Heavy Fermions and Superconductivity: Ten Years After F

Heavy Fermions and Superconductivity: Ten Years After F. Steglich, Darmstadt (Institut für Festkörperphysik, Technische Hochschule, D-6100 Darmstadt) The 1989 Hewlett-Packard Europhysics Prize was awarded jointly to Professor Frank Steglich, Professor Hans R. Ott of ΕΤΗ Zurich and Dr. Gilbert Lonzarich of the Cavendish Laboratory, Cambridge, for their pioneering investigations of heavy fermion metals. In addi­ tion to the intrinsic interest of these unforeseen materials they are ideal for testing models of other physical phenomena such as band magnetism and superconductivity.

In 1979, superconductivity was dis­ correlations, give rise to magnetic phe­ large specific heat jump height and the covered below Tc  0.6 K for the com­ nomena, whereas conduction-electron very steep slope of the curve of the pound CeCu2Si2 [1]. Since the trivalent systems of more delocalized (s-, p-, 5d-, upper critical magnetic field Bc2(T), Ce ions in this material carry local mo­ 4d-) electrons favour a superconducting separating the superconducting and the ments associated with the singly oc­ state. normal phases (Fig. 1b), prove that the cupied 4f-shells, a magnetically ordered CeCu2Si2 was the first example ex­ Cooper pairs in CeCu2Si2 are formed by rather than a superconducting ground hibiting "superconductivity of strongly the heavy fermions. Consequently, the state was to be expected. However, correlated electrons", in that its 4f elec­ homologue LaCu2Si2 with no 4f elec­ upon cooling CeCu2Si2 to below a cha­ trons are responsible not only for the trons and no heavy-fermion properties racteristic temperature, T*  15 K, the local magnetic moments but also for is not a superconductor. local moments become quenched, and superconductivity. This can be inferred the system approaches a metallic ("Fer­ from Fig. 1a, which shows that the mi-liquid") low-temperature phase. specific heat discontinuity at T is of Kondo Lattice These observations are reminiscent the same gigantic size as the normal- Although for all heavy-fermion com­ and, in fact, closely related [2] to the state specific heat, Cn(Tc), which is of pounds the separation of neighbouring so-called "Kondo effect" as frequently electronic origin since the phonon con­ f-ions is much too large to allow a sig­ observed in dilute magnetic alloys like tribution can be completely neglected nificant f-wavefunction overlap, hybridi­ CuFe or LaCe. at these low temperatures. The asymp­ zation with ligand orbitals tends to des­ The finding of bulk superconductivity totic specific heat behaviour in the nor­ tabilize the magnetic configuration of in a trivalent cerium intermetallic was mal state as extrapolated from above Tc the f-shell. In the case of weak hybridi­ surprising for most experimentalists, as to T → 0 is Cn =  T, where the coeffi­ zation, the Kondo effect is relevant [2], usually, less than 1 at% Ce3+ dopant cient  (  1 J/K2mol) exceeds that of while stronger hybridization may give suffices to suppress superconductivity simple metals like Al or Pb by two to rise to instabilities of the f-occupation, in a BCS superconductor: an exchange three orders of magnitude,  indicates i.e. via charge or "valence" fluctuations interaction between the conduction- the effective masses of the low-energy [4]. For most heavy-fermion com­ electron spins and the spins of the loca­ excitations ("quasiparticles") of the pounds, however, inter-site couplings lized 4f-shells at the Ce sites breaks the normal Fermi-liquid phase which are ap­ compete with the single-site Kondo Cooper pairs. The pair-breaking capabi­ proximately 300 times larger than the interaction and cause a magnetically lity of paramagnetic ions demonstrates free-electron mass, me . These "heavy ordered state to develop in the range of the antagonistic relationship between fermions" can be interpreted as 4f elec­ liquid helium temperatures. The arche­ magnetism and classical superconduc­ trons that have been weakly delocalized typical systems CeAI2 and NpSn3, both tivity: sufficiently localized (3d-, 5f- and by a Kondo-type effect and, thus, have studied in the 1970s, are characterized 4f-> electrons, whose motion is domi­ become itinerant states right at the by reduced ordered magnetic moments nated by their intra-atomic Coulomb Fermi energy, EF [2], Both the extremely (compared with those of the correspon­ ding free f-ions) and strongly enhanced electronic specific heats (compared with their non-f homologues, e.g., LaAI2 and LaSn3), see Figs. 2a and b: the Kondo coupling sets in already at eleva­ ted temperatures and gradually reduces the local moments upon cooling. This process is associated with a gradual piling up of a resonance at the Fermi energy, indicating an increasing effec­ tive mass of the quasiparticles [2]. For an illustration, the increase of the coef­ ficient y (being proportional to m*) is shown in Fig. 2a, as it would be expec­ Fig. 1 — Specific heat as C/T vs T (a) and upper critical field as Sc2 vs T (b) for the same ted within the Kondo-impurity model for sample of CeCu2Si2 (Steglich F. et al., Physics 126B (1984) 82). Dashed line in (b) indi­ fictitious paramagnetic CeAI2. In reali­ cates the giant Bc2(T) slope at Tc (-13 T/K). ty, the magnetic interactions dominate 159 lity against superconducting and ma­ gnetic phase transitions.

Cooperative Phenomena In contrast to experimentalists, theo­ rists seemed to look with favour on the observation of heavy-fermion supercon­ ductivity in CeCu2Si2: the ratio Tc/TF*, where kBTF*is the effective width of the heavy-fermion band at EF, greatly ex­ ceeds the ratio Tc/TF for classical super­ conductors, whose conduction-band width is of the order of the Fermi energy Fig. 2 — C/T vs T for CeAI2 (Bredl C.D. et al., J. Magn. Magn. Mat. 9 (1978) 60) (a) and EF = kBTF. It is even considerably larger NpSn3 (Trainor R.J. et al., Phys. Rev. Lett. 34 (1976) 1019) (b). Solid lines mark specific heat coefficient y (as T — 0). Dashed curve in (a) represents result of S = 1/2 Kondo-impu- than Tc/TF for the cuprate supercon­ rity model (the Kondo temperature TK is chosen to equal the characteristic temperature ductors. Therefore, heavy-fermion com­ T* = 3.5 K). pounds may be considered "high-Tc superconductors" in a figurative sense. over the Kondo interaction at low T, cientA ~ (m*)2, the quadratic term for In particular, an estimate of the renor­ so that long-range antiferromagnetism a "Kondo lattice" like CeCu6 exceeds malized Fermi velocity reveals the same develops below the Néel temperature the corresponding one for simple metals order of magnitude as for the velocity of TN = 3.9 K. by several orders of magnitude. This sound. As a result, the phase space For a relatively small number of com­ unique property is shown in Fig. 3b for for the BCS-type coupling mechanism pounds, this type of ordering between CeAI3 (T*  4 K), the prototype sys­ appears to be strongly reduced, so that local moments does not occur since the tem that was discovered by Klaus the latter came under early suspicion. demagnetizing interaction finally wins. Andres and his collaborators in 1975. By analogy to the Fermi liquid 3He, Even in the absence of long-range ma­ The new band of coherent heavy- which turns superfluid in the mK-range, gnetic order, however, heavy-fermion fermion states at EF can interact with a non-phononic Cooper-pairing mecha­ compounds behave qualitatively diffe­ the ordinary conduction bands, even­ nism was presupposed. In 3He, short­ rently from their related dilute alloys. tually leading to structure in the reso­ lived magnetic excitations ("spin fluc­ This is demonstrated in Fig. 3a which nant density of states as a function of tuations") instead of phonons are as­ gives the resistivity results by Yoshio energy. Accordingly, a low-temperature sumed to be essential for the formation Onuki and Takemi Komatsubara for the maximum may show up in  (T), as first of Cooper pairs. In addition, the on-site quasi-binary system La1 xCexCu6. At noticed in 1981 by Jacques Flouquet Coulomb repulsion between heavy fer­ low Ce concentration, the Ce's are dis­ and his collaborators for CeAl3 (max  mions, which resembles the hard-core tributed at random on the La sites and 1.6 J/K2mol). This phenomenon is de­ repulsion between two 3 He atoms, is are subject to a Kondo effect. The elec­ monstrated in Fig. 1a for normal-state invoked in arguments against an s-wave trical resistivity which approaches its CeCu2Si2 as measured at B = 2 T. The Cooper-pair state as in a BCS supercon­ maximum as T approaches absolute most convincing verification of a Fermi ductor. A node of the pair-wave func­ zero and decreases continuously upon surface built up by the heavy fermions tion (or the order parameter) is expected warming, demonstrates the action of a in a lattice of f-ions is due to the obser­ to occur at the origin of the relative resonance scattering: heavy fermions vation of de-Haas - van-Alphen oscilla­ coordinate [2], So continuing the analo­ become formed locally, as can also be tions both in UPt3 and CeCu6. They gy to 3He, whose Cooper pairs are in a inferred from the anomalous rise in y(T) prove the existence of effective masses, spin-triplet state, while BCS supercon­ displayed in Fig. 2a. For the compound m*, up to 90 me and 40 me , respective­ ductors exhibit singlet pairs, CeCu2Si2 CeCu6 a qualitatively similar resistivity ly (see the articles by Gil Lonzarich and was assumed to be the first triplet su­ curve is found well above the charac­ Mike Springford in [5]). Though very perconductor. teristic temperature T*  4 K, whereas interesting in itself, most excitement However, this possibility was quickly a quite different behaviour is observed about the low-T Fermi-liquid state of a discarded by the subsequent investiga­ at low temperatures. Most importantly, Kondo lattice is derived from its instabi­ tions: while the large initial slope of the resistivity reaches a maximum near T* rather than as T → 0. This demon­ strates that, as a consequence of the translational symmetry in the Ce sub­ system, the incoherent resonance scat­ tering disappears. No matter how ano­ malous the scattering potential of the single Ce ions, upon building a periodic lattice, at sufficiently low temperature they give rise to "coherent scattering": the scattering processes between the heavy fermions, represented by exten­ ded ("Bloch") states well below T = Fig. 3 — a: Ce increment of the electrical resistivity for the dilute Kondo alloy La0.906 T*, give rise to a contribution AT2 to Ce0.o94Cu6 and the "Kondo lattice" CeCu6 (Onuki Y. and Komatsubara T., J. Magn. Magn. the resistivity, which is a hallmark of a Mat. 63 & 64 (1987) 281). b: Resistivity vs T2 for CeAI3 below T = 0.1 K (Andres K. et al., (charged) Fermi liquid. Since the coeffi- Phys. Rev. Lett. 35 (1975) 1779). 160 Bc2(T) results from a “diamagnetic effect", which acts on the momentum pairing of the Cooper pairs via the Lorentz force, the pronounced flattening of the Bc2(T) curve at lower tempera­ tures (Fig. 1b) proves the importance of a second (paramagnetic) pair-breaking effect of the external magnetic field, which acts on the spin state of the Cooper pairs. In a rotational-invariant system like 3He this so-called "Pauli limiting" should be absent for triplet pairs. Measurements of the orientation Fig. 4 — UPt3: (a) C/T vs T results reveal two closely spaced superconducting transitions dependence of the Pauli-limit for at 0.49 K and 0.43 K. If they are replaced by a single transition, this would be located at CeCu2Si2 single crystals revealed that 0.47 K. Extrapolation of the superconducting data to T → 0 suggests a dependence Cs = triplet pairing in this material is most  ST +  sT2 (Fisher R.A. et al., Phys. Rev. Lett. 62 (1989) 1411). (b) Neutron-scattering- unlikely. Additional support for this con­ intensity results for the (1/2,0,1) reflex as a function of temperature indicate interference clusion was derived from careful mea­ between superconductivity and antiferromagnetic order (Aeppli G. et al., Phys. Rev. Lett. surements of the DC-Josephson cur­ 60 (1988) 615). rent through weak links between below Tc by a T2 dependence of the tration depth shown in Fig. 5a. On the CeCu2Si2 and clear-cut BCS super­ specific heat. According to Fig. 4a, other hand, a T3-dependence of the conductors like Al and Nb. Whereas in UPt3 may be a d-wave superconductor. nuclear spin-lattice relaxation rate was the case of triplet-superconducting The second class of unconventional found for this material and interpreted CeCu2Si2 no Josephson effect would order parameters is characterized by as evidence for d-wave superconduc­ be expected prior to ordering, a Joseph­ gap zeros at certain symmetry points on tivity. This illustrates that the observa­ son current of the size typical for sin­ the Fermi surface as expected with, tion of "simple power laws" in heavy- glet-singlet junctions was observed. e.g., (triplet) p-wave superconductors. fermion superconductors may be fortui­ Though presumably a singlet super­ Such materials should show a T3 beha­ tous. Sometimes a thorough analysis conductor, CeCu2Si2 as well as the viour of the specific heat as T — 0 in of experimental data yields no power three U-based heavy-fermion supercon­ apparent agreement with experimental laws at all, whilst they are often re­ ductors UBe13, UPt3 and URu2Si2 (as results for UBe13. A triplet state of gistered close to Tc (cf. Fig. 4a) when discovered in 1983 and 1984 by Hans UBe13 has also been inferred from the they ought to occur far below Tc [3]. In Ott, Greg Stewart, Wolfram Schlabitz T2-dependence of the magnetic pene­ the presence of lattice defects, more and their collaborators) seem to exhibit highly anisotropic order parameters. This is inferred, for example, from the non-exponential temperature depen­ dence of the low-temperature specific Physica heat (see Fig. 4a) and related transport properties. Small values of the super­ conducting energy gap in certain direc­ Scripta tions lead to an increased density of An international journal for excited states at low energies which experimental and theoretical physics Vol. T29 1989 raises the low-T specific heat above the (exponential) BCS result. No agreement can presently be achieved, however, as Proceedings of the 9th General Conference to the type of order-parameter aniso­ of the EPS Condensed Matter Division tropy. Arguments have been presented Nice, 6-9 March 1989 in favour of both "unconventional" Edited by: J. Friedel, J.P. Laheurte, J.P. Romagnan and "conventional" order parameters In addition to the texts of six plenary talks the issue contains 54 whose symmetry is either lower than or papers submitted to the Conference. These are arranged in the equal to the symmetry of the Fermi sur­ order in which they were presented. face. Since the de-Haas - van-Alphen Although the section on superconductors is particularly signifi­ measurements as well as theoretical cant, of equal importance are the contributions on charge density considerations [3] reveal strong Fermi- waves, surfaces and interfaces, semiconductors (e.g. MBE struc­ tures and optical phenomena), industrial applications and mate­ surface anisotropies, conventional or­ rials science, phonons and fractons, liquids, polymers and quan­ der parameters in HFS would also have tum fluids. to be highly anisotropic. A volume of 310 pages of topical information. Among unconventional order para­ meters, two different general classes Orders to Physica Scripta at the address shown below; are distinguished according to group- Price: SKR 460,- or US$ 70- theoretical considerations: the first Published by the class involves zeros of the energy gap ROYAL SWEDISH ACADEMY OF SCIENCES along lines on the Fermi surface and is Box 50005 ISSN 0281-1847 represented by, e.g., a (singlet) d-wave S-10405 Stockholm / Sweden ISBN 91-87308-56-8 function. It should be manifested well 161 mode" of the lattice, which is associa­ ted with large volume changes, couples strongly to the heavy fermions. Experimental evidence for this parti­ cular electron-phonon coupling is found in striking anomalies of the elastic cons­ tants (as determined in Bruno Lüthi's group), thermal expansion and longi­ tudinal sound absorption (see Fig. 5b). Outlook Heavy-fermion systems have provi­ ded a number of exciting phenomena Fig. 5 — UBe13: (a) increment of the magnetic field penetration depth as a function of which were quite unexpected within (T/Tc)2. For comparison, data for the BCS superconductor Sn are also shown (Einzel D. et al., Phys. Rev. Lett. 56 (1986) 2513). (b) Longitudinal ultrasonic attenuation data taken as traditional concepts of metal physics. a function of temperature at 1.7 GHz for B = 0 and 2 T display the superconducting phase They allow us to investigate, for exam­ transition (Golding B. et al., Phys. Rev. Lett. 55 (1985) 2479). ple, how band magnetism develops in a material characterized at elevated tem­ complex T-dependencies are predicted the pairing interaction is presently not peratures by a dense array of local mo­ theoretically [4]. Simple power-law de­ known. Arguments in favour of magne­ ments. In addition, they are ideal testing pendencies have been established also tic coupling, in addition to the seeming grounds for models concerned with for classical superconductors. For analogy to superfluid 3He, were provi­ superconductivity in strongly correlated example, the specific heat of HfV2 (TC ded by the Los Alamos group who electron systems and they may have = 9 K) is well described by a T3 depen­ noticed that superconductivity in UPt3 some impact on the future understan­ dence over a large temperature regime. "possibly coexists" with the pronoun­ ding of the phenomenon of high-Tc Anisotropies of the order parameter ced spin-fluctuation phenomena that superconductivity in Cu-oxides. should be smeared out by potential had been previously discovered for this scattering. In fact, non-magnetic do­ compound by Jaap Franse and his col­ REFERENCES leagues. In the meantime, it has been pants like Th, La and Y partially substi­ [1] For an early review, see: Stewart G.R., tuted for the f-ions in CeCu2Si2 and found using sensitive microscopic pro­ Rev. Mod. Phys. 56 (1984) 755. UBe13 act qualitatively in a way similar bes like muon-spin rotation, nuclear [2] See, e.g. : Fulde P., Europhys. News 16 to magnetic dopants in a BCS supercon­ magnetic resonance and neutron-dif­ (1985) 14. ductor. On the other hand, the “pair fraction measurements that magnetic [3] For recent reviews, see: Lee P.A., Rice breaking" by non-magnetic scatterers correlations dominate the low-7 "Fer­ T.M., Serene J.W., Sham L.J. and Wilkins is surprisingly weak if estimated on mi-liquid" properties of certain heavy- J.W., Comments on Condensed Matter the assumption of an isotropic potential fermion compounds like CeCu6 and Phys. 12 (1986) 99; Ott H.R., Progr. Low scattering. This difficulty adds to CeRu2Si2 and even give rise to a coope­ Temp. Phys. XI (1987) 217; Fulde P, Keller J. another serious problem: for both rative antiferromagnetic state in others and Zwicknagl G., Solid State Phys. 41 (1988) 1; Grewe N. and Steglich F., Hand­ CeCu2Si2 and UBe13, the mean free like URu2Si2, UPt3 and U1-xThxBe13 book on the Physics and Chemistry of Rare path of the heavy fermions estimated (0.01 < x < 0.05). Since for these sys­ Earths, Vol. 14, Eds. K.A. Gschneidner Jr. from the measured normal-state resisti­ tems the ordered moments are usually and L. Eyring, forthcoming. vities are considerably shorter than the extremely small ( 10-2µB/f-ion), a ma­ [4] See, e.g. : Müller-Hartmann E., Europhys. superconducting coherence lengths. gnetic instability of the coherent heavy- News 13 (1982) 7, 9. These observations highlight a strong fermion band seems to lead to "band [5] Proceedings of the 6th Int. Conf. on anisotropy of the scattering rate in magnetism" in the Fermi liquid; this Crystal-Field Effects and Heavy-Fermion Kondo lattices [3] which has to be un­ effect should be distinguished from the Physics, Eds. W. Assmus, P. Fulde, B. Lüthi ravelled by future work. "local-moment ordering" in heavy-fer­ and F. Steglich, Frankfurt (1988), published As with the shape of the supercon­ mion systems like CeAI2. The coexis­ in J. Magn. Magn. Mat. 76 & 77 (1988). ducting order parameter, the nature of tence between this form of band an­ tiferromagnetism and superconducti­ Hewlett-Packard vity is illustrated in Figs. 4b and 6 for both UPt3 and URu2Si2. More than one Europhysics Prize superconducting phase transition seem To celebrate the 50th anniversary of to occur in UPt3 (Fig. 4a) and in the company's foundation and the 30th Ul-xThxBe13 and they have been asso­ anniversary of its establishment in Eu­ ciated with both unconventional and rope, Hewlett-Packard Europe decided conventional order parameters. to double the value of the 1989 Hew­ Alternatively, an electron-phonon lett-Packard Europhysics Prize from the coupling unique for Kondo lattices had previous 20 000 SFR to 40 000 SFR. been proposed to explain superconduc­ We are delighted to announce that the tivity with Tc  1 K in CeCu2Si2 [2, 3], company proposes to take this as a precedent and will maintain the figure in Fig. 6 — URu2Si2:  vs T (on a logarithmic It is based upon a pronounced volume scale). Arrows mark the onsets of anti­ dependence of kBT*, the characteristic the coming years. ferromagnetism (TN) and superconductivity energy or heavy-fermion band width The winner(s) of the 1990 Hewlett- (Tc). Both phenomena are coexisting at low and can be expressed by a giant "Grün- Packard Europhysics Prize will be an­ temperatures (Schlabitz W. et al., Z. Phys. eisen parameter",  = -d ln (kBT*)/ nounced in the February issue of Euro­ B62 (1986) 171). dlnV  100. Therefore, the "breathing physics News. 162