet with triplet state.Plot taken from (Fujimoto, 2007). symmetric spin-orbit coupling leads to a splitting of theFermi surface along (cid:126)k F × ∇ φ . In magnetic materials theasymmetric spin-orbit coupling also generates a contribu-tion to the exchange interaction that is akin to superex-change, where the role of the nonmagnetic atom is playedby an empty orbital (Moriya, 1963). This superexchangeis also known as Dzyaloshinsky-Moriya interaction.In a simple-minded view the asymmetric spin-orbitcoupling leads to a highly unusual chiral exchange split-ting of the Fermi surface (see e.g. (Fujimoto, 2007)).A qualitative depiction is shown in Fig. 15, where ∇ φ isalong the z-axis (the x- and y-axis are in the plane). Theexchange splitting translates into dispersion curves thatenergetically favor a precessional motion of the electronspin with a particular handedness, where the axis of theprecession is denoted by the gray arrows in Fig. 15. Fora Fermi surface with chiral exchange splitting a Cooperpair forming between electrons with momentum (cid:126)k andspin ↑ and momentum − (cid:126)k and ↓ do not correspond to aspin singlet state, because | (cid:126)k ↑(cid:105)| − (cid:126)k ↓(cid:105) and | (cid:126)k ↓(cid:105)| − (cid:126)k ↑(cid:105) form on different Fermi surface sheets. Instead it has longbeen predicted (Edelstein, 1989; Gor’kov and Rashba,2001), that superconductive pairing requires an admix-ture of a spin singlet with a spin triplet state.The formation of parity violating Cooper pairs in termsof the singlet-triplet mixing applies also in more generalcases with more complicated forms of ∇ φ . To quantifythe absence of inversion symmetry it is convenient to in-troduce a vector α(cid:126)g (cid:126)k , where α is the Rashba parame-ter and (cid:104)| (cid:126)g (cid:126)k | (cid:105) = 1 is the normalization condition interms of the average over the Fermi surface. The vec-tor (cid:126)g (cid:126)k may be determined by symmetry arguments. Forthe analysis of the allowed superconducting pairing sym-metry (cid:126)g (cid:126)k is compared with (cid:126)d ( (cid:126)k ). For simple cases, likeCePt Si, the gap function ∆ ± may then be expressed as∆ ± ( (cid:126)k ) = ( ψ ± d | (cid:126)g (cid:126)k | ) where ψ corresponds to the wavefunction of the singlet condensate and d | (cid:126)g (cid:126)k | to the tripletstate (Agterberg et al. , 2006; Gor’kov and Rashba, 2001).The discovery of the non-centrosymmetric heavy-fermion superconductors reviewed in the following hasrevived the interest in non-centrosymmetric supercon-0ductors with weak or modest electronic correlations. Forinstance, the boride superconductor Li (Pd,Pt) B (Bad-ica et al. , 2005) displays conventional BCS behavior forLi Pd B (Togano et al. , 2004) and unconventional su-perconductivity for Li Pt B. Interestingly the evolutionfrom conventional to unconventional superconductivitymay be studied as function composition where super-conductivity is observed for all compositions; i.e., thedisorder introduced by the alloying does not affect thesuperconductivity.The gap symmetry is explained as s -wave withspin-singlet and spin-triplet admixtures (Yuan et al. ,2006). Another example is the Pyrochlore oxide systemCdRe O (Hanawa et al. , 2001; Sakai et al. , 2001). Forthis system a structural phase transition leads to a loss ofinversion symmetry at low temperatures, so that strongspin-orbit coupling effects change the electronic structure(Hanawa et al. , 2002). Old examples from the literatureare rare earth sesquicarbides, R C − y , where R=La orY. For instance Y C − y displays a high T s = 18 K, butonly weak spin-orbit coupling (Amano et al. , 2004; Giorgi et al. , 1970; Krupta et al. , 1969). On a more general notewe also mention that amorphous superconductors haveno centre of inversion. However, they are characterizedby a very low diffusivity and associated large Ginzburg-Landau parameter κ (cid:29)
1, i.e., in contrast to the ma-terials listed so far they are in the extreme dirty limit.Likewise thin superconducting films lack inversion sym-metry, i.e., they also fall into a different category.
1. CePt Si The compound CePt Si was the first non-centrosymmetric compound in which heavy-fermionsuperconductivity was discovered (Bauer et al. , 2004a).Recent reviews may be found in (Bauer et al. , 2005a,b,2007; Settai et al. , 2007b). In the following we firstintroduce the structural and physical properties of thenormal metallic state. We then proceed to presentthe magnetic properties. This sets the stage for thesuperconducting properties presented at the end of thissection.CePt Si crystallizes in the tetragonal CePt B typestructure with space group P mm (No. 99). The latticeconstants are a = 4 . c = 5 . C V the mirror plane z → − z is missing.The crystal structure of CePt Si may be derived fromthe cubic AuCu -type crystal structure of CePt , whichis isostructural to CeIn . In contrast to the series ofCe n M m In n +2 m compounds discussed above, which arerelated to CeIn in terms of additional MIn layers, thestructure of CePt Si evolves from the AuCu structureby filling a void with Si. It is interesting to note that fill-ing a void in cage like structures, as for the skutteruditesdiscussed in this review, plays a key role for their proper- TABLE IV Key properties of on-centrosymmetric intermetal-lic superconductors. Missing table entries may reflect morecomplex behavior discussed in the text. H c represents theextrapolated value for zero temperature. References are givenin the text. *Samples with much sharper antiferromagneticand superconducting transitions.CePt Si CeIrSi CeRhSi CeCoGe structure tetrag. tetrag. tetrag. tetrag.space group P4mm I4mm I4mm I4mm a (˚A) 4.072(1) 4.252 4.244 - c (˚A) 5.442(1) 9.715 9.813 - c/a , Γ , Γ Γ , Γ , Γ Γ , Γ , Γ Γ , Γ , Γ ∆ (meV) 13 13.7 22.4or 1.4∆ (meV) 24 43 23.3state AF, SC AF, SC AF, SC AF, SC T N (K) 2.2 5.0 1.6 21, 12, 8 (cid:126)Q (0,0, ) - ( . , , ) - µ ord ( µ B ) 0.2 - - - γ (J / molK ) 0.39 0.12 0.11 0.0320.35* A ( µ Ωcm / K ) - 0.33 0.2 0.011 T K (K) 7–11 - 50 - p c (kbar) 8 25 20 55 T s,max (K) ∼ .
75 1.6 ( p c ) 0.72 0.69 ∼ . C/γ n T s ∼ .
25 - - H abc (T) ∼ . ∼ . dH abc /dT -8.5 -13 --7.2* H cc (T) ∼ > > ∼ dH cc /dT -8.5 -20 ( p N ) -23 -20 ξ ab (˚A) ∼
82 - - ξ c (˚A) ∼
90 54 70year 2004 2006 2007 2008 ties in terms of soft rattling modes. Concerning CePt Sithis does not seem to be the case. Rather, there is animportant indirect role played by the Si atom in gener-ating the lack of inversion symmetry and a considerabletetragonal distortion with c/a = 1 . T N = 2 . Si displays long range antifer-romagnetic order followed by the superconducting tran-sition at T s (Bauer et al. , 2004a). Recent studies suggestthat samples either exhibit T s = 0 .
75 K or T s ≈ .
45 K,where the samples with lower T s show much sharpermagnetic and superconducting transitions (Settai et al. ,2007b; Takeuchi et al. , 2007). Neutron scattering in sam-1ples with T s = 0 .
75 K revealed subtle metallurgical segre-gations and a broad distribution of lattice constant (Pflei-derer et al. , 2008). Similar conclusions were drawn fromthe pressure dependence of T N and T s , i.e., the larger T s seems related to a distribution of lattice constant (Aoki et al. , 2008). A systematic study of small specimens thatwere all cut from the same polycrystalline ingot showedthe same T N for all pieces. However, samples either dis-played T s = 0 .
45 K or T s = 0 .
75 K (Motoyama et al. ,2008a), characteristic of two different superconductingstates. This may be consistent with the earlier obser-vation of a superconducting double transition (Scheidt et al. , 2005). To date the majority of studies have beencarried out in samples with the larger T s .The antiferromagnetism and superconductivity inCePt Si emerge from a normal state above T N that istypical of f -electron heavy-fermion systems. The spe-cific heat exhibits an enhanced Sommerfeld contribution C/T = γ ∼ .
39 J / mol K as extrapolated from T > T N .The resistivity varies quadratically just above T N with A = 2 . µ Ωcm / K , where the samples exhibiting super-conductivity have low residual resistivities ρ of a few µ Ωcm.The low-temperature properties of CePt Si may be at-tributed to an interplay of three energy scales. RKKYinteractions as the origin of the magnetic order, Kondoscreening as the origin of strong correlations and finallycrystal electric fields. The Kondo temperature may bededuced in various different ways, where values are in therange 7 to 11 K (Bauer et al. , 2007). This uncertainty israther typical for f-electron systems. Nevertheless, T N clearly falls below T K . The CEFs lift the 2 j + 1 = 6-folddegenerate ground state of the j = 5 / et al. , 2005b), where the Γ first excitedstate and Γ second excited state are separated from theΓ ground state by 13 meV and 24 meV, respectively. Theproposed CEF level scheme has been compared with thebulk properties, where the sibling La- compound LaPt Siand substitutional doping with La have been consideredadditionally. Because LaPt Si is metallic with a Debyetemperature Θ D ≈
160 K, the 2nd excited CEF level can-not be identified quantitatively in the bulk properties.It is also interesting that the ratio
A/γ ≈ . × − m mol K / J is consistent with a small degeneracy ofthe ground state as lifted by the CEFs. This differs fromanother group of materials in the Kadowaki-Woods plotwith large degeneracy and small CEF splitting, where A/γ ≈ . × − m mol K / J (Bauer et al. , 2005a).The CEF assignment given by (Bauer et al. , 2007) iscontrasted by inelastic neutron scattering measurementsproviding evidence of excitations at 1.4 meV and 24 meV(Metoki et al. , 2004). The associated CEF level schemeis a Γ ground state and Γ and Γ first and second ex-cited states, respectively, where the lower two doublets originate from a Γ quartet in the cubic point symme-try. The CEF scheme with the low lying Γ was foundto account for the magnetization as measured up to 50 T(Takeuchi et al. , 2005). We return to a discussion of thehigh-field magnetization below.The magnetic properties of CePt Si are dominated bya strong Curie-Weiss susceptibility at high temperatureswith an effective fluctuating moment µ eff = 2 . µ B ofthe free Ce ion and a Curie temperature Θ p = −
46 Kcharacteristic of antiferromagnetic exchange coupling ofthe moments. The Curie-Weiss susceptibility extendsdown to ∼
11 K, where a broad maximum in χ signalsKondo-type screening of the fluctuating moments. At T N = 2 .
25 K a λ -anomaly in the specific heat shows theonset of long range antiferromagnetic order. The sizeof the specific heat anomaly implies a release of entropythrough T N of ∆ S ≈ .
22 R ln 2, characteristic of smallordered moments. Under pulsed magnetic fields up to50 T the magnetization increases almost linearly up to ∼
23 T for H (cid:107) [100] and [110], respectively. Above 23 Tthe magnetization levels of and settles around 0 . µ B / Ce.This implies a rather small in-plane magnetic anisotropy.Microscopic evidence for antiferromagnetic order be-low T N has been obtained in neutron diffraction (Metoki et al. , 2004) and µ -ion spin rotation (Amato et al. , 2005)experiments. Neutron diffraction is consistent with amagnetic ordering vector (cid:126)k = (0 , , /
2) and small or-dered moments µ ≈ . µ B at T = 1 . c -axis, where the ordered moments are orientedin-plane. By comparison to the slightly reduced mo-ment expected of a degenerate Γ CEF quartet, µ (Γ ) =1 . µ B , the ordered moment is strongly reduced. Thismay by due to the doublet ground state, but suggests alsosignificant Kondo-screening. µ -SR measurements showthat the small ordered moments exist throughout the en-tire sample volume (Amato et al. , 2005).Keeping in mind what is presently known about theantiferromagnetic order of CePt Si, it is interesting toconsider the spin-orbit splitting of the Fermi surface dueto the lack of inversion symmetry. Quantum oscilla-tory studies in CePt Si have shown a small number ofbranches (Hashimoto et al. , 2004). Cyclotron masses upto 20 times of the bare electron mass have been observed,where masses of up to 65 times of the bare electron massare expected. While this is not a definitive identifica-tion, it represents nevertheless strong evidence for a fairlyconventional heavy fermion state. The lack of inversionsymmetry generates a spin-orbit splitting consistent withthe de Haas – van Alphen data in LaPt Si (Hashimoto et al. , 2004). In principle this splitting may generate chi-ral components of the magnetization.The crystal structure and magnetic properties ofCePt Si set a stage where no superconductivity is ex-pected. Yet, superconductivity emerges in CePt Si wellbelow T N . A number of properties suggest unconven-tional superconductivity. For instance, under substitu-tional doping of Ce by La superconductivity is suppressed2in La x Ce − x Pt Si for x ≥ .
02 (Young et al. , 2005). Ahigh sensitivity to nonmagnetic impurities is considereda hall mark of unconventional pairing. However, remov-ing a magnetic atom from the structure may not qualifyas nonmagnetic impurity, rather than a magnetic defect.Near T s the upper critical field varies strongly as dH c /dT | T s = − . / K for samples with T s = 0 .
75 K(Bauer et al. , 2007) and with dH c /dT | T s = − . / Kfor samples with T s = 0 .
45 K (Takeuchi et al. , 2007).This is characteristic of heavy-fermion superconductiv-ity and consistent with the mass enhancement inferredfrom the normal state specific heat. The upper crit-ical field reaches H c ∼ T s ≈ .
75 K (Bauer et al. , 2007). The value of H c exceeds by a large margin the Pauli-Clogston limit, H P C ∼ . H c does not dis-play a sizable anisotropy, namely H cc /H abc ≈ . Sishow a large Rashba parameter α = 100 meV (Samokhin,2004). This implies that spin-orbit splitting of the Fermisurface plays an important role for the superconductivity.An analysis of the Rashba splitting in CePt Si in termsof group theory for the space group P mm and generat-ing point group C V suggests that (cid:126)g (cid:126)k = k − F ( k y , − k x , et al. , 2004a). The most stable spin pairingstate expected for (cid:126)d ( (cid:126)k ) (cid:107) (cid:126)g (cid:126)k corresponds then to a p -state (cid:126)d ( (cid:126)k ) = ˆ xk y − ˆ yk x . This state is characterized bypoint nodes. However, for mixing of this triplet state witha singlet state the experimental properties are expectedto display the behavior of line nodes. Alternatively aBalian-Werthamer (BW) state (cid:126)d ( (cid:126)k ) = ˆ xk x + ˆ yk y + ˆ zk z ispossible, which would have no nodes. However, the BWstate is expected to be less stable (Frigeri et al. , 2004a).In fact, one may consider the ˆ xk y − ˆ yk x p-state as beingprotected by the Rashba exchange splitting.A key characteristic of the ˆ xk y − ˆ yk x state for magneticfield parallel to the c -axis is the absence of paramagneticlimiting, while there would be considerable paramagneticlimiting for magnetic field perpendicular to the c-axis. Toaccount for the nearly isotropic behavior of H c it hasbeen suggested that the superconducting wave functiondevelops a helical phase factor exp( i(cid:126)q (cid:126)R ) in applied mag-netic fields (Agterberg et al. , 2006; Kaur et al. , 2005). Itis, however, also important to take into account the in-terplay of antiferromagnetic order with the superconduc-tivity, which accounts in parts for the reduced anisotropy(Yanase and Sigrist, 2007).The superconducting transition in samples with larger T s is accompanied by a fairly broad anomaly in thespecific heat with ∆ C/γT s ≈ .
25. This value isstrongly reduced as compared with the isotropic BCSvalue ∆
C/γT s ≈ .
43 (Bauer et al. , 2004a). It deviates inparticular from strong coupling behavior frequently ob-served in heavy fermion superconductors. The reducedspecific heat anomaly may be explained by a vanishingof the superconducting gap on parts of the Fermi sur- face, e.g., due to an unconventional Cooper pair symme-try. It also raises the question how the antiferromagneticorder and superconductivity coexist microscopically. Insingle-crystal samples with T s ≈ . C e /T = γ s + β s T with γ s = 34 . / mol K and β s = 1290 mJ / mol K below 0 . et al. ,2007). Moreover, the specific heat displays a nonlinearmagnetic field dependence γ s ∝ √ H . When taken to-gether this suggests line-nodes of the superconductinggap.The presence of line nodes has also been inferred frommeasurements of the temperature and magnetic field de-pendence of the thermal conductivity κ ( T, H ) in singlecrystals (Izawa et al. , 2005). The key results of thisstudy are (i) a residual term in κ ( T,
0) for T → et al. , 1996; Lee, 1993; Sun and Maki, 1995), (ii)a linear temperature dependence κ ( T, ∝ T at low T , and (iii) a magnetic field dependence that exhibitsone-parameter scaling of the form T / √ H . This behav-ior is taken as evidence that the magnetic field depen-dence is due to a Doppler shift of the quasiparticles(Volovik effect) (Hussey, 2002; K¨ubert and Hirschfeld,1998; Vekhter and Houghton, 1999; Volovik, 1993).Measurements of the penetration depth λ ( T ) representthe most direct probe of the superfluid density. They arenot connected with any other preponderant interactionprocess, notably the long range antiferromagnetic order.The experimental data in polycrystals and single crys-tals displays a broad transition with a point of inflec-tion around 0.5 K (Bonalde et al. , 2005, 2007). Below ∼ . T s changes of the penetration depth are linearin temperature, characteristic of line nodes in the gap(Hayashi et al. , 2006b).While the low temperature specific heat, thermal con-ductivity and penetration depth only shed light on thebehavior well below T s , NMR measurements of the spin-lattice relaxation rate 1 /T and the Knight shift provideinsights into the full temperature dependence (Yogi et al. ,2004, 2006). For the spin-lattice relaxation rate the be-havior appears to be a mixture of differing contributions.Near T s a Hebel-Slichter peak is observed. The variationbelow T s clearly deviates from conventional exponentialactivation, but does not fully settle into a power-law de-pendence either. The data have been interpreted in twodifferent ways. In the first scenario the temperature de-pendence is attributed to a mixing of the singlet andtriplet states (Hayashi et al. , 2006a). This accounts forthe Hebel-Slichter peak and shows that the low temper-ature data essentially limits to the T dependence ex-pected of line nodes. In the second scenario the interplayof the antiferromagnetic order with the triplet contribu-tion to the superconducting pairing symmetry is consid-ered (Fujimoto, 2006). In particular it is pointed out thatthe signatures of line nodes may be found even in fullygaped triplet superconductors in the presence of suitablychosen magnetic order.The NMR Knight shift as measured at a field of 2 T3perpendicular and parallel to the field does not changewhen entering the superconducting state (Yogi et al. ,2006). The Knight shift may be taken as a probe of thespin susceptibility χ . The experimental result are con-trasted by the theoretical prediction for a spin-singlet andspin-triplet state in the presence of asymmetric spin-orbitinteractions (Frigeri et al. , 2004b). For the spin-singletstate both χ ⊥ and χ (cid:107) are expected to decrease in thesuperconducting state with some anisotropy, where thedecrease gets smaller with increasing α . For the spin-triplet state the susceptibility becomes independent ofthe size of the spin-orbit coupling. Here χ (cid:107) shows nodecrease, while χ ⊥ shows a modest decrease. Thus theexperimental absence of any decrease is taken as evidencefor the dominant spin-triplet component, where the in-plane behavior awaits further clarification.In a phase diagram that combines the effects of pres-sure and substitutional Ge-doping assuming a bulk mod-ulus, B = 1000 kbar(Bauer et al. , 2007; Nicklas et al. ,2005; Takeuchi et al. , 2007; Tateiwa et al. , 2005; Yasuda et al. , 2004) two features may be noticed. First the anti-ferromagnetism vanishes for pressures in excess of p N ≈ p s ≈
15 kbar. Interestingly the supercon-ducting transition broadens considerable in the range be-tween p N and p s in samples with higher T s (Nicklas et al. ,2005) (note that the crystal structure of CePt Si − x Ge x is not stable for x > . T s . In samples with larger T s the DC susceptibility israpidly suppressed (Motoyama et al. , 2008b). This mightbe the result of the large distribution of lattice constantsin these samples, mentioned above.In any case the phase diagram suggests the vicinity ofa quantum critical point. This is underscored by tenta-tive evidence for critical fluctuations in the specific heatand inelastic neutron scattering of pure CePt Si at am-bient pressure. Above T N electronic contributions to thespecific heat decrease as ∆ C/T ∼ log( T ). Preliminaryinelastic neutron scattering measurements suggest thatthere is Q -independent quasi-elastic scattering at hightemperatures typical of conventional heavy-fermion sys-tems. At low temperatures, however, short range corre-lations are observed for Q − ≈ . Si and CeIn the circumstan-tial evidence suggests that quantum critical spin fluctua-tions may be a key ingredient. However, due to the lackof inversion symmetry these may include complex tex-tures, like the skyrmion ground states observed recently(M¨uhlbauer et al. , 2009; Neubauer et al. , 2009).It is finally interesting to point out that several LaMX compounds are superconducting. In particular, LaRh Si,LaIr Si and LaPd Si show superconductivity with T s FIG. 16 (a) Neel temperature T N versus unit cell volume V inthe series CeMX (M: Rh, Ir, Co; X: Si, Ge). (b) Sommerfeldcoefficient γ of the specific heat versus unit cell volume V inthe series CeMX . Plot taken from (Kawai et al. , 2008b). of 1.9, 2.7 and 2.6 K respectively (Muro, 2000). ForLaRh Si the upper critical field is low H c = 0 .
03 T.
2. CeMX Following the discovery of superconductivity in the an-tiferromagnetic state of the non-centrosymmetric heavy-fermion system CePt Si superconductivity was also dis-covered in the non-centrosymmetric systems CeRhSi (Kimura et al. , 2005), CeIrSi (Suginishi and Shima-hara, 2006) and CeCoGe (Kawai et al. , 2008b; Settai et al. , 2007a). These compounds belong to the class ofisostructural CeMX systems, where M=Co, Ru, Pd, Os,Ir, Pt, Fe, Rh and X=Si and Ge. The BaNiSn crystalstructure, space group I mm (No. 107), of the CeMX series derives from the body-centered tetragonal BaAl crystal structure, space group I4/mm. We note thatBaAl is also the parent structure of the body-centeredThCr Si systems of the heavy fermion superconductorsCeCu Si , CeCu Ge , CePd Si , CeRh Si and URu Si (cf Fig. 2). It is moreover the parent structure of the se-ries of CaBe Ge body-centered tetragonal ternary sys-tems, none of which have so far been found to be su-perconducting. As shown in Fig. 2 the BaSnNi struc-ture is composed of a sequence of planes along the c-axisR-M-X(1)-X(2)-R-M-X(1)-X(2)-R, where X(1) and X(2)denote different lattice positions of the X atom. Thusthe structure lacks inversions symmetry along the c-axis.The generating point group C V is identical to that ofCePt Si.4For a brief review of the properties of the series CMX we refer to (Kawai et al. , 2008b). This review includesconsiderations on the crystal electric fields, which play aprominent role in the ground state properties. Amongstthe systems studied, CeCoGe has the highest antiferro-magnetic ordering temperature. Interestingly those com-pounds with low values of T N have the a-axis as easymagnetic axis, while in CeCoGe the c-axis is magnet-ically soft. It is interesting to note, that the antiferro-magnetic transition temperatures, their pressure depen-dence and the Sommerfeld coefficient of the specific heatas function of decreasing unit cell volume are consistentwith a Doniach phase diagram (Fig. 16). So far, super-conductivity near a magnetic quantum phase transitionhas been observed in those systems that are on the righthand border of the phase diagram.In passing we note that the system CeNiGe , whichalso displays superconductivity when antiferromagnetismis suppressed at high pressure (Kotegawa et al. , 2006;Nakashima et al. , 2004), crystallizes in a centro-symmetric orthorhombic structure (see section II.A.3). a. CeRhSi The first system in this class for which su-perconductivity was observed is CeRhSi . For a recentreview we refer to (Kimura et al. , 2007b). The latticeconstants are a = 4 .
244 ˚A and c = 9 .
813 ˚A and thesingle crystals studied had very low residual resistivi-ties of a few tenths of a µ Ω cm. At ambient pressureCeRhSi orders antiferromagnetically below T N = 1 . (cid:126)Q = ( ± . , , .
5) (Aso et al. , 2007). Themagnetic field dependence of the magnetization suggestsan anisotropy of about 2. Only a small magnetization isseen up to 8 T. At high temperatures an isotropic Curie-Weiss susceptibility is observed with a fluctuating mo-ment µ eff = 2 . µ B as expected of the full Ce mo-ment.The specific heat is interpreted in terms of a Schot-tky anomaly around 100 K and Kondo temperature oforder T K ≈
50 K. The Kondo screening is affected bythe CEF level scheme, where inelastic neutron scatteringhas been interpreted as three doublets with a Γ groundstate and Γ and Γ first and second excited state at260 and 270 K, respectively (Muro et al. , 2007). The lowtemperature specific heat above T N exhibits a stronglyenhanced Sommerfeld contribution γ = 0 .
110 J / mol K .This suggests that a heavy fermion state forms despite alarge Kondo temperature, in which incommensurate anti-ferromagnetism stabilizes at very low temperatures. TheFermi surface has been investigated by means of quantumoscillations (Kimura et al. , 2007b, 2001) and comparedto LaRhSi . Substantial differences are interpreted as ev-idence for an itinerant f-electron and spin density wavetype of antiferromagnetism. Moreover, several branchesshow a small splitting with similar angular dependences. This is seen as evidence for Rashba splitting.The pressure dependence of T N in CeRhSi is quiteunique. Up to 9 kbar T N increases moderately beforedecreasing again gradually up to 20 kbar. For pressureabove 2 kbar (Kimura et al. , 2005, 2007b) superconduc-tivity emerges in the antiferromagnetic state, where T s increases up to 30 kbar, the highest pressure measured.The superconducting dome is exceptionally wide. TheAC susceptibility shows susperconducting screening withadditional features that require further clarification. To-gether with the zero resistance state the susceptibilityis a strong indication of superconductivity. However, itdoes not establish spontaneous Meissner flux expulsionand thus volume superconductivity. The initial slope of H c is strongly enhanced and becomes anomalously largearound 26 kbar with dH c /dT | T s = −
23 T / K. This sug-gests that H c is exceptionally large and may even exceed30 T (Kimura et al. , 2007a). b. CeIrSi The compound CeIrSi is isostructural toCeRhSi with lattice constants a = 4 .
252 ˚A and c =9 .
715 ˚A (Muro et al. , 1998). The ambient pressure prop-erties of CeIrSi are characteristic of a heavy-fermion sys-tem with an enhanced Sommerfeld contribution to thenormal state specific heat γ = 0 .
12 J / mol K and anti-ferromagnetic order below T N = 5 . doublet and the first and second ex-cited states are Γ and Γ doublets at 149 K and 462 K,respectively (Okuda et al. , 2007). The magnetization isanisotropic by a factor of about two, where the a-axisis the easy axis. Quantum oscillatory studies of LaIr Sisuggest show that the Fermi surface is similar to that ofLaCoGe , characteristic of a compensated metal wherebranches with an exchange splitting of 1000 K exhibit anangular dependence that track each other rather closely.This suggests the presence of Rashba splitting due to thelack of inversion symmetry.The antiferromagnetism in CeIrSi vanishes for pres-sures in excess of p N = 22 . et al. , 2007) and a superconductingdome emerges, with T maxs = 1 .
65 K for pressure in ex-cess of p N as shown in Fig. 17. H c exhibits a strongtemperature dependence near T s . For the basal plane dH abc /dp = − T /K at p N with H abc ( T →
0) = 9 . dH c /dp = − T /K and H cc reaches 18 Tjust below 1 K, suggesting a extremely large value in ex-cess of 30 T (Settai et al. , 2008). This is strikingly similarto CeRhSi . Recent specific heat and AC susceptibilitymeasurements up to 35 kbar show distinct specific heatanomalies for both the antiferromagnetic and supercon-ducting transitions, i.e. they may be tracked very well asa function of pressure using an AC method, but quanti-tative information is not available (Tateiwa et al. , 2007).Above p N the specific heat anomaly is particularly pro-nounced and suggests strong coupling superconductivity.NMR studies show the absence of a coherence peak in5 FIG. 17 (a) Temperature versus pressure phase diagram ofCeIrSi . At the border of antiferromagnetic order a wide su-perconducting dome emerges. Note that the pressure axisbegins at 19 kbar. (b) Superconducting specific heat anomalyas a function of pressure. (c) Extrapolated zero temperatureupper critical field. Plot taken from (Settai et al. , 2008). the spin lattice relaxation rate and a cubic temperaturedependence characteristic of line nodes (Mukuda et al. ,2008). The normal state spin-lattice relaxation rate isthereby characteristic of an abundance of antiferromag-netic spin fluctuations, which are likely to be implicatedin the superconducting pairing.Rather remarkable is the behavior observed under sub-stitutional doping in CeIr − x Co x Si (Okuda et al. , 2007).Replacing Ir with Co represents to leading order a reduc-tion of unit cell volume equivalent to the application ofpressure. For x = 0 . x = 0 .
35 the N´eel temperatureis reduced and superconductivity is observed. Metallur-gical tests suggest that the compound for x = 0 .
35 is notsingle phase with a dominant contribution also of the x = 0 . c. CeCoGe Amongst the CeMX compounds CeCoGe has the highest magnetic ordering temperature T N =21 K, followed by two more transitions at T N = 12 Kand T N = 8 K (Kawai et al. , 2008b; Settai et al. , 2007a).The metallic state is well described as a moderately en- hanced Fermi liquid with a Sommerfeld coefficient of thespecific heat γ = 0 .
032 J / mol K and a coefficient ofthe quadratic temperature dependence of the resistivity A = 0 . µ Ωcm / K . The easy magnetic axis is the c-axis, as opposed to other members of the CeMX series,where the a-axis is the easy axis. Under pressure T N decreases and vanishes around 55 kbar, where the rate ofsuppression drops around 30 kbar. Superconductivity isobserved in the range 54 to 75 kbar with T s = 0 .
69 K at apressure around 65 kbar. For this pressure the H c alongthe c-axis, as extrapolated from the very large increasenear T s , given by dH c = −
20 K / T, is exceptionally largeand may reach 45 T.The Fermi surface of the series LaTGe (T: Fe, Co,Rh, Ir) has been reported in (Kawai et al. , 2008a). Allsystems exhibit strong Rashba spin-orbit splitting. Itwill be interesting to see how the characteristics of thesesuperconductors relate to those of the Ce-systems. Forinstance, the La-compounds may display the singlet statesuperconductivity to which the triplet state pairing getsadmixed in the Ce-systems.We finally note that superconductivity has also beenreported in CeCoSi at 0.5 K (Haen et al. , 1985). How-ever this observation could not be confirmed down to50 mK in a subsequent study (Eom et al. , 1998).
3. UIr
Superconductivity in non-centrosymmetric heavy-fermion systems also exists at the border of ferromag-netism in UIr (Akazawa et al. , 2004a,b). The struc-ture of UIr is monoclinic of PbBi-type (space group P2 )and lacks inversion symmetry (Dommann and Hullinger,1988). Four different uranium sites may be distin-guished. In the paramagnetic state the susceptibility fol-lows a Curie-Weiss dependence with an effective moment µ eff = 2 . µ B / U. Below a Curie temperature T C = 46 KIsing ferromagnetism develops with a reduced orderedmoment of 0 . µ B /U , characteristic of itinerant electronmagnetism. The easy axis is [10¯1]. The properties aresummarized in table III.A recent review of the temperature-pressure-magneticfield phase diagram of UIr may be found in (Kobayashi et al. , 2007). Several samples of varying quality havebeen studied so far, where an indenter pressure cell wasused. The pressure technique leaves room for uncertain-ties regarding the possible role of non-hydrostatic condi-tions. As shown in Fig. 18, the resistivity, AC suscepti-bility and magnetization establish, that three magneticphases may be distinguished under pressure. Data weremostly collected for the [10¯1] easy axis and [010] hardaxis. The nature of the magnetic states has not beenidentified by means of microscopic probes yet. Based onthe available bulk data the phases are referred to as fer-romagnetic states.Under pressure the FM1 state vanishes for pressurein excess of p c = 17 kbar. The transition at T c may6 FIG. 18 Temperature versus pressure phase diagram of UIr.Three ferromagnetic phases have been identified. Supercon-ductivity is only observed at the border of the FM3 phasewith very low superconducting transition temperatures. Plottaken from (Kobayashi et al. , 2007). be readily seen in the resistivity, AC susceptibility andmagnetization. The ordered moment decreases graduallybetween 0 . µ B / U and 0 . µ B / U before dropping discon-tinuously at p c . In the limit T → p c and p c = 21 kbar. As a function oftemperature the FM2 transition may be seen in the ACsusceptibility, but not in the resistivity. The ordered mo-ment in the FM2 state is strongly reduced and not largerthan 0 . µ B / U. The FM3 phase exists in the limit T → p c = 27 . p c , where thebehavior between p c and p c is complex with the possi-bility of a metamagnetic transition from the FM2 to theFM3 phase. The magnetic ordering temperature of theFM3 phase at T c may be seen in the resistivity and ACsusceptibility. As a rather peculiar feature of the FM3phase T c ( p ) is not directly connected with either T c ( p )or T c ( p ), but begins in the middle of the paramagneticregime as shown in Fig. 18. Clearly, based on symme-try considerations there must be another transition linealong which the symmetry breaking takes place.Superconductivity is observed in the FM3 phase of UIrfor pressures in the range 26 kbar < p < p c (Akazawa et al. , 2004a,b), reaching T s = 0 .
15 K and H c = 0 . p c . Also,the superconductivity is only observed for samples withfairly high residual resistivity ratios ( > T s of Ir,which does not match or track the behavior observed ex-perimentally. We finally return to the question of the nature of theFM1, FM2 and FM3 phases. The FM1 phase appears tobe a straight-forward Ising ferromagnet. In contrast, thedominant feature of the FM2 phase is a 25-fold increaseof the residual resistivity for the magnetically hard [010]axis (Hori et al. , 2006) and a strongly reduced sponta-neous moment. Moreover, quantum oscillations vanishoutside the FM1 phase (Shishido et al. , 2006). This letto the speculation of a multilayer-like phase separationalong the [010] axis. It is presently not clear, whetherthis structure is related to a structural modification, sofar not supported by high pressure x-ray diffraction. Theeasy and hard axis of the magnetization are unchanged inthe FM3 phase, which supports the superconducivity atlow temperatures (Kobayashi et al. , 2007). Finally, theFM3 phase again appears to be a straight forward Isingferromagnet with strongly reduced ordered moment. Ithas been argued that there is no additional modulationin the FM3 state, because the easy and hard axis areunchanged. Finally, it appears unlikely that the crys-tal structure has recovered the centro-symmetric sym-metry under pressure, because this would require ma-jor rearrangements of the atomic positions. The orderedmagnetic moment in the FM3 phase ( ∼ . µ B / U) cor-responds to a fairly small internal field, also consistentwith conventional superconductivity. Also, the coherencelength of ξ = 1100 ˚A as inferred from H c is comparableto the charge carrier mean free path of l = 1240 ˚A. Therole of the different U-sites has not been addressed atall. Clearly the interplay of magnetism and supercon-ductivity in UIr poses a large number of experimentaland theoretical challenges for the future. B. Superconductivity near electron localization
The degree of itineracy of the f-electrons in intermetal-lic compounds provides a major source of scientific de-bate. The transition from an itinerant to a localizedstate creates variations in the charge density that alsodrive strong correlations in the spin density. Interest-ingly, heavy fermion superconductivity is found in ma-terials at the border of such a localization transition.This suggests that the nature of the superconductive in-teractions is related to charge density fluctuations as anew route to superconductivity. The interplay of thesefluctuations with spin fluctuations and further degrees offreedom is an open issue.
1. Border of valence transitions
It has recently been suggested that the superconduc-tivity maximum in CeCu Si at high pressures is relatedto a Ce to Ce +4 valence transition (cf Fig.3), wherethe 4f electron is delocalized in the high pressure Ce state (Holmes et al. , 2004; Yuan et al. , 2004). This type ofQPT transition is non-symmetry breaking in the spirit of7itinerant-electron metamagnetism. The suggestion wasinspired by the analogy of the temperature versus pres-sure phase diagrams of CeCu Ge and CeCu Si . InCeCu Ge x-ray diffraction suggests a valence transitionat a pressure p c ≈
15 GPa (Onodera et al. , 2002). How-ever, there is no microscopic evidence for a valence tran-sition in CeCu Si − x Ge x except for faint features seenin the L III -x-ray absorption (Roehler et al. , 1988) andchanges of the metallic state notably the electrical resis-tivity.Studies of the magnetic phase diagram under pressureestablish, that the T coefficient of the resistivity qual-itatively tracks dH c /dT up to ∼ . ∼ . et al. , 1998). Further studies es-tablished, that the T coefficient of the resistivity dropsabruptly, when the characteristic temperature scale T max1 varies under pressure or Ge-doping reaches a value of ∼
70 K (Holmes et al. , 2004). Under the same condi-tions a five-fold enhancement of the residual resistivity isobserved and a tiny maximum in the specific heat coeffi-cient.It is conceivable that the superconductivity inCeCu Si at high pressure develops with a rather differ-ent pairing symmetry. A microscopic pairing mechanismhas been proposed in which the pairing is dominantlymediated by the exchange of charge fluctuations betweenthe conduction bands and the f-site (Onishi and Miyake,2004). In the limit of a spherical Fermi surface and weakcoupling this model predicts a d-wave superconductingstate, where the value of T c scales with the slope of thecontinuous valence transition as a function of the f-levelenergy.
2. Plutonium and neptunium based systems
Another surprise in recent years has been the discov-ery of heavy-fermion superconductivity in the actinidecompounds PuCoGa (Sarrao et al. , 2002), PuRhGa (Wastin et al. , 2003) and NpPd Al (Aoki et al. , 2007a).The properties of these systems are summarized in ta-ble II. Status reports for PuCoGa and PuRhGa havebeen given by (Haga et al. , 2007; Sarrao and Thompson,2007; Thompson et al. , 2006a,b,c). The striking featureabout the superconductivity in PuCoGa , PuRhGa andNpPd Al are values of T s of 18.5, 8.7 and 4.9 K, respec-tively, which are the highest of all f-electron systems.It seems natural to assume that the key ingredients re-sponsible for the high transition temperatures in thesesystems are related to the special electronic properties ofthe 5f electrons in the elements.First, plutonium is delicately placed at the border be-tween a large and small Wigner-Seitz radius characteris-tic of the transition between delocalized and localized f-electrons. Second, because Coulomb screening is strongerfor 4f than 5f electrons, the typical band width of 5f sys-tems is intermediate between 3d and 4f systems. More- over, the effects of spin-orbit coupling in 5f systems varyquite strongly along the series and change from weak forU to very strong for Pu, Am and Cm (Moore and van derLaan, 2009). Qualitatively this suggests that certain 5fsuperconductors are intermediate between the traditional4f heavy fermion superconductors and 3d high- T c super-conductors. This conjecture is strongly supported by theexperimentally observed properties, especially when plot-ting T s , versus a temperature characteristic of the elec-tronic correlations T (cf. the band width). a. PuCoGa & PuRhGa Both PuCoGa and PuRhGa crystallize in the tetragonal HoCoGa structure, spacegroup P4/mmm (Sarrao et al. , 2002; Wastin et al. , 2003).The structure is identical to the series of Ce n M m In n +2 m compounds reviewed in section II.A.2 and derives fromthe cubic HoGa in terms of MGa layers stacked se-quentially along the [100] axis (for further informationsee (Wastin et al. , 2003)). The normal state of bothof PuCoGa and PuRhGa is characterized by a Curie-Weiss susceptibility with an effective fluctuating moment µ eff ∼ . µ B / Pu, respectively. The effective momentis close to the 5f (Pu ) configuration of 0 . µ B . InPuCoGa the Curie-Weiss temperature, Θ = − et al. , 2002). Above ∼
100 K theeffective moment in PuRhGa assumes the free ion value(Haga et al. , 2007). The susceptibility exhibits Curie-Weiss behavior throughout the normal state. The tem-perature dependence of the electrical resistivity is anoma-lous, showing a power law dependence ∝ T n with n ∼ .
35 instead of the conventional quadratic temperaturedependence of an enhanced Fermi liquid. In both systemsthe specific heat is well described as that of a heavy Fermiliquid state plus lattice term C ( T ) = γT + βT , where γ = 0 .
077 J / mol K and γ = 0 .
07 J / mol K for PuCoGa and PuRhGa , respectively. The value for β correspondsto a Debye temperature Θ D ∼
240 K for PuCoGa andPuRhGa .Below T s = 18 . exhibits superconductiv-ity. The initial change of the upper critical field near T s in polycrystals is unusually large dH c /dT = − . / K.This implies H c = 74 T (Werthamer et al. , 1966), whichexceeds the estimated Pauli limit ( H P = 34 T) by a fac-tor of two. The estimated value of H c corresponds toa Ginzburg-Landau coherence length ξ GL ≈
21 ˚A. Theheat capacity confirms bulk supercoductivity, where thesize of the anomaly, ∆
C/γT s = 1 .
4, is consistent withconventional BCS superconductivity. Further specificheat studies in single-crystals confirm these conjecturesand show a quadratic temperature dependence, consis-tent with an axial gap symmetry with line nodes (Ja-vorsk´y et al. , 2007). This study also establishes the pos-sibility of an FFLO state in PuCoGa , where a large Makiparameter α is inferred. The magnetization is character-istic of strong type II superconductivity.It has been noticed that the anisotropy of the super-conductivity in PuCoGa and PuRhGa to an applied8magnetic field qualitatively matches the anisotropy of theantiferromagnetism in NpCoGa and NpRhGa to an ap-plied magnetic field (Colineau et al. , 2006; Wastin et al. ,2006). This supports the notion that the magnetic in-teractions arise on the same grounds than the supercon-ductivity. However, using polarized neutron scatteringorbital and spin contributions to the Curie-Weiss sus-ceptibility have been discriminated (Hiess et al. , 2008).While the microscopic magnetization in NpCoGa agreeswith the bulk susceptibility, there is a large discrepancyin PuCoGa . In fact, the polarized neutron scatteringdata imply that orbital contributions to the fluctuatingmoment are dominant. In turn this suggests that the su-perconductivity is not straight forwardly due to the moretraditional versions of spin fluctuation mediated pairing.Microscopic evidence for unconventional superconduc-tivity has been inferred from measurements of the , Gaand Co Knight shift K s and nuclear spin-lattice re-laxation rate T in PuCoGa (Curro et al. , 2005; Sakai et al. , 2006) and PuRhGa (Bang et al. , 2006; Sakai et al. ,2005). The Knight shift provides information on theorbital susceptibility χ o , which is essentially constant,and the spin susceptibility, χ s , which decreases in thesuperconducting state. This clearly identifies PuCoGa and PuRhGa as spin-singlet d-wave superconductors.The spin-lattice relaxation rate in both systems dropsabruptly when entering the superconducting state with-out evidence of a Hebel-Slichter peak. Below T s the re-laxation rate initially varies as T − ∝ T and settles intoa dependence T − ∝ T at the lowest temperatures, pre-sumably due to impurity scattering.The spin-lattice relaxation in PuCoGa and PuRhGa differs markedly from conventional electron-phonon me-diate superconductivity observed in Al or MgB , corre-sponding to the predictions of antiferromagnetically me-diated superconductive pairing and scales with the be-havior observed in CeCoIn and YBa Cu O . Thus theobserved form of T suggests common microscopic fea-tures of the superconductivity for materials with vastlydifferent values of T s which, however, are all strong con-tenders for antiferromagnetic pairing. In fact, when plot-ting T s versus spin fluctuation temperature, which mea-sures the effective band width, PuCoGa and PuRhGa are found to be intermediate to the class of 4f heavyfermion superconductors and the 3d high T c cuprates asshown in Fig. 19. Interestingly the temperature depen-dence of T in the normal state of PuRhGa deviates fromthat observed in PuCoGa , CeCoIn and YBa Cu O .This has been interpreted as a pseudogap consistent withthe canonical phase diagram of a superconducting domesurrounding a quantum phase transition.The strong radioactivity of Pu imposes several experi-mental constraints. Self-heating generates a considerableheat load that does not allow to perform experiments atvery low temperatures. A typical value is ∼ . µ Wper mg for PuRhGa . More important is the structuraldamage incurred by the radioactive decay of Pu, whichresults in a uranium nucleus and a high-energy alpha par-
FIG. 19 Comparison of superconducting transition tempera-ture with the characteristic spin fluctuation temperature. Thelatter is essentially a band width and may be insensitive tothe precise microscopic nature of the correlations. Plot from(Curro et al. , 2005) as shown in (Sarrao and Thompson, 2007). ticle. The uranium nucleus displaces by a mean distanceof 120 ˚A and creates on average of 2300 Frenkel pairsof vacancies and displaced interstitials distributed over arange of 75 ˚A (Wolfer, 2000).Several studies have addressed the effects of self-irradiation (Booth et al. , 2007; Jutier et al. , 2006,2005; Ohishi et al. , 2007, 2006), which may be seenas an unique opportunity to study the evolution of asuperconducting state as a function of increasing de-fect concentration. Experimentally it is observed that T s decreases in both compounds under self-irradiation,where ∆ T s / ∆ t ∼ − .
39 K / month for PuRhGa and∆ T s / ∆ t ∼ − .
24 K / month for PuCoGa (Jutier et al. ,2005). For doped samples with PuCo . Rh . Ga andPuCo . Rh . Ga the rates of decrease are intermedi-ate (Jutier et al. , 2006). H c and the critical currentdensity show more complex behavior. The initial varia-tion of H c near T s increases in 553 days for PuCoGa from dH c /dT = − . / K to dH c /dT = −
13 T / K,while is decreases strongly for PuRhGa from dH c /dT = − . / K to dH c /dT = − . / K. The same trendsare reflected in the critical currents. These studies sug-gest that self-irradiation generates point defects, wheredefects of the size of the coherence length are known torepresent effective pinning centers (Campbell and Evetts,1972).The nature of the damage caused by self-irradiationhas been studied microscopically by µ -SR (Ohishi et al. ,2007, 2006). The µ SR line widths are found to narrowdramatically with increasing self-irradiation. This is seenas the result of an abundance of pinning centers thattrap flux lines thereby reducing the internal field distri-bution. The absolute value of the penetration depth asinferred from the µ SR data strongly depends on the de-fect concentration. Yet, the low temperature variation, λ ∝ T , consistently shows d-wave behavior for the pres-tine and the irradiated samples. When taken togetherthis suggests that the superconducting state is rather ro-9bust against the damages incurred by self-irradiation.Finally, when monitoring the consequences of self-irradiation over a period of four years, the degradationof the superconductivity actually deviates from a strictlylinear behavior (Jutier et al. , 2008). This deviation hasbeen explained in the framework of an Eliashberg the-ory of a ’dirty’ d-wave superconductor, consistent withthe NMR measurements. These authors point out that aphononic mechanism reproduces the experimental data,leaving open the role of the spin and orbital fluctuations.We now turn to the possible nature of the supercon-ductivity in PuCoGa and PuRhGa . The Curie-Weissdependence may be taken as evidence of localized 5f elec-trons. Yet, PuIn , shows a similar strong Curie-Weisssusceptibility, but quantum oscillatory studies establishthat the 5f electrons are in an itinerant state. An itin-erant f-electron state in PuCoGa and PuRhGa is alsosupported by the temperature dependence of the resistiv-ity. This is supported further by band structure calcu-lations for PuCoGa in the local density approximationwhich suggest that the origin of the high value of T s in-deed lies in the 5f electrons (Maehira et al. , 2003; Opahleand Openeer, 2003; Szajek and Morkowski, 2003).Also, a comparison of the resistivity of the seriesACoGa (A=U, Np, Pu and Am) establishes that the re-sistivity for the systems CeCoIn , PuCoGa and UCoGa scale with each other characteristic of a single spin fluc-tuation energy. Moreover, the physical properties of theACoGa systems suggests that PuCoGa resides near anitinerant to localized crossover of the 5f electrons thatis reminiscent of the itinerant to localized crossover thatoccur near Pu in the actinide series (Moore and van derLaan, 2009). The peculiar emergence of the supercon-ductivity out of a metallic state with strong Curie-Weisssusceptibility has inspired theoretical considerations con-cerning the symplectic symmetry of the spin in PuCoGa and NpCoGa and how a coupling of local spins withthe conduction electrons may promote superconductiv-ity (Flint et al. , 2008).The specific heat of PuCoGa suggests a Debye tem-perature Θ D = 240 K, which, using the McMillan equa-tion with a Coulomb pseudopotential µ ∗ = 0 . λ = 0 . λ = 1, sug-gests T s ≈ . ∼
14 K, respectively (Thompson et al. , 2004). Thus conventional electron-phonon medi-ated pairing cannot be ruled out. However, it is difficultto reconcile it with the large fluctuating magnetic mo-ments seen in the normal-state susceptibility. Moreover,because the temperature dependence of the resistivity isbest explained in terms of scattering by antiferromag-netic spin fluctuations it has been concluded that super-conductivity in PuCoGa is unconventional. In fact, tak-ing into account the presence of defects as measured bythe residual resistivity ρ = 20 µ Ωcm transition temper-atures as high as ∼
40 K may be expected (Bang et al. ,2004).The lattice dynamics of PuCoGa was studied experi-mentally by room temperature inelastic x-ray scattering (Raymond et al. , 2006) and compared to first principlescalculations using the generalized gradient approxima-tion (GGA) in density functional theory (Piekarz et al. ,2005). Excellent quantitative agreement was obtainedwhen the on-site Coulomb repulsion was taken into ac-count with U = 3 eV (GGA+U) and Hund’s rule ex-change. The estimated averaged electron-phonon con-stant is calculated to be λ = 0 . et al. , 2005).In the Allen-Dynes or equivalently McMillan formalismthis value of λ , when taken together with the Debyetemperature and a pseudo-Coulomb interaction µ ∗ be-low 0.1, implies T s to be in the range 7 to 14 K . In otherwords electron-phonon coupling alone cannot be respon-sible for the superconductivity in PuCoGa . However,the detailed understanding of electron-phonon interac-tions in PuCoGa and CeCoIn requires to resolve alsowhy UCoGa is not superconducting even though thephonon spectra are similar.A dual nature of the 5f electrons was inferred froma photoemission study of PuCoGa (Joyce et al. , 2003),where excellent agreement with a so-called mixed-levelcalculation (MLL) in density functional theory was ob-served. In this calculation one f-electron is in an itinerantstate and four f-electrons are localized 1.2 eV below E F .The data are in stark contrast with the predictions ofpurely itinerant f electrons in a generalized gradient ap-proximation (GGA). The conclusion of the MLL calcula-tion has been questioned by a first principles calculationof the ground state (S¨oderlind, 2004). It transpires thatthe photoemission spectra can be accounted for by fullyitinerant f electrons when the spin and orbital degrees offreedom are allowed to be correlated.Using relativistic linear augmented-plane-waves theFermi surface was found to be dominated by several largecylindrical f-electron sheets in fair agreement with theFermi surface of CeMIn (Maehira et al. , 2003). In par-ticular, the band width of the 5f electrons is intermediateto typical 3d and 4f systems. While the calculated Fermisurface of PuCoGa and CeMIn (M=Co, Ir, Rh) is simi-lar it differs from the calculated Fermi surface of the pairof actinide systems UCoGa and NpCoGa , which con-sists of several small sheets plus a single large sheet forthe case of NpCoGa .The similarities of the Fermi surface in PuCoGa andCeMIn can be explained in terms of a tight-bindingcalculation taking into account j - j coupling (Hotta andUeda, 2003; Maehira et al. , 2003). The analogy may betraced to the pseudo-spin representation of the j - j cou-pling, where one electron exists in the j = 5 / , while there is one hole for the five electronsof Pu in the sextett. Thus Pu may be viewed asthe hole analogue of the one electron state of Ce . Theincreased value of T s may then be attributed to the in-creased width of the 5f bands, where an additional roleof the orbital structure of the Pu systems is likely.The role of the transition metal element in controllingthe nature of the ground state in PuCoGa and relatedcompounds has been explored experimentally by means0substitutional replacement of Pu by U and Np and of Coby Fe, Rh and Ni (Boulet et al. , 2005). Superconductivityis most dramatically suppressed for U and Np substitu-tion, while isoelectronic substitution is the least destruc-tive. These results are theoretically underpinned by DFTcalculations in the full-potential linear-muffin-tin-orbital(FP LMTO) approximation, where the transition metalelement does not contribute directly to the density ofstates at the Fermi level (Oppeneer et al. , 2006). Ratherthe transition metal effectively hole- or electron-dopesthe Pu atom.Ab initio total energy calculations in the local spindensity approximation suggest antiferromagnetic groundstates for PuCoGa and PuRhGa (Opahle et al. , 2004).When taking into consideration that LSDA calculationsdo not treat correlation effects properly these results sug-gest that PuCoGa and PuRhGa are at least close toantiferromagnetic order. The effects of Coulomb correla-tions have been addressed in a study using the relativisticLSDA+U (Shick et al. , 2005). This study unexpectedlyshows a considerable reconstruction of the LSDA resultssuggesting j - j like coupling for the Pu 5f manifold similarto what is observed for pure Pu metal. The dynamicalmean field theory (DMFT), finally, suggests an impor-tant role of van Hove singularities in the (cid:126)k -resolved spec-tral density that may provide strong enhancements ofthe magnetic susceptibility leading to d-wave supercon-ductivity (Pourovskii et al. , 2006).The analogy of PuMGa and CeMGa has been ex-plored experimentally in several studies. Besides the ev-idence for an important role of critical antiferromagneticfluctuations and the general considerations based on thecalculated band structure given above, there is strikingsimilarity concerning the dependence of T s on the ratio c/a of the lattice parameters as shown in Fig. 10 in sec-tion II.A.2 (Bauer et al. , 2004b). This trend is consistentwith trends predicted for magnetically mediated pairing(Monthoux and Lonzarich, 2001, 2002). However, the ex-perimental investigation of the lattice parameters underhigh pressure establishes that for none of the PuMGa and CeMGa systems the value of T s scales with c/a .This suggests that there are other important aspects be-sides the c/a ratio (Normile et al. , 2005). On anothernote it has been suggested, that the normalized pressuredependence of the superconductivity is consistent with adome, which may be qualitatively viewed in a commonphase diagram (Thompson et al. , 2006c). b. NpPd Al We next turn to the question of furtheractinide superconductors that are neither based on ura-nium nor plutonium. An important element in this re-spect is neptunium which is adjacent to plutonium. TheWigner-Seitz radius thereby suggests that the f-electronsin Np are in an itinerant state. Examination of spec-troscopy and physical properties shows that the 5f statesof Np are beginning to show the effects of localization,however, the metal is still fairly delocalized (Moore and van der Laan, 2009).Recently heavy fermion superconductivity has alsobeen discovered in NpPd Al (Aoki et al. , 2007a; Griveau et al. , 2008). This represents the first Np-based super-conducting system. It is interesting to compare the prop-erties of this system with the Pu-based heavy fermionsuperconductors. The crystal structure of NpPd Al is ZrNi Al type body-centered tetragonal, space groupI4/mmm with atomic positions Np (0 , , , / , . , , /
2) and Al (0 , , . et al. , 2008).The normal state is characterized by a Fermi liquidspecific heat with γ = 0 . / mol K . In contrast, themagnetic susceptibility is temperature independent forthe c-axis, but diverges all the way until superconductiv-ity sets in. This and the linear temperature dependenceof the electrical resistivity for the a-axis clearly signalNFL properties.The normal state susceptibility shows a Curie-Weisstemperature dependence with a fluctuating moment µ abeff = 3 . µ B / Np, µ ceff = 3 . µ B / Np for the ab-planeand c-axis, respectively, that is intermediate to the freeNp 5 f free ion value of 3 . / Np and the Np 5 f con-figuration with 2 . / Np. The Curie-Weiss susceptibilityextends all the way down to the onset of superconductiv-ity at T s = 4 . ∼ µ Ωcm down to T s . Just above T s the resistivity is linear in temperature, characteristic ofcharge carrier scattering by critical fluctuations consis-tent with the Curie Weiss susceptibility. The extrapo-lated residual resistivity is ρ ≈ µ Ωcm. Despite the ev-idence for strongly temperature dependent fluctuationsin the normal state specific heat show the behavior of aFermi liquid with an enhanced γ = 0 . / mol K .The superconducting transition is accompanied bya pronounced λ anomaly in the specific heat, where∆ C/γT s = 2 .
33. This is characteristic of strong-couplingsuperconductivity. The temperature dependence of thespecific heat in the superconducting state is highly un-conventional, following initially a T dependence thatsettles into a T dependence below ∼ . T dependence of the low temperature specific heat is con-sistent with point nodes in the superconducting gap. Incombination with the antiferromagnetic fluctuations in-ferred from the normal state susceptibility this suggestsa d -wave state with point nodes.The initial slopes of H c near T s are anomalously largewith dH abc /dT = − . / K and dH cc /dT = −
31 T / K,as for the Pu based superconductors. However, H c ishighly anisotropic and in comparison the Pu-based sys-tems reduced, where H abc ( T →
0) = 3 . H cc ( T →
0) = 14 . H c . The d.c. magnetization shows that1the lower critical field H c = 0 .
008 T, coherence length ξ = 94 ˚A, penetration depth λ = 2600 ˚A and Ginzburg-Landau parameter κ = 28. For the c-axis the magneti-zation suggests first order behavior at low temperatures,akin CeCoIn (for the a-axis H c is too large). This im-plies also the possibility for an FFLO state. Al NMR in single crystal NpPd Al (Chudo et al. ,2008) shows a broadening of the NMR spectra when en-tering the superconducting state, consistent with a fluxline lattice. Further, there is no coherence peak and thespin-lattice relaxation rate, 1 /T shows a cubic temper-ature dependence. Both, the spin-lattice relaxation rateas well as the Knight shift point at line-nodes and strongcoupling d-wave superconductivity.Changes of the temperature dependence of the resistiv-ity under magnetic field suggests the vicinity to a quan-tum critical point; the T coefficient decrease as if itis singular at H c . Interestingly pressure suppresses T s above 57 kbar, reminiscent of a superconducting dome(Honda et al. , 2008b). This is also consistent with avicinity to quantum criticality.As for the Pu-based superconductors it is not clear,where the entropy of the magnetic fluctuations is dumpedin the superconducting state. Based on the striking sim-ilarity of the Pu and Np superconductor it interesting tospeculate on the possible implications of the paramag-netic limiting as the only difference. Since Pu is closer tothe localization of the f electron, this may suggest an im-portant role of charge fluctuations (Schlottmann, 1989).In fact, similar considerations as for the vicinity of a va-lence instability discussed above may also apply here andcharge density fluctuations may promote the supercon-ductive pairing (Monthoux and Lonzarich, 2004; Onishiand Miyake, 2000). C. Border of polar order
For systems where the quasiparticle dressing cloud isdominated by excitations of the crystal electric fieldsan interesting question concerns, whether the quasiparti-cle interactions also include attractive components thatmay stabilize superconductivity. A scenario of this kindhas been proposed for UPd Al as discussed in sectionII.B.1.a. However, for UPd Al the superconductivitycoexists with large-moment antiferromagnetism where T s (cid:28) T N . In turn the interplay of the crystal field ex-citations with the antiferromagnetic order is of consid-erable complexity and essentially not accessible directlyexperimentally due to the strong hybridization of the 5felectrons with the conduction electrons.In comparison to U-based compounds, Pr-based com-pounds generally show distinct crystal electric field exci-tations. The quasiparticle dressing clouds in the Fermiliquid regime in pure Pr were, for instance, identifiedas being excitonic (Lonzarich, 1988). In recent yearsheavy-fermion superconductivity has been discovered inPrOs Sb and related compounds (cf table V). There TABLE V Key properties of Pr-based heavy-fermion super-conductors and siblings exhibiting conventional superconduc-tivity. Missing table entries may reflect more complex behav-ior discussed in the text. References are given in the text.PrOs Sb PrRu Sb PrRu P structure cubic cubic cubicspace group Im ¯3 Im ¯3 Im ¯3 a (˚A) 9.302 -∆ CEF (meV) 7 64state SC, AFQ SC IN, SC T c (K) 1.3 (at 9 T) 62 (cid:126)Q (0 , ,
1) - µ ord ( µ B ) 0.085 - γ (J / molK ) 0.5 0.059 - T s (K) 1.85 1.3 1.8( p >
110 kbar)∆
C/γ n T s > H c (T) 2.3 0.2 2 dH c /dT (T/K) -1.9 - ξ (˚A) 120 400 - λ (˚A) 3440 3650 - κ GL (˚A) 28 9year of discovery 2002 2005 2004 is now growing consensus that the superconductivityin PrOs Sb may be mediated by the exchange ofquadrupolar fluctuations. In the following we first reviewthe properties of PrOs Sb . For more detailed reviewswe refer to (Aoki et al. , 2007b; Hassinger et al. , 2008;Maple, 2005; Maple et al. , 2008). The section concludeswith a paragraph on PrRu P , in which superconduc-tivity emerges, when an insulating state is suppressed at110 kbar. c. PrOs Sb PrOs Sb belongs to the rare-earth-filledskutterudites, a class of systems with an exceptionallyrich spectrum of vastly different ground states. Exam-ples include insulating and metallic behavior, long rangemagnetic and polar order as well as conventional and un-conventional superconductivity (Aoki et al. , 2005; Maple,2005; Maple et al. , 2008; Sales, 2003; Sato et al. , 2007).The large variety of electronic behaviors may be tracedto the unusual crystal structure, which for the case ofPrOs Sb consists of a stiff icosahedron Sb cage typicalof binary skutterudites, filled with a loosely bound Pr ion.The Pr ion is presumably in an off-center position (Goto et al. , 2004). The space group of the crystal structure is Im ¯3, where the local point symmetry of the rare-earthion is tetrahedral, T h ( m J = 4 multiplet of the Pr ions into a Γ and twotriplets Γ , (Takeuchi et al. , 2001). As for all rare-earthfilled skutterudites the Pr-ion exhibits ’rattling’ modes,leading to almost dispersionless low-energy phonons asseen in Raman scattering (Ogita et al. , 2008). In neutronscattering the rattling modes result in large Debye-Wallerfactors and in Raman scattering a second order phononpeak has been observed (Goto et al. , 2004; Kaneko et al. ,2006). Even though the Pr ion is only loosely bound, thep-f hybridization is expected to be large because of thecage of Sb atoms surrounding it.The resistivity of PrOs Sb decreases monotonicallywith temperature and displays a roll-off around 10 K fol-lowed by a superconducting transition at T s = 1 .
85 K(Bauer et al. , 2002). The susceptibility displays a broadmaximum around 3 K and the specific heat exhibits a pro-nounced Schottky anomaly. The features in the resistiv-ity, susceptibility and specific heat are due to thermallypopulated CEF-split Pr energy levels. Two differingcrystal field schemes were initially proposed, a Γ sin-glet ground state and Γ triplet first excited state (Aoki et al. , 2002) and vice versa (Maple et al. , 2002; Vollmer et al. , 2003). Inelastic neutron scattering (Goremychkin et al. , 2004) and detailed measurements of the magneticfield dependence (Aoki et al. , 2002) have settled this is-sue and it is now accepted (Bauer et al. , 2006b), that theground state is a Γ singlet, followed by a Γ triplet firstexcited state.Quantum oscillatory studies show Fermi surface sheetsconsistent with localized 4f electrons (Sugawara et al. ,2008, 2002). In comparison with other systems in thisseries the Fermi surface lacks nesting and compares wellwith that of LaOs Sb . The similarity of the Fermisurface topology is underscored by Hall effect and ther-mopower measurements, which are similar for both com-pounds (Sugawara et al. , 2005).It was immediately recognized that PrOs Sb repre-sents the first example of a Pr-based heavy-fermion su-perconductor (Bauer et al. , 2002). Although the lowtemperature specific heat is dominated by a Schottkyanomaly around 2 K, it is possible to infer a stronglyenhanced linear term in the normal state specific heat C/T ≈ . .
75 J / mol K (for a comprehensive dis-cussion of the analysis of C ( T ) see (Grube et al. , 2006)and references therein). A related large anomaly is ob-served in the specific heat at the superconducting transi-tion, ∆ C/T s ≈ . / mol K , which, depending on thestrength of the coupling, also points to a large valueof γ . Finally, H c ∼ . H orbc = 2 . dH c /dT ≈ − . / K near T s . The large valueof dH c /dT also supports heavy fermion superconductiv-ity.An increasing number of experimental data suggestthat the superconductivity in PrOs Sb is unconven-tional. µ -SR shows that the superconductivity is accom-panied by time reversal symmetry breaking (Aoki et al. ,2003). The penetration depth measurements show a tem- FIG. 20 Magnetic field versus temperature phase diagram ofPrOs Sb . In high magnetic field an ordered state is stabi-lized that is driven by the level crossing of the crystal elec-tric fields under magnetic field. Plot taken from (Aoki et al. ,2007a). perature dependence of the penetration depth λ ∝ T and superfluid density ρ s ∝ T down to 0 . T s (Chia et al. , 2003). The zero temperature penetration depth λ = 3440 ˚A is comparatively short. The data for λ and ρ s are consistent with point nodes of strong-coupling su-perconductivity with ∆(0) /k B T s = 2 .
6. This is con-trasted by Sb-NMR of the spin-lattice relaxation rate,which lacks a coherence peak and shows a temperaturedependence consistent with an isotropic energy gap ofa very strong-coupling state (Kotegawa et al. , 2003).A well-developed superconducting gap, which is nearlyisotropic is also observed in tunneling spectroscopy (Sud-erow et al. , 2004). Small angle neutron scattering inPrOs Sb has revealed an asymmetry of the flux line lat-tice that suggests a p-wave superconducting state (Hux-ley et al. , 2004).The case for unconventional superconductivity inPrOs Sb is underscored by the observation of con-ventional superconductivity in the Pr-filled skutteruditesPrRu Sb (Frederick et al. , 2005) and PrRu As ( T s =2 . et al. , 1997), as well as the La-filledskutterudites LaRu As ( T s = 10 . P ( T s =4 . P ( T s = 7 . Sb ( T s = 3 . Sb ( T s = .
74 K) (Maple et al. , 2002; Miyake et al. , 2004; Sato et al. , 2003). Remarkably, upon dop-ing PrOs Sb by La on the Pr site and Ru on the Ossite the heavy-fermion superconductivity gradually turnsinto conventional superconductivity. This suggests, thata certain stability of the heavy fermion superconductivityagainst defects exists.A controversial question in PrOs Sb concerns,whether the superconductivity consists of multiple super-3conducting phases and/or multiband superconductivity.The specific heat and thermal expansion display a dou-ble superconducting transition (Aoki et al. , 2002; Bauer et al. , 2002; Measson et al. , 2004; Oeschler et al. , 2004;Rotundu et al. , 2004; Vollmer et al. , 2003). The simi-larity of the observed behavior across a large number ofdifferent samples seems to suggest that the behavior is in-trinsic. However, recent studies of a very high quality sin-gle crystal show only a single transition (M´easson et al. ,2008; Seyfarth et al. , 2006). A detailed study of sampleswith a double transition in the specific heat using micro-Hall probe and magneto-optical imaging reveal consider-able inhomogeneities that question a bulk nature of thedouble transition (Kasahara et al. , 2008). The doubletransition is also reflected in the susceptibility (Cichorek et al. , 2005; Frederick et al. , 2004; Grube et al. , 2006; Me-asson et al. , 2004; M´easson et al. , 2008) and resistivity(Measson et al. , 2004), which points at an extrinsic origin.Multiband superconductivity has been suggested on thebasis of thermal conductivity measurements, which read-ily return to the normal state behavior in small magneticfields (Seyfarth et al. , 2006, 2005). Further, H c showspositive curvature near T s (Measson et al. , 2004). Multi-band superconductivity has also been inferred from Sb-NQR studies (Yogi et al. , 2008), which supports a fullygapped large Fermi surface that drives strong-couplingsuperconductivity accompanied by a small Fermi surfacewith line nodes.Additional transitions to further superconductingstates have been inferred from magnetothermal trans-port (Izawa et al. , 2003), the low field magnetization (Ci-chorek et al. , 2005) and Andreev reflections (Turel et al. ,2008). As for the magnetothermal transport, a changeof symmetry is observed at fairly high fields ∼ ∼ . et al. , 2005; Raymond et al. , 2008b). Aclear dispersion is found for the transition Γ to Γ (2)4 for (cid:126)Q = ( ζ, ,
0) in zero magnetic field. Both the excita-tion energy and scattering intensity exhibit a minimum at (cid:126)q = (1 , , Sb .A pronounced phase transition emerges above ∼ et al. ,2002; Oeschler et al. , 2004; Rotundu et al. , 2004; Sug-awara et al. , 2005; Tayama et al. , 2003; Vollmer et al. ,2003). The large entropy released at this phase tran-sition clearly shows that the 4f electrons are involvedin the ordering process. Neutron diffraction revealsa small antiferromagnetic modulation in the high fieldphase (Kaneko et al. , 2007; Kohgi et al. , 2003). For field (cid:126)H (cid:107) [0 , ,
1] and (cid:126)H (cid:107) [1 , ,
0] the superlattice has wavevector (cid:126)q = (1 , , µ ord = 0 . µ B / Pr represents only a few % ofthe uniform magnetization.It is possible to show that this modulation resultsfrom Γ -type antiferroquadrupolar interactions (Shiina,2004; Shiina and Aoki, 2004). Within this scenario theanisotropy of the field induced ordered phase is due tothe tetrahedral point symmetry T h of the Pr ion. Theantiferroquadrupolar order is driven by the Zeeman split-ting and the crossing of the lower triplet with the sin-glet level at 9 T. It is interesting to note that the order-ing wave-vector corresponds to the nesting wave-vectorin PrRu P (Hao et al. , 2004; Lee et al. , 2001) andPrFe P (Iwasa et al. , 2002), which display anomalousordering transitions. d. PrRu P The Pr filled skutterudite compoundsPrRu P exhibits a metal insulator transition at T MI =62 K, that defies an explanation in terms of magnetic orcharge ordering (Sekine et al. , 1997). Under hydrostaticpressure T MI varies only weakly, but additional anoma-lies emerge below T MI that suggest further ordering tran-sitions. Above 110 kbar PrRu P turns metallic with asuperconductivity below T s ∼ . et al. , 2004).The upper critical field of this superconducting state israther high H c ≈ T s and H c with PrOs Sb isinteresting to note. V. MULTIPLE PHASESA. Order parameter transitions
Many of the superconducting phases of intermetalliccompounds reviewed in this paper are candidates for un-conventional superconductivity with complex supercon-ducting order parameters. They may in turn display var-ious symmetry broken superconducting phases. In thefollowing we summarize the evidence for such multiple su-perconducting phases. At present the only stoichiometricsuperconductor, where multiple superconducting phasesare observed beyond doubts is the archetypical heavy-4fermion system UPt , which will be addressed first. Thisis followed by short summaries on further candidates forsuch phases, where prominent examples are PrOs Sb and U − x Th x Be .
1. Superconducting phases of UPt The normal state properties of UPt have been in-troduced in section II.B.2.a. At low temperature UPt displays a peculiar form of commensurate antiferromag-netic order below T N = 5 K with tiny magnetic moments,that appears to be related to a highly dynamic magneticground state. The antiferromagnetic order is only ob-served in neutron scattering and the metallic state sharesthe properties of a strongly renormalized Fermi liquid. Inthis heavy fermion ground state superconductivity ap-pears below T s = 0 .
54 K. While heavy-fermion super-conductivity in its own right would already be quite re-markable, it is the observation of three superconductingphases that has attracted tremendous scientific interest.In the following we briefly review the superconductingphase diagram in UPt . A detailed account may be foundin (Joynt and Taillefer, 2002).The first indication for multiple superconductingphases was observed in the ultrasound attenuation in ap-plied magnetic fields and in H c . The bulk property thatexhibits the most distinct evidence of multiple supercon-ducting phases is the specific heat, where two transitionsare seen. The transition temperatures are T + s = 0 .
530 Kand T − s = 0 .
480 K (Fisher et al. , 1989). Thus the split-ting is of the order ∼
10% of T s and rather small. Withrespect to the linear term of the normal state specificheat, γ n = 0 .
44 J / mol K , the anomalies at T s are givenby ∆ C + /γT + s = 0 .
545 and ∆ C − /γT − s = 0 . C/T below T s down to 0.1 K, below which apronounced upturn is observed (Brison et al. , 1994).Applied magnetic field has been found to reduce T + s and T − s at different initial rates without significantbroadening for field parallel and perpendicular to the c-axis as shown in Fig. 21 (Hasselbach et al. , 1989, 1990).The transition merges at a tetracritical point ( H ∗ , T ∗ H ),where for H (cid:107) ˆ c : H ∗ = 0 . T ∗ H = T + s − . H ⊥ ˆ c : H ∗ = 0 . T ∗ H = T + s − .
15 K. Tetracritical-ity has been confirmed by ultrasound attenuation (Aden-walla et al. , 1990; Bruls et al. , 1990), dilatometry (vanDijk et al. , 1993) and the magnetocaloric effect (Bogen-berger et al. , 1993). The general consensus has become,that UPt exhibits three superconducting phases referredto as A, B and C. Phases A and B support a Meissner anda Shubnikov phase below and above H c . As a functionof temperature H c shows a sudden increase in slope at T − s (Vincent et al. , 1991). Qualitatively the three compo- FIG. 21 Superconducting phases of UPt as a function ofmagnetic field. The insets show the nodal structures of theE u representations, proposed on the basis of small angle neu-tron scattering of the flux line lattice. Plot taken from (Hux-ley et al. , 2000). nent phase diagram contrasts an extrinsic origin, wherethe phase transition lines may be expected to have simi-lar field dependences.In general H c in UPt exceeds Pauli limiting. Theanisotropy of H (cid:107) ˆ c and H ⊥ ˆ c changes at around 0.2 Kwith H (cid:107) c < H ⊥ c at low temperatures and H (cid:107) c > H ⊥ c near T s (Shivaram et al. , 1986). The presence of thethree superconducting phases requires to distinguish co-herence lengths and penetration depths according tothese phases. On the one hand, the zero temperaturevalue of H c is characteristic of the C phase, where H (cid:107) c ( T →
0) = 2 . H ⊥ c ( T →
0) = 2 . H c may be accounted for by an anisotropicmass enhancement. The coherence length inferred from H c then is ξ ≈
120 ˚A. On the other hand, the initialslope of H c with temperature near T + s is characteristicof the A phase, where dH (cid:107) c /dT | T + s = − . ± . / Kand dH ⊥ c /dT | T + s = − . ± . / K. When accountingfor this anisotropy also in terms of the effective mass en-hancement, it is possible to obtain an estimate of theGinzburg-Landau parameter κ GL = 44. In other wordsUPt is a strong type 2 superconductor. Some simpleestimates arrive at values of the penetration depth of theorder λ (cid:107) ( T →
0) = 4500 ˚A and λ ⊥ ( T →
0) = 7400 ˚A,consistent with the short coherence length estimated forthe C phase. It can finally be shown that weak couplingtheory yields the same value of κ GL . This implies thatUPt is still fairly well described in a weak coupling ap-proximation.The effect of hydrostatic pressure on the superconduct-ing transitions and the antiferromagnetic order stronglysuggests, that the antiferromagnetic order is instrumental5for the symmetry breaking between the different super-conducting phases. In the specific heat the two super-conducting transitions are found to decrease at differentrates, eventually merging into a single transition above ∼ et al. , 1991). At the same time neu-tron scattering establishes that the ordered moment de-creases under pressure and vanishes above ∼ T N is essentially not affected by pressure (Hayden et al. ,1992).Numerous other experimental probes suggest uncon-ventional pairing and provide important hints as to theprecise nature of the gap symmetry. For instance, in arecent small angle neutron scattering study the magneticfield dependence of the flux line lattice has been estab-lished. The upshot of this study is that the three super-conducting phases belong to the E u symmetry (Huxley et al. , 2000) (see also (Champel and Mineev, 2001) fortheoretical considerations on the flux line lattice). Foran extended review and critical discussion of the vari-ous theoretical scenarios we refer to (Joynt and Taillefer,2002). Despite the large body of studies the search forthe correct order parameter symmetry has not been en-tirely conclusive so far.
2. Further candidates
Nearly all of the systems covered in this review in oneway or the other may be candidates for multiple super-conducting phases. The nature of these phases may bequite different, representing either different order param-eter symmetries or real space modulations with differentordered state. In the following we draw attention to can-didates, which await further clarification. a. CeCu Si As reviewed in sections II.A.1 and IV.B.1recent high pressure studies in pure and Ge dopedCeCu Si reveal the presence of two superconductingdomes (Fig. 3). At low pressures this material is a can-didate for magnetically mediated pairing driven by thevicinity to an antiferromagnetic quantum critical point.At high pressures a second dome emerges and it hasbeen argued that this superconducting phase is related tofluctuations in the charge density of a valence transition(Holmes et al. , 2004; Yuan et al. , 2004). b. CeNi Ge At ambient pressure CeNi Ge displays anincipient form of superconductivity. It has been ar-gued that the ambient pressure behavior is reminiscent ofCePd Si in the vicinity of the critical pressure. Underpressure the signatures of superconductivity vanish. Athigh pressures an additional superconducting transitionemerges as shown in Fig. 4 (Grosche et al. , 1997b). Inprinciple this second superconducting dome may hint atan additional superconducting phase, but little is knownabout this state. c. CeIrIn Pure single crystals of CeIrIn display a dif-ference of the temperature of a zero resistance transition, T s = 0 .
75 K and the bulk superconducting transition inthe specific heat, T s = 0 . d. UPd Al In UPd Al single crystals grown with anAl-rich starting composition showed particularly sharpsuperconducting transitions at T s in the resistivity(Sakon et al. , 1993). This suggested an improved samplequality. Remarkably the specific heat, thermal expan-sion and elastic constants in these samples revealed anadditional anomaly around 0.6 K well below T s (Matsui et al. , 1994; Sakon et al. , 1994; Sato et al. , 1994). Thenature of this transition has so far not been settled. Ei-ther it signals an additional superconducting transitionakin the double transition observed in UPt , or it cor-responds to another ordering transition. For the firstcase, it is conceivable that the antiferromagnetic order ofUPd Al represents the symmetry breaking field. In thelatter case, it is possible that the emerging order leadsto an additional symmetry breaking of the superconduct-ing order that may stabilize additional superconductingphases. e. URu Si Early studies of the specific heat of thesuperconducting transition in URu Si showed featuresreminiscent of the double transition in UPt (Hasselbach et al. , 1991). Detailed studies in high quality single crys-tals did not confirm the first findings. Keeping in mindthat the tiny-moment antiferromagnetism in UPt rep-resents the symmetry breaking field, that stabilizes thedifferent superconducting phases, it seems plausible thatthe same might occur in URu Si . However, the anti-ferromagnetism in URu Si seems to be related to animpurity phase. Moreover, under pressure the super-conductivity vanishes, when large moment antiferromag-netism appears. The observed change of curvature in H c of URu Si has motivated considerations of the possibleformation of a FFLO state. However, as discussed be-low URu Si does not develop a FFLO state. In turnit is currently accepted that URu Si does not supportadditional superconducting phases. f. UBe Doping UBe with Th results in the phasediagram shown in Fig. 12 (Ott et al. , 1986). For x =0 . < x < x = 0 .
042 two successive transitions at T s > T s are observed in the specific heat. The on-set of superconductivity is thereby at T s . The pres-sure dependence of Th-doped samples also suggest theexistence of two superconducting phases (Lambert et al. ,61986), where an investigation of the lower critical fieldsuggests that T s indeed marks the onset of another su-perconducting phase (Rauchschwalbe et al. , 1987). Agroup theoretical analysis of these properties has been re-ported in (Luk’yanchuk and Mineev, 1989; Makhlin andMineev, 1992). However, it still seems unsettled whetherthe lower transition at T s indeed represents another su-perconducting transition (Kumar and W¨olfle, 1987; Mar-tisovits et al. , 2000; Sigrist and Rice, 1989) or the onsetof a defect induced form of magnetic order as suggestedby µ -SR (Heffner et al. , 1986). g. UGe Pressure and magnetic field studies suggestthat the superconductivity in UGe is driven by the firstorder transition between the FM1 and FM2 ferromag-netic phases (Fig. 13). The superconductivity hence ex-ists in the presence of two different forms of ferromagneticorder. Theoretical considerations have shown, that theorder parameter symmetry in ferromagnetic supercon-ductors depends on the orientation of the ferromagneticmoment. Experimental evidence that tentatively sup-ports different superconducting phases in the FM1 andFM2 state may be seen in the discontinuity of T s at p x and the reentrance of H c for pressures just above p x andfield applied along the crystallographic a-axis. However,as discussed in section III.A.1, the magnetic anisotropyof UGe remains unchanged under pressure. It thereforeappears unlikely that the superconducting phases in theFM1 and FM2 state are fundamentally different. Furtherstudies will have clarify this issue. h. URhGe One of the most unusual phase diagramsamongst all f-electron superconductors is observed inURhGe. As a function of magnetic field superconduc-tivity is at first suppressed, but reappears at high mag-netic fields, when the ordered moment is forced to rotatefrom the c-axis to the b-axis. The phase diagram yieldsup to three different superconducting phases: the zerofield state, the high field state below H R , where the mo-ment is almost rotated into the b-axis and finally above H R , where the moment is essentially aligned with the b-axis. As for UGe the allowed order parameter symme-tries have been worked out for the orthorhombic crystalstructure. i. CePt Si & CeMX The pressure versus temperaturephase diagram of the four non-centrosymmetric heavy-fermion superconductors is dominated by a decreases ofthe Neel temperature. The transiton line crosses the su-perconducting dome in the middle, so that the phase di-agram is comprised of a regime where T N > T s and aregime where T N has vanished. These two regimes are inprinciple candidates for differences in the order parame-ter. FIG. 22 Superconducting phase diagram of PrOs Sb . Plottaken from (Grube et al. , 2006). j. PrOs Sb The superconducting state of PrOs Sb exhibits several features that have been interpreted astentative evidence for multiple superconducting phases.The specific heat of PrOs Sb displays two supercon-ducting transitions (Huxley et al. , 2004; Measson et al. ,2004; Vollmer et al. , 2003), where doping by Ru and Lastabilizes the upper transition while mechanical thinningstabilizes the lower transition. However, the origin of thedouble transition is a controversial issue, where recentstudies suggest that it may of extrinsic origin (Kasa-hara et al. , 2008; M´easson et al. , 2008; Seyfarth et al. ,2006) (for details see section IV.C). As shown in Fig. 22tentative transition lines in the susceptibility and a va-riety of other quantities may be traced all the way tozero temperature. Studies of the thermal conductivity(Izawa et al. , 2003) also suggest multiple superconduct-ing phases, but with a different phase diagram that is notmatched by any other property. Finally, high precisionmeasurements of the magnetization suggest the possibleexistence of yet another transition line at very low fields(Cichorek et al. , 2005). A comprehensive discussion alongwith detailed measurements of the specific heat and ACsusceptibility have been given by (Grube et al. , 2006). B. Textures
An important fundamental and technological questionin condensed matter systems are weak interactions thatcause the formation of intermediate- and long-scale tex-tures. The f-electron superconductors reviewed here ex-hibit several forms of electronic order and thus posses dif-ferent types of characteristic rigidities. As far as the su-perconducting state is concerned these are the coherencelength and penetration depth, while the magnetic state ischaracterized by the spin-wave stiffness, spin-orbit cou-pling, CEF pinning potential and dipolar interactions.As a first example the competition of exchange splittingwith superconducting pairing is addressed. This compe-tition may result in real-space modulations of the super-conductivity and spin polarization as reviewed in section[V.B.1]. The possible interplay of ferromagnetic domain7structures and superconductivity is briefly addressed insection [V.B.2].
1. Fulde-Ferrell-Larkin-Ovchinnikov states
The novel forms of superconductivity of interest in thepresent review are characterized by real space modula-tions and anisotropies of the superconducting gap func-tion that are caused by a loss of symmetries beyond thoseof the underlying crystal structure. In turn the phaserigidity of the superconducting condensate in these su-perconductors yields changes of sign in momentum space.An entirely different class of novel superconducting stateswas predicted by Fulde, Ferrel, Larkin and Ovchinnikov(FFLO) (Fulde and Ferrell, 1964; Larkin and Ovchin-nikov, 1965). As opposed to changes in momentum spacein the FFLO state the order parameter changes sign inreal space. In its original version the FFLO state consid-ered superconductivity in the presence of a strong uni-form exchange field. The Cooper pairs thereby form be-tween Zeeman split parts of the Fermi surface, so thatpairing with a finite momentum (cid:126)q is stabilized, where( (cid:126)k ↑ , − (cid:126)k + (cid:126)q ↓ ). In the following we will briefly reviewthe current status of FFLO states in the f-electron su-perconductors addressed in this paper. Detailed reviewsmay be found, e.g, in (Buzdin et al. , 1997; Casalbuoniand Nardulli, 2004; Matsuda and Shimahara, 2007); forrecent theory see (Houzet and Mineev, 2006, 2007).Despite intense efforts, only a small number of can-didate materials could be identified that may supportan FFLO state, notably heavy fermion superconductorsand quasi-two-dimensional organic superconductors forfields parallel to the layers (Burkhardt and Rainer, 1994;Buzdin and Kachkachi, 1997; Dupuis, 1995; Gloos et al. ,1993; Gruenberg and Gunther, 1966; Shimahara, 1994;Tachiki et al. , 1996; Yin and Maki, 1993). This may betraced back to the rather severe conditions under whichthe FFLO state is expected to form. As a first precon-dition, pair breaking in applied magnetic fields must bedominated by paramagnetic limiting and not orbital lim-iting (Gruenberg and Gunther, 1966). Second, impuritiesare detrimental to the FFLO state, making high-puritysamples a key requirement (Aslamazov, 1969; Takada andIzuyama, 1969). Third, large anisotropies of the Fermisurface are favorable to the FFLO state.FFLO considered the effects of a uniform exchangefield on s-wave superconductors. In the presence of pureorbital limiting the superconducting transition is secondorder at all magnetic fields and the superconductivityis unchanged by the exchange field. In contrast, in thepresence of pure Pauli limiting the superconducting tran-sition changes in finite fields from second to first order fortemperatures below T † = 0 . T s (Ketterson and Song,1999; Saint-James et al. , 1969). Below T † an inhomoge-neous form of superconductivity stabilizes, in which theCooper pairs support a finite momentum ( (cid:126)k ↑ , − (cid:126)k + (cid:126)q ↓ ).In the bulk properties the signature of the FFLO state is an increase of H c below T † , that may be accompaniedby a change of curvature. The size of this increase de-pends sensitively on the anisotropy of the Fermi surfaceranging from 7% of the Pauli limit for three dimensions(Fulde and Ferrell, 1964; Larkin and Ovchinnikov, 1965;Saint-James et al. , 1969; Takada and Izuyama, 1969),over 42% for two dimensions (Aoi et al. , 1974; Bulaevskii,1974; Burkhardt and Rainer, 1994; Shimahara, 1994) toa divergence for one dimension (Machida and Nakanishi,1984; Suzumura and Ishino, 1983). Microscopically theFFLO state consists in spatial modulations of the super-conductivity in real space, for which the order parametermay be given in general as ∆( (cid:126)r ) = (cid:80) Mm =1 ∆ m exp i(cid:126)q m · (cid:126)r (Bowers and Rajagopal, 2002; Combescot and Tonini,2005; Fulde and Ferrell, 1964; Larkin and Ovchinnikov,1965; Mora and Combescot, 2004, 2005; Shimahara, 1998;Wang et al. , 2006). The superposition of degenerate com-ponents then yields a rich variety of symmetries of thereal space modulations, e.g., hexagonal, square, triangu-lar and one-dimensional modulations.It has long been appreciated that the stringent re-quirements for an FFLO state may be satisfied in su-perconductors with short coherence length, because theorbital limiting field diverges as H orbc ∝ /ξ so thatPauli limiting may dominate. Prime examples are theheavy fermion superconductors reviewed here. The situ-ation for an FFLO state then involves (i) a finite admix-ture of orbital limiting, (ii) the coexistence of antiferro-or ferromagnetic order, and (iii) anisotropic (unconven-tional) order parameter symmetries. The exploration ofthese issues has stimulated a large number of theoreti-cal studies (Adachi and Ikeda, 2003; Buzdin and Brison,1996a,b; Gruenberg and Gunther, 1966; Houzet and Mi-neev, 2006; Klein et al. , 2000; Shimahara et al. , 1996;Shimahara and Rainer, 1997; Suginishi and Shimahara,2006; Tachiki et al. , 1996; Yang and MacDonald, 2004).For a recent review of these studies we refer to (Matsudaand Shimahara, 2007).The question, whether FFLO states exist in heavyfermion superconductors has been explored in a numberof systems. For instance, the AC susceptibility, magneti-zation, ultrasound velocity and thermal expansion near H c in CeRu and UPd Al exhibit the characteristics ofa peak effect (Gegenwart et al. , 1996; Haga et al. , 1996;Steglich et al. , 1996; Tachiki et al. , 1996; Takahashi et al. ,1996). It is now broadly accepted that these features donot yield microscopic characteristics related to a FFLOstate, but instead may be due to subtle forms of defectrelated pinning. Further candidates for an FFLO stateare URu Si and UBe , which display additional contri-butions in H c (Brison et al. , 1995; Buzdin and Brison,1996a,b; Gl´emot et al. , 1999). For URu Si this contri-bution is seen for the c-axis and rather small. In con-trast UBe displays a change of curvature in H c ( T ). Ithas been shown that these features are consistent with avicinity to the FFLO formation, but the FFLO state doesnot form. Possible explanations include the sample pu-rity, which is very good but does not reach the exception-8 FIG. 23 Superconducting phase diagram of CeCoIn . In thelow temperature limit a body of evidence suggests the for-mation of a FFLO state (pink shading). Plot taken from(Matsuda and Shimahara, 2007). ally clean limit required. Candidates for a FFLO statethat have been identified recently in specific heat studiesunder magnetic field are PuCoGa , PuRhGa (Javorsk´y et al. , 2007) and NpPd Al (Aoki et al. , 2007a).The perhaps best candidate of an FFLO state knownto date has been identified in CeCoIn (Fig. 23). Sev-eral features in the superconducting phase diagram havebeen observed uniquely in CeCoIn . The specific heat(Bianchi et al. , 2003a; Radovan et al. , 2003), magneti-zation, (Gratens et al. , 2006) magnetostriction (Correa et al. , 2007), thermal conductivity (Capan et al. , 2004),penetration depth (Martin et al. , 2005), ultrasound ve-locity (Watanabe et al. , 2004b) and NMR Knight shift(Kakuyanagi et al. , 2005; Kumagai et al. , 2006; Mitrovi´c et al. , 2006) show that the transition at H c is first orderfor T < . T s and T < . T s for field parallel and per-pendicular to the c-axis, respectively. This is the behav-ior expected for paramagnetic limiting of H c , where thesamples studied were readily in the ultra-pure limit, i.e.,the coherence length is only a small fraction of the chargecarrier mean free path. It is thereby helpful to note thatthe orbital limit H orbc ,ab = 38 . H orbc ,c = 11 . H c near T s substantiallyexceeds the experimentally observed values of H c . Thecorresponding values of the Maki parameter near 5 ex-ceed by a large margin the threshold of 1.8, above whicha FFLO state may be expected.Specific heat and torque magnetization first identifieda second order phase transition line in the superconduct-ing state that branches off from H c ( T ) at a temperaturewell below that of the change from second to first orderand decreases with decreasing temperature (see Fig. 23)(Bianchi et al. , 2002; Miclea et al. , 2006). The presenceof this line was confirmed in subsequent measurementsof the penetration depth (Martin et al. , 2005), thermalconductivity (Capan et al. , 2004), ultrasound velocity(Watanabe et al. , 2004b), magnetization (Gratens et al. ,2006), magnetostriction (Correa et al. , 2007) and NMR(Kakuyanagi et al. , 2005; Mitrovi´c et al. , 2006). The re-sulting phase pocket is a strong contender for a FFLOstate. The size of the novel phase pocket is anisotropic andconsiderably smaller in a field perpendicular to the ab-plane. The transition field is weakly temperature depen-dent for field direction perpendicular to the ab-plane andstrongly field dependent for field direction parallel to theab-plane. The anisotropy suggests that the FFLO stateis more stable for field parallel to the ab-plane. This maybe related to the two-dimensional character of the Fermisurface and the anisotropy of the spin fluctuation spec-tra, where the latter appear to be involved in the pairinginteractions as discussed in section II.A.2.Key characteristics observed for the novel phase pocketmay be summarized as follows. The penetration depthin the ab-plane increases, consistent with a decrease ofthe superfluid density (Martin et al. , 2005). The ther-mal conductivity, providing a directional probe of thequasiparticle spectrum, is highly anisotropic. For heatcurrent parallel to the applied field the thermal conduc-tivity is enhanced, while it has not been possible to clar-ify changes of the thermal conductivity for heat currenttransverse to the applied field. As this behavior is some-what counterintuitive, it has been proposed that the in-terplay of vortex lattice, spatial order parameter mod-ulation and nodal gap structure results in an effectiveincrease of vortex cores in the nodal plane (Capan et al. ,2004).The flux line lattice in CeCoIn has been studied, e.g.,by the ultrasound velocity (Ikeda, 2006; Watanabe et al. ,2004b), which provides information on the pinning of thevortex cores by defects. Notably it is possible to extractinformation on the c dispersive flux line tilt mode. Acareful analysis of the decrease observed in c implies adecrease of the superconducting volume fraction. Smallangle neutron scattering reported so far did not meet thescattering condition necessary to probe the FFLO state(Bianchi et al. , 2008). Microscopic information on theFFLO regime is also provided by NMR spectra of theIn(1) and In(2) sites in the CeIn - and CoIn layers, re-spectively (Kakuyanagi et al. , 2005; Kumagai et al. , 2006;Mitrovi´c et al. , 2006; Singleton et al. , 2001; Young et al. ,2007). In the FFLO regime a key feature for both fielddirections is the appearance of a second resonance linein the superconducting state, where the lines are close tothe values of the normal and superconducting state, re-spectively. It is currently unresolved if the NMR spectrafor field parallel to the ab-plane also reveal antiferromag-netic components of the vortex cores (Kakuyanagi et al. ,2005; Miclea et al. , 2006; Singleton et al. , 2001; Young et al. , 2007). Moreover, Cd doping of CeCoIn leads toa rapid suppression of the first order behavior of H c ,but Hg doping only smears out the phase pocket with-out change of characteristic temperatures (Tokiwa et al. ,2008). These studies support a nonmagnetic origin of thephase pocket, in the spirit of the original FFLO proposal.9
2. Magnetic domains versus flux lines
An issue that has not yet been explored experimen-tally concerns the interplay of the length scales charac-teristic of superconductivity with those characteristic ofcompeting or coexisting forms of order. For the caseof the superconducting ferromagnets several papers haveexplored this question from a theoretical point of view,e.g., (Buzdin and Mel’nikov, 2003; Sonin, 2002; Sonin andFelner, 1998).
VI. PERSPECTIVES
Even though the first example of a heavy-fermion su-perconductor, CeCu Si , was discovered nearly 30 yearsago, an impressive series of new systems with surpris-ing combinations of properties have come to light onlyrecently This has resulted in two developments. First,more systems are different and we are only beginning todistinguish new classes of systems that are outside thesegeneral patterns. Second, the more general experimentalingredients controlling unconventional superconductivityare finally becoming apparent. In the following we brieflysummarize these new developments.Dominant interactions that control the properties off-electron compounds may be summarized as follows:(i) the degree of f-electron localization, (ii) crystal elec-tric fields, (iii) spin and orbital degrees of freedom andtheir coupling, and (iv) electron-phonon interactions.Amongst the large variety of f-electron superconductorsthat have been discovered in recent years, there are can-didates where any one of the first three interaction chan-nels appears to dominate the pairing interactions. Forinstance most of the members of the series CeM X andCe n M m In n +2 m are candidates for antiferromagneticallymediated pairing. The U-compounds UGe , URhGe andUCoGe are candidates for ferromagnetically mediatedpairing. Systems like PrOs Sb are candidates for pair-ing by quadrupolar fluctuations, while CeCu Si at highpressure or the Pu-based superconductors are candidatesfor valence fluctuations of the f-electrons and thus elec-tron density. For instance, DMFT calculations reveal thefluctuating valence of Pu between 4, 5 and 6, ending inan average f-occupancy of 5.2. Despite their great micro-scopic differences all of these systems may be combinedin a single graph shown in Fig. 19, where the supercon-ducting transition temperature (here denoted as T c ) iscompared on logarithmic scales with characteristic tem-perature scale T of the correlations (Sarrao and Thomp-son, 2007). Note that because T represents essentiallyan effective band width, this does not capture just spinfluctuation mediated pairing.Regarding the bulk properties of the f-electrons super-conductors reviewed here a host of characteristics sug-gests unconventional superconductivity with a complexnodal structure of the superconducting gap. A particu-larly remarkable property concerns the large upper criti- cal field. In the immediate vicinity of a quantum criticalpoint these upper critical fields become additionally en-hanced. Examples include URhGe, CeRhSi and CeIrSi .It will be every interesting to learn more about the mech-anism underlying this exceptional enhancement.A common theme for many of the systems covered inthis review is the vicinity of the superconductivity toinherent Fermi surface instabilities. In the bulk prop-erties this may be seen in the observed deviations fromFermi liquid behavior. As a rather remarkable micro-scopic piece of information quantum oscillatory studiesunder pressure reveal changes of the Fermi surface topol-ogy precisely where the superconductivity is most pro-nounced. Examples include CeRh Si , CeIn , CeRhIn and UGe . This contrasts the traditional Ansatz to treatsuperconductivity as a property of stable Fermi liquids.It may therefore be highly instructive to investigate boththeoretically and experimentally scenarios of supercon-ductivity in the vicinity of Fermi surface reconstructions.For the case of the high- T c cuprates this question hasbeen explored extensively in a variety of scenarios, suchas Pomeranchuk instabilities, preformed pairs, orbitalcurrents and stripes. In this context it is interesting toconsider, whether the recent discovery of electron-pocketsof the Fermi surface in a hole doped cuprate actually hitson yet another analogy of the superconducting phases off-electron compounds (Pfleiderer and Hackl, 2007).Many compounds discovered so far exhibit supercon-ductivity in the vicinity of zero temperature instabilities.Examples are the systems like CeM X , Ce n M m In n +2 m ,UGe , URhGe, UCoGe, UIr, CePt Si and CeMX . It hasthereby been noticed that moderate anisotropies of theelectronic and crystal structure promote the occurrenceof superconductivity, while full inversion symmetry ofthe crystal structure does not seem to be a precondition.These studies suggest as a requirement for superconduc-tive pairing the need to balance stronger interactions thatwould otherwise lead to other forms of order such as mag-netism. Although this is an important theme, it is alsoimportant to keep in mind that the recent discoverieswere made by following this approach experimentally tostart with. It is then interesting to note that a number ofcompounds are also quite insensitive to pressure. Exam-ples are CeCu Ge above p c , UPd Al , UNi Al , UBe .This implies either, that we do not have an appropri-ate control parameter to change the particular balancein these compounds, or it suggests that unconventionalforms of superconductivity exist, that are much more ro-bust and do not require the vicinity to a zero temperatureinstability.Experimentally the types of f-electron superconductorsobserved so far enforce the question why heavy-fermionsuperconductivity has only been observed in systems con-taining Ce, Pr, U, Pu and Np? There is a priori no rea-son, why compounds based on other f-electron elementsshould not also exhibit unconventional forms of super-conductivity. Clearly, as concerns the electronic proper-ties of these compounds the understanding must be far0from complete. For instance, superconductivity has re-cently been reported in the Yb-boride β -YbAlB (Nakat-suji et al. , 2008) and pure Eu metal (Debessai et al. ,2009).Last but not least, the importance of high sample qual-ity cannot be emphasized enough. It is not just that theunconventional superconductivity tends to be extraordi-narily sensitive to defects. Well characterized high qual-ity single crystals are also essential to unravel the precisenature of the superconductivity alongside any other elec-tronic properties in these compounds. Once high qualitysamples are available, controlled experimental techniquesto systematically screen the evolution of these materialsas a function of a non-thermal control parameter havebecome the outstanding tool.We conclude this review with the remark, that it isgenerally very difficult to assign unambiguously the pos-sible pairing interactions to a single interaction channelin a number of the f-electron compounds. For examplein UPd Al both an antiferromagnetically mediated andexcitonic pairing mechanism have been proposed. Thisunderscores quite generally the need for a descriptionbased on a coupling of two or more correlated subsys-tems. From a purely esthetic point of view complex cou-pled systems tend to appear less beautiful, because theyare generally over-parametrized and less tractable. How-ever, the very need to consider these complexities alsoemphasizes the enormous potential for new and entirelyunexpected phenomena, many of which are yet to be dis-covered. Acknowledgments
I wish to thank P. B¨oni. R. Hackl and M. Sigrist forcarefully reading the manuscript. Comments by S. Fu-jimoto, T. Kobayashi, V. P. Mineev, K. Moore, J. D.Thompson and F. Steglich are gratefully acknowledged.
References
Aarts, J., A. P. Volodin, A. A. Menovsky, G. J. Nieuwenhuys,and J. A. Mydosh, 1994, Europhys. Lett. , 203.Abrikosov, A. A., 1952, Dokl. Adad. Nauk SSSR , 489.Abrikosov, A. A., 2001, J. Phys.: Condens. Matter , L943.Abrikosov, A. A., and L. P. Gor’kov, 1961, Sov. Phys. JETP , 1243.Adachi, H., and R. Ikeda, 2003, Phys. Rev. B (18), 184510.Adenwalla, S., S. W. Lin, Q. Z. Ran, Z. Zhao, J. B. Ketterson,J. A. Sauls, L. Taillefer, D. G. Hinks, M. Levy, and B. K.Sarma, 1990, Phys. Rev. Lett. (18), 2298.Aeppli, G., E. Bucher, C. Broholm, J. K. Kjems, J. Baumann,and J. Hufnagl, 1988, Phys. Rev. Lett. (7), 615.Aeppli, G., E. Bucher, A. I. Goldman, D. Shirane, C. Bro-holm, and J. K. Kjems, 1987, J. Magn. Magn. Mater. , 385.Agterberg, D. F., P. A. Frigeri, R. P. Kaur, A. Koga, andM. Sigrist, 2006, Physica B , 351. Akazawa, T., H. Hidaka, T. Fujiwara, T. C. Kobayashi, E. Ya-mamoto, Y. Haga, R. Settai, and Y. Onuki, 2004a, J. Phys.:Condens. Matter , L29.Akazawa, T., H. Hidaka, H. Kotegawa, T. C. Kobayashi,T. Fujiwara, E. Yamamoto, Y. Haga, R. Settai, andY. Onuki, 2004b, J. Phys. Soc. Jpn. , 3129.Allen, P., and R. C. Dynes, 1975, Phys. Rev. B , 905.Amano, G., S. Akutagawa, T. Muranaka, Y. Zenitani, andJ. Akimitsu, 2004, J. Phys. Soc. Jpn. , 530.Amato, A., D. Andreica, F. N. Gygax, M. Pinkpank, N. K.Sato, A. Schenck, and G. Solt, 2000, Physica B ,447.Amato, A., E. Bauer, and C. Baines, 2005, Phys. Rev. B ,092501.Amato, A., C. Geibel, F. Gygax, R. Heffner, E. Knetsch,D. Maclaughlin, C. Schank, A. Schenck, F. Steglich, andM. Weber, 1992, Z. Phys. B - Cond. Matt. , 159.Amitsuka, H., K. Matsuda, I. Kawasaki, K. Tenya,M. Yokoyama, C. Sekine, N. Tateiwa, T. C. Kbayashi,S. Kawarazaki, and H. Yoshizawa, 2006.Amitsuka, H., M. Sato, N. Metoki, M. Yokoyama, K. Kuwa-hara, T. S. nand H. Morimoto, S. Kawarazaki, Y. Miyako,and J. A. Mydosh, 1999, Phys. Rev. Lett. , 5114.Amitsuka, H., K. Tenya, M. Yokoyama, A. Schenck, D. An-dreica, F. N. Gygax, A. Amato, Y. Miyako, Y. K. Huang,and J. A. Mydosh, 2003, Physica B , 418.Amitsuka, H., M. Yokoyama, K. Tenya, T. Sakakibara,K. Kuwahara, M. Sato, N. Metoki, T. Honma, Y. Onuki,S. Kawarazaki, Y. Miyako, S. Ramakrishnan, et al. , 2000,J. Phys. Soc. Jpn. (Suppl. A) , 5.Anderson, P. W., 1959, J. Phys. Chem. Solids , 26.Anderson, P. W., 1984, Phys. Rev. B (7), 4000.Andrei, N., 1982, Phys. Lett. A , 299.Andres, K., J. E. Graebner, and H. R. Ott, 1975, Phys. Rev.Lett. (26), 1779.Aoi, K., W. Dieterich, and P. Fulde, 1974, Z. Phys. , 223.Aoki, D., Y. Haga, T. D. Matsuma, N. Tateiwa, S. Ikeda,Y. Homma, H. Sakai, Y. Shiokawa, E. Yamamoto, A. Naka-mura, R. Settai, and Y. Onuki, 2007a, J. Phys. Soc. Jpn. , 063701.Aoki, D., A. Huxley, E. Ressouche, D. Braithwaite, J. Flou-quet, J.-P. Brison, E. Lhotel, and C. Paulsen, 2001, Nature , 613.Aoki, H., T. Sakakibara, H. Shishido, R. Settai, Y. Onuki,P. Miranovic, and K. Machida, 2004, J. Phys. Condens.Matter , L13.Aoki, Y., T. Namiki, S. Ohsaki, S. R. Saha, H. Sugawara, andH. Sato, 2002, J. Phys. Soc. Jpn. , 2098.Aoki, Y., H. Sugawara, H. Harima, and H. Sato, 2005, J.Phys. Soc. Jpn. , 209.Aoki, Y., A. Sumiyama, G. Motoyama, Y. Oda, T. Yasuda,R. Settai, and Y. Onuki, 2008, J. Phys. Soc. Jpn. ,114708.Aoki, Y., T. Tayama, T. Sakakibara, K. Kuwahara, K. Iwasa,M. Kohgi, W. Higemoto, D. E. MacLaughlin, H. Sugawara,and H. Sato, 2007b, J. Phys. Soc. Jpn. , 051006.Aoki, Y., A. Tsuchiya, T. Kanayama, S. R. Saha, H. Sug-awara, H. Sato, W. M. Higemoto, A. Koda, K. Ohishi,K. Nishiyama, and R. Kadono, 2003, Phys. Rev. Lett. ,067003.Araki, S., M. Nakashima, R. Settai, T. C. Kobayashi, andY. ¯Onuki, 2002a, J. Phys.: Condens. Matter , L377.Araki, S., R. Settai, T. C. Kobayashi, H. Harima, andY. ¯Onuki, 2001, Phys. Rev. B (22), 224417. Araki, S., R. Settai, M. Nakashima, H. Shishido, S. Ikeda,H. Nakawaki, Y. Haga, N. Tateiwa, T. C. Kobayashi,H. Harima, H. Yamagami, Y. Aoki, et al. , 2002b, J. Phys.Chem. Solids , 1133.Aronson, M. C., J. D. Thompson, J. L. Smith, Z. Fisk, andM. W. McElfresh, 1989, Phys. Rev. Lett. (20), 2311.Aslamazov, L. G., 1969, Sov. Phys. JETP , 773.Aso, N., H. Miyano, H. Yoshizawa, N. Kimura, T. Komat-subara, and H. Aoki, 2007, J. Magn. Magn. Mater. ,602.Aso, N., H. Nakane, G. Motoyama, N. K. Sato, Y. Uwatoko,T. Takeuchi, Y. Homma, Y. Shiokawa, and K. Hirota, 2005,Physica B , 1051.Aso, N., K. Ohwada, T. Watanuki, A. Machida, A. Ohmura,T. Inami, Y. Homma, Y. Shiokawa, K. Hirota, and N. K.Sato, 2006, J. Phys. Soc. Jpn. , Suppl. 88.Aso, N., B. Roessli, N. Bernhoeft, R. Calemczuk, N. K. Sato,Y. Endoh, T. Komatsubara, A. Hiess, G. H. Lander, andH. Kadowaki, 2000, Phys.. Rev. B , 11867.Assmus, W., M. Herrmann, U. Rauchschwalbe, S. Riegel,W. Lieke, H. Spille, S. Horn, G. Weber, F. Steglich, andG. Cordier, 1984, Phys. Rev. Lett. (6), 469.Badica, P., T. Kondo, and K. Togano, 2005, J. Phys. Soc.Jpn. , 1014.Bakker, K., A. de Visser, E. Br¨uck, A. A. Menovsky, andJ. J. M. Franse, 1991, J. Magn. Magn. Mater. , 63.Bang, Y., A. V. Balatsky, F. Wastin, and J. D. Thompson,2004, Phys. Rev. B , 104512.Bang, Y., M. J. Graf, N. J. Curro, and A. V. Balatsky, 2006,Phys. Rev. B (5), 054514.Bao, W., G. Aeppli, J. W. Lynn, P. G. Pagliuso, J. L. Sarrao,M. F. Hundley, J. D. Thompson, and Z. Fisk, 2002a, Phys.Rev. B (10), 100505.Bao, W., A. Chritianson, P. Pagliuso, J. Sarrao, J. Thompson,A.H.Lacerda, and J.W.Lynn, 2002b, Physica B ,120.Bao, W., P. G. Pagliuso, J. L. Sarrao, J. D. Thompson,Z. Fisk, and J. W. Lynn, 2001, Phys. Rev. B (2), 020401.Bao, W., P. G. Pagliuso, J. L. Sarrao, J. D. Thompson,Z. Fisk, J. W. Lynn, and R. W. Erwin, 2000, Phys. Rev. B (22), R14621.Bao, W., P. G. Pagliuso, J. L. Sarrao, J. D. Thompson,Z. Fisk, J. W. Lynn, and R. W. Erwin, 2003, Phys. Rev. B (9), 099903.Bardeen, J., L. N. Cooper, and J. R. Schrieffer, 1957, Phys.Rev. (5), 1175.Barzykin, V., and L. P. Gorkov, 1993, Phys. Rev. Lett. ,2479.Bauer, E., I. Bonalde, and M. Sigrist, 2005a, Fiz. Niz. Temp. , 984.Bauer, E., G. Hilscher, H. Michor, C. Paul, E.-W. Scheidt,A. Gribanov, Y. Seropegin, H. Noel, M. Sigrist, andP. Rogl, 2004a, Phys. Rev. Lett. , 027003.Bauer, E., G. Hilscher, H. Michor, M. Sieberer, E. Scheidt,A. Gribanov, Y. Seropegin, P. Rogl, A. Amato, W. Y. Song,J.-G. Park, D. T. Adroja, et al. , 2005b, Physica B ,360.Bauer, E., H. Kaldarar, A. Prokofiev, E. Royanian, A. Amato,J. Sereni, W. Br¨amer-Escamilla, and I. Bonalde, 2007, J.Phys. Soc. Jpn. , 051009.Bauer, E., D. Mixon, F. Ronning, N. Hur, R. Movshovich,J. Thompson, J. Sarrao, M. Hundley, P. Tobash, andS. Bobev, 2006a, Physica B , 142.Bauer, E. D., C. Capan, F. Ronning, R. Movshovich, J. D. Thompson, and J. L. Sarrao, 2005c, Phys. Rev. Lett. (4),047001.Bauer, E. D., R. P. Dickey, V. S. Zapf, and M. B. Maple,2001, J. Phys.: Condens. Matter , L759.Bauer, E. D., N. A. Frederick, P. C. Ho, V. S. Zapf, and M. B.Maple, 2002, Phys. Rev. B , 100506(R).Bauer, E. D., P.-C. Ho, M. B. Maple, T. Schauerte, D. L. Cox,and F. B. Anders, 2006b, Phys. Rev. B (9), 094511.Bauer, E. D., J. D. Thompson, J. L. Sarrao, L. A. Morales,F. Wastin, J. Rebizant, J. C. Griveau, P. Javorsky,P. Boulet, E. Colineau, G. H. Lander, and G. R. Stewart,2004b, Phys. Rev. Lett. (14), 147005.Bel, R., K. Behnia, Y. Nakajima, K. Izawa, Y. Matsuda,H. Shishido, R. Settai, and Y. Onuki, 2004, Phys. Rev.Lett. (21), 217002.Belitz, D., and T. R. Kirkpatrick, 2004, Phys. Rev. B (18),184502.Belitz, D., T. R. Kirkpatrick, and T. Vojta, 2005, Rev. Mod.Phys. , 579.Bellarbi, B., A. Benoit, D. Jaccard, J. M. Mignot, and H. F.Braun, 1984, Phys. Rev. B (3), 1182.Benoit, A., J. X. Boucherle, P. Convert, J. Flouquet, J. Pal-leau, and J. Schweitzer, 1980, Solid State Commun. ,293.Berk, N. F., and J. R. Schrieffer, 1966, Phys. Rev. Lett. ,433.Bernhoeft, N., B. Roessli, N. Sato, N. Aso, A. Hiess, G. H.Lander, Y. Endoh, and T. Komatsubara, 1999, Physica B , 614.Bernhoeft, N., N. Sato, B. Roessli, N. Aso, A. Hiess, G. H.Lander, Y. Endoh, and T. Komatsubara, 1998, Phys. Rev.Lett. , 4244.Bernhoeft, N. R., G. H. Lander, M. J. Longfield, S. Langridge,D. Mannix, E. Lidstrom, E. Colineau, A. Hiess, C. Vettier,F. Wastin, J. Rebizant, and P. Lejay, 2003, Acta Phys. Pol.B , 1367.Bernhoeft, N. R., and G. G. Lonzarich, 1995, J. Phys.: Con-dens. Matter , 7325.Bianchi, A., R. Movshovich, C. Capan, P. G. Pagliuso, andJ. L. Sarrao, 2003a, Phys. Rev. Lett. , 187004.Bianchi, A., R. Movshovich, M. Jaime, J. D. Thompson, P. G.Pagliuso, and J. L. Sarrao, 2001, Phys. Rev. B (22),220504.Bianchi, A., R. Movshovich, N. Oeschler, P. Gegenwart,F. Steglich, J. D. Thompson, P. G. Pagliuso, and J. L.Sarrao, 2002, Phys. Rev. Lett. (13), 137002.Bianchi, A., R. Movshovich, I. Vekhter, P. G. Pagliuso, andJ. L. Sarrao, 2003b, Phys. Rev. Lett. (25), 257001.Bianchi, A. D., M. Kenzelmann, L. DeBeer-Schmitt, J. S.White, E. M. Forgan, J. Mesot, M. Zolliker, J. Kohlbrecher,R. Movshovich, E. D. Bauer, J. L. Sarrao, Z. Fisk, et al. ,2008, Science , 177.Blackburn, E., A. Hiess, N. Bernhoeft, and G. H. Lander,2006a, Phys. Rev. B (2), 024406.Blackburn, E., A. Hiess, N. Bernhoeft, M. C. Rheinst¨adter,W. H. ler, and G. H. Lander, 2006b, Phys. Rev. Lett. (5),057002.Bogenberger, B., H. v. L¨ohneysen, T. Trappmann, andL. Taillefer, 1993, Physica B , 248.Bonalde, I., W. Bramer-Escamilla, and E. Bauer, 2005, Phys.Rev. Lett. , 207002.Bonalde, I., W. Br¨amer-Escamilla, C. Rojas, Y. Haga,E. Bauer, T. Yasuda, and Y. Onuki, 2007, Physica B - ,in print, proceedings of M2S Dresden, 2006. Bonn, D. A., J. D. Garrett, and T. Timusk, 1988, Phys. Rev.Lett. , 1305.Booth, C. H., E. D. Bauer, M. Daniel, R. E. Wilson, J. N.Mitchell, L. A. Morales, J. L. Sarrao, and P. G. Allen, 2007,Phys. Rev. B , 064530.Borth, R., E. Lengyl, P. Pagliuso, J. Sarrao, G. Sparn,F. Steglich, and J. Thompson, 2002, Physica B , 136.Boulet, P., E. Colineau, F. Wastin, J. Rebizant, P. Javorsk´y,G. H. Lander, and J. D. Thompson, 2005, Phys. Rev. B (10), 104508.Boulet, P., A. Daoudi, M. Potel, H. No¨el, G. M. Gross,G. Andr´e, and F. Bour´ee, 1997, J. Alloys and Comp. ,104.Bourdarot, F., A. Bombardi, P. Burlet, M. Enderle, J. Flou-quet, P. Lejay, N. Kernavanois, V. P. Mineev, L. Paolasini,M. E. Zhitomirsky, and B. F˚ak, 2005, Physica B ,986.Bourdarot, F., B. F˚ak, K. Habicht, and K. Prokes, 2003, Phys.Rev. Lett. , 067203.Bowers, J. A., and K. Rajagopal, 2002, Phys. Rev. D (6),065002.Brison, J.-P., N. Keller, P. Lejay, A. Huxley, L. Schmidt,A. Buzdin, N. R. Bernhoeft, I. Mineev, A. N. Stepanov,J. Flouquet, D. Jaccard, S. R. Julian, et al. , 1994, PhysicaB , 70.Brison, J.-P., N. Keller, A. Verniere, P. Lejay, L. Schmidt,A. Buzdin, J. Flouquet, S. R. Julian, and G. G. Lonzarich,1995, Physica C , 128.Broholm, C., J. K. Kjems, W. J. L. Buyers, P. Matthews,T. T. M. Palstra, A. A. Menovsky, and J. A. Mydosh, 1987,Phys. Rev. Lett. , 1467.Broholm, C., H. Lin, P. T. Matthews, T. E. Mason, W. J. L.Buyers, M. F. Collins, A. A. Menovsky, J. A. Mydosh, andJ. K. Kjems, 1991, Phys. Rev. B , 12809.Bruls, G., D. Weber, B. Wolf, P. Thalmeier, B. L¨uthi,A. Visser, and A. Menovsky, 1990, Phys. Rev. Lett. ,2294.Bruls, G., B. Wolf, D. Finsterbusch, P. Thalmeier,I. Kouroudis, W. Sun, W. Assmus, B. L¨uthi, M. Lang,K. Gloos, F. Steglich, and R. Modler, 1994, Phys. Rev.Lett. (11), 1754.Bucher, E., J. P. Maita, and A. S. Cooper, 1972, Phys. Rev.B (7), 2709.Bucher, E., J. P. Maita, G. W. Hull, R. C. Fulton, and A. S.Cooper, 1975, Phys. Rev. B (1), 440.Budko, S., and P. C. Canfield, 2006, C. R. Physique , 56.Bulaevskii, L. N., 1974, Sov. Phys. JETP , 634.Bulaevskii, L. N., A. I. Buzdin, M. L. Kulic, and S. V. Pan-jukov, 1985, Adv. Phys. , 175.Burkhardt, H., and D. Rainer, 1994, Ann. Physik , 181.Buschow, K. H. J., 1979, Rep. Prog. Phys. , 1373.Buschow, K. H. J., and H. J. van Daal, 1970, Solid StateCommun. , 363.Buzdin, A. I., and J.-P. Brison, 1996a, Phys. Lett. A ,359.Buzdin, A. I., and J.-P. Brison, 1996b, Europhys. Lett. ,707.Buzdin, A. I., and H. Kachkachi, 1997, Phys. Lett. A ,341.Buzdin, A. I., and A. S. Mel’nikov, 2003, Phys. Rev. B ,020503.Buzdin, J. P. B. A., L. Glemont, F. Thomas, and J. Flouquet,1997, Physica B , 406.Campbell, A. M., and J. E. Evetts, 1972, Adv. Phys. , 199. Capan, C., A. Bianchi, F. Ronning, A. Lacerda, J. D. Thomp-son, M. F. Hundley, P. G. Pagliuso, J. L. Sarrao, andR. Movshovich, 2004, Phys. Rev. B (18), 180502.Casalbuoni, R., and G. Nardulli, 2004, Reviews of ModernPhysics (1), 263.Caspary, R., P. Hellmann, M. Keller, G. Sparn, C. Wassilew,R. K¨ohler, C. Geibel, C. Schank, F. Steglich, and N. E.Phillips, 1993, Phys. Rev. Lett. (13), 2146.Champel, T., and V. P. Mineev, 2001, Phys. Rev. Lett. (21), 4903.Chandra, P., P. Coleman, J. A. Mydosh, and V. Tripathi,2002, Nature , 831.Chandra, P., P. Coleman, J. A. Mydosh, and V. Tripathi,2003, J. Phys.: Condens. Matter , S1965.Chandrasekhar, B. S., 1962, Appl. Phys. Lett. , 7.Chang, J., I. Eremin, P. Thalmeier, and P. Fulde, 2007, Phys.Rev. B (2), 024503.Chen, G. F., K. Matsubayashi, S. Ban, K. Deguchi, and N. K.Sato, 2006, Phys. Rev. Lett. (1), 017005.Chevalier, B., and J. Etourneau, 1999, J. Mag. Mag. Mater. , 880.Chia, E. E. M., M. B. Salamon, H. Sugawara, and H. Sato,2003, Phys. Rev. Lett. , 247003.Chiao, M., B. Lussier, B. Ellman, and L. Taillefer, 1997, Phys-ica B , 370.Christianson, A. D., E. D. Bauer, J. M. Lawrence, P. S. Rise-borough, N. O. Moreno, P. G. Pagliuso, J. L. Sarrao, J. D.Thompson, E. A. Goremychkin, F. R. Trouw, M. P. Hehlen,and R. J. McQueeney, 2004, Phys. Rev. B (13), 134505.Christianson, A. D., A. Llobet, W. Bao, J. S. Gardner, I. P.Swainson, J. W. Lynn, J.-M. Mignot, K. Prokes, P. G.Pagliuso, N. O. Moreno, J. L. Sarrao, J. D. Thompson, et al. , 2005, Phys. Rev. Lett. (21), 217002.Chudo, H., H. Sakai, Y. Tokunaga, S. Kambe, D. Aoki,Y. Homma, Y. Shiokawa, Y. Haga, S. Ikeda, T. D. Mat-suda, Y. Onuki, and H. Yasuoka, 2008, J. Phys. Soc. Jpn. , 77.Cichorek, T., A. C. Mota, F. Steglich, N. A. Frederick, W. M.Yuhasz, and M. B. Maple, 2005, Phys. Rev. Lett. ,107002.Clogston, A. M., 1962, Phys. Rev. Lett. (6), 266.Coad, S., A. Hiess, D. F. McMorrow, G. H. Lander, G. Aeppli,Z. Fisk, G. R. Stewart, S. M. Hayden, and H. A. Mook,2000, Physica B , 764.Colineau, E., F. Wastin, P. Javorsky, and J. Rebizant, 2006,Physica B , 1015.Combescot, R., and G. Tonini, 2005, Phys. Rev. B (9),094513.Cordes, H. G., K. Fischer, and F. Pobell, 1981, , 531.Cornelius, A. L., A. J. Arko, J. L. Sarrao, M. F. Hundley, andZ. Fisk, 2000, Phys. Rev. B (21), 14181.Cornelius, A. L., P. G. Pagliuso, M. F. Hundley, and J. L.Sarrao, 2001, Phys. Rev. B (14), 144411.Correa, V. F., T. P. Murphy, C. Martin, K. M. Purcell, E. C.Palm, G. M. Schmiedeshoff, J. C. Cooley, and S. W. Tozer,2007, Phys. Rev. Lett. (8), 087001.Curro, N. J., T. Caldwell, E. D. Bauer, L. A. Morales, M. J.Graf, Y. Bang, A. V. Balatsky, J. D. Thompson, andJ. J. L. Sarrao, 2005, Nature , 622.van Daal, H. J., and K. H. L. Buschow, 1969, Solid StateCommun. , 217.Dalichaouch, Y., M. C. de Andrade, D. A. Gajewski, R. Chau,P. Visani, and M. B. Maple, 1995, Phys. Rev. Lett. (21),3938. Dalichaouch, Y., M. C. de Andrade, and M. B. Maple, 1992,Phys. Rev. B (13), 8671.Daniel, M., E. D. Bauer, S.-W. Han, C. H. Booth, A. L.Cornelius, P. G. Pagliuso, and J. L. Sarrao, 2005, Phys.Rev. Lett. (1), 016406.De Wilde, Y., J. Heil, A. G. M. Jansen, P. Wyder, R. Deltour,W. Assmus, A. Menovsky, W. Sun, and L. Taillefer, 1994,Phys. Rev. Lett. (14), 2278.DeBeer-Schmitt, L., C. D. Dewhurst, B. W. Hoogenboom,C. Petrovic, and M. R. Eskildsen, 2006, Phys. Rev. Lett. (12), 127001.Debessai, M., T. Matsuoka, J. Hamlin, J. Schilling, andK. Shimizu, 2009, condmat/0903.1808.van Dijk, N. H., B. F˚ak, T. Charvolin, P. Lejay, and J. M.Mignot, 2000, Phys. Rev. B (13), 8922.van Dijk, N. H., A. de Visser, J. J. M. Franse, and A. A.Menovsky, 1995, Phys. Rev. B (18), 12665.van Dijk, N. H., A. de Visser, J. J. M. Franse, and L. Taillefer,1993, J. Low Temp. Phys. , 101.Divis, M., L. M. Sandratskii, M. Richter, P. Mohn, and P. No-vak, 2002, J. Alloys and Comp. , 48.Dommann, E., and F. Hullinger, 1988, Solit State Communic. , 1093.Donath, J. G., F. Steglich, E. D. Bauer, J. L. Sarrao, andP. Gegenwart, 2008, Physical Review Letters (13),136401.Dresselhaus, G., 1955, Phys. Rev. , 580.Dungate, D. G., 1990, unknown , Ph.D. thesis.Dupuis, N., 1995, Phys. Rev. B (14), 9074.Durivault, L., F. Bour!ee, B. Chevalier, G. Andr!e, andJ. Etourneau, 2002, J. Mag. Mag. Mater. , 366.Edelstein, V. M., 1989, Sov. Phys. JETP , 1244.Efremov, D. V., N. Hasselmann, E. Runge, P. Fulde, andG. Zwicknagl, 2004, Phys. Rev. B (11), 115114.Einzel, D., P. J. Hirschfeld, F. Gross, B. S. Chandrasekhar,K. Andres, H. R. Ott, J. Beuers, Z. Fisk, and J. L. Smith,1986, Phys. Rev. Lett. (23), 2513.Ejima, T., S. Suzuki, S. Sato, N. Sato, S. i. Fujimori, M. Ya-mada, K. Sato, T. Komatsubara, T. Kasuya, Y. Tezuka,S. Shin, and T. Ishi, 1994, J. Phys. Soc. Jpn. , 2428.Elgazzar, S., I. Opahle, R. Hayn, and P. M. Oppeneer, 2004,Phys. Rev. B (21), 214510.Eliashberg, G. M., 1960, Sov. Phys. JETP , 696.Ellman, B., M. Sutton, B. Lussier, R. Br¨unig, L. Taillefer,and S. M. Hayden, 1997.Ellman, B., A. Zaluska, and L. Taillefer, 1995, Physica B ,346.Eom, D., M. Ishikawa, J. Kitagawa, and N. Takeda, 1998, J.Phys. Soc. Jpn. , 2495.Eskildsen, M. R., C. D. Dewhurst, B. W. Hoogenboom,C. Petrovic, and P. C. Canfield, 2003, Phys. Rev. Lett. (18), 187001.F˚ak, B., B. Farago, and N. H. van Dijk, 1999, Physica B ,644.F˚ak, B., J. Flouquet, G. Lapertot, T. Fukuhara, and H. Kad-owaki, 2000, J. Phys. Condens. Matter , 5423.Fawcett, E., 1988, Rev. Mod. Phys. , 209.Fay, D., and J. Appel, 1980a, Phys. Rev. B , 3173.Fay, D., and J. Appel, 1980b, Phys. Rev. B , 2325.Felsch, W., and K. Winzer, 1973, Solid State Commun. ,569.Ferreira, L. M., T. Park, V. Sidorov, M. Nicklas, E. M. Bit-tar, R. Lora-Serrano, E. N. Hering, S. M. Ramos, M. B.Fontes, E. Baggio-Saitovich, H. Lee, J. L. Sarrao, et al. , 2008, Physical Review Letters (1), 017005.Fertig, W. A., D. C. Johnston, L. E. DeLong, R. W. McCal-lum, M. B. Maple, and B. T. Matthias, 1977, Phys. Rev.Lett. (17), 987.Feyerherm, R., A. Amato, F. N. Gygax, A. Schenck,C. Geibel, F. Steglich, N. Sato, and T. Komatsubara, 1994,Phys. Rev. Lett. (13), 1849.Fischer, Ø., 1990, Ferromagnetic Materials (Elsevier, NewYork), volume 5 of
Topics in Current Physics .Fischer, Ø., and M. B. Maple (eds.), 1982,
Superconductivityin Ternary Compounds I (Springer-Verlag, Berlin, Heidel-berg, New York), volume 32 of
Topics in Current Physics .Fisher, R. A., S. Kim, B. F. Woodfield, N. E. Phillips,L. Taillefer, K. Hasselbach, J. Flouquet, A. L. Giorgi, andJ. L. Smith, 1989, Phys. Rev. Lett. (12), 1411.Fisher, R. A., S. Kim, Y. Wu, N. E. Phillips, M. W. McEl-fresh, M. S. Torikachvili, and M. B. Maple, 1990, PhysicaB , 419.Fisher, R. A., B. F. Woodfield, S. Kim, N. E. Phillips,L. Taillefer, A. L. Giorgi, and J. L. Smith, 1991, Solid StateCommunic. , 263.Flint, R., M. Dzero, and P. Coleman, 2008, Nature Physics ? ,nphys1024.Flouquet, J., 2006, J. Prog. Low Temp. Phys. , 139, cond-mat/0501602.Flouquet, J., G. Knebel, D. Braithwaite, D. Aoki, J. Brison,F. Hardy, A. Huxley, S.Raymond, B. Salce, and I. Sheikin,2006, Comptes Rend. Phys. , 22, condmat/0505713.Fort, D., 1987, J. Less Comm. Metals , 45.Frazer, B. H., Y. Hirai, M. L. Schneider, S. Rast, M. Onellion,I. Nowik, I. Felner, S. Roy, N. Ali, A. Reginelli, L. Perfetti,D. Ariosa, et al. , 2001, Eur. Phys. J. B , 177.Frederick, N. A., T. D. Do, P.-C. Ho, N. P. Butch, V. S. Zapf,and M. B. Maple, 2004, Phys. Rev. B (2), 024523.Frederick, N. A., T. A. Sayles, and M. B. Maple, 2005, Phys.Rev. B (6), 064508.Frigeri, P. A., D. F. Agterberg, A. Koga, and M. Sigrist,2004a, Phys. Rev. Lett. , 097001.Frigeri, P. A., D. F. Agterberg, and M. Sigrist, 2004b, NewJnl. Phys. , 115.Frings, P. H., J. J. M. F. F. R. de Boer, and A. Menovsky,1983, J. Magn. Magn. Mater. , 240.Fujimoto, S., 2006, J. Phys. Soc. Jpn. , 083704.Fujimoto, S., 2007, J. Phys. Soc. Jpn. , 034712.Fukazawa, H., and K. Yamada, 2003, J. Phys. Soc. Jpn. ,2449.Fulde, P., and R. Ferrell, 1964, Phys. Rev. , A550.Gaulin, B. D., M. Mao, C. R. Wiebe, Y. Qiu, S. M. Shapiro,C. Broholm, S.-H. Lee, and J. D. Garrett, 2002, Phys. Rev.B (17), 174520.Gegenwart, P., M. Deppe, M. Koppen, F. Kromer, M. Lang,R. Modler, M. Weiden, C. Geibel, F. Steglich, T. Fukase,and N. Toyota, 1996, Ann. Phys. , 307.Gegenwart, P., F. Kromer, M. Lang, G. Sparn, C. Geibel, andF. Steglich, 1999, Phys. Rev. Lett. (6), 1293.Gegenwart, P., C. Langhammer, C. Geibel, R. Helfrich,M. Lang, G. Sparn, F. Steglich, R. Horn, L. Donnevert,A. Link, and W. Assmus, 1998, Phys. Rev. Lett. (7),1501.Gegenwart, P., C. Langhammer, R. H. N. O. M. L. J. S. Kim,G. R. Stewart, and F. Steglich, 2004, Physica C ,157.Geibel, C., C. Schank, S. Thies, H. Kitazawa, C. D. Bredl,A. Bohm, M. Rau, A. Grauel, R. Caspary, R. Helfrich, U. Ahlheim, G. Weber, et al. , 1991a, Z. Phys. B - Cond.Matt. , 1.Geibel, C., S. Thies, D. Kazorowski, A. Mehner, A. Grauel,B. Seidel, U. Ahlheim, R. Helfrich, K. Petersen, C. D.Bredl, and F. Steglich, 1991b, Z. Phys. B - Cond. Matt. , 305.de Gennes, P. G., 1989, Superconductivity of Metals and Al-loys (Addison-Wesley Publishing Company, Reading MA,USA), reprint of 1966 edition.Ginzburg, V. L., 1957, Sov. Phys. JETP , 153.Giorgi, A., E. G. Szkalrz, M. Krupta, and N. H. Krikoria,1970, J. Less-Comm. Metals , 121.Gl´emot, L., J. P. Brison, J. Flouquet, A. I. Buzdin, I. Sheikin,D. Jaccard, C. Thessieu, and F. Thomas, 1999, Phys. Rev.Lett. , 169.Gloos, K., R. Modler, H. Schimanski, C. D. Bredl, C. Geibel,F. Steglich, A. I. Buzdin, N. Sato, and T. Komatsubara,1993, Phys. Rev. Lett. (4), 501.Goldman, A. I., S. M. Shapiro, D. E. Cox, J. L. Smith, andZ. Fisk, 1985, Phys. Rev. B (9), 6042.Goldman, A. I., S. M. Shapiro, G. Shirane, J. L. Smith, andZ. Fisk, 1986, Phys. Rev. B (3), 1627.Goremychkin, E. A., R. Osborn, E. D. Bauer, M. B. Maple,N. A. Frederick, W. M. Yuhasz, F. M. Woodward, andJ. W. Lynn, 2004, Phys. Rev. Lett. (15), 157003.Gor’kov, L. P., and E. I. Rashba, 2001, Phys. Rev. Lett. (3),037004.Goto, T., Y. Nemoto, K. Sakai, T. Yamaguchi, M. Akatsu,T. Yanagisawa, H. Hazama, K. Onuki, H. Sugawara, andH. Sato, 2004, Phys. Rev. B , 180511.Graf, M. J., S.-K. Yip, J. A. Sauls, and D. Rainer, 1996, Phys.Rev. B (22), 15147.Graf, T., J. D. Thompson, M. F. Hundley, R. Movshovich,Z. Fisk, D. Mandrus, R. A. Fisher, and N. E. Phillips,1997, Phys. Rev. Lett. (19), 3769.Gratens, X., L. M. Ferreira, Y. Kopelevich, N. F. O. jr., P. G.Pagliuso, R. Movshovich, R. R. Urbano, J. L. Sarrao, andJ. D. Thompson, 2006, condmat/0608722.Grauel, A., A. B¨ohm, H. Fischer, C. Geibel, R. K¨ohler,R. Modler, C. Schank, F. Steglich, G. Weber, T. Komat-subara, and N. Sato, 1992, Phys. Rev. B (9), 5818.Grewe, N., and F. Steglich, 1991, Heavy fermions (North-Holland, Amsterdam, Holland), volume 14 of
Handbook ofthe Physics and Chemistry of Rare Earths , chapter 97.Grier, B. H., J. M. Lawrence, V. Murgai, and R. D. Parks,1984, Phys. Rev. B (5), 2664.Griveau, J.-C., K. Gofryk, and J. Rebizant, 2008, Phys. Rev.B (21), 212502.Grosche, F. M., P. Argarwa, S. R. Julian, N. J. Wilson,R. K. W. Haselwimmer, S. J. S. Lister, N. D. Mathur, F. V.Carter, S. S. Saxena, and G. G. Lonzarich, 2000, J. Phys.:Condens. Matter , L533.Grosche, F. M., S. R. Julian, N. D. Mathur, F. V. Carter, andG. G. Lonzarich, 1997a, Physica B , 197.Grosche, F. M., S. R. Julian, N. D. Mathur, and G. G. Lon-zarich, 1996, Physica B , 50.Grosche, F. M., S. J. S. Lister, F. Carter, S. S. Saxena,R. K. W. Haselwimmer, N. D. Mathur, S. R. Julian, andG. G. Lonzarich, 1997b, Physica B , 62.Gross, F., B. Chandrasekhar, K. Andres, U. Rauchschwalbe,E. Bucher, and B. L¨uthi, 1988, Physica C , 439.Gross, F., B. S. Chandrasekhar, D. Einzel, K. Andres, P. K.Hirschfeld, H. R. Ott, J. Beurs, Z. Fisk, and J. L. Smith,1986, Z. Phys. B , 175. Groß, W., K. Knorr, A. P. Murani, and K. H. J. Buschow,1980, Z. Physik B , 123.Grube, K., S. Drobnik, C. Pfleiderer, H. v. L¨ohneysen, E. D.Bauer, and M. B. Maple, 2006, Phys. Rev. B (10),104503.Gruenberg, L. W., and L. Gunther, 1966, Phys. Rev. Lett. , 996.Haen, P., P. Lejay, B. Chevalier, B. Lloret, J. Etourneau, andP. Hagenmuller, 1985, Mater. Res. Bull. , 115.Haga, Y., T. Honma, E. Yamamoto, H. Ohkuni, Y. Onuki,M. Ito, and N. Kimura, 1998, Jpn. J. Appl. Phys. , 3604.Haga, Y., Y. Inada, H. Harima, K. Oikawa, M. Murakawa,H. Nakawaki, Y. Tokiwa, D. Aoki, H. Shishido, S. Ikeda,N. Watanabe, and Y. Onuki, 2001, Phys. Rev. B (6),060503.Haga, Y., M. Nakashima, R. Settai, S. Ikeda, T. Okubo,S. Araki, T. Kobayashi, N. Tateiwa, and Y. Onuki, 2002,J. Phys.: Condens. Matter , L125.Haga, Y., H. Sakai, and S. Kambe, 2007, Jnl. Phys. Soc. Jpn. , 051012.Haga, Y., E. Yamamoto, Y. Inada, D. Aoki, K. Tenya,M. Ikeda, T. Sakakibara, and Y. Onuki, 1996, J. Phys.Soc. Jpn. , 3646.Hagmusa, I. H., K. Prokes, Y. Echizen, T. Takabatake, T. Fu-jita, J. C. P. Klaasse, E. Br¨uck, V. Sechovsky, and F. R.de Boer, 2000, Physica B , 223.Hall, D., E. C. Palm, T. P. Murphy, S. W. Tozer, C. Petrovic,E. Miller-Ricci, L. Peabody, C. Q. H. Li, U. Alver, R. G.Goodrich, J. L. Sarrao, P. G. Pagliuso, et al. , 2001, Phys.Rev. B (6), 064506.Hanawa, M., Y. Muraoka, T. Tayama, T. Sakakibara, J. Ya-maura, and Z. Hiroi, 2001, Phys. Rev. Lett. (18), 187001.Hanawa, M., J. Yamaura, Y. Muraoka, F. Sakai, and Z. Hiroi,2002, J. Phys. Chem. Solids , 1027.Hao, L., K. Iwasa, K. Kuwahara, M. Kohgi, H. Sugawara,Y. Aoki, H. Sato, C. Sekine, C. H. Lee, and H. Harima,2004, J. Magn. Magn. Mater. , e271.Hardy, F., and A. D. Huxley, 2005, Phys. Rev. Lett. ,247006.Hardy, F., A. D. Huxley, J. Flouquet, B. Salce, G. Knebel,D. Braithwaite, D. Aoki, M. Uhlarz, and C. Pfleiderer,2005, Physica B , 1111.Hardy, F., M. Uhlarz, A. D. Huxley, and C. Pfleiderer, 2004,unpublished.Harrison, N., M. Jaime, and J. A. Mydosh, 2003, Phys. Rev.Lett. , 096402.Harrison, N., K. H. Kim, M. Jaime, and J. A. Mydosh, 2004,Physica B , 92.Hashimoto, S., T. Yasuda, T. Kubo, H. Shishido, T. Ueda,R. Settai, T. D. Matsuda, Y. Haga, H. Harima, andY. Onuki, 2004, J. Phys.: Condens. Matter , L287.Hasselbach, K., J. R. Kirtley, and J. Flouquet, 1993, Phys.Rev. B (1), 509.Hasselbach, K., J. R. Kirtley, and P. Lejay, 1992, Phys. Rev.B (9), 5826.Hasselbach, K., P. Lejay, and J. Flouquet, 1991, Phys. Lett.A , 313.Hasselbach, K., L. Taillefer, and J. Flouquet, 1989, Phys. Rev.Lett. (1), 93.Hasselbach, K., L. Taillefer, and J. Flouquet, 1990, PhysicaB , 357.Hassinger, E., D. Aoki, and J. Flouquet, 2008, J. Phys. Soc.Jpn. , 073703.Hayashi, N., K. Wakabayashi, P. A. Frigeri, and M. Sigrist, (9), 092508.Hayashi, N., K. Wakabayashi, P. A. Frigeri, and M. Sigrist,2006b, Phys. Rev. B (2), 024504.Hayden, S. M., L. Taillefer, C. Vettier, and J. Flouquet, 1992,Phys. Rev. B (13), 8675.Heffner, R. H., D. W. Cooke, Z. Fisk, R. L. Hutson, M. E.Schillaci, J. L. Smith, J. O. Willis, D. E. MacLaughlin,C. Boekema, R. L. Lichti, A. B. Denison, and J. Oostens,1986, Phys. Rev. Lett. (10), 1255.Hegger, H., C. Petrovic, E. G. Moshopoulou, M. F. Hundley,J. L. Sarrao, Z. Fisk, and J. D. Thompson, 2000, Phys.Rev. Lett. (21), 4986.Hein, R. A., R. L. Falge, B. T. Matthias, and C. Corenzwit,1959, Phys. Rev. Lett. (12), 500.Herring, C., 1958, Physica , S 184.Hertz, J. A., 1976, Phys. Rev. B , 1165.Hewson, A. C., 1993, The Kondo problem to heavy fermions (Cambridge University Press, Cambridge).Hiess, A., P. J. Brown, E. Leli`evre-Berna, B. Roessli, N. Bern-hoeft, G. H. Lander, N. Aso, and N. K. Sato, 2001, Phys.Rev. B (13), 134413.Hiess, A., R. H. Heffner, J. E. Sonier, G. H. Lander, J. L.Smith, and J. C. Cooley, 2002, Phys. Rev. B (6), 064531.Hiess, A., A. Stunault, E. Colineau, J. Rebizant, F. Wastin,R. Caciuffo, and G. H. Lander, 2008, Physical Review Let-ters (7), 076403.Hill, H. H., 1970 (The Metallurgical Society of the AIME,New York), Plutonium and Other Actinides.Hiroi, M., M. Sera, N. Kobayashi, Y. Haga, E. Yamamoto,and Y. Onuki, 1997, J. Phys. Soc. Jpn. , 1595.Holmes, A. T., D. Jaccard, and K. Miyake, 2004, Phys. Rev.B , 024508.Honda, F., M.-A. Measson, Y. Nakano, N. Yoshitani, E. Ya-mamoto, Y. Haga, T. Takeuchi, H. Yamagami, K. Shimizu,R. Settai, and Y. Onuki, 2008a, J. Phys. Soc. Jpn. ,043701.Honda, F., R. Settai, D. Aoki, Y. Haga, T. Matsuda,N. Tateiwa, S. Ikeda, Y. Homma, H. Sakai, Y. Shiokawa,E. Yamamoto, A. Nakamura, et al. , 2008b, J. Phys. Soc.Jpn. (Suppl. A) , 77.Honma, T., Y. Haga, E. Yamamoto, N. Metoki, Y. Koike,H. Ohkuni, N. Suzuki, and Y. Onuki, 1999, J. Phys. Soc.Jpn. , 338.Hori, A., H. Hidaka, H. Kotegawa, T. Kobayashi, T. Akazawa,S. Ikeda, E. Yamamoto, Y. Haga, R. Settai, and Y. Onuki,2006, J. Phys. Soc. Jpn. , 82.Horn, S., E. Holland-Moritz, M. Loewenhaupt, F. Steglich,H. Scheuer, A. Benoit, and J. Flouquet, 1981, Phys. Rev.B (7), 3171.Hossain, Z., S. Hamashima, K. Umeo, T. Takabatake,C. Geibel, and F. Steglich, 2000, Phys. Rev. B (13),8950.Hotta, T., and K. Ueda, 2003, Phys. Rev. B (10), 104518.Houzet, M., and V. P. Mineev, 2006, Phys. Rev. B (14),144522.Houzet, M., and V. P. Mineev, 2007, Phys. Rev. B (22),224508.Hu, R., Y. Lee, J. Hudis, V. F. Mitrovic, and C. Petrovic,2008, Physical Review B (Condensed Matter and MaterialsPhysics) (16), 165129.Hu, T., H. Xiao, T. A. Sayles, M. B. Maple, K. Maki, B. D´ora,and C. C. Almasan, 2006, Phys. Rev. B (13), 134509.Hussey, N. E., 2002, Adv. Phys. , 1685.Huxley, A., P. Rodiere, D. M. Paul, N. van Dijk, R. Cubitt, and J. Flouquet, 2000, Nature , 160.Huxley, A. D., M. A. Measson, K. Izawa, C. D. Dewhurst,R. Cubitt, B. Grenier, H. Sugawara, J. Flouquet, Y. Mat-suda, and H. Sato, 2004, Phys. Rev. Lett. , 187005.Huxley, A. D., V. Mineev, B. Grenier, E. Ressouche, D. Aoki,J. P. Brison, and J. Flouquet, 2005, Physica C , 9.Huxley, A. D., S. Raymond, and E. Ressouche, 2003a, Phys.Rev. Lett. (20), 207201.Huxley, A. D., E. Ressouche, B. Grenier, D. Aoki, J. Flouquet,and C. Pfleiderer, 2003b, J.Phys.: Condens. Matter ,S1945.Huxley, A. D., I. Sheikin, E. Ressouche, N. Kernavanois,D. Braithwaite, R. Calemczuk, and J. J. Flouquet, 2001,Phys. Rev. B , 144519.Huy, N., A. Gasparini, J. Klaase, A. de Visser, S. Sakarya,and N. van Dijk, 2007, cond-mat/0704.2116v1.Huy, N. T., D. E. de Nijs, Y. K. Huang, and A. de Visser,2008, Physical Review Letters (7), 077002.Ikeda, H., and Y. Ohashi, 1998, Phys. Rev. Lett. , 3723.Ikeda, R., 2006, condmat/0610796.Ikeda, S., H. Shishido, M. Nakashima, R. Settai, D. Aoki,Y. Haga, H. Harima, Y. Aoki, T. Namiki, H. Sato, andY. Onuki, 2001, J. Phys. Soc. Jpn. , 2248.Inada, Y., H. Yamagami, Y. Haga, K. Sakurai, Y. Tokiwa,T. Honma, E. Yamamoto, Y. Onuki, and T. Yanagisawa,1999, J. Phys. Soc. Jpn. , 3643.Ishida, K., Y. Kawasaki, K. Tabuchi, K. Kashima, Y. Kitaoka,K. Asayama, C. Geibel, and F. Steglich, 1999, Phys. Rev.Lett. (26), 5353.Ishida, K., D. Ozaki, T. Kamatsuka, H. Tou, M. Kyogaku,Y. Kitaoka, N. Tateiwa, N. K. Sato, N. Aso, C. Geibel,and F. Steglich, 2002, Phys. Rev. Lett. (3), 037002.Ishiguro, A., A. Sawada, Y. Inada, J. Kimura, M. Suzuki,N. Sato, and T. Komatsubara, 1995, J. Phys. Soc. Jpn. ,378.Ishikawa, I., and O. Fischer, 1977, Solid State Commun. ,37.Ito, T., H. Kumigashira, S. Souma, T. Takahashi, Y. Haga,and Y. Onuki, 2002, J. Phys. Soc. Jpn. , Suppl. 262.Iwasa, K., Y. Watanabe, K. Kuwahara, M. Kohgi, H. Sug-awara, T. D. Matsuda, Y. Aoki, and H. Sato, 2002, PhysicaB , 834.Izawa, K., K. Behnia, Y. Matsuda, H. Shishido, R. Settai,Y. Onuki, and J. Flouquet, 2007, Physical Review Letters (14), 147005.Izawa, K., Y. Kasahara, Y. Matsuda, K. Behnia, T. Yasuda,R. Settai, and Y. Onuki, 2005, Phys. Rev. Lett. , 197002.Izawa, K., Y. Nakajima, J. Goryo, Y. Matsuda, S. Osaki,H. Sugawara, H. Sato, P. Thalmeier, and K. Maki, 2003,Phys. Rev. Lett. , 117001.Izawa, K., H. Yamaguchi, Y. Matsuda, H. Shishido, R. Settai,and Y. Onuki, 2001, Phys. Rev. Lett. (5), 057002.Jaccard, D., K. Behnia, and J. Sierro, 1992, Phys. Lett. A , 475.Jaccard, D., A. T. Holmes, G. Behr, Y. Inada, and Y. Onuki,2002, Phys. Rev. B , 282.Jaccarino, V., and M. Peter, 1962, Phys. Rev. Lett. (7), 290.Javorsk´y, P., E. Colineau, F. Wastin, F. Jutier, J.-C. Griveau,P. Boulet, R. Jardin, and J. Rebizant, 2007, Phys. Rev. B (18), 184501.Jeffries, J. R., N. P. Butch, B. T. Yukich, and M. B. Maple,2007, Physical Review Letters (21), 217207.Jo, Y. J., L. Balicas, C. Capan, K. Behnia, P. Lejay, J. Flou-quet, J. A. Mydosh, and P. Schlottmann, 2007, Phys. Rev. Lett. (16), 166404.Jourdan, M., M. Huth, and H. Adrian, 1999, Nature , 47.Jourdan, M., A. Zakharov, M. Foerster, and H. Adrian, 2004,Phys. Rev. Lett. (9), 097001.Joyce, J. J., J. M. Wills, T. Durakiewicz, M. T. Butterfield,E. Guziewicz, J. L. Sarrao, L. A. Morales, A. J. Arko, andO. Eriksson, 2003, Phys. Rev. Lett. , 176401.Joynt, R., and L. Taillefer, 2002, Rev. Mod. Phys. (1), 235.Jutier, F., J. Griveau, E. Colineau, F. Wastin, J. Rebizant,P. Boulet, and E. Simon, 2006, J. Phys. Soc. Jpn. ,Suppl. 47.Jutier, F., J. C. Griveau, E. Colineau, J. Rebizant, P. Boulet,and F. Wastin, 2005, Physica B , 1078.Jutier, F., G. A. Ummarino, J.-C. Griveau, F. Wastin,E. Colineau, J. Rebizant, N. Magnani, and R. Caciuffo,2008, Physical Review B (Condensed Matter and Materi-als Physics) (2), 024521.Kadowaki, K., and S. B. Woods, 1986, Solid State Communic. , 507.Kakuyanagi, K., M. Saitoh, K. Kumagai, S. Takashima,M. Nohara, H. Takagi, and Y. Matsuda, 2005, Phys. Rev.Lett. (4), 047602.Kaneko, K., N. Metoki, T. D. Matsuda, and M. Kohgi, 2006,J. Phys. Soc. Jpn. , 034701.Kaneko, K., N. Metoki, R. Shiina, T. D. Matsuda, M. Kohgi,K. Kuwahara, and N. Bernhoeft, 2007, Phys. Rev. B (9),094408.Karchev, N., 2003, Phys. Rev. B , 054416.Kasahara, S., K. Hirata, H. Takeya, T. Tamegai, H. Sug-awara, D. Kikuchi, and H. Sato, 2008, J. Phys. Soc. Jpn.,Suppl. A , 327.Kasahara, Y., T. Iwasawa, H. Shishido, T. Shibauchi,K. Behnia, Y. Haga, T. D. Matsuda, Y. Onuki, M. Sigrist,and Y. Matsuda, 2007, Physical Review Letters (11),116402.Kasahara, Y., Y. Nakajima, K. Izawa, Y. Matsuda,K. Behnia, H. Shishido, R. Settai, and Y. Onuki, 2005,Phys. Rev. B (21), 214515.Kaur, R. P., D. F. Agterberg, and M. Sigrist, 2005, Phys.Rev. Lett. , 137002.Kawai, T., H. Muranaka, T. Endo, N. D. Dong, Y. Doi,S. Ikeda, T. D. Matsuda, Y. Haga, H. Harima, R. Settai,and Y. Onuki, 2008a, J. Phys. Soc. Jpn. , 064717.Kawai, T., H. Muranaka, M.-A. Measson, T. Shimoda, Y. Doi,T. D. Matsuda, Y. Haga, G. Knebel, G. Lapertot, D. Aoki,J. Flouquet, T. Takeuchi, et al. , 2008b, J. Phys. Soc. Jpn. , 064716.Kawarazaki, S., M. Sato, Y. Miyako, N. Chigusa, K. Watan-abe, N. Metoki, Y. Koike, and M. Nishi, 2000, Phys. Rev.B (6), 4167.Kawasaki, S., T. Mito, Y. Kawasaki, G.-q. Zheng, Y. Kitaoka,H. Shishido, S. Araki, R. Settai, and Y. Onuki, 2002, Phys.Rev. B (5), 054521.Kawasaki, S., T. Mito, G.-q. Zheng, C. Thessieu,Y. Kawasaki, K. Ishida, Y. Kitaoka, T. Muramatsu, T. C.Kobayashi, D. Aoki, S. Araki, Y. Haga, et al. , 2001, Phys.Rev. B (2), 020504.Kawasaki, S., M. Yashima, Y. Mugino, H. Mukuda, Y. Ki-taoka, H. Shishido, and Y. ¯Onuki, 2006, Phys. Rev. Lett. (14), 147001.Kawasaki, S., G. qing Zheng, H. Kan, Y. Kitaoka, H. Shishido,and Y. ¯Onuki, 2005, Phys. Rev. Lett. (3), 037007.Kawasaki, Y., K. Ishida, S. Kawasaki, T. Mito, G. q. Zheng,Y. Kitaoka, C. Geibel, and F. Steglich, 2004, J. Phys. Soc. Jpn. , 194.Kawasaki, Y., S. Kawasaki, M. Yashima, T. Mito,G. q. Zheng, Y. Kitaoka, H. Shishido, R. Settai, Y. Haga,and Y. Onuki, 2003, J. Phys. Soc. Jpn. , 2308.Kernavanois, N., J.-X. Boucherle, P. D. de R´eotier, F. Givord,E. Lelievre-Barna, E. Ressouche, A. Rogalev, J. Sanchez,N. Sato, and A. Yaouanc, 2000, J. Phys.: Condens. Matter , 7857.Kernavanois, N., S. Raymond, E. Ressouche, B. Grenier,J. Flouquet, and P. Lejay, 2005, Phys. Rev. B (6),064404.Kernavonis, N., B. Grenier, A. Huxley, E. Ressouche, J.-P.Sanchez, and J. Flouquet, 2001, Phys. Rev. B , 174509.Ketterson, J. B., and S. N. Song, 1999, Superconductivity (Cambridge University Press, Cambridge).Kim, J. S., J. Alwood, G. R. Stewart, J. L. Sarrao, and J. D.Thompson, 2001a, Phys. Rev. B (13), 134524.Kim, J. S., D. Hall, P. Kumar, and G. R. Stewart, 2003a,Phys. Rev. B , 014404.Kim, J. S., N. K. Sato, and G. R. Stewart, 2001b, J. LowTemp. Phys. , 527.Kim, K. H., N. Harrison, H. Amitsuka, G. A. Jorge, M. Jaime,and J. A. Mydosh, 2004, Phys. Rev. Lett. , 206402.Kim, K. H., N. Harrison, M. Jaime, G. S. Boebinger, andJ. A. Mydosh, 2003b, Phys. Rev. Lett , 256401.Kimura, N., K. Ito, H. Aoki, S. Uji, and T. Terashima, 2007a,Phys. Rev. Lett. , 197001.Kimura, N., K. Ito, K. Saitoh, Y. Umeda, H. Aoki, andT. Terashima, 2005, Phys. Rev. Lett. (24), 247004.Kimura, N., Y. Muro, and H. Aoki, 2007b, J. Phys. Soc. Jpn. , 051010.Kimura, N., R. Settai, Y. Onuki, H. Toshima, E. Yamamoto,K. Mezawa, H. Aoki, and H. Harima, 1995, J. Phys. Soc.Jpn. , 3881.Kimura, N., Y. Umeda, T. Asai, T. Terashima, and H. Aoki,2001, Physica B , 280.Kirkpatrick, T. R., and D. Belitz, 2003, Phys. Rev. B ,024515.Kirkpatrick, T. R., and D. Belitz, 2004, Phys. Rev. Lett. (3), 037001.Kirkpatrick, T. R., D. Belitz, T. Vojta, and R. Narayanan,2001, Phys. Rev. Lett. , 127003.Kiss, A., and P. Fazekas, 2005, Phys. Rev. B , 054415.Kita, H., A. D¨onni, Y. Endoh, K. Kakurai, N. Sato, andT. Komatsubara, 1994, J. Phys. Soc. Jpn. , 726.Klamut, P. W., B. Dabrowski, S. Kolesnik, M. Maxwell, andJ. Mais, 2001, Phys. Rev. B (22), 224512.Klein, U., D. Rainer, and H. Shimahara, 2000, J. Low Temp.Phys. , 91.Knafo, W., S. Raymond, B. F˚ak, G. Lapertot, P. C. Canfield,and J. Flouquet, 2003, J. Phys.: Condensed Matter ,3741.Knebel, G., D. Aoki, D. Braithwaite, B. Salce, and J. Flou-quet, 2006, Phys. Rev. B (2), 020501.Knebel, G., D. Aoki, J.-P. Brison, and J. Flouquet, 2008, J.Phys. Soc. Jpn. , 114704.Knebel, G., D.Braithwaite, P. C. Canfield, G. Lapertot, andJ. Flouquet, 2001, Phys. Rev. B (2), 024425.Knebel, G., M.-A. M´easson, B. Salce, D. Aoki, D. Braith-waite, J. P. Brison, and J. Flouquet, 2004, J. Phys. Con-dens. Matter , 8905.Kn¨opfle, K., A. Mavromaras, L. M. Sandratskii, andJ. K¨ubler, 1996, J. Phys.: Condens. Matter , 901.Knopp, G., A. Loidl, K. Knorr, L. Pawlak, M. Duczmal, R. Caspary, U. Gottwick, H. Spille, F. Steglich, and A. P.Murani, 1989, Z. Phys. B , 95.Kobayashi, T., A. Hori, S. Fukushima, H. Hidaka, H. Kote-gawa, T. Akezawa, K. Takeda, Y. Ohishi, and E. Ya-mamoto, 2007, J. Phys. Soc. Jpn. , 051007.Kohgi, M., K. Iwasa, M. Nakajima, N. Metoki, S. Araki,N. Bernhoeft, J.-M. Mignot, A. Gukasov, H. Sato, Y. Aoki,and H. Sugawara, 2003, J. Phys. Soc. Jpn. , 1002.Kohori, Y., K. Matsuda, and T. Kohara, 1994, Solid StateCommunic. , 121.Kohori, Y., K. Matsuda, and T. Kohara, 1996, J. Phys. Soc.Jpn. , 1083.Kohori, Y., Y. Yamato, Y. Iwamoto, T. Kohara, E. D. Bauer,M. B. Maple, and J. L. Sarrao, 2001, Phys. Rev. B (13),134526.Koitzsch, A., S. V. Borisenko, D. Inosov, J. Geck, V. B.Zabolotnyy, H. Shiozawa, M. Knupfer, J. Fink, B. B¨uchner,E. D. Bauer, J. L. Sarrao, and R. Follath, 2008, Physi-cal Review B (Condensed Matter and Materials Physics) (15), 155128.Kotegawa, H., A. Harada, S. Kawasaki, Y. Kawasaki, Y. Ki-taoka, Y. Haga, E. Yamamoto, Y. Onuki, K. M. Itoh, E. E.Haller, and H. Harima, 2005, J. Phys. Soc. Jpn. , 705.Kotegawa, H., K. Takeda, T. Miyoshi, S. Fukushima,H. Hidaka, T. C. Kobayashi, T. Akazawa, Y. Ohishi,M. Nakashima, A. Thamizhavel, R. Settai, and Y. Onuki,2006, J. Phys. Soc. Jpn. , 044713.Kotegawa, H., M. Yogi, Y. Imamura, Y. Kawasaki, G. Q.Zheng, Y. Kitaoka, S. Ohsaki, H. Sugawara, and Y. A. H.Sato, 2003, Phys. Rev. Lett. , 027001.Krimmel, A., P. Fischer, B. Roessli, H. Maletta, C. Geibel,C. Schank, A. Grauel, A. Loidl, and F. Steglich, 1992, Z.Phys. B - Cond. Matt. , 161.Krimmel, A., and A. Loidl, 1997, Physica B , 877.Krimmel, A., A. Loidl, R. Eccleston, C. Geibel, andF. Steglich, 1996, J. Phys.: Condens. Matter , 1677.Krimmel, A., A. Loidl, P. Fischer, B. Roessli, A. D¨onni,H. Kita, N. Sato, Y. Endoh, T. Komatsubara, C. Geibel,and F. Steglich, 1993, Sol. State Communic. , 829.Krimmel, A., A. Loidl, K. Knorr, B. Buschinger, C. Geibel,C. Wassilew, and M. Hanfland, 2000, J. Phys.: Condens.Matter , 8801.Krimmel, A., A. Loidl, H. Schober, and P. C. Canfield, 1997,Phys. Rev. B , 6416.Kromer, F., R. Helfrich, M. Lang, F. Steglich, C. Langham-mer, A. Bach, T. Michels, J. S. Kim, and G. R. Stewart,1998, Phys. Rev. Lett. (20), 4476.Kromer, F., M. Lang, N. Oeschler, P. Hinze, C. Langhammer,F. Steglich, J. S. Kim, and G. R. Stewart, 2000, Phys. Rev.B (18), 12477.Krupta, M., A. Giorgi, N. H. Krikoria, and E. G. Szkalrz,1969, J. Less-Comm. Metals , 91.K¨ubert, C., and P. J. Hirschfeld, 1998, Phys. Rev. Lett. (22), 4963.Kumagai, K., M. Saitoh, T. Oyaizu, Y. Furukawa,S. Takashima, M. Nohara, H. Takagi, and Y. Matsuda,2006, Phys. Rev. Lett. (22), 227002.Kumar, P., and P. W¨olfle, 1987, Phys. Rev. Lett. (17),1954.Kuwahara, K., K. Iwasa, M. Kohgi, K. Kaneko, N. Metoki,S. Raymond, M.-A. M´easson, J. Flouquet, H. Sugawara,Y. Aoki, and H. Sato, 2005, Phys. Rev. Lett. (10),107003.Kuwahara, K., H. Sagayama, K. Iwasa, M. Kohgi, Y. Haga, Y. Onuki, K. Kakurai, M. Nishi, K. Nakajima, N. Aso, andY. Uwatoko, 2002, Physica B , 106.Kwok, W. K., L. E. DeLong, G. W. Crabtree, D. G. Hinks,and R. Joynt, 1990, Phys. Rev. B (16), 11649.Kyogaku, M., Y. Kitaoka, K. Asayama, C. Geibel, C. Schank,and F. Steglich, 1993, J. Phys. Soc. Jpn. , 4016.Lake, B., H. Ronnow, N. B. Christensen, G. Aeppli, K. Lef-mann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl,N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takaga, et al. , 2002, Nature , 299.Lambert, S. E., Y. Dalichaouch, M. B. Maple, J. L. Smith,and Z. Fisk, 1986, Phys. Rev. Lett. (13), 1619.Lander, G. H., E. S. Fischer, and S. D. Bader, 1994, Adv.Phys. , 1.Lander, G. H., S. M. Shapiro, C. Vettier, and A. J. Dianoux,1992, Phys. Rev. B (9), 5387.Lang, M., R. Modler, U. Ahlheim, R. Helfrich, P. Reinder,F. Steglich, W. Assmus, W. Sun, G. Bruls, D. Weber, andB. L¨uthi, 1991, Physica Scripta T39 , 135.Larkin, A. I., and Y. N. Ovchinnikov, 1965, Sov. Phys. JETP , 762.Laver, M., E. M. Forgan, S. P. Brown, D. Charalambous,D. Fort, C. Bowell, S. Ramos, R. J. Lycett, D. K. Christen,J. Kohlbrecher, C. D. Dewhurst, and R. Cubitt, 2006, Phys.Rev. Lett. (16), 167002.Lawrence, J., 1979, Phys. Rev. B (9), 3770.Lawrence, J. M., and S. M. Shapiro, 1980, Phys. Rev. B (9),4379.Layzer, A., and D. Fay, 1971, Int. J. Mag. , 135.Lee, C. H., M. Matsuhata, A. Yamamoto, T. Ohta,H. Takazawa, K. Ueno, C. Sekine, I. Shirotani, and T. Hi-rayama, 2001, J. Phys.: Condens. Matter , L45.Lee, P. A., 1993, Phys. Rev. Lett. (12), 1887.Leggett, A. J., 1975, Rev. Mod. Phys. , 331.Lengyel, E., J. Sarrao, G. Sparn, F. Steglich, and J. Thomp-son, 2004, J. Mag. Mag. Materials , 52.L´evy, F., I. Sheikin, B. Grenier, and A. D. Huxley, 2005,Science , 1343.L´evy, F., I. Sheikin, and A. Huxley, 2007, Nature Physics ,460.Lhotel, E., C. Paulsen, and A. D. Huxley, 2003, Phys. Rev.Lett. (20), 209701.Link, P., D. Jaccard, C. Geibel, C. Wassilew, and F. Steglich,1995, J. Phys.: Condens. Matter , 373.Llobet, A., J. S. Gardner, E. G. Moshopoulou, J.-M. Mignot,M. Nicklas, W. Bao, N. O. Moreno, P. G. Pagliuso, I. N.Goncharenko, J. L. Sarrao, and J. D. Thompson, 2004,Phys. Rev. B (2), 024403.v. L¨ohneysen, H., A. Rosch, M. Vojta, and P. W¨olfle, 2007,Reviews of Modern Physics (3), 1015.Lonzarich, G. G., 1980, Fermi surface studies of groundstateand magnetic excitations in itinerant electron ferromagnets (Cambridge University Press, Cambridge, UK), Electronsat the Fermi Surface, pp. 224–277.Lonzarich, G. G., 1987, J. Mag. Mag. Materials , 445.Lonzarich, G. G., 1988, J. Mag. Mag. Materials , 1.Lonzarich, G. G., 1997, The magnetic electron (CambridgeUniversity Press, Cambridge, UK), Electron.Lonzarich, G. G., and L. Taillefer, 1985, J. Phys. C: SolidState Physics , 4339.Luk’yanchuk, I. A., and V. P. Mineev, 1989, Sov. Phys. JETP , 402.Lussier, B., B. Ellman, and L. Taillefer, 1994, Phys. Rev. Lett. (24), 3294. Lussier, J. G., M. Mao, A. Schr¨oder, J. D. Garrett, B. D.Gaulin, S. M. Shapiro, and W. J. L. Buyers, 1997, Phys.Rev. B (18), 11749.L¨uthi, B., B. Wolf, D. Finsterbusch, and G.Bruls, 1995, Phys-ica B , 228.L¨uthi, B., B. Wolf, P. Thalmeier, M. G¨unther, W. Sixl, andG. Bruls, 1993, Phys. Lett. A , 237.Machida, K., and H. Nakanishi, 1984, Phys. Rev. B (1),122.Machida, K., and T. Ohmi, 2001, Phys. Rev. Lett. , 850.Mackenzie, A. P., R. K. W. Haselwimmer, A. W. Tyler, G. G.Lonzarich, Y. Mori, S. Nishizaki, and Y. Maeno, 1998,Phys. Rev. Lett. (1), 161.MacLaughlin, D. E., M. D. Lan, C. Tien, J. M. Moore, G. C.W, H. R. Ott, Z. Fisk, and J. L. Smith, 1987, J. Mag. Mag.Mater.
63 & 64 , 455.MacMillan, W. L., 1968, Phys. Rev. , 331.Maehira, T., T. Hotta, K. Ueda, and A. Hasegawa, 2003, J.Phys. Soc. Jpn. , 854.Majumdar, S., G. Balakrishnan, M. R. Lees, D. M. Paul, andG. J. McIntyre, 2002, Phys. Rev. B (21), 212502.Makhlin, Y., and V. P. Mineev, 1992, J. Low Temp. Phys. , 49.Maki, K., B. Dora, B. Korin-Hamzic, M. Basletic, A. Vi-rosztek, and M. V. Kartsovnik, 2002, J. Phys. IV , 49.Malinowski, A., M. F. Hundley, C. Capan, F. Ronning,R. Movshovich, N. O. Moreno, J. L. Sarrao, and J. D.Thompson, 2005, Phys. Rev. B (18), 184506.Maple, M. B., 1969, Ph.D. thesis.Maple, M. B., 1970, Solid State Commun. , 1915.Maple, M. B., 1976, Appl. Phys. , 179.Maple, M. B., 1995, Physica B , 110.Maple, M. B., 2005, J. Phys. Soc. Jpn. , 222.Maple, M. B., J. W. Chen, Y. Dalichaouch, T. Kohara,C. Rossel, M. S. Torikachvili, M. W. McElfresh, and J. D.Thompson, 1986, Phys. Rev. Lett. , 185.Maple, M. B., J. W. Chen, S. E. Lambert, Z. Fisk, J. L.Smith, H. R. Ott, J. S. Brooks, and M. J. Naughton, 1985,Phys. Rev. Lett. (5), 477.Maple, M. B., W. Fertig, A. Mota, L. DeLong, D. Wohlleben,and R. Fitzgerald, 1972, Solid State Commun. , 829.Maple, M. B., and Ø. Fischer (eds.), 1982, Superconductivityin Ternary Compounds II (Springer-Verlag, Berlin, Heidel-berg, New York), volume 34 of
Topics in Current Physics .Maple, M. B., and Z. Fisk, 1968, Proceedings 11th Int. Conf.Low Temp. Phys., San Andrews , 1288.Maple, M. B., Z. Henkie, R. Baumbach, T. Sayles, N. Butch,P.-C. Ho, T. Yanagisawa, W. Yuhasz, R. Wawryk, T. Ci-chorek, and A. Peitraszko, 2008, J. Phys. Soc. Jpn., Suppl.A , 7.Maple, M. B., P.-C. Ho, V. S. Zapf, N. A. Frederick, E. D.Bauer, W. M. Yuhasz, F. M. Woodward, and J. W. Lynn,2002, J. Phys. Soc. Jpn., suppl. , 23.Martin, C., C. C. Agosta, S. W. Tozer, H. A. Radovan, E. C.Palm, T. P. Murphy, and J. L. Sarrao, 2005, Phys. Rev. B (2), 020503.Martisovits, V., G. Zar´and, and D. L. Cox, 2000, Phys. Rev.Lett. (25), 5872.Mason, T. E., W. J. L. Buyers, T. Petersen, A. A. Menovsky,and J. D. Garrett, 1995, J. Phys. Condens. Matter , 5089.Mason, T. E., B. D. Gaulin, J. D. Garrett, Z. Tun, W. J. L.Buyers, and E. D. Isaacs, 1990, Phys. Rev. Lett. , 3189.Mathur, N. D., F. M. Grosche, S. R. Julian, I. R. Walker,D. M. Freye, R. K. W. Haselwimmer, and G. G. Lonzarich, 1998, Nature , 39.Matsuda, K., Y. Kohori, and T. Kohara, 1996, J. Phys. Soc.Jpn. , 679.Matsuda, K., Y. Kohori, T. Kohara, H. Amitsuka, K. Kuwa-hara, and T. Matsumoto, 2003, J. Phys.: Condens. Matter , 2363.Matsuda, K., Y. Kohori, T. Kohara, K. Kuwahara, andH. Amitsuka, 2001, Phys. Rev. Lett. , 087203.Matsuda, T. D., D. Aoki, S. Ikeda, E. Yamamoto, Y. Haga,H. Ohkuni, R. Settai, and Y. Onuki, 2008, J. Phys. Soc.Jpn., Suppl. A , 362.Matsuda, Y., and H. Shimahara, 2007, J. Phys. Soc. Jpn. ,051005.Matsui, H., T. Goto, N. Sato, and . Komatsubara, 1994, Phys-ica B , 140.Matthias, B. T., H. Suhl, and E. Corenzwit, 1958a, Phys.Rev. Lett. (12), 449.Matthias, B. T., H. Suhl, and E. Corenzwit, 1958b, Phys.Rev. Lett. (3), 92.Mayer, H. M., U. Rauchschwalbe, C. D. Bredl, F. Steglich,H. Rietschel, H. Schmidt, H. W¨uhl, and J. Beuers, 1986,Phys. Rev. B (5), 3168.Mazin, I. I., D. A. Papaconstantopoulos, and M. J. Mehl,2002, Phys. Rev. B , 100511.Mazumdar, C., and R. Nagarajan, 2005, Current Science ,83.Mazumdar, C., R. Nagarajan, S. K. Dhar, L. C. Gupta,R. Vijayaraghavan, and B. D. Padalia, 1992, Phys. Rev.B (14), 9009.McCollam, A., R. Daou, S. R. Julian, C. Bergemann, J. Flou-quet, and D. Aoki, 2005, Physica B , 1.McDonough, J., and A. D. Huxley, 1996, Rev. Sci. Instrum. , 1105.McElfresh, M. W., M. B. Maple, J. O. Willis, Z. Fisk, J. L.Smith, and J. D. Thompson, 1990, Phys. Rev. B (10),6062.McElfresh, M. W., J. D. Thompson, J. O. Willis, M. B. Maple,T. Kohara, and M. S. Torikachvili, 1987, Phys. Rev. B ,43.McHale, P., P. Fulde, and P. Thalmeier, 2004, Phys. Rev. B , 014513.McMullan, G. J., P. M. C. Rourke, M. R. Norman, A. D.Huxley, N. Doiron-Leyraud, J. Flouquet, G. G. Lonzarich,A. McCollam, and S. R. Julian, 2008, condmat/0803.1155.Measson, M. A., D. Braithwaite, J. Flouquet, G. Seyfarth,J. P. Brison, E. Lhotel, C. Paulsen, H. Sugawara, andH. Sato, 2004, Phys. Rev. B , 064516.M´easson, M.-A., D. Braithwaite, G. Lapertot, J.-P. Brison,J. Flouquet, P. Bordet, H. Sugawara, and P. C. Canfield,2008, Physical Review B (Condensed Matter and MaterialsPhysics) (13), 134517.Meissner, W., and R. Ochsenfeld, 1933, Naturwissenschaften , 787.Metoki, N., Y. Haga, Y. Koike, and Y. Onuki, 1998, Phys.Rev. Lett. (24), 5417.Metoki, N., K. Kaneko, T. D. Matsuda, A. Galatanu,T. Takeuchi, S. Hashimoto, T. Ueda, R. Settai, Y. Onuki,and N. Bernhoeft, 2004, J. Phys.: Condens. Matter ,L207.Miclea, C. F., M. Nicklas, D. Parker, K. Maki, J. L. Sarrao,J. D. Thompson, G. Sparn, and F. Steglich, 2006, Phys.Rev. Lett. (11), 117001.Mihalik, M., F. E. Kayzel, T. Yoshida, K. Kuwahara,H. Amitsuka, T. Sakakibara, A. A. Menovsky, J. A. My- dosh, and J. J. M. Franse, 1997, Physica B , 364.Millis, A. J., 1993, Phys. Rev. B , 7183.Mineev, V., 2006, Comptes. Rend. Phys. Soc. Jpn. , 35.Mineev, V. P., 2002a, Phys. Rev. B , 134504.Mineev, V. P., 2002b, Phys. Rev. B (13), 134504.Mineev, V. P., 2004, Int. J. Mod. Phys. B , 2963.Mineev, V. P., 2005a, Phys. Rev. B , 012509.Mineev, V. P., 2005b, Int. J. Mod. Phys. B , 2963.Mineev, V. P., and T. Champel, 2004, Phys. Rev. B (14),144521.Mineev, V. P., and K. V. Samokhin, 1999, Introduction tounconventional superconductivity (Gordon and Breach Sci-ence Publishers, Australia).Mineev, V. P., and M. E. Zhitomirsky, 2005, Phys. Rev. B , 014432.Misiorek, H., J. Mucha, R. Troc, and B. Coqblin, 2005, J.Phys.: Condens. Matter , 679.Mito, T., S. Kawasaki, Y. Kawasaki, G. q. Zheng, Y. Kitaoka,D. Aoki, Y. . Haga, and Y. Onuki, 2003, Phys. Rev. Lett. (7), 077004.Mito, T., S. Kawasaki, G.-q. Zheng, Y. Kawasaki, K. Ishida,Y. Kitaoka, D. Aoki, Y. Haga, and Y. Onuki, 2001, Phys.Rev. B (22), 220507.Mitrovi´c, V. F., M. Horvati´c, C. Berthier, G. Knebel,G. Lapertot, and J. Flouquet, 2006, Phys. Rev. Lett. (11), 117002.Miyake, A., K. Shimizu, C. Sekine, K. Kihou, and I. Shirotani,2004, J. Phys. Soc. Jpn. , 2370.Miyake, K., and N. K. Sato, 2001, Phys. Rev. B (5), 052508.Modler, R., K. Gloos, H. Schimanski, C. Geibel, M. G¨unther,G. Bruls, B. L¨uthi, T. Komatsubara, N. Sato, C. Schank,and F. Steglich, 1993, Physica B , 294.Moncton, D. E., D. B. McWhan, J. Eckert, G. Shirane, andW. Thomlinson, 1977, Phys. Rev. Lett. (18), 1164.Monthoux, P., and G. G. Lonzarich, 2001, Phys. Rev. B ,054529.Monthoux, P., and G. G. Lonzarich, 2002, Phys. Rev. B ,224504.Monthoux, P., and G. G. Lonzarich, 2004, Phys. Rev. B (6),064517.Monthoux, P., D. Pines, and G. G. Lonzarich, 2007, Nature , 1177.Moore, K. T., and G. van der Laan, 2009, Rev. of Mod. Phys. , 235.Mora, C., and R. Combescot, 2004, Eur. Phys. Lett. , 833.Mora, C., and R. Combescot, 2005, Phys. Rev. B (21),214504.Morel, P., and P. W. Anderson, 1962, Phys. Rev. (4),1263.Morin, P., C. Vettier, J. Flouquet, M. Konczykowski, Y. Las-sailly, J.-M. Mignot, and U. Welp, 1988, J. Low Temp.Phys. , 377.Moriya, T., 1963, Weak ferromagnetism (Academic Press,New York), volume 1 of
Magnetism , chapter 3.Moriya, T., 1985,
Spin fluctuations in itinerant electron mag-netism , volume 56 of
Solid-State Sciences (Springer, BerlinHeidelberg New York).Morris, G. D., R. H. Heffner, N. O. Moreno, P. G. Pagliuso,J. L. Sarrao, S. R. Dunsiger, G. J. Nieuwenhuys, D. E.MacLaughlin, and O. O. Bernal, 2004, Phys. Rev. B (21), 214415.Motoyama, G., K. Maeda, and Y. Oda, 2008a, J. Phys. Soc.Jpn. , 044710.Motoyama, G., Y. Yamaguchi, K. Maeda, A. Sumiyama, and Y. Oda, 2008b, J. Phys. Soc. Jpn. , 075004.Movshovich, R., T. Graf, J. D. Thompson, J. L. Smith, andZ. Fisk, 1996, Phys. Rev. B , 8241.Movshovich, R., M. Jaime, J. D. Thompson, C. Petrovic,Z. Fisk, P. G. Pagliuso, and J. L. Sarrao, 2001, Phys. Rev.Lett. (22), 5152.M¨uhlbauer, S., B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,A. Neubauer, R. Georgii, and P. B¨oni, 2009, Science ,915.M¨uhlbauer, S., C. Pfleiderer, P. B¨oni, M. Laver, E. M. Forgan,D. Fort, U. Keiderling, and G. Behr, 2009, Phys. Rev. Lett. , 136408.Mukuda, H., T. Fuji, T. Ohara, A. Harada, M. Yashima,Y. Kitaoka, Y. Okuda, R. Settai, and Y. Onuki, 2008, Phys.Rev. Lett. , 107003.M¨uller, V., C. Roth, D. Maurer, E. W. Scheidt, K. L¨uders,E. Bucher, and H. E. B¨ommel, 1987, Phys. Rev. Lett. (12), 1224.M¨uller-Hartmann, E., and J. Zittartz, 1971, Phys. Rev. Lett. (8), 428.Muramatsu, T., N. Tateiwa, T. C. Kobayashi, K. Shimizu,K. A. D. Aoki, H. Shishido, Y. Haga, and Y. Onuki, 2001,J. Phys. Soc. Jpn. , 3362.Murani, A. P., A. D. Taylor, R. Osborn, and Z. A. Bowden,1993, Phys. Rev. B (14), 10606.Muro, Y., 2000, Ph.D. thesis.Muro, Y., D. Eom, N. Takeda, and M. Ishikawa, 1998, J.Phys. Soc. Jpn. , 3601.Muro, Y., M. Ishikawa, K. Hirota, Z. Hiroi, N. Takeda,N. Kimura, and H. Aoki, 2007, J. Phys. Soc. Jpn. ,033706.Naidyuk, Y. G., H. v. L¨ohneysen, G. Goll, I. K. Yanson, andA. A. Menovsky, 1996, Europhys. Lett. , 557.Nair, S., S. Wirth, M. Nicklas, J. L. Sarrao, J. D. Thompson,Z. Fisk, and F. Steglich, 2008, Physical Review Letters (13), 137003.Nakajima, Y., K. Izawa, Y. Matsuda, S. Uji, T. Terashima,H. Shishido, R. Settai, Y. Onuki, and H. Kontani, 2004, J.Phys. Soc. Jpn. , 5.Nakajima, Y., H. Shishido, H. Nakai, T. Shibauchi, M. Hedo,Y. Uwatoko, T. Matsumoto, R. Settai, Y. Onuki, H. Kon-tani, and Y. Matsuda, 2008, Physical Review B (CondensedMatter and Materials Physics) (21), 214504.Nakashima, M., H. Kohara, A. Thamizhavel, T. DMatsuda,Y. Haga, M. Hedo, Y. Uwatoko, R. Settai, and Y. Onuki,2005, J. Phys.: Condens. Matter , 4539.Nakashima, M., H. Ohkuni, Y. Inada, R. Settai, Y. Haga,E. Yamamoto, and Y. Onuki, 2003, J. Phys.: Condens.Matter , S2011.Nakashima, M., K. Tabata, A. Thamizhavel, T. Kobayashi,M. Hedo, Y. Uwatoko, K. Shimizu, R. Settai, and Y. Onuki,2004, J. Phys.: Condens. Matter , L255.Nakatsuji, S., K. Kuga, Y. Machida, T. Tayama, T. Sakak-ibara, Y. Karaki, H. Ishimoto, S. Yonezawa, Y. Maeno,E. Pearson, G. G. Lonzarich, L. Balicas, et al. , 2008, Na-ture Physics , 603.Nakatsuji, S., S. Yeo, L. Balicas, Z. Fisk, P. Schlottmann,P. G. Pagliuso, N. O. Moreno, J. L. Sarrao, and J. D.Thompson, 2002, Phys. Rev. Lett. (10), 106402.Neubauer, A., C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G.Niklowitz, and P. B¨oni, 2009, Phys. Rev. Lett. , 186602.Nicklas, M., R. Borth, E. Lengyel, P. G. P. J. L. Sarrao, V. A.Sidorov, G. Sparn, F. Steglich, and J. D. Thompson, 2001,J. Phys.: Condens. Matter , L905. Nicklas, M., V. A. Sidorov, H. A. Borges, P. G. Pagliuso,C. Petrovic, Z. Fisk, J. L. Sarrao, and J. D. Thompson,2003, Phys. Rev. B (2), 020506.Nicklas, M., G. Sparn, R. Lackner, E. Bauer, and F. Steglich,2005, Physica B , 386.de Nijs, D. E., N. T. Huy, and A. de Visser, 2008, Physi-cal Review B (Condensed Matter and Materials Physics) (14), 140506.Nishioka, T., G. Motoyama, S. Nakamura, H. Kadoya, andN. K. Sato, 2002, Phys. Rev. Lett. , 237203.Norman, M. R., 1993, Phys. Rev. Lett. , 3391.Norman, M. R., W. E. Pickett, H. Krakauer, and C. S. Wang,1987, Phys. Rev. B (7), 4058.Normile, P. S., S. Heathman, M. Idiri, P. Boulet, J. Rebizant,F. Wastin, G. H. Lander, T. L. Bihan, and A. Lindbaum,2005, Phys. Rev. B (18), 184508.Oda, K., T. Kumada, K. Sugiyama, N. Sato, T. Komatsubara,and M. Date, 1994, J. Phys. Soc. Jpn. , 3115.Oda, K., K. Sugiyama, N. K. Sato, T. Komatsubara,K. Kindo, and Y. Onuki, 1999, J. Phys. Soc. Jpn. , 3115.Oeschler, N., P. Gegenwart, F. Weickert, I. Zerec,P. Thalmeier, F. Steglich, E. D. Bauer, N. A. Frederick,and M. B. Maple, 2004, Phys. Rev. B , 235108.Ogita, N., R. Kojima, T. Hasegawa, T. Takasu, M. Uda-gawa, T. Kondo, S. Narazu, T. Takabatake, N. Takeda,Y. Ishikawa, H. Sugawara, T. Ikeno, et al. , 2008, J. Phys.Soc. Jpn., Suppl. A , 251.Oh, Y. S., K. H. Kim, P. A. Sharma, N. Harrison, H. Amit-suka, and J. A. Mydosh, 2007, Phys. Rev. Lett. (1),016401.Ohashi, M., F. Honda, T. Eto, S. Kaji, I. Minamidake,G. Oomi, S. Koiwai, and Y. Uwatoko, 2002, Physica B , 443.Ohishi, K., R. H. Heffner, G. D. Morris, E. D. Bauer, M. J.Graf, J. Zhu, I. A. Morales, J. L. Sarrao, M. J. Fluss, D. E.MacLaughlin, L. Shu, W. Higemoto, et al. , 2007, cond-mat/0706.4367.Ohishi, K., T. U. Ito, W. Higemoto, and R. H. Heffner, 2006,J. Phys. Soc. Jpn. , Suppl. 53.Ohkawa, F. J., and H. Shimizu, 1999, J. Phys.: Condens.Matter , L519.Ohta, T., Y. Nakai, Y. Ihara, K. Ishida, K. Deguchi, N. K.Sato, and I. Satoh, 2008, J. Phys. Soc. Jpn. , 023707.Oikawa, K., T. Kamiyama, H. Asano, Y. Onuki, and M. Ko-hgi, 1996, J. Phys. Soc. Jpn. , 3229.Okane, T., J. Okamoto, K. Mamiya, S. Fujimori, Y. Takeda,Y. Saitoh, Y. Muramatsu, A. Fujimori, Y. Haga, E. Ya-mamoto, A. Tanaka, T. Honma, et al. , 2006, J. Phys. Soc.Jpn. , 024704.Okazaki, R., Y. Kasahara, H. Shishido, M. Konczykowski,K. Behnia, Y. Haga, T. D. Matsuda, Y. Onuki,T. Shibauchi, and Y. Matsuda, 2008, Physical Review Let-ters (3), 037004.Okuda, Y., Y. Miyauchi, Y. Ida, Y. Takeda, C. Tonohiro,Y. Oduchi, T. Yamada, N.-D. Dung, T. D. Matsuda,Y. Haga, T. Takeuchi, M. Hagiwara, et al. , 2007, J. Phys.Soc. Jpn. , 044708.Okuno, Y., and K. Miyake, 1998, J. Phys. Soc. Jpn. , 2469.Onishi, Y., and K. Miyake, 2000, J. Phys. Soc. Jpn. , 3955.Onishi, Y., and K. Miyake, 2004, J. Phys. Soc. Jpn. , 3955.Onnes, H. K., 1911a, Leiden Comm. .Onnes, H. K., 1911b, Leiden Comm. .Onnes, H. K., 1911c, Leiden Comm. .Onodera, A., S. Tsuduki, Y. Ohishi, T. Watanuki, K. Ishida, Y. Kitaoka, and Y. Onuki, 2002, Solid State Communica-tions , 113.Onuki, Y., 1993, Phys. Prop. Act. Rare Earth Comp. JJAPseries 8 , 149.Onuki, Y., and A. Hasegawa, 1995,
Fermi Surfaces of In-termetallic Compounds (North-Holland, Amsterdam, Hol-land), volume 20 of
Handbook of the Physics and Chemistryof Rare Earths .Onuki, Y., R. Settai, K. Sugiyama, T. Takeuchi, T. C.Kobayashi, Y. Haga, and E. Yamamoto, 2004, J. Phys.Soc. Jpn. , 769.Onuki, Y., I. Ukon, S. W. Yun, I. Umehara, K. Satoh,T. Kukuhara, H. Sato, S. Takayanagi, M. Shikama, andA. Ochiai, 1992, J. Phys. Soc. Jpn. , 293, note that thecrystal structure is incorrect.Oomi, G., K. Kagayama, K. Nishimura, S. W. Yun, andY. Onuki, 1995, Physica B
206 & 207 , 515.Oomi, G., K. Nishimura, Y. Onuki, and S. W. Yun, 1993,Physica B , 758.Opahle, I., S. Elgazzar, K. Koepernik, and P. M. Oppeneer,2004, Phys. Rev. B , 104504.Opahle, I., and P. M. Openeer, 2003, Phys. Rev. Lett. ,157001.Oppeneer, P. M., S. Lesb´egue, O. Eriksson, I. Ophale, andA. B. Shick, 2006, J. Phys. Soc. Jpn. , Suppl. 215.Ormeno, R. J., A. Sibley, C. E. Gough, S. Sebastian, and I. R.Fisher, 2002, Phys. Rev. Lett. (4), 047005.Osheroff, D. D., R. C. Richardson, and D. M. Lee, 1972, Phys.Rev. Lett. (14), 885.Ott, H. R., E. Felder, C. Bruder, and T. M. Rice, 1987, Eu-rophys. Lett. , 1123.Ott, H. R., H. Rudiger, Z. Fisk, and J. L. Smith, 1983, Phys.Rev. Lett. , 1595.Ott, H. R., H. Rudigier, E. Felder, Z. Fisk, and J. L. Smith,1986, Phys. Rev. B (1), 126.Ott, H. R., H. Rudigier, Z. Fisk, and J. L. Smith, 1984a, Moment formation in solids (Plenum, New York, USA).Ott, H. R., H. Rudigier, T. M. Rice, K. Ueda, Z. Fisk, andJ. L. Smith, 1984b, Phys. Rev. Lett. (21), 1915.Otzschi, K., T. Mizukami, T. Hinouchi, J. Shimoyama, andK. Kishio, 1999, J. Low. Temp. Physics , 885.¨Ozcan, S., D. M. Broun, B. Morgan, R. K. W. Haselwim-mer, J. L. Sarrao, S. Kamal, C. P. Bidinosti, P. J. Turner,M. Raudsepp, and J. R. Waldram, 2003, Europhys. Lett. , 412.Paglione, J., P.-C. Ho, M. B. Maple, M. A. Tanatar, L. Taille-fer, Y. Lee, and C. Petrovic, 2008, Physical Review B (Con-densed Matter and Materials Physics) (10), 100505.Paglione, J., T. A. Sayles, P.-C. Ho, and J. R. Jeffries, 2007,Nature Physics , 703.Paglione, J., M. A. Tanatar, D. G. Hawthorn, E. Boaknin,R. W. Hill, F. Ronning, M. Sutherland, L. Taillefer,C. Petrovic, and P. C. Canfield, 2003, Phys. Rev. Lett. (24), 246405.Pagliuso, P., N. Curro, N. Moreno, M. F. Hundley, J. D.Thompson, J. L. Sarrao, and Z. Fisk, 2002a, Physica B , 370.Pagliuso, P., R. Movshovich, A. Bianchi, M. Nicklas,N. Moreno, J. Thompson, M. Hundley, J. Sarrao, andZ. Fisk, 2002b, Physica B , 129.Pagliuso, P. G., N. O. Moreno, N. J. Curro, J. D. Thompson,M. F. Hundley, J. L. Sarrao, Z. Fisk, A. D. Christianson,A. H. Lacerda, B. E. Light, and A. L. Cornelius, 2002c,Phys. Rev. B (5), 054433. Pagliuso, P. G., C. Petrovic, R. Movshovich, D. Hall, M. F.Hundley, J. L. Sarrao, J. D. Thompson, and Z. Fisk, 2001,Phys. Rev. B (10), 100503.Palstra, T., A. A. Menovsky, J. Vandenberg, A. J. Dirkmaat,P. H. Kes, G. J. Nieuwenhuys, and J. A. Mydosh, 1985,Phys. Rev. Lett. , 2727.Paolasini, L., J. A. Paixao, G. Lander, A. Delapalme, N. Sato,and T. Komatsubara, 1993, J. Phys.: Condens. Matter ,8905.Park, J.-G., K. A. McEwen, S. deBrion, G. Chouteau,H. Amitsuka, and T. Sakakibara, 1997, J. Phys.: Condens.Matter , 3065.Park, T., F. Ronning, H. Yuan, M. Salamon, R. Movshovich,J. Sarrao, and J. Thompson, 2006, Nature , 65.Park, W. K., J. L. Sarrao, J. D. Thompson, and L. H. Greene,2008, Physical Review Letters (17), 177001.Parks, R. D. (ed.), 1969, Superconductivity (Marcel Dekker,New York, USA).Pearson, W. B., 1958,
Handbook of lattice spacings and struc-tures of metals , volume 3 (Pergamon, New York, USA).Petersen, T., T. E. Mason, G. Aeppli, A. P. Ramirez,E. Bucher, and R. N. Kleinman, 1994, Physica B ,151.Petrovic, C., , R. Movshovich, M. Jaime, P. G. Pagliuso, M. F.Hundley, J. L. Sarrao, Z. Fisk, and J. D. Thompson, 2001a,Europhys. Lett. , 354.Petrovic, C., P. G. Pagliuso, M. F. Hundley, R. Movshovich,J. L. Sarrao, J. D. Thompson, Z. Fisk, and P. Monthoux,2001b, J. Phys.: Condens. Matter , L337.Pfleiderer, C., 2005, J. Phys. Cond. Matter , 887.Pfleiderer, C., E. Bedin, and B. Salce, 1997a, Rev. Sci. In-strum. , 3120.Pfleiderer, C., R. H. Friend, G. G. Lonzarich, N. R. Bernhoeft,and J. Flouquet, 1993, Int. J. Mod. Phys. B , 887.Pfleiderer, C., and R. Hackl, 2007, Nature , 492.Pfleiderer, C., and A. D. Huxley, 2002, Phys. Rev. Lett. ,147005.Pfleiderer, C., A. D. Huxley, and S. M. Hayden, 2005, J.Phys.: Condens. Matter , S3111.Pfleiderer, C., S. R. Julian, and G. G. Lonzarich, 2001a, Na-ture , 427.Pfleiderer, C., G. J. McMullan, S. R. Julian, and G. G. Lon-zarich, 1997b, Phys. Rev. B , 8330.Pfleiderer, C., J. A. Mydosh, and M. Vojta, 2006, Phys. Rev.B (10), 104412.Pfleiderer, C., D. Reznik, L. Pintschovius, H. v. L¨ohneysen,M. Garst, and A. Rosch, 2004, Nature , 227.Pfleiderer, C., R. Ritz, S. M¨uhlbauer, P. Niklowitz, T. Keller,E. M. Forgan, M. Laver, J. White, R. Cubitt, and E. Bauer,2008, unpublished.Pfleiderer, C., M. Uhlarz, S. M. Hayden, R. Vollmer,H. v. Lonheysen, N. R. Bernhoeft, and G. G. Lonzarich,2001b, Nature , 58.Pham, L. D., T. Park, S. Maquilon, J. D. Thompson, andZ. Fisk, 2006, Phys. Rev. Lett. (5), 056404.Phillips, N. E., and B. T. Matthias, 1961, Phys. Rev. (1),105.Piekarz, P., K. Parlinski, P. T. Jochym, A. M. Ole´s, J.-P. Sanchez, and J. Rebizant, 2005, Phys. Rev. B (1),014521.Ponchet, A., J. M. Mignot, A. de Visser, J. J. M. Franse, andA. Menovsky, 1986, J. Magn. Magn. Mater. , 399.Pourovskii, L. V., M. I. Katsnelson, and A. I. Lichtenstein,2006, Phys. Rev. B (6), 060506. Prokes, K., T. Tahara, Y. Echizen, T. Takabatake, T. Fujita,I. H. Hagmusa, J. C. P. Klaasse, E. Br¨uck, F. R. de Boer,M. Divis, and V. Sechosky, 2002, Physica B , 220.Qian, Y. J., M.-F. Xu, A. Schenstrom, H.-P. Baum, J. B.Ketterson, D. Hinks, M. Levy, and B. K. Sarma, 1987,Solid State Communic. , 599.Radovan, H. A., R. Movshovich, I. Vekhter, P. G. Pagliuso,and J. L. Sarrao, 2003, Nature , 51.Rashba, E. I., 1960, Sov. Phys. Solid State , 1109.Rauchschwalbe, U., U. Ahlheim, F. Steglich, D. Rainer, andJ. J. M. Franse, 1985, Z. Phys. B , 379.Rauchschwalbe, U., W. Lieke, C. D. Bredl, F. Steglich,J. Aarts, K. M. Martini, and A. C. Mota, 1982, Phys. Rev.Lett. (19), 1448.Rauchschwalbe, U., F. Steglich, G. R. Stewart, A. L. Giorgi,P. Fulde, and K. Maki, 1987, Europhys. Lett. , 751.Raymond, S., and D. Jaccard, 2000, Phys. Rev. B (13),8679.Raymond, S., G. Knebel, D. Aoki, and J. Flouquet, 2008a,Physical Review B (Condensed Matter and MaterialsPhysics) (17), 172502.Raymond, S., K. Kuwahara, K. Kaneko, K. Iwasa, M. Kohgi,A. Hiess, M.-A. Measson, J. Flouquet, N. Metoki, H. Sug-awara, Y. Aoki, and H. Sato, 2008b, J. Phys. Soc. Jpn.,Suppl. A , 25.Raymond, S., P. Piekarz, J. P. Sanchez, J. Serrano, M. Krisch,B. Janouˇsov´a, J. Rebizant, N. Metoki, K. Kaneko, P. T.Jochym, A. M. Ole´s, and K. Parlinski, 2006, Phys. Rev.Lett. (23), 237003.Remenyi, G., D. Jaccard, J. Flouquet, A. Briggs, Z. Fisk, J. L.Smith, and H. R. Ott, 1986, J. Phys. , 367.Riblet, G., and K. Winzer, 1971, Solid State Commun. ,1663.Roehler, J., J. Klug, and K. Keularz, 1988, J. Magn. Magn.Materials , 340.Ronning, F., C. Capan, A. Bianchi, R. Movshovich, A. Lac-erda, M. F. Hundley, J. D. Thompson, P. G. Pagliuso, andJ. L. Sarrao, 2005, Phys. Rev. B (10), 104528.Rotundu, C. R., H. Tsujii, Y. Takano, B. Andraka, H. Sug-awara, Y. Aoki, and H. Sato, 2004, Phys. Rev. Lett. ,037203.Rueff, J.-P., S. Raymond, A. Yaresko, D. Braithwaite,P. Leininger, G. Vank´o, A. Huxley, J. Rebizant, andN. Sato, 2007, Phys. Rev. B (8), 085113.Sa, D., 2002, Phys. Rev. B , 140505.Sachdev, S., 1999, Quantum Phase Transitions (CambridgeUniversity Press, Cambridge, USA).Saint-James, D., G. Sarma, and E. J. Thomas (eds.), 1969,
Type II Superconductivity (Pergamon, New York, USA).Sakai, H., Y. Tokunaga, T. Fujimoto, S. Kambe, R. E. Wal-stedt, H. Yasuoka, D. Aoki, Y. Homma, E. Yamamoto,A. Nakamura, Y. Shiokawa, K. Nakajima, et al. , 2005, J.Phys. Soc. Jpn. , 1710.Sakai, H., Y. Tokunaga, T. Fujimoto, S. Kambe, R. E. Wal-stedt, H. Yasuoka, D. Aoki, Y. Homma, E. Yamamoto,A. Nakamura, Y. Shiokawa, K. Nakajima, et al. , 2006,Physica B , 1005.Sakai, H., K. Yoshimura, H. Ohno, H. Kato, S. Kambe,R. Walstedt, T. Matsuda, and Y. Haga, 2001, J. Phys.Cond. Matter , L785.Sakakibara, T., A. Yamada, J. Custers, K. Yano, T. Tayama,H. Aoki, and K. Machida, 2007, J. Phys. Soc. Jpn. ,051004.Sakarya, S., N. H. van Dijk, and E. Br¨uck, 2005, Phys. Rev. B (17), 174417.Sakarya, S., N. H. van Dijk, A. de Visser, and E. Br¨uck, 2003,Phys. Rev. B (14), 144407.Sakon, T., K. Imamura, N. Koga, N. Sato, and T. Komatsub-ara, 1994, Physica B , 154.Sakon, T., K. Imamura, N. Takeda, N. Sato, and T. Komat-subara, 1993, Physica B , 297.Sakon, T., T. Namiki, and M. Motokawa, 2001, J. Phys. Soc.Jpn. , 3046.Sales, B. C., 2003, Filled skutterudites (Elsevier Science, Am-sterdam), volume 33 of
Handbook of the Physics and Chem-istry of Rare Earths , pp. 1–34.Samokhin, K. V., 2002, Phys. Rev. B (21), 212509.Samokhin, K. V., 2004, Phys. Rev. B , 104521.Samokhin, K. V., and M. B. Walker, 2002, Phys. Rev. B ,174501.Samokhin, K. V., E. S. Zijlstra, and S. K. Bose, 2004, Phys.Rev. B , 094514.Sandeman, K., G. G. Lonzarich, and A. J. Schofield, 2003,Phys. Rev. Lett. , 167005.Sandratskii, L. M., J. K¨ubler, P. Zahn, and I. Mertig, 1994,Phys. Rev. B (21), 15834.Santini, P., and G. Amoretti, 1994, Phys. Rev. Lett. , 1027.Santini, P., G. Amoretti, R. Caciuffo, F. Bourdarot, andB. F˚ak, 2000, Phys. Rev. Lett. , 654.Sarrao, J. L., L. A. Morales, J. D. Thompson, B. L. Scott,G. R. Stewart, F. Wastin, J. Rebizant, P. Boulet, E. Col-ineau, and G. H. Lander, 2002, Nature , 297.Sarrao, J. L., and J. D. Thompson, 2007, J. Phys. Soc. Jpn. , 051013.Sato, H., D. Kikuchi, K. Tanaka, H. Aoki, K. Kuwahara,Y. Aoki, M. Kohgi, H. Sugawara, and K. Iwasa, 2007, J.Magn. Magn. Mater. , 188.Sato, H., H. Sugawara, T. Namiki, S. R. Saha, S. Osaki, T. T.Matsuda, Y. Aoki, Y. Inada, H. Shishido, R. Settai, andY. Onuki, 2003, J. Phys.: Condens. Matter , S2063.Sato, N., 1999, Physica B , 634.Sato, N., N. Aso, G. H. Lander, B. Roessli, T. Komatsubara,and Y. Endoh, 1997a, J. Phys. Soc. Jpn. , 2981.Sato, N., N. Aso, G. H. Lander, B. Roessli, T. Komatsubara,and Y. Endoh, 1997b, J. Phys. Soc. Jpn. , 1884.Sato, N., K. Imamura, T. Sakon, Y. Inada, A. Sawada, T. Ko-matsubara, H. Matsui, and T. Goto, 1994, Physica B , 122.Sato, N., N. Koga, and T. Komatsubara, 1996, J. Phys. Soc.Jpn. , 1555.Sato, N., T. Sarkon, N. Takeda, T. Komatsubara, C. Geibel,and F. Steglich, 1992, J. Phys. Soc. Jpn. , 32.Sato, N. K., N. Aso, K. Miyake, R. Shiina, P. Thalmeier,G. Varelogiannis, C. Geibel, F. Steglich, P. Fulde, andT. Komatsubara, 2001, Nature , 340.Sauls, J. A., 1994, Adv. Phys. , 113.Saxena, S. S., P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W.Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R.Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, et al. ,2000, Nature , 587.Saxena, S. S., and P. B. Littlewood, 2001, Nature , 290.Scheffler, M., M. Drssel, M. Jourdan, and H. Adrian, 2005,Nature , 1135.Scheidt, E. W., F. Mayr, G. Eickerling, P. Rogl, and E. Bauer,2005, J. Phys.: Condens. Matter , L121.Schenck, A., D. Andreica, F. N. Gygax, D. Aoki, andY. Onuki, 2002, Phys. Rev. B (14), 144404.Schenck, A., N. K. Sato, G. Solt, D. Andreica, F. N. Gygax, M. Pinkpank, and A. Amato, 2000, Euro. Phys. J. B ,245.Schenstrom, A., M. F. Xu, Y. Hong, D. Bein, M. Levy, B. K.Sarma, S. Adenwalla, Z. Zhao, T. Tokuyasu, D. W. Hess,J. B. Ketterson, J. A. Sauls, et al. , 1989, Phys. Rev. Lett. (3), 332.Schlabitz, W., J. Baumann, B. Pollit, U. Rauchschwalbe,H. M. Mayer, U. Ahlheim, and C. D. Bredl, 1984, unpub-lished, ICVF Cologn .Schlabitz, W., J. Baumann, B. Pollit, U. Rauchschwalbe,H. M. Mayer, U. Ahlheim, and C. D. Bredl, 1986, Z. Phys.B , 171.Schlottmann, P., 1989, Phys. Rep. , 1.Schmidt, F. A., and O. N. Carlson, 1976, Metallurgical Trans-actions A , 127.Schr¨oder, A., J. G. Lussier, B. D. Gaulin, J. D. Garrett,W. J. L. Buyers, L. Rebelsky, and S. M. Shapiro, 1994,Phys. Rev. Lett. (1), 136.Sechovsky, V., and L. Havella, 1998 (North-Holland, Amster-dam), volume 11 of Handbook of magnetic materials , p. 7.Sekine, C., T. Uchiumi, I. Shirotani, and T. Yagi, 1997, Phys.Rev. Lett. (17), 3218.Settai, R., T. Kubo, T. Shirmoto, D. Honda, H. Shishido,K. Sugiyama, Y. Haga, T. Matsuda, K. Betsuyaku,H. Harima, T. Kobayashi, and Y. Onuki, 2005, J. Phys.Soc. Jpn. , 3016.Settai, R., A. Misawa, S. Araki, M. Kosaki, K. Sugiyama,T. Takeuchi, K. Kindo, Y. Haga, E. Yamamoto, andY. Onuki, 1997, J. Phys. Soc. Jpn , 2260.Settai, R., Y. Miyauchi, T. Takeuchi, F. Levy, I. Sheikin, andY. Onuki, 2008, J. Phys. Soc. Jpn. , 073705.Settai, R., Y. Okuda, I. Sugitani, Y. Onuki, T. D. Matsuda,Y. Haga, and H. Harima, 2007a, Int. J. Mod. Phys. B ,3238.Settai, R., H. Shishido, S. Ikeda, Y. Murakawa,M. Nakashima, D. Aoki, Y. Haga, H. Harima, andY. Onuki, 2001, J. Phys.: Condens. Matter , L627.Settai, R., T. Takeuchi, and Y. Onuki, 2007b, J. Phys. Soc.Jpn. , 051003.Seyfarth, G., J. P. Brison, G. Knebel, D. Aoki, G. Lapertot,and J. Flouquet, 2008, Physical Review Letters (4),046401.Seyfarth, G., J. P. Brison, M.-A. M´easson, D. Braithwaite,G. Lapertot, and J. Flouquet, 2006, Phys. Rev. Lett. (23), 236403.Seyfarth, G., J. P. Brison, M.-A. M´easson, J. Flouquet,K. Izawa, Y. Matsuda, H. Sugawara, and H. Sato, 2005,Phys. Rev. Lett. (10), 107004.Shakeripour, H., M. A. Tanatar, S. Y. Li, C. Petrovic, andL. Taillefer, 2007, Physical Review Letters (18), 187004.Sharma, P. A., N. Harrison, M. Jaime, Y. S. Oh, K. H. Kim,C. D. Batista, J. A. Mydosh, and H. Amitsuka, 2005, cond-mat/ , 0507545.Sheikin, I., E. Steep, D. Braithwaite, J. P. Brison, S. Ray-mond, D. Jaccard, and J. Flouquet, 2001, J. Low Temp.Phys. , 591.Shick, A. B., 2002, Phys. Rev. B , 180509.Shick, A. B., V. Janiˇs, V. Drchal, and W. E. Pickett, 2004,Phys. Rev. B (13), 134506.Shick, A. B., V. Janiˇs, and P. M. Oppeneer, 2005, Phys. Rev.Lett. (1), 016401.Shick, A. B., and W. E. Pickett, 2001, Phys. Rev. Lett. ,300.Shiina, R., 2004, J. Phys. Soc. Jpn. , 2257. Shiina, R., and Y. Aoki, 2004, J. Phys. Soc. Jpn. , 541.Shimahara, H., 1994, Phys. Rev. B (17), 12760.Shimahara, H., 1998, J. Phys. Soc. Jpn. , 736.Shimahara, H., S. Matsuo, and K. Nagai, 1996, Phys. Rev. B (18), 12284.Shimahara, H., and D. Rainer, 1997, J. Phys. Soc. Jpn. ,3591.Shimizu, K., T. Kimura, S. Furomoto, K. Takeda, K. Kontani,Y. Onuki, and K. Amaya, 2001, Nature , 316.Shirotani, I., T. Uchiumi, K. Ohno, C. Sekine, Y. Nakazawa,K. Kanoda, S. Todo, and T. Yagi, 1997, Phys. Rev. B (13), 7866.Shishido, H., R. Settai, H. Harima, and Y. Onuki, 2005, J.Phys. Soc. Jpn. , 1103.Shishido, H., T. Ueda, S. Hashimoto, T. Kubo, R. Settai,H. Harima, and Y. Onuki, 2003, J. Phys.: Condens. Matter , L499.Shishido, H., E. Yamamoto, Y. Haga, S. Ikeda,M. Nakashima, R. Settai, and Y. Onuki, 2006, J.Phys. Soc. Jpn. , 119.Shivaram, B. S., Y. H. Jeong, T. F. Rosenbaum, and D. G.Hinks, 1986, Phys. Rev. Lett. (10), 1078.Sidorov, V. A., M. Nicklas, P. G. Pagliuso, J. L. Sarrao,Y. Bang, A. V. Balatsky, and J. D. Thompson, 2002, Phys.Rev. Lett. (15), 157004.Sigrist, M., 2005, AIP Conf. Proc. , 165.Sigrist, M., and T. M. Rice, 1989, Phys. Rev. B (4), 2200.Sigrist, M., and K. Ueda, 1991, Rev. Mod. Phys. , 239.Singh, S., C. Capan, M. Nicklas, M. Rams, A. Gladun, H. Lee,J. F. DiTusa, Z. Fisk, F. Steglich, and S. Wirth, 2007, Phys.Rev. Lett. (5), 057001.Singleton, J., J. A. Symington, M.-S. Nam, A. Ardavan,M. Kurmoo, and P. Day, 2001, Physica B , 418.S¨oderlind, P., 2004, Phys. Rev. B (9), 094515.Sonin, E. B., 2002, Phys. Rev. B , 100504(R).Sonin, E. B., and I. Felner, 1998, Phys. Rev. B , 14000(R).Souptel, D., W. Loser, and G. Behr, 2007, J. Crystal Growth , 538.Spalek, J., 2001, Phys. Rev. B , 104513.Sparn, G., R. Borth, E. Lengyel, P. G. Paglusio, J. Sarrao,F. Steglich, and J. Thompson, 2002, Physica B , 262.Sparn, G., O. Stockert, F. Grosche, H. Yuan, E. Faul-haber, C. Geibel, M. Deppe, H. Jeevan, M. Loewenhaupt,G. Zwicknagl, and F. Steglich, 2006, J. Phys. Chem. Solids , 529.Steeman, R. A., E. Frikkee, R. B. Helmholdt, A. A. Menovsky,J. van den Berg, G. J. Nieuwenhuys, and J. A. Mydosh,1988, Solid State Communic. , 103.Steglich, F., J. Aarts, C. D. Bredl, W. Lieke, D. Meschede,W. Franz, and H. Sch¨afer, 1979, Phys. Rev. Lett. , 1892.Steglich, F., P. Gegenwart, C. Geibel, P. Hinze, M. Lang,C. Langhammer, G. Sparn, T. Tayama, O. Trovarelli,N. Sato, T. Dahm, and G. Varelogiannis, 2001, More isdifferent- fifty years of condensed matter physics (Prince-ton University Press, USA).Steglich, F., R. Modler, P. Gegenwart, M. Deppe, M. Weiden,M. Lang, C. Geibel, T. Luhmann, C. Paulsen, J. L. Tho-lence, Y. Onuki, M. Tachiki, et al. , 1996, Physica C ,498.Stewart, G. R., 2001, Rev. Mod. Phys. (4), 797.Stewart, G. R., 2006, Rev. Mod. Phys. (3), 743.Stewart, G. R., Z. Fisk, J. O. Willis, and J. L. Smith, 1984,Phys. Rev. Lett. , 679.Stock, C., C. Broholm, J. Hudis, H. J. Kang, and C. Petrovic, 2008, Physical Review Letters (8), 087001.Stockert, O., E. Faulhaber, G. Zwicknagl, N. St¨ußer,M. Deppe, R. Borth, R. K¨uchler, M. Loewenhaupt,C. Geibel, and F. Steglich, 2004, Phys. Rev. Lett. ,136401.Suderow, H., J. P. Brison, A. Huxley, and J. Flouquet, 1997,J. Low Temp. Phys. , 11.Suderow, H., S. Vieira, J. D. Strand, S. Bud’ko, and P. C.Canfield, 2004, Phys. Rev. B , 060504.Sugawara, H., Y. Iwahashi, K. Magishi, T. Sato, K. Koyama,H. Harima, D. Kikuchi, H. Sato, T. Endo, R. Settai,Y. Onuki, N. Wada, et al. , 2008, J. Phys. Soc. Jpn., Suppl.A , 108.Sugawara, H., M. Kobayashi, S. Osaki, S. R. Saha, T. Namiki,Y. Aoki, and H. Sato, 2005, Phys. Rev. B (1), 014519.Sugawara, H., S. Osaki, S. R. Saha, Y. Aoki, H. Sato, Y. In-ada, H. Shishido, R. Settai, Y. Onuki, H. Harima, andK. Oikawa, 2002, Phys. Rev. B , 220504.Suginishi, Y., and H. Shimahara, 2006, Phys. Rev. B (2),024518.Sugiyama, K., T. Inoue, T. Ikeda, N. Sato, T. Komatsubara,A. Yamagishi, and M. Date, 1993, Physica B ,723.Sugiyama, K., T. Inoue, K. Oda, T. Kumada, N. Sato, T. Ko-matsubara, A. Yamagishi, and M. Date, 1994, Physica B , 227.Suhl, H., 2001, Phys. Rev. Lett. , 167007.Suhl, H., and B. T. Matthias, 1959, Phys. Rev. (4), 977.S¨ullow, S., B. Becker, A. de Visser, M. Mihalik, G. J.Nieuwenhuys, , A. A. Menovsky, and J. A. Mydosh, 1997,J. Phys.: Condens. Matter , 913.S¨ullow, S., B. Janossy, G. L. E. van Vliet, G. J. Nieuwen-huys, A. A. Menovsky, and J. A. Mydosh, 1996, J. Phys.:Condens. Matter , 729.Sun, Y., and K. Maki, 1995, Europhys. Lett. , 355.Suzumura, Y., and K. Ishino, 1983, Prog. Theor. Phys. ,654.Szajek, A., and J. A. Morkowski, 2003, J. Phys.: Condens.Matter , L155.Tachiki, M., S. Takahashi, P. Gegenwart, M. Weiden,M. Lang, C. Geibel, F. Steglich, R. Modler, C. Paulsen,and Y. Onuki, 1996, Z. Phys. B , 369.Taillefer, L., and G. G. Lonzarich, 1988, Phys. Rev. Lett. (15), 1570.Taillefer, L., R. Newbury, G. G. Lonzarich, Z. Fisk, and J. L.Smith, 1987, J. Magn. Magn. Mater.
63& 64 , 372.Takada, S., and T. Izuyama, 1969, Prog. Theor. Phys. ,635.Takahashi, S., M. Tachiki, R. Modler, P. Gegenwart, M. Lang,and F. Steglich, 1996, Physica C , 30.Takahashi, T., N. Sato, T. Yokoya, A. Chainani, T. Morimoto,and T. Komatsubara, 1995, J. Phys. Soc. Jpn. , 156.Takegahara, K., H. Harima, and T. Kasuya, 1986, J. Phys. F:Metal Phys. , 1691.Takegahara, M., and H. Harima, 2000, Physica B ,764.Takeuchi, T., S. Hashimoto, T. Yasuda, H. Shishido, T. Ueda,M. Yamada, Y. Obiraki, M. Shiimoto, H. Kohara, T. Ya-mamoto, K. Sugiyama, K. Kindo, et al. , 2005, J. Phys.Condens. Matter , L333.Takeuchi, T., T. Inoue, K. Sugiyama, D. Aoki, Y. Tokiwa,Y. Haga, K. Kindo, and Y. Onuki, 2001, J. Phys. Soc. Jpn. , 877.Takeuchi, T., T. Yasuda, M. Tsujino, H. Shishido, R. Settai, H. Harima, and Y. Onuki, 2007, J. Phys. Soc. Jpn. ,014702.Tanatar, M. A., J. Paglione, S. Nakatsuji, D. G. Hawthorn,E. Boaknin, R. W. Hill, F. Ronning, M. Sutherland,L. Taillefer, C. Petrovic, P. C. Canfield, and Z. Fisk, 2005,Phys. Rev. Lett. (6), 067002.Tateiwa, N., Y. Haga, T. D. Matsuda, S. Ikeda, E. Yamamoto,Y. Okuda, Y. Miyauchi, R. Settai, and Y. Onuki, 2007,condmat/0706.4398.Tateiwa, N., Y. Haga, T. D. Matsuda, S. Ikeda, T. Yasuda,T. Takeuchi, R. Settai, and Y. Onuki, 2005, J. Phys. Soc.Jpn. , 1903.Tateiwa, N., K. Hanazono, T. C. Kobayashi, K. Amaya, T. In-oue, K. Kindo, Y. Koike, N. Metoki, Y. Haga, R. Settai,and Y. Onuki, 2001a, J. Phys. Soc. Jpn. , 2876.Tateiwa, N., T. C. Kobayashi, K. Amaya, Y. Haga, R. Settai,and Y. Onuki, 2004, Phys. Rev. B (18), 180513.Tateiwa, N., T. C. Kobayashi, K. Hanazono, K. Amaya,Y. Haga, R. Settai, and Y. Onuki, 2001b, J. Phys.: Con-dens. Matter , L17.Tateiwa, N., N. Sato, and T. Komatsubara, 1998, Phys. Rev.B (17), 11131.Tayama, T., A. Harita, T. Sakakibara, Y. Haga, H. Shishido,R. Settai, and Y. Onuki, 2002, Phys. Rev. B (18),180504.Tayama, T., Y. Namai, T. Sakakibara, M. Hedo, Y. Uwatoko,H. Shishido, R. Settai, and Y. Onuki, 2005, J. Phys. Soc.Jpn. , 1115.Tayama, T., T. Sakakibara, H. Sugawara, Y. Aoki, andH. Sato, 2003, J. Phys. Soc. Jpn. , 1516.Tenya, K., I. Kawasaki, K. Tameyasu, S. Yasuda,M. Yokoyama, H. Amitsuka, N. Tateiwa, and T. C.Kobayashi, 2005, Physica B , 1135.Tenya, K., K. Kuwahara, H. Amitsuka, T. Sakakibara,H. Ohkuni, Y. Inada, E. Yamamoto, Y. haga, andY. Onuki, 2000, Physica B , 991.Terashima, T., K. Enomoto, T. Konoike, T. Matsumoto,S. Uji, N. Kimura, M. Endo, T. Komatsubara, H. Aoki,and K. Maezawa, 2006, Phys. Rev. B (14), 140406.Terashima, T., C. Haworth, M. Takashita, H. Aoki, N. Sato,and T. Komatsubara, 1997, Phys. Rev. B (20), R13369.Terashima, T., T. Matsumoto, C. Terakura, S. Uji,N. Kimura, M. Endo, T. Komatsubara, and H. Aoki, 2001,Phys. Rev. Lett. , 166401.Terashima, T., T. Matsumoto, C. Terakura, S. Uji,N. Kimura, M. Endo, T. Komatsubara, H. Aoki, andK. Maezawa, 2002, Phys. Rev. B , 174501.Thalmeier, P., and B. L¨uthi, 1991, The electron-phonon in-teraction in intermetallic compounds (North-Holland), vol-ume 14 of
Handbook of the Physics and Chemistry of RareEarths .Thalmeier, P., and G. Zwicknagl, 2005,
Unconventional Su-perconductivity and Magnetism in Lanthanide and Ac-tinide Intermetallic Compounds (North-Holland, Amster-dam, Holland), volume 34 of
Handbook of the Physics andChemistry of Rare Earths , chapter 219.Thalmeier, P., G. Zwicknagl, O. Stockert, G. Sparn, andF. Steglich, 2005,
Unconventional Superconductivity andMagnetism in Lanthanide and Actinide Intermetallic Com-pounds (Springer Verlag, Berlin), volume XXXII of
Fron-tiers in Magnetic Materials/Frontiers in SuperconductiveMaterials , chapter 109–182, condmat/0409363.Thomas, F., J. Thomasson, C. Ayache, C. Geibel, andF. Steglich, 1993, Physica B , 303. Thomas, F., B. Wand, T. L¨uhmann, P. Gegenwart, G. R.Stewart, F. Steglich, J. P. Brison, A. Buzdin, L. Gl´emont,and J. Flouquet, 1995, J. Low Temp. Phys. , 117.Thompson, J. D., A. A. Ekimov, V. A. Sidorov, E. D. Bauer,L. A. Morales, F. Wastin, and J. L. Sarrao, 2006a, J. Phys.Chem. Solids , 557.Thompson, J. D., M. Nicklas, V. A. Sidorov, E. D. Bauer,R. Movshovich, N. J. Curro, and J. L. Sarrao, 2006b, J.Alloys and Comp. , 16.Thompson, J. D., T. Park, N. J. Curro, and J. L. Sarrao,2006c, J. Phys. Soc. Jpn. , Suppl. 1.Thompson, J. D., R. D. Parks, and H. Borges, 1986, J. Mag.Mag. Mater. , 377.Thompson, J. D., J. L. Sarrao, L. A. Morales, F. Wastin, andP. Boulet, 2004, Physica C , 10.Tinkham, M., 1969, Introduction to Superconductivity (Mar-cel Dekker, New York, USA).Togano, K., P. Badica, Y. Nakamori, S. Orimo, H. Takeya,and K. Hirata, 2004, Phys. Rev. Lett. (24), 247004.Tokiwa, Y., R. Movshovich, F. Ronning, E. D. Bauer,P. Papin, A. D. Bianchi, J. F. Rauscher, S. M. Kau-zlarich, and Z. Fisk, 2008, Physical Review Letters (3),037001 (pages 4), URL http://link.aps.org/abstract/PRL/v101/e037001 .Tou, H., Y. Kitaoka, K. Asayama, N. Kimura, Y. ¯Onuki,E. Yamamoto, and K. Maezawa, 1996, Phys. Rev. Lett. (7), 1374.Tou, H., Y. Kitaoka, T. Kamatsuka, K. Asayama, C. Geibel,C. Schank, F. Steglich, S. S¨ullow, and J. Mydosh, 1997,Physica B , 360.Tran, V. H., S. Paschen, R. Tro´c, M. Baenitz, and F. Steglich,2004, Phys. Rev. B (19), 195314.Tran, V. H., R. Troc, and G. Andr´e, 1998, J. Magn. Magn.Mater. , 81.Trappmann, T., H. v. L¨ohneysen, and L. Taillefer, 1991, Phys.Rev. B (16), 13714.Trovarelli, O., M. Weiden, R. M¨uller-Reisener, M. G´omez-Berisso, P. Gegenwart, M. Deppe, C. Geibel, J. G. Sereni,and F. Steglich, 1997, Phys. Rev. B (2), 678.Turel, C. S., J. Y. T. Wei, W. . Yuhasz, R. Baumbach, andM. B. Maple, 2008, J. Phys. Soc. Jpn., Suppl. A , 21.Uemura, S., G. Motoyama, Y. Oda, T. Nishioka, and N. K.Sato, 2005, J. Phys. Soc. Jpn. , 2667.Urbano, R. R., B.-L. Young, N. J. Curro, J. D. Thompson,L. D. Pham, and Z. Fisk, 2007, Physical Review Letters (14), 146402.Ushida, Y., H. Nakane, T. Nishioka, G. Motoyama, S. Naka-mura, and N. K. Sato, 2003, Physica C , 525.Vargoz, E., D. Jaccard, J. Y. Genoud, J. P. Brison, andJ. Flouquet, 1998, Solid State Communications , 631.Varma, C. M., and L. Zhu, 2006, Phys. Rev. Lett. , 036405.Vekhter, I., and A. Houghton, 1999, Phys. Rev. Lett. (22),4626.Villaume, A., D. Aoki, Y. Haga, G. Knebel, R. Boursier, andJ. Flouquet, 2007, contmat/0707.3029.Vincent, E., J. Hammann, L. Taillefer, K. Behnia, andN. Keller, 1991, J. Phys.: Condens. Matter , 3517.de Visser, A., K. Bakker, L. T. Tai, A. A. Menovsky,S. Mentink, G. Nieuwenhuys, and J. Mydosh, 1993, PhysicaB , 291.de Visser, A., A. Menovsky, and J. J. M. Franse, 1987, PhysicaB , 81.de Visser, A., H. Nakotte, L. Tai, A. A. Menovsky, S. Mentink,G. Nieuwenhuys, and J. A. Mydosh, 1992, Physica B , , 2069.Vollhardt, D., and P. W¨olfle, 1990, The superfluid phases of He (Taylor & Francis, New York, USA).Vollmer, R., A. Faisst, C. Pfleiderer, H. von Lohneysen, E. D.Bauer, P. C. Ho, V. Zapf, and M. B. Maple, 2003, Phys.Rev. Lett. , 057001.Vollmer, R., C. Pfleiderer, H. v. L¨ohneysen, E. D. Bauer, andM. B. Maple, 2002, Physica B , 112.Volovik, G. E., 1993, JETP Lett. , 469.Waldram, J. R., 1996, Superconductivity of Metals andCuprates (IoP Publishing Ltd., London, UK).Walker, I. R., F. M. Grosche, D. M. Freye, and G. G. Lon-zarich, 1997, Physica C , 303.Wang, Q., H.-Y. Chen, C.-R. Hu, and C. S. Ting, 2006, Phys.Rev. Lett. (11), 117006.Wassermann, A., and M. Springford, 1994, Physica B , 1801.Wastin, F., P. Boulet, J. Rebizant, E. Colineau, and G. H.Lander, 2003, J. Phys.: Condens. Matter , S2279.Wastin, F., E. Colineau, P. Javorsky, and J. Rebizant, 2006,J. Phys. Soc. Jpn. , Suppl. 10.Watanabe, S., and K. Miyake, 2002, J. Phys. Soc. Jpn. ,2489.Watanabe, T., K. Izawa, Y. Kasahara, Y. Haga, Y. Onuki,P. Thalmeier, K. Maki, and Y. Matsuda, 2004a, Phys. Rev.B (18), 184502.Watanabe, T., Y. Kasahara, K. Izawa, T. Sakakibara, Y. Mat-suda, C. J. van der Beek, T. Hanaguri, H. Shishido, R. Set-tai, and Y. Onuki, 2004b, Phys. Rev. B (2), 020506.Wei, W., Z. Tun, W. J. L. Buyers, B. D. Gaulin, T. E. Mason,J. D. Garrett, and E. D. Isaacs, 1992, J. Magn. Magn.Mater. , 77.Weickert, F., P. Gegenwart, H. Won, D. Parker, and K. Maki,2006, Phys. Rev. B (13), 134511.Werthamer, N. R., E. Helfand, and P. C. Hohenberg, 1966,Phys. Rev. (1), 295.Wheatly, J. C., 1975, Rev. Mod. Phys. , 415.White, R. M., and T. H. Geballe, 1979, Long range order insolids (Academic Press, New York, USA).Wiebe, C. R., J. A. Janik, G. J. MacDougall, G. M. Luke,J. D. Garrett, H. D. Zhou, Y.-J. Jo, L. Balicas, Y. Qiu,J. R. D. Copley, Z. Yamani, and W. J. L. Buyers, 2007,Nature Physics , 96.Wilhelm, H., S. Raymond, D. Jaccard, O. Stockert, H. vonL¨ohneysen, and A. Rosch, 2001, J. Phys.: Condens. Matter , L329.Willis, J. O., J. D. Thompson, Z. Fisk, A. de Visser, J. J. M.Franse, and A. Menovsky, 1985, Phys. Rev. B (3), 1654.Wolfer, W. G., 2000, Los Alamos Sci. , 274.W¨uchner, S., N. Keller, J. L. Tholence, and J. Flouquet, 1993,Solid State Communic. , 355.Xiao, H., T. Hu, C. C. Almasan, T. A. Sayles, and M. B.Maple, 2008, Physical Review B (Condensed Matter andMaterials Physics) (1), 014510.Yamagami, H., D. Aoki, Y. Haga, and Y. Onuki, 2008, J.Phys. Soc. Jpn. , 76.Yanase, Y., and M. Sigrist, 2007, J. Phys. Soc. Jpn. ,043712.Yang, K., and A. H. MacDonald, 2004, Phys. Rev. B (9),094512.Yang, S.-H., H. Kumigashira, T. Yokoya, A. Chainani, T. Takahashi, S.-J. Oh, N. Sato, and T. Komatsubara,1996, J. Phys. Soc. Jpn. , 2685.Yano, K., T. Sakakibara, T. Tayama, M. Yokoyama, H. Amit-suka, Y. Homma, P. Miranovi´c, M. Ichioka, Y. Tsutsumi,and K. Machida, 2008, Physical Review Letters (1),017004.Yaouanc, A., P. Dalmas de R´eotier, G. van der Laan, A. Hiess,J. Goulon, C. Neumann, P. Lejay, and N. Sato, 1998, Phys.Rev. B (13), 8793.Yashima, M., S. Kawasaki, Y. Kawasaki, G. q. Zheng, Y. Ki-taoka, H. Shishido, R. Settai, Y. Haga, and Y. Onuki, 2004,J. Phys. Soc. Jpn. , 2073.Yasuda, T., H. Shishido, T. Ueda, S. Hashimoto, R. Settai,T. Takeuchi, T. D. Matsuda, Y. Haga, and Y. Onuki, 2004,J. Phys. Soc. Jpn. , 1657.Yelland, E. A., S. M. Hayden, S. J. C. Yates, C. Pfleiderer,M. Uhlarz, R. Vollmer, H. v. Lohneysen, N. R. Bernhoeft,R. P. Smith, S. S. Saxena, and N. Kimura, 2005, cond-mat/0502341.Yin, G., and K. Maki, 1993, Phys. Rev. B (1), 650.Yogi, M., Y. K. S. Hashimoto, T. Yasuda, R. Settai, T. D.Matsuda, Y. Haga, Y. Onuki, P. Rogl, and E. Bauer, 2004,Phys. Rev. Lett. , 027003.Yogi, M., H. Mukuda, Y. Kitaoka, S. Hashimoto, T. Yasuda,R. Settai, T. D. Matsuda, Y. Haga, Y. Onuki, P. Rogl, andE. Bauer, 2006, J. Phys. Soc. Jpn. , 013709.Yogi, M., T. Nagai, Y. Imamura, H. Mukuda, Y. Kitaoka,D. Kikuchi, H. Sugawara, Y. Aoki, H. Sato, and H. Harima,2008, J. Phys. Soc. Jpn., Suppl. A , 31.Yokoyama, M., H. Amitsuka, K. Tenya, K. Watanabe,S. Kawarazaki, H. Yoshizawa, and J. A. Mydosh, 2005,Phys. Rev. B , 214419.Yokoyama, M., H. Amitsuka, K. Watanabe, S. Kawarzarki,H. Yoshizawa, and J. A. Mydosh, 2002, J. Phys. Soc. Jpn.(Suppl.) , 264.Yokoyama, M., N. Oyama, H. Amitsuka, S. Oinuma,I. Kawasaki, K. Tenya, M. Matsuura, K. Hirota, and T. J.Sato, 2008, Physical Review B (Condensed Matter and Ma-terials Physics) (22), 224501.Young, B.-L., R. R. Urbano, N. J. Curro, J. D. Thompson,J. L. Sarrao, A. B. Vorontsov, and M. J. Graf, 2007, Phys.Rev. Lett. (3), 036402.Young, D. P., M. Moldovan, X. S. Wu, P. W. Adams, andJ. Y. Chan, 2005, Phys. Rev. Lett , 107001.Yuan, H. Q., D. F. Agterberg, N. Hayashi, P. Badica, D. Van-dervelde, K. Togano, M. Sigrist, and M. B. Salamon, 2006,Phys. Rev. Lett. (1), 017006.Yuan, H. Q., F. M. Grosche, M. Deppe, C. Geibel, G. Sparn,and F. Steglich, 2004, Science , 2104.Zapf, V. S., E. J. Freeman, E. D. Bauer, J. Petricka, C. Sir-vent, N. A. Frederick, R. P. Dickey, and M. B. Maple, 2001,Phys. Rev. B (1), 014506.Zheng, G., N. Yamaguchi, H. Kan, Y. Kitaoka, J. L. Sarrao,P. G. Pagliuso, N. O. Moreno, and J. D. Thompson, 2004,Phys. Rev. B (1), 014511.Zheng, G.-q., K. Tanabe, T. Mito, S. Kawasaki, Y. Kitaoka,D. Aoki, Y. Haga, and Y. Onuki, 2001, Phys. Rev. Lett. (20), 4664.Zwicknagl, G., and U. Pulst, 1993, Physica B , 895.Zwicknagl, G., A. N. Yaresko, and P. Fulde, 2002, Phys. Rev.B65