Experimental study of Ce-based heavy-fermion compounds

Journal of Magnetism and Magnetic Materials 100 (1991) 186-203 North-Holland

Experimental study of Ce-based heavy-fermion compounds * F. Steglich lnstitut fiir FestkiJrperphysik, TechnischeHochschule Darmstadt, W-6100Darmstadt, Germany

In the first part of the paper, a brief survey is given on the development of heavy-fermionphysicswith special emphasis on Ce-based intermetallic compounds. Subsequently, selected topics of current interest are discussed, e.g., heavy-fermionband magnetism, heavy-fermionsuperconductivity and its relationship to a novel kind of lattice instability.

I. Early work 1.1. Dilute Kondo alloys and intermediate-valence compounds Heavy-fermion physics, which is in the focus of current investigations of strongly correlated charge-carrier systems, has developed out of two roots, the Kondo-impurity [1] and the intermediate-valence (IV) [2,3] problems, respectively. Originally, experimental studies of the Kondoimpurity problem were mainly devoted to 3d dopants in noble metals [1]. Because of the rather large spatial extent of the 3d wavefunction, however, inter-ionic correlations were not easy to separate from the more essential single-ion effects; these difficulties becoming extremely serious because of the relatively poor solubility of 3d ions in noble metals. The breakthrough in this field occurred at the end of the 1960s, when rare-earth (RE) elements of sufficient purity became available: Due to the strong localization of the 4f electrons inside the Xe core (with full 5s25p 6 shells), the direct 4f-wavefunction overlap between R E ions on adjacent lattice sites is negligible. In addition, R E generally can be substituted in a wide composition range for La, as well as for Lu and Y. Consequently, the single-ion

* Dedicated to Professor Paul Kienle on the occasion of his 60th birthday.

effects can be studied reliably in systems like La~_xRExA12 [4] at RE-concentration levels up to several at%. Like in insulators, the diluted 4f shells in a metallic environment are characterized by the intra-shell Hund's rule correlations, yielding a 2S+ILj ( J = L _+ S) ground state with a magnetic m o m e n t # = - g j i x B J . H e r e S is the spin, L the orbital angular momentum, J the total angular momentum, gj the Land6 g-factor of the 4f shell and ~B the Bohr magneton. U n d e r the influence of the crystalline-electrical field (CF) (AcF, the CF-excitation energy being smaller than the spin-orbit energy by typically one order of magnitude) the (2J + 1)-fold degeneracy of the ground state is partly removed. The majority of the R E ions occur in the trivalent state only and are frequently labeled " n o r m a l " (or "stable-moment") RE. Some of the RE, i.e., Ce, Sm, Pr, Eu, T m and Yb, with electronic configurations close to the stable 4f °, 4f 7 and 4f 14 configurations, however, show an ambivalent behavior: in insulators, they occur not only in the trivalent but also in an adjacent (tetra and divalent, respectively) state. If dissolved in a metallic environment, ambivalent R E ions can give rise to Kondo and IV phenomena. Experiments performed on dilute alloys, particularly with Ce, but also with Yb, Sm and Pr dopants [5] revealed a richness of low-temperature anomalies, the so-called Kondo anomalies: upon cooling to below the Kondo temperature

0304-8853/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved

F. Stegfich / Ce-based heavy-ferrnion compounds

TK, for example, the resistivity increment increases, the "effective" local moment disappears and "heavy" quasiparticles are formed locally. The latter have 4f symmetry and can thus be considered weakly delocalized 4f-electrons [6]. It is reassuring that the aforementioned experimental efforts helped to develop a profound understanding of the Kondo-impurity problem [7]. As an especially exciting property of superconducting Kondo alloys we mention the possibility of more than one transition temperature in one sample. According to Miiller-Hartmann and Zittartz [8], this can happen by reason of a strongly temperature- and energy-dependent pair-breaking rate of the Kondo ions in a superconducting matrix. The first example showing a re-entrant Tc vs. x phase boundary was the quasi-binary system La~_xCe x A12 [9,10]. Later on, Winzer [11] was able to establish even three different Tc'S, i.e., in (La0. 8 Y0.2)l - x C e x •

When Ce-impurities are exerted to sufficiently high pressure (external or internal), a transition into a "non-magnetic" state is induced, which essentially means a transition into an IV state [12-14]. Here the hybridization energy between 4f and conduction-band States, A = avV2NF (V: average over hybridization matrix elements, NF: conduction-band density of states at the Fermi energy, EF), is at least comparable with, if not larger than, the valence-excitation energy e. By contrast, e/A = 3-5 for typical Kondo ions and e / A >> 1 for "stable-moment" RE. The strong 4f-conduction electron hybridization in the IV compounds containing ambivalent R E causes real charge fluctuations between the local 4f states and the conduction band ("valence fluctuations") which, well below the characteristic fluctuation temperature (100-1000 K), form a "homogeneous" IV state. Heavy-fermion compounds are found at the concentrated limit of the Kondo alloys and are, therefore, frequently labeled Kondo-lattice systems. The characteristic Kondo-lattice temperature T * replacing the single-ion Tg ranges between about a few tenths and a few tens of kelvin. Phenomenological differences between heavy-fermion and IV compounds have been established, e.g., for the magnetic neutron cross

187

section [15,16], thermopower [17,18], thermal expansion [19,20] and de Haas-van Alphen (dHvA) effect [21].

1.2. Kondo-lattice systems The experimental studies of Kondo-lattice systems began with resistivity measurements on Ce intermetallic compounds like CeA12 [22]. Typically, a negative temperature coefficient of the resistivity, p(T), similar to that of dilute Kondo alloys is found for T > T*. At low temperatures, however, the resistivity drops and reaches a residual value which is comparable to that of transition metals, of course depending on the disorder in the sample. The low-T resistivity behavior, therefore, manifests a major difference between the Kondo lattice and the Kondo alloy (fig. 1): Whereas the maximum resistivity (as T---, 0) in the latter reflects a resonance scattering off the single Kondo impurities at EF, the coupling of the phases of the scattered carriers in the former is visible as a "freezing out" of incoherent scattering below the maximum temperature Tmax ( = T*). Extended ("Bloch") states are formed by the weakly delocalized 4f electrons in the coherent low-temperature phase of the Kondo lattice [24], i.e., at T < Tooh << T*. In this low-T regime the resistivity is dominated by a contribution quadratic in temperature as first discovered by Andres et al. [25] for CeA13 (fig. 2). The gigantic coefficient of this electron-electron scattering law, A T 2, indicates a strong renormalization of the conduction-band states at E F. A is proportional to ( m * ) 2, where the effective carrier mass

........

-

~

........

i

........

i

300

L) E C~ 200

........

I

........

I

.......

CexLal_xCu6 x = 0.091. 0.29

........................

JIIb

............................... .~...

0.50

O,,,

loo o

................................. ~ "'"'7"~ 0.73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.90~ ............,~,:..... 0.99. - .................. ,,"" ,, :o,,,~ ............ ~ ............................ j 0.01 0.1 1 10 100 T(K)

Fig. 1. Temperature dependence of electrical resistivity increment per mole of cerium for Ce x Lai_xCu 6 [23].

F. Steglich / Ce-based heaLy-fermion compounds

188

is of the order of several hundred me~, the free electron mass. The gigantic magnitude of the A T e term is the hallmark for a very slow propagation of the 4f electrons under the dominant influence of the intra-atomic correlation energies [26,27]. Correspondingly gigantic values were observed for the Sommerfeld coefficient in the electronic specific heat, 3' = 1.6 J / K 2 tool, and the Pauli spin susceptibility, X0 = 4.5 × 10 -7 m3/mol [25], both being proportional to m* and exceeding the 3' and X0 of simple metals like Na by about three orders of magnitude. At sufficiently high temperatures, where p(T) shows a negative temperature coefficient, the susceptibility follows a Curie-Weiss law, characteristic of Ce-derived magnetic moments weakly coupled to the conduction electrons, and the Sommerfeld coefficient becomes as small as in ordinary d-band metals. In the intermetallic compounds, the demagnetizing Kondo interaction (of dominant on-site character) is competing with the inter-site indirect exchange interaction, the R u d e r m a n - K i t t e l Kasuya-Yosida interaction [28]. Using the coupling constant for the exchange between the local spin and the conduction-electron spins, g = NFJ ( J < 0: exchange integral), TRKKY~g2, whereas T * - T K ~ exp(1/g). Because of their different dependences on g, kBTRKKY exceeds kBT K for sufficiently low values of I g l, but becomes dominated by it, if I g[ is sufficiently large. Doniach [29] has derived from the competition of the Kondo and RKKY interactions a generalized

A

E t.a

1.1

:z 1.0 Q.

0.9 08

0.7

,

t

i

2

t+

6 8 T 2 (K 2 )

t

10 x10"3

Fig. 2. Electrical resistivity of C e A I 3 below I00 m K as p vs.

T 2 [25].

0

Igcl Igl

Fig. 3. Fundamental energies kBTRKKY ~ g2 and kBT K e x p ( - 1 / I g I) (upper part) as well as magnetic ordering temperature Tm and Kondo temperature T K (lower part) as a function of microscopic coupling parameter I g I, according to Doniach [29].

"phase diagram" for a one-dimensional Kondo lattice as schematically reproduced in fig. 3. There exists a critical value I gc I of I g l , below which the systems show long-range magnetic order and above which they have a non-magnetic ground state. The maximum in the dependence of the magnetic ordering temperature Tm on I gl reflects the competition between the increase in the RKKY interaction and the Kondo-derived reduction of the "effective" ordered m o m e n t / x s, Tm bt2sTRKKY.

Most of the Ce-based heavy-fermion compounds show an antiferromagnetically ordered ground state, in which somewhat reduced 4f moments are coupled below the Nfiel temperature TN = Tm via the RKKY interaction. Such a state will be labeled "local-moment magnetism" (LMM) in the following. As was first demonstrated for CeAl2, the moderate moment reduction [30] is accompanied by an also moderate carrier renormalization: The Sommerfeld coefficient 3' = 0.135 J / K 2 mol of CeAI 2 [31] exceeds the 3' value of the ferromagnet Fe by about a factor of twenty, but is smaller than the estimate for fictitious paramagnetic CeAI2 by more than a factor of ten, see fig. 4. As another early example for LMM we mention CeB 6 [32,33]. The oscillatory nature of the RKKY interaction may result, under fortunate conditions, in ferro- rather than antiferromagnetic order. Examples are CeSix, x < 1.85 [34,35], CeCu 2 [36] and CeRu2Ge 2 [37]. Disorder on the non-f ligand sites or on intersti-

F. Steglich / Ce-basedheavy-fermion compounds

0

5

10

T [KI Fig. 4. Specific heat as C / T vs. T for CeAI 2 [31]. Solid line t h r o u g h data points is guide to the eye. Thin horizontal line indicates l o w - t e m p e r a t u r e value, 3'0 = 135 m J / K 2 mol. D a s h e d line is Bethe-ansatz result for S = 1 / 2 K o n d o impurity with T K = 3.5 K [7].

tials surrounding Ce sites can even lead to a spin-glass type of ordering on a periodic Celattice, as was established first for C e C u 6 . 5 / k ] 6 . 5 [38] and subsequently confirmed for CePd3B0. 3 [39] and CePtGa 3 [40]. Transitions from local-moment magnetism to a non-magnetic heavy-fermion state have been induced by application of pressure [14] or suitable alloying [5,12,13], i.e., by increasing I gl. Only a few Ce compounds behave as non-magnetic heavy Fermi liquids already at ambient pressure, e.g., CeCu 6 [41,42] and CeRu2Si 2 [43], besides CeA13 [25].

189

first heavy-fermion superconductor CeCu2Si 2. Both the values of the specific-heat jump height [45] and the initial Bc2(T) slope [46] scale with the huge Sommerfeld coefficient 3' "-~0.7 J / K 2 mol. This proves that the heavy fermions (with effective mass m* = 200-300m 0 [46]) form the Cooper pair states in this material. Consequently, no superconductivity is found in the non-f homolog LaCu2Si 2 [45]. The observation that the typical heavy-fermion energy, kBT*, is much smaller than the typical phonon energy, kBO D (OD: Debye temperature), led to the suggestion that in CeCu2Si 2 the BCS-type deformation-potential coupling cannot be efficient for the Cooper-pair formation [45]. On the other hand, the seeming analogy [47] to the Fermi liquid 3He and its superfluid transition stimulated suggestions of an unconventional order parameter, having a symmetry lower than the crystal [26], as well as of a magnetic pairing mechanism. For a recent survey on the superconducting properties of heavy-fermion compounds, see ref. [48]. The discovery of bulk superconductivity in CeCu2Si 2 was received with much scepticism which, in fact, hindered a more rapid development of this field in the following years. One argument of the critics involved the complex ternary Ce-Cu-Si phase diagram [49] which made it difficult to disentangle intrinsic from extrinsic effects [50-53]. For example, single crystals grown from stoichiometric melts were not superconducting [51,54-56], whereas bulk-superconducting single crystals of small dimensions (1 × 1 × 0.1 mm 3)

1.3. Instabilities of the heauy Fermi liquid 1,5

Heavy-Fermi liquid phases, though being of much interest on their own right, have become a favored subject of modern condensed-matter physics when their inherent tendency towards instabilities against superconducting and (itinerant) magnetic phase transitions was recognized. The corresponding transition temperatures lie below T*, i.e., the driving interactions between the heavy fermions must be limited by the characteristic (on-site) energy kBT*. In fig. 5, specific-heat results, along with upper-critical-field, Bc2(T) , data are shown for the

....

I ' ' ' ~ '

T ,

(J/KZ m°l)1 - B ( Tesla1/,/ ~.~ ~-[

,

I

'

2

%,,,,~

( Teslo )

-

O/ o

0

/,

CeCu2Si2 .

,I°,,,F, 0.5

0

!

, ,~, I ,~o

0.5 T (K) Fig. 5. Specific heat as C / T vs. T (a) and u p p e r critical field, Bc2 vs. T (b), of CeCu2Si 2. Solid line in (b) is guide to the eye, and dashed line indicates Bc2 (T) slope at Tc ( - 1 3 T / K ) . D a t a in (a) and (b) taken from the same polycrystalline sample [44].

190

F. Steglich / Ce-based heavy-fermion compounds

became available when melts with a Cu excess were used [51,52]. Apart from such concerns about sample quality, the strong pair-breaking capability of dilute Ce 3+ ions in a BCS superconductor [9-11] made it difficult to believe in bulk superconductivity of a trivalent Ce-based intermetallic

compound. Heavy-fermion superconductivity attracted much greater interest, however, after it became confirmed in the three U-based compounds UBe ~3 [57], UPt 3 [58] and URu2Si 2 [59]. The recent discovery [60] of superconductivity above 30 K in some cuprates, whose transport properties seem to be strongly influenced by the intra-atomic correlations on the Cu-3d shell, initiated additional motivation to understand superconductivity of the most h i g h l y correlated electrons, the heavy fermions. The present state of the art may be best characterized by a continuing lack of a unified microscopic theory [6,61,62,26,27], but an increasing manifold of experimental discoveries [63,64, 53,65,27], e.g., phase diagrams with more than one superconducting phase in U1_xThxBe13 [66,67] and UPt 3 [68], the coexistence between heavy-fermion superconductivity and antiferromagnetic ordering with Tn = 10Tc in both U R u E S i 2 [59,69-71] and UPt 3 [72] as well as the new heavy-fermion superconductors UNi2AI3 [73] and UPdEA13 [74]. A magnetic (Stoner-type) instability in a heavy-fermion compound was first reported for NpSn 3 [75]. Later examples a r e U E Z n I 7 [76], NpBe13 [77], URu2Si 2 [70,71] and UPt 3 [72]. Though the classification as "heavy-fermion band magnets" (HFBM) is controversial for the first two compounds [78,27], the extremely low ordered moments /z~ = (2-3) × 10-2/zB/U-atom found in both URu2Si 2 and UPt 3 appears to justify this assignment for the latter: According to the theoretical treatment of a Kondo lattice utilizing an idealized hybridized band model, such a low value for ~s is expected in an itinerant magnetic state [79]. Among the Ce-based compounds, only CePb 3 has been identified so far as HFBM: Its Sommerfeld coefficient well below T N = 1 K is as high.as 1 J / K 2 mol [80], and its ordered moment is reduced by about a factor of ten

compared to that of the free Ce 3+ ion in the F 7 CF ground state [81]. In the remainder of this paper, we wish to discuss typical ground-state properties and recent developments in the study of Ce-based heavyfermion compounds. We address in section 2 the Pauli paramagnets, before turning to local-moment magnets (LMM) (section 3) as well as to a system in which an alloying-induced transition between LMM and H F B M can be monitored (section 4). Recent results on CeCu2Si2, emphasizing an intimate relationship between heavyfermion superconductivity, magnetism and lattice instability are the subject of section 5. The paper is concluded in section 6 by a short perspective.

2. Strongly enhanced Pauli paramagnetism Before discussing in some detail the low-temperature properties of the prototypical systems CeAI 3 [25], CeCu 6 [41,42] and CeRuzSi 2 [43], we wish to emphasize again the dominating role of on-site effects in determining the thermodynamic properties of a Kondo-lattice. This is demonstrated in figs. 6 and 7, showing the specific-heat and magnetic-susceptibility results [23] up to T = 30 and 300 K, respectively, for a series of CexLo1-x [ u 6

-

-6 1.0 E

- 2° r - - ~ - - - - ~

I_

,

I

\. 0 Fig. heat with data

10

'~ T (K]

100

6. T e m p e r a t u r e dependence of the 4f-derived specific per mole of cerium as C m / T vs. T for CexLa I xCu6 x = 1 (o), 0.8 ( R ) and 0.5 ( × ) [23]. Inset shows low-T for CeCu 6 [82]. D a s h e d curve indicates result for S = 1 / 2 Kondo impurity with T K = 4.2 K [7].

191

F. Steglich / Ce-basedheavy-fermion compounds 10"1L

_

,

, ,,,I,,

I

I

I ''''''1

I

X = 1------~. f 0 9>~L__.--=..~ ",.

'~" t n0' 37~3./ ~ ' ' ' '~--| v.~/~

r °-r

-~

I ,I,,I,~

I

r ''fqJt

"6 E

[ex Lo.l_xCU 6

- : ..'::"":::.... ".. "... '-~

~%

1.5

I--

-

°%%e e + %

0 100

1000 T(K)

Fig. 7. Temperature dependence of the magnetic susceptibility per mole of formula unit for CexLa 1_xCu6 [23].

CexLa ] _xCu6 alloys. For both quantities, a rather uniform t e m p e r a t u r e dependence of the Ce-derived contribution is found, in wide ranges of the L a - C e composition. In the inset of fig. 6, the low-T specific-heat data [82] of the compound CeCu 6 are c o m p a r e d to the numerical result for an S = 1 / 2 Kondo impurity (T K = 4.2 K) as obtained by the Bethe-ansatz calculation [7]. A p a r t from deviations seen at the lowest temperatures - which can be more pronounced in other compounds, as we shall see below - the Ce-derived part C m / T vs. T is found to be in close agreement with single-ion theory. The same holds for the susceptibility results in fig. 7 (for the sake of clarity displayed per mole of the formula unit, rather than per mole of Ce). Obviously, the large effective carrier mass m * , being proportional to both the Sommerfeld coefficient y = C m / T and the Pauli spin susceptibility X0, (i) is of local origin and (ii) forms only upon cooling to way below the Kondo-lattice t e m p e r a t u r e T*. Because of inter-site correlations, the latter is expected to slightly exceed T K [26,27] as is found, e.g., for CexLat_xCu 6. On the other hand, in CexLa]_~A12 the reduction of the unit-cell volume on going from the dilute alloy to the CeAJ2 compound causes a ratio T * / T K = 10 [31]. Clear phenomenological differences between the dilute Kondo alloy and the Kondo lattice, which are so pronounced in the t e m p e r a t u r e dependence of the electrical resistivity (fig. 1), can, under favorable circumstances, also be observed

0.4

0.8

-'1

cO

-2

+++ "~. +~* Q

• • 0 Teslo + 4 Teslo

1.0

, ~ L,,, . . . . . .

10

Y

0>

" '...i?:::)\.

, ,,,,,,a

*2

,A

BIIc

" '""-...,

10-3/ , , ,,,,,,I 0.1 I

CeAI 3

b

0

T (K)

1

Fig. 8. Low-temperature specific-heat (a), as C / T vs. T at B = 0 and 4 T [83], and thermopower (b), as S vs. T [44], for CeA13. in the low-T specific heat. A maximum in C / T vs. T is found, e.g., in CeAl 3 (fig. 8a) and normal-state CeCu2Si 2 (fig. 5) near Tooh --- 0.4 K [83], the t e m p e r a t u r e below which p = A T 2 (cf. fig, 2). The C ( T ) / T maximum has been ascribed [83] to pseudogap formation in the resonant 4f density of states (DOS) at E F due to 4f-conduction band hybridization [84-86]. A corresponding change of sign in the thermoelectric power [44], which probes the DOS slope at EF, demonstrates the asymmetry of the pseudogap with respect to the Fermi energy [82] (fig. 8b). In order to get this kind of fine structure in the DOS, a coherent 4f band has to form, which is the case below Tooh. A direct manifestation for coherent heavy quasiparticles in the low-T phase of heavy-fermion compounds is provided by d H v A oscillations in the magnetization [87,88]. Effective masses up to

.\.~1~

-

~x3

0

I

~

°'°'~°'~.o~

i

m* = 6me!

i

x

t

"- 1

-1

E e CU 6

m*=6"Omet ~ = ' k

0

]

0

50

r

+ ~

100

I

150

z

J

i

200

I

250

t

300

T [K)

Fig. 9. De Haas-van Alphen amplitude as a function of temperature for two oscillatory components of the magnetization of CeCu6, indicating two different effective masses [87]. X = 2"rr2kBT/htoc (toc/2at: cyclotron frequency).

F. Steglich / Ce-based heavy-fermion compounds

192

40mel have been observed in CeCu 6 [87] (fig. 9) and up to 90reel in UPt 3 [88]. To build up a non-magnetic ground state in a dense Kondo system such as CeA13, antiferromagnetic short-range correlations are prerequisite: As was pointed out by Nozi~res [89], in order to screen the moments of only a small fraction of Kondo ions, one would already have to exhaust the whole system of conduction electrons. Thus, antiferromagnetic correlations develop upon decreasing temperature and support the quenching of the local moments, so that a macroscopic singlet state is formed. Fig. 10 shows that for CeCu 6 the correlation length for these antiferromagnetic couplings is anisotropic and increases upon cooling, until below T* it saturates at a finite value of the order of the nearest-neighbor distance. If the short-range correlations are sufficiently strong, an external magnetic field can cause a "metamagnetic" type of transition. This has been discovered for UPt 3 [91] and was confirmed later for the non-magnetic heavy-fermion compound CeRu2Si 2 [92]: For the magnetic field applied along the tetragonal c-axis, an upturn occurs in the magnetization curve near 8 T (fig. 11). Mignot et al. [92] were able to demonstrate clearly, via pressure experiments, that the energy scale for the short-range antiferromagnetic correlations is

15

I

I

I

I

CeCu 6 10 r

~Q

@

u

5

0 |

I

1

1

L

0

1

2

3

t+

5

TIKI

Fig. 10. Magnetic correlation lengths along a and c directions as a function of temperature for CeCu 6 [90]. Lines through data points are guides to the eye.

1.5

CeRu2'Si2

-

~'-~

J~P=O~/05 /12kbor

0.5

0

0

I

I

I

5

10

1.5 B (T)

Fig. 11. Pressure dependence of the magnetization curve, M vs. B (Bl[c), of CeRu2Si2 at 1.5 K. Dots (squares): experimental data at p = 0 (2 kbar). Closed triangles: deduced from the latter using an approximate linearized Maxwell equation for ~p = 0.5 kbar. Solid lines: scaled curves for 0.5 and 2.0 kbar using a l n ( k B T * ) / a p = 171 Mbar-~. The value for the "metamagnetic field" BM indicated corresponds to a pronounced peak in the magnetoresistance as measured at 1.2 K

[92].

identical with the characteristic Kondo-lattice energy, k BT*.

3. Local-moment magnetism LMM in heavy-fermion compounds has to be considered intermediate between magnetic ordering in stable-moment rare-earth systems, like Gd-metal, and itinerant magnetism, e.g., of the Fe-group metals. Intermediate values of the Sommerfeld coefficient 3' ( = a few 100 m J / K z mol) above TRK~V ( > T * ) indicate moderate renormalizations due to the Kondo effect already at these elevated temperatures. Usually, the y value in the magnetically ordered state (T<< Tm) is somewhat reduced compared to the one in the paramagnetic state. This reflects a reduction of the heavy-fermion DOS at E v caused by magnetic ordering. As already mentioned in subsection 1.2, in realistic compounds the RKKY interaction can be strongly anisotropic. As a consequence, the magnetic structures in these materials are often rather complex. We wish to illustrate this point below by discussing two exemplary cases, CeAI 2 and CeCu2Ge 2, the latter being doped with low Ni concentrations. In CeA12, a type-II antiferro-

F. Stegfich / Ce-basedheauy-fermioncompounds

sponding jumps ACpl and ACp2 both are positive and, therefore, are not so easy to separate t h o u g h their superposition can be inferred from the c h a n g e of slope seen in the Cp(T)/T data of fig. 12. With the Ehrenfest relation

ct/T . . . . . . | ZeAl2 potycrysfo[ [ [eAt 2 _ single crystn[I (10"K'2) r ~ t ! [1001 1 2 F .,,f"~ a, + ,-4

'r"

L--4

4

0I . . . . . . . . . . . . .

tp = -~N 0

I

I

I

3

t

/~

I

I

I

3

T (K)

I

/4

T (K)

5 one gets a positive (hydrostatic) pressure coefficient for the ordering t e m p e r a t u r e TN1, but a negative one for TN2. T h e f o r m e r has b e e n observed t h r o u g h calorimetric [94] and the latter one t h r o u g h susceptibility [95,96] and neutrondiffraction [97] experiments, cf. figs. 13a and b. T h e s e two coexisting magnetic structures, which react quite differently to lattice defects [93], seem to realize the limiting cases of weak and strong local exchange coupling (fig. 3), a fact which could as yet not be explained. Let us now turn to the c o m p o u n d C e C u 2 G e 2, a L M M with TN I = TNh = 4.1 K [98]. TRKKy = 7 K, the t e m p e r a t u r e below which short-range magnetic correlations show up [99], nearly coincides with T * = (8 + 2) K [99]. In fact, this c o m p o u n d is found near the critical value of the coupling constant I gc I in D o n i a c h ' s schematic phase diagram (fig. 3), a modification of which is displayed for CeM2Si 2 and C e M 2 G e 2 homologs (M = Cu, Ag, Au, Ni, Ru) in fig. 14. H e r e the microscopic

magnetic structure, involving a ferromagnetic coupling within and an antiferromagnetic one between adjacent (111) planes, exists which is m o d u l a t e d by an i n c o m m e n s u r a t e wave vector 1 (½ + r , ~1 - r , ~), r = 0.11 [30]. T h e thermal-expansion results in fig. 12, especially on the polycrystalline sample, d e m o n s t r a t e two closelyspaced phase transition anomalies [93]: T h e b r o a d e r transition at TNI shows a positive j u m p Aa~, whereas the sharper one at TN2 is characterized by A a 2 < 0. In the specific heat the correi

1

i

i)Ph ]Ph--'O = 3Vm°' ACp'

I

Fig. 12. Linear thermal-expansion coefficient as a / T vs. T (top) and specific heat as Co / T vs. T (bottom) for polycrystalline (left) and single-crystal (right) CeAl 2. a measurement on single crystal was done along [100]. Dashed lines give interpolations between high- and low-temperature data points, and solid lines are used to replace broad transitions by idealized sharp ones at TNI (with positive jumps Aa I and ACpt, respectively). A second sharper transition at TN2 is characterized by Aa 2 < 0 and ACt,2 > 0 [93].

i

193

i

i

_8 o E

CeAt 2 ,,13=o op =(7± 1)kbor

--, 6

=g

7.5

¢-

~ 4 .m

2

I%-L"-"

/

TN(K) F

4,

3,8

361

N3

I~T

t~

I

I

I

[

I

2

/,

6

8 10 TCKI

0

I

12

Cs.9,,.C

, b ,

1

3

2

t~

0

5

6 T (K)

!'57

8

Fig. 13. Temperature dependences of the specific heat [94], at ambient pressure and p = 7 kbar (a), as well as of the magnetic susceptibility [95]; at p = 0, 1.96, 4.00 and 5.98 kbar (b), for CeAI 2.

F. Steglich / Ce-based heavy-fermion compounds

194

coupling p a r a m e t e r I gl is simulated by the inverse of the C e - M distance r. From its location close to the m a g n e t i c - n o n magnetic transition one would expect CeCu2Ge 2 to exhibit a N6el temperature with negative pressure coefficient: compression of the volume should further destabilize the magnetic 4f-configuration. Most surprisingly, the thermal-expansion jump associated with the formation of antiferromagnetism is positive [101], as is the specific-heat jump (fig. 15). Ehrenfest's relation thus yields tph = (1/TNh)(OTNh/OPh)O> O. U p to now pressure experiments have not been performed on this compound. Substituting Ni on Cu sites one can achieve a reduced average unit-cell volume as well. The results for a 2 at% Ni sample also shown in fig. 15 furnish a rather complex behavior: Compared to C e C u z G e 2, (i) T N h is reduced, although (ii) Aa h remains positive, (iii) a second phase transition, established to be an intrinsic property of the Ce(Cu l_xNix)zGe 2 systems with 0.02 < x < 0.3 [102], occurs at TNe < TNh. The corresponding thermal-expansion jump (i.e. tpe) is negative. According to Tachiki [102] the lower transition at TNe in the low-Ni doped systems may reflect a superposition of two incommensurate spin structures. These results, which are far from being understood, underline that the C e - M separation is not the only p a r a m e t e r determining

/~0

ol600K)~ ~'

/

Ni

|

t

Ni

Au

/T

"-'°

[

I

CeM2X2 X- Sl

o {::] T* Tm

X=- 6e

•

•

t

I

t

I

Ee ( EUl_x Nix)zl3e2 o x : 002

ff~

i...." o

o

e •

~" 0

'o ,o

~

I 2

~

'

'

'

I t~

1

ee

'o ..

_

o...

,, 0 " ~ . . . . . . . . . .

"3

~

I I

TIK)

.°4

o,,"

•

oo % I,I

o

oo

pooo~a~

e--l----

l 2

t

I

b

/+ T ( K }

Fig. 15. Low-temperature phase transitions in CeCu2Ge2 and Ce(Cu0.98Ni0.0e)eGe2 as revealed by the specific heat in a plot C / T vs. T (a) [102] and the linear thermal expansion, a vs. T (b) [101]. For the 2 at% Ni alloy, N~el temperatures TNh and TNc are indicated, see text. the magnetic state of these systems. Band-structure effects will certainly play an important role, since for the dopant Ni the 3d states are close to E F, whereas they lie way below E v for Cu. In addition, two very different types of magnetic structure seem to develop in the pure CeCu2Ge 2 and in dilute Ce(Cu, Ni)2Ge 2 on the one hand and for Ni concentrations x > 0.2 on the other, as is discussed in the following section.

4. Transition from local-moment to heavy-fermion band magnetism

0 3.1

, 3.2

3.3

3.k

,,

• 3.5 r(A)

Fig. 14. Kondo-lattice temperature T * , from low-T quasielastic neutron line width (circles) as well as entropy, thermal expansion and thermopower (bars), and magnetic ordering t e m p e r a t u r e T m in C e M 2 S i 2 a n d C e M 2 G e 2 vs. r, t h e C e - M d i s t a n c e (M: C u , A g , A u , Ni, R u ) . L i n e s a r e g u i d e s to t h e eye

[1OO].

To study the transition between L M M and heavy-Fermi liquid in a Ce-based compound like CeA12, application of hydrostatic pressure is an appropriate tool [14]. Because of the obvious difficulties in determining various properties like specific heat, transport coefficients and thermal expansion in a pressure cell, alloying experiments have been frequently performed. Usually, the f-

F. Steglich / Ce-based heavy-fermion compounds L

T

TN

\

TNI(O'~'~~

-~

~

n

d+

4

~ ~T.

'

/

\

/ f

*~

,7_,.._

05 ~@~,~;~~ ~

•o

:._.;...o-t,r-

F 0 0

2

Ce( CUI_xNix)2 Ge2 ,

,

5

=

,

I 0.5

,

~

,

1 ,

0

x

Fig. 16. N6el temperatures TN1, THe (left) and Kondo-lattice temperature T* (right), normalized by TNI (x = 0), vs. Niconcentration for Ce(CUl_xNix)2Ge 2. Results are shown from specific heat ( v , x7, +), thermal expansion ( * , A, × ) and dc susceptibility (e, ©) (left) as well as quasielastic neutron scattering (e), thermal expansion ( • ) and resistivity ( 0 ) measurements (right). Lines are guides to the eye. Hatched region marks systems with two phase transitions at TNh and TNe , cf. fig. 15 [107].

ion C e 3+ w a s replaced by a non-magnetic trivalent dopant of smaller ionic radius, e.g., y3+. However, in such a situation disorder in the Cesublattice can lead to spin-glass-type ordering which complicates the magnetic phase diagram [13]. Therefore, in the last years magnetic-nonmagnetic phase transitions have mostly been monitored for such compounds in which non-f ligand atoms were partially substituted by dopants [103]. As a recent example we address the system Ce(CUl_xNi~)2Ge 2. Whereas CeCuEGe 2 is a LMM, CeNi2Ge 2 [104] belongs to the class of non-magnetic heavy-Fermi-liquid systems, with T* = 3 0 K and y = 0 . 4 J / K e tool [104]. The quasi-binary alloys have been investigated in the whole concentration range by specific heat, thermal expansion, thermopower, resistivity, susceptibility [105] and magnetization [106] experiments. The dependence of the magnetic ordering temperatures TNI and TN2 on the Ni concentration x is shown in fig. 16. In accord with Doniach's phase diagram (fig. 3), substitution of the smaller Ni for Cu results in a precipitous depression of TNI(X) for x < 0.2. Instead of the expected nonmagnetic heavy-Fermi-liquid state, however, a second type of antiferromagnetic ordering below TNE(X) develops near x = 0.15. By contrast, the Kondo-lattice temperature T* as determined by

195

different experimental techniques exhibits a continuous increase, in agreement with expectation. By neutron powder diffractometry utilizing the multidetector instrument D l b at the high-flux reactor of the Institut Laue-Langevin (Grenoble), Ix)idl et al. [107] could characterize the magnetic structures below TNI(X) and Ti~2(x) , respectively. Like for the pure CeCu2Ge 2 compound [99], an incommensurate spiral spin arrangement with relatively long ordering wave vector, q0 = (0.28, 0.28, 0.41), was obtained for the 10 at% Ni alloy. On the other hand, an also incommensurate spiral structure, but with a much shorter qo, i.e. (0, 0, 0.14), was inferred from these data for Ce(Cu0. 5 Ni0.5)EGe 2. The ordered Ce-moment is 0.3/z B for x = 0.5. No magnetic Bragg reflections can be resolved from the powder spectra for the x -- 0.65 alloy, although long-range antiferromagnetic order is clearly visible in the bulk measurements. This implies that ~s < 0.2k~B, since moments of this size cannot be extracted from powder diffraction. The larger modulation vectors found for low Ni concentrations are of the typical order of Fermi-surface diameters and, thus, indicative of RKKY interactions. Kondo-reduced local Ce moments Izs = (0.5-0.74)/.t a characterize CeCu2Ge 2 and the lightly Ni-doped systems as members of the class of LMM discussed in section 4. The short propagation vector and the smaller ordered moment determined for x = 0.5 describe a modulated spin arrangement which extends over almost ten unit cells. Correspondingly short q0 vectors - along with small ordered moments iz s have to be considered hallmarks of HFBM [79]: By comparing the theoretical predictions by Grewe and Welslau [79] with the experimental T*(x) and TN(X) dependences of fig. 16, Loidl et al. [107] could establish a surprisingly good agreement for x _> 0.5. Further support for a magnetically ordered state developing out of a heavy Fermi liquid derives from the magnetic neutron cross section which is dominated for Ce(Cul_ x Nix)2Ge 2 by single-site relaxation processes, while for the CeCu2Ge 2 compound additional inter-site (RKKY) interactions between local Ce moments are found [99]. The magnetic phase diagram of fig. 16 suggests

F. Steglich / Ce-based heauy-fermion compounds

196

that HFBM is suppressed near x = 0.75. Arguments for a non-magnetic Fermi-liquid state at higher Ni concentrations involve the lack of any phase-transition anomalies in the bulk properties, i.e., the specific heat and transport coefficients [105]. However, in such a situation H F B M associated with a very small /~s value cannot be ruled out completely [100], as has been demonstrated in an exemplary way for UPt 3 [72]. Neutron-diffraction work on single crystals of Ni-rich Ce(Cul_ x Nix)2Ge 2 alloys are in progress to decide if longrange ordering with correspondingly tiny moments exists up to x - - 0 . 8 5 , the limiting value extrapolated from the TN2(X) dependence (fig. 16). In ref. [100] it was noticed that, combining this extrapolation with the T * (x) curve for x > 0.75, one gets TN2=0.6-0.8 K for T * = 15 K, the Kondo-lattice temperature for the heavy-fermion superconductor CeCu2Si 2. Indeed, for the latter a number of experimental techniques, such as magneto-resistivity [108], Cu-NMR [109] and I~SR [110] reveal - in just this temperature window anomalies which have been tentatively ascribed [109,111,100] to the development of some longrange antiferromagnetic order. The ordered Cemoment involved was estimated [110] to be quite

2

i

I

t

[

i

i

I

i

i

I

Ce Cu2Si 2 ...'~

E

6

i

i

small,/x s < 0.1 /z B. A survey on the low-temperature magnetic phase diagram of C e f u 2 S i 2 is given in the following section.

5. Heavy-fermion superconductivity, magnetism and lattice instability in CeCuzSi 2 We begin this section by discussing calorimetric and dilatometric results that were obtained recently [112,113] on a "new generation" of CeCu2Si 2 single crystals grown by the cold-boat technique and subsequently annealed in a Cu atmosphere [114]. Their high crystalline perfection could be demonstrated convincingly by measurements of dHvA oscillations in the magnetization as a function of the applied field [115], as well as by sharp superconducting transitions at Tc = 0.63 K. On the other hand, before annealing the same crystals show only spurious superconductivity with low Tc (0.2-0.3 K) and a small Meissner volume ( < 0.4%), which reflects inhomogeneities in the Cu concentration. For true bulk experiments, e.g., specific heat and thermal expansion, these "as grown" crystals can, therefore, serve as "non-superconducting" reference samples.

i

cta T

I

eI

t

i

(106 KZ)/~,)w~,,.~ .. ,#°/

•.

i--

I

4'

i

i

i

i

i

1

I

."

.....

R

°o

%c~,~,~^

i

2

i

O?

I

-2 m-

i I

3 o

0

3

1

-2

3

15

:2

l

I 1.5"~

i

,.'~ ,'"

."

7

3:

-2 :2

2 I

I

I

I

I

0.5

I

I

i

,

I ~

I

I

I

0.5

I

I

I

,

I

0

,

,

,

I

0.5

,

,

i

i

T (K

Fig. 17. Specific heat as C / T vs. T (a), linear-thermal-expansion coefficient along a-axis as a a / T vs. T (b) and along c-axis as a c / T vs, T (c) of CeCu2Si 2 single crystal at B = 0 (e), 0.5 T (©), 1.5 T ( ~ ) , 2 T ( I ) , 3 T ( v ) , 5 T (D), 6 T ( - ) and 6.5 T (zx) [113]. The anomaly at 0.35 K in the 2 T data of (a) indicates superconducting transition.

F. Steglich / Ce-based heay-fermion

Figs. 17a-c display the specific-heat results in a plot C/T vs. T as well as the thermal-expansion results taken along both the a- and c-axis, as a,/T vs. T and a,/T vs. T, on bulk superconducting CeCu,Si, single crystals for B = 0 and several values of the external magnetic field [113]. The data for T > T, are typical for a heavy Fermi liquid: In relation to the already very large Sommerfeld coefficient (as T + 0), y = 770 mJ/ K’mol, the coefficient a/T is enhanced by almost two more orders of magnitude, as is expressed by the “volume Griineisen parameter”, r = -a In T*/a In I/= V,o,~B(2~a + a,)/C = 63 (I&, = 50.3 cm33 molar volume, cn = 1200 kbar: bulk modulus [116]). This characterizes a very strong coupling of the heavy fermions to the breathing mode (“Griineisen-parameter coupling”), which shows a 50% anisotropy: The “orientation-dependent Griineisen parameters” r’ and r, amount to 80 and 40, respectively [117]. The superconducting transition of the new high-quality crystals is characterized by pronounced jump anomalies in C(T) as well as in a,(T) and a,(T). The idealized reduced specificheat-jump height, AC/&CT,) = 1.48 (obtained by replacing, in the usual way [45], the broadened transition as measured by a sharp one), exceeds the BCS value (1.43). The specific-heat data points well below T, (T 2 0.1 K) lie above the exponential BCS law. Entropy conservation requires that a fictitious superconductor with the same T,, but a “BCS-like” C(T) dependence, would show an even larger ratio AC/C,(T,). Thus CeCuzSi,, like UBe,, [57], must be considered a strong-coupling superconductor. The thermal-expansion jumps measured along the respective u- and c-axes are of opposite sign. Comparing them with AC, one gets the uniaxial pressure derivatives of T, (as pi + 0; i: a, c) via Ehrenfest’s relation

The hydrostatic pressure derivative (aT,_i3p,), = 2@T,/ap,), + (aT,/ap,), = 3.5 mK/kbar. This value is nearly identical with the one directly measured calorimetrically on polycrystalline sam-

197

compounds

ples in a pressure cell [1181. Employing the appropriate elastic constants cij as determined on such new single crystals by Liithi’s group, one can calculate [113] the uniaxial strain dependencies (as li + 0)

aT, -= i

1

a%

-hl+4

5),-c13(

Z),

0

= - (20 + 9) K

and (Z),=

-%3(

5)p33(

= +(10*8)

Z),

K.

The large uncertainties are caused by the uncertainty in the bulk modulus cn [116] which enters the calculation of the transverse mode ci3. We conclude that negative strain applied along the a-axis, namely a reduction of the Ce-Ce separation, gives rise to a T, increase, whereas the opposite effect, though much less pronounced, is found along the c-axis. Since the intra-plane coupling between Ce ions is dominated by the 4f-5d hybridization, which tends to destabilize the magnetic 4f-configuration of Ce3+ via raising T *, the observed strain dependence (aT&J, demonstrates clearly that a sufficient compensation of the local 4f moments is prerequisite for heavy-fermion superconductivity in CeCu,Si,. The systematic investigation of high-quality single crystals revealed that the upper-criticalmagnetic-field curves determined resistively [52] on Bridgman-grown single crystals for the field along the a- and c-axes is an intrinsic property of CeCuzSi,, as was also found above for the hydrostatic pressure derivative of T,. The B,,(T) values read off the thermal-expansion results of figs. 17b and c coincide with the previous results by Assmus et al. 1521, see fig. 18: The pronounced anisotropy in B,,(T) can well explain the anisotropy of the thermal-expansion jump Acwas a function of the magnetic field, applied along the same direction used for the length measurement, i.e.: Aa’(B) Aa,

=27

atB=15T . .

.

F. Steglich / Ce-basedheacy-ferrnion compounds

198 F

I

I

I

0.8 ~-

[e [u~ Siz

~-

\\ 0.2

\\ i

B"ll°°lH

,~loo@ I

0

L

0

I[

I ,x.tl

2

."o.

4

\ ,t0011

"~" ,x

~

tl

.

b

t~

I

6

0

B (T)

Fig. 18. B-T diagram for CeCu2Si2, containing: B,.2-curves, defining the superconducting (SC) to normal transition, for fields parallel to the a- and c-axes (solid lines), temperatures at which Cu-NMR intensity on a polycrystallinesample drops from 90 to 10% upon cooling (+), magnetic fields where the isothermal longitudinal magnetoresistance (B I[J IIa) shows a minimum (×), onset of magnetic correlations inferred from IxSR (A), peak positions in dHvA oscillations for B IIa (o) and BIIc (*), step-like phase transition anomalies in the longitudinal elastic constants CII(T, Blla) (v) and C33 (T, Bllc) (v) and phase-transition-like anomalies in the coefficient of thermal expansion a,(T, Blla) (o). Also included are a(T, B IIa)-derived transition temperatures Tt(B) ( • ) of a non-bulk-superconducting crystal [113]. Starting from the appropriate thermodynamic relation [113]

Vmo,.O t-D7-- S,.,t-D-PTP,/ (i: a, c) and assuming that both the uniaxial pressure derivatives (STc/OPi) B and the susceptibility (OM/OB)pi,r,Bc2 = [1.16(2K 2 - 1)]-1 (K: G i n z b u r g Landau parameter) are independent of the orientation of the external field with respect to the symmetry axes of the crystal, one finds

A =

~

~}

]B&="&:l's'r = 2.9 + 1,

in close agreement to the experimental value. We now turn to the second phase transition which apparently exists, besides superconductivity, in CeCu2Si 2 at low temperatures. Fig. 18 indicates those B-fields (applied along both the a- and c-axis) at which (i) isothermal magnetoresistivity measurements reveal a "kink" [108], (ii) d H v A oscillations show maxima [115] and (iii) the elastic c o n s t a n t s c al(T) and c33(T) exhibit "steps"

[113]. In addition, we have marked in the figure those temperatures T((B) at which the intensity of the C u - N M R of polycrystalline samples disappears [109] as well as the one below which internal magnetic fields develop in the (B = 0) IxSR m e a s u r e m e n t [110]. As is seen in fig. 17b, anomalies develop in the thermal expansion for B > 5 T, whose positions are close to the ones obtained by the techniques listed above. Interestingly enough, corresponding anomalies are neither found in a ( T ) for B < 5 T [113] nor in the specific heat up to B = 6.5 T (fig. 17a). Since Cll ~ F z C and a ~ FC, a very large and field-dependent Griineisen p a r a m e t e r associated with the second phase transition may be concluded [113]. In order to shed more light on the nature of this transition documented for bulk-superconducting CeCuzSi2 samples by so different probes and tentatively ascribed [111] to long-range antiferromagnetic ordering, Lang et al. [113] have studied several of the "as-grown" single crystals, that are lacking bulk superconductivity. Most surprisingly, mean-field-type anomalies in C(T) and a ( T ) are found at T 1 = 625 m K for zero-field (figs. 19a and b), although no related features can be resolved in the presence of bulk superconductivity. Clearly T 1 coincides with T~', the critical temperature of the second phase transition in the superconducting samples. U p o n increasing external B-field, T l is found to be depressed more rapidly than T[, see fig. 18. Employing the negative jumps in aa(T) and ac(T) at T 1 (fig. 19b) one gets for the volume-expansion coefficient, /3 = 2 a a + a¢, a change of sign from plus to minus upon cooling through T I. This goes along with a gigantic negative hydrostatic pressure derivative, @T1/Oph) o = --200 m K / k b a r , as calculated from the Ehrenfest relation which links Aa~ and A a c with AC (fig. 19a). Also, the strain dependences, (0T~/0Ea)0= +(290 _+ 50) K and (~TI/aEc) o = (160 _+ 70) K are much larger compared to the analogous relations for Tc. This, along with the different signs of (OTJ~%) o and (OTJO%)o, clearly distinguish the new transition from the superconducting one. An only minor change in the low-field susceptibility x ( T ) at T t makes a magnetic transition unlikely as well [113]. Therefore, Lang et al. [113] have

199

F. Steglich / Ce-based heavy-fermion c o m p o u n d s

I

i

CeCu2Si 2

A

E

I

,ff

,

,

,

I

i

'

'

~

-

"7

• e ~°e

/

II

-'1

I" 0./,

q

• o

'v-

l_.J

I

e*mo •

"ns grown"

o

i

- 2 'o T--

L--":

,,

...--' i W~ _

TI

,%%,

1 I'- T2 oLT. ~ ,

II [1001 .,,t001]

•

o'~ o6~ I

0 0

-

I

b

I/(

0.5

I

I

I

I

0

I

0.5

I

I

I

-2

I

T(K)

Fig. 19. Low-temperature properties of "as grown", i.e., non-bulk-superconducting, CeCu2Si 2 single crystal. (a) specific heat, C vs. T; (b) linear thermal-expansion coefficients measured along [100] and [001] directions, a a vs. T (e) and a c vs. T ( • ) . Inset: field dependences of the phase transition anomalies as derived from a a vs. T measured at different B-fields [113].

ascribed the phase-transition anomalies of fig. 19, related ones being observed in the elastic constants [113], to a "lattice instability". From the quite low value of T 1 and its dependence on magnetic field (inset of fig. 19) this structural transition was suggested to be driven by some mode (e.g., an anisotropic volume mode), which couples strongly to the heavy electronic degrees of freedom. The microscopic origin of the new phase transition is presently not known. Quadrupolar ordering is most unlikely, in view of both the temperature dependence of the various elastic constants [113] and the CF-level scheme of Ce 3+ in CeCu2Si 2 [119]. However, the possibility of a charge-density wave in the heavy-Fermi-liquid phase should be seriously considered, since it has already been established theoretically under fortunate conditions [120]. Also a spin-density wave associated with a very small ordered moment can presently not be ruled out definitively, though being in seeming conflict with outward appearance. Irrespective of its underlying mechanism, the new phase transition is closely related to both magnetic and superconducting phenomena: In bulk superconducting samples, strong "dynamical

magnetic correlations" [121] develop below T , ' presumably due to changes in the 4f-ligand hybridization. The expansion of the volume observed upon cooling for "as grown" crystals is expected to enhance the stability of the magnetic 4f-configuration. Therefore, the anomaly found in a(T) at T2 = 115 mK (fig. 19b) should be ascribed to an antiferromagnetic transition, which appears suppressed in the presence of bulk superconductivity. Most importantly, the transition temperature T 1 coincides with the highest Tc that can be achieved in CeCu2Si 2 by optimal treatment similar to recent observations made for certain high-Tc superconductors [117,122]. Hopefully, a deeper insight in the nature of this new phase transition will help to better understand the pairing mechanism in the prototypical heavy-fermion superconductor CeCu2Si 2.

6. Outlook

Ce-based heavy-fermion compounds have played an outstanding role to establish the behavior of the Kondo lattice and several of its novel,

200

F. Steglich / Ce-based heacy-fermion compounds

partly unexpected, g r o u n d states: the c o h e r e n t heavy-Fermi-liquid phase, local (reduced) mom e n t m a g n e t i s m , heavy-fermion b a n d m a g n e t i s m a n d heavy-fermion superconductivity. Since their characteristic energies ( < k BT* a n d = k BTRKKV) are c o m p a r a b l e , these different g r o u n d - s t a t e properties c o m p e t e for stability a n d even can coexist. R e c e n t investigations on C e C u 2 S i 2 single crystals reveal the existence of an as yet unexp l a i n e d lattice instability as an i n h e r e n t property of the heavy-fermion system. A p a r t from the "classical" heavy-fermion comp o u n d s with usually high carrier c o n c e n t r a t i o n discussed in this report, metallic Ce [21,123] a n d Yb [124] systems with very low carrier c o n c e n t r a tion a n d even s e m i c o n d u c t i n g S m - c o m p o u n d s [125], all of which showing heavy-fermion-type of p h e n o m e n a , are presently receiving increasing attention. T h e s e investigations p r o m i s e an improved u n d e r s t a n d i n g of the local processes u n derlying the f u n d a m e n t a l K o n d o interaction. T o conclude, it can be h o p e d that the recent progress m a d e in the materials d e v e l o p m e n t on the o n e h a n d a n d the theoretical m e t h o d s o n the o t h e r will furnish a m u c h more detailed knowledge of the C e - b a s e d heavy-fermion c o m p o u n d s in the next few years. This will p r e s u m a b l y have great impact o n the whole field of the physics of strongly c o r r e l a t e d electrons.

Acknowledgements I would like to t h a n k my colleagues P. Fulde, N. Grewe, B. Liithi, A. Loidl, G. W e b e r , U. A h l h e i m , W. Assmus, C.D. Bredl, R. Caspary, C. Geibel, M. L a n g a n d G. S p a r n for a fruitful c o o p e r a t i o n over the years o n the topics described in this article. F i n a n c i a l s u p p o r t by the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t , partly u n d e r the auspices of the SFB 252, is gratefully acknowledged.

References [1] For a review, see, e.g., G. Griiner and A. Zawadowski, Rep. Progr. Phys. 37 (1974) 1497.

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