Fermi liquid aspects in valence fluctuating systems

Journal of Magnetism and Magnetic Materials 52 (1985) 85-90 North-Holland, Amsterdam

INVITED

85

PAPER

FERMI LIQUID J. F L O U Q U E T ,

ASPECTS

IN VALENCE FLUCTUATING

P. H A E N , C. M A R C E N A T ,

SYSTEMS

P. L E J A Y

Centre de Recherches sur les Trbs Basses Tempbratures, CNRS, 166 X, 38042 Grenoble-Cbdex. France

A. A M A T O ,

D. J A C C A R D

a n d E. W A L K E R

Dbpartment de Physique de la Matibre Condensbe, Universitb de Genbve, 24 Quai Ansermet, 1211 Geneve 4, Switzerland

Low temperature properties of quasi-trivalent non-cubic compounds of cerium (CeA13, CeCu 6 and CeRu 2Si 2 ) are discussed with special emphasis on the role of the degeneracy of the ground state and of the symmetry of the lattice in the interactions between f electrons. In CeRu~Si 2, an interesting new phenomenon, the occurrence of a polarized phase in a high magnetic field, is reported. The entrance in the intermediate valence regime ~,ad the possibility of couplings characteristic of a lattice are discussed.

1. Introduction

In a dilute alloy, the interaction of the angular m o m e n t u m J of a rare earth ion with the surrounding (crystal field effect with characteristic splitting Ccv) and its resonant coupling with the conduction electrons (Kondo energy k B T K ) give rise to a large variety of competing phenomena. For example, studies of dilute cerium impurities in a hexagonal lattice of yttrium have shown that, in spite of the large value of the Kondo temperature ( T K ~-20 K), the magnetic anisotropy of the J z = [ -+ 1 / 2 ) doublet ground state is preserved [1]. Another illustrating case is that of dilute AuYb alloys [2] where a breakdown of the Kondo scattering through the formation of an electronuclear singlet in 171yb takes place; the B r e i t - R a b i effect is observed since the hyperfine coupling is stronger than the Kondo coupling. On the other hand, for k B T ~ / C c v > 1, the consequence of recovering 2 J + 1 channels of the f" configuration is to favor the mixing with the f" +1 configurations leading to the so-called intermediate valence (IV) state. For a periodic array of anomalous ions with f electrons, magnetic ordering or very narrow band effects with strong interactions can arise; the strong localization of the f electrons and their weak delocalization through the mixing with the itinerant electrons lead to strongly correlated systems comparable to liquid 3He [3]. However, the already mentionned specific character of f electrons and the existence of two types of particles prior to mixing differentiates these so-called f heavy fermion system from quantum liquid 3He. In particular,

the duality between light and heavy particles may play a main role in collective phenomena such as superconductivity or opening of a gap. Here we will focus on the properties of quasi trivalent cerium compounds (CeAI 3, CeCu 6 and CeRuzSi2) which stay in a normal non-magnetic phase down to 0 K. An extension of the phenomenological Kondo model with a single Lorentzian resonance to two splitted Lorentzians describes the experimental data rather well and supports the idea that even for magnetically ordered compounds, like TmS or CeB6, a temperature of coherence T* may appear below the magnetic ordering temperature T N.

2. Lorentz number of CeAI 3

Specific heat (C), thermal expansion (a) and transport measurements of CeAI 3, have clearly shown that two temperatures T K and T* must be considered [3-5]. T~ ( = 5 K) corresponds to the delocalization of a f electron by its Kondo-like coupling with the itinerant electrons, while T* represents coherence effects (interactions) among the heavy carriers, i.e. a regime where disordered paramagnetic scattering has disappeared. Experimentally, T~: can be evaluated from the width of the quasi elastic neutron line [6] while T* ~ 350 m K is given by the temperature of the maximum of C / T or of the minimum of a [3 5]. The susceptibility X presents a shallow maximum at T x -- 700 m K [7] while the thermoelectric power Q crosses zero for T = T* and reaches

86

J. Flouquet et al. / Fermi liquid systems

a maximum for T = 200 mK [5]. Accurate measurements of the electrical resistivity (p) and the thermal conductivity (•) were simultaneously performed on a long rod of CeAI 3 down to 20 m K (fig 1). The first information is that below T*, p follows a T 2 dependence, i.e. P = P0 + A T 2 with P0 = 1.985 ~ c m and A = 32 ~ c m K 2. By defining a Lorentz number L = K p / T , it appears that, above I K, L is larger than L 0, the free electron Lorentz number. A possible explanation for this departure is that the phonon contribution becomes important at high temperature. However, the same observation was made for other heavy fermion compounds (UBel3 or CeB6) above T* [5]. Below 0.9 K, the phonon contribution to K is negligible, and the thermal conductivity increases on cooling down to 200 mK where it reaches a maximum. An usual electronic thermal conductivity (K = T), due to the limitation of the electronic mean free path by static impurities, is recovered only below 50 mK. The Lorentz number goes through a minimum at T = 500 mK. An extrapolation to T = 0 K of the data taken above 500 m K gives L = 0.75L0; this deviation from L 0 emphasizes the importance of inelastic processes. It is worthwhile to mention that in the coherent regime of light and heavy particles (basically the Baber mechanism), L is equal to 0.65L 0 [8]. Qualitatively, below 500 mK, the increase of L is due to the entrance in a regime where the contribution of static impurities to scattering increases in proportion as T decreases (P0 > A T 2 ) : L should tend to L 0 when T ~ 0 K. However, the interesting feature is that L reaches a weak maximum L M = 1.10 L 0 at 50 m K and

"H.(mW/Kcm)

eLjj/'"" I

i

I

I

i

I

1.1

0.5

0.9

0:i 0

I 1

T(K) T(K)

I 2

Fig. 1. Temperature dependence of the thermal conductivity (1¢) and of the Lorentz number L for CeAI3.

then a value 1.07L o at the lowest temperature. L > L 0 has been already found in Kondo systems like C uFe [9]. However, the deviation from L 0 can also reflect indirectly the special difficulties encountered in the metallurgy of CeAI3 which is obtained by a peritectoid transformation of CeAI 2 and C%Alal composite phases. A part of the residual electrical resistivity may be due to the presence of parasitic phases. The possibility of such spurious effects is reinforced by the observation that the present measurements, performed on a sample with P0 = 1.9 ~ 2 c m , indicates a maximum in • at nearly the same temperature as previous experiments on a sample with P0 = 0.7 F~2cm [7,10]. Our data show that accurate simultaneous measurements of K, p and thermoelectric power Q reveal different regimes which are not directly observed in the measurement of only one property. The main difficulty is that CeAI 3 presents " e n d e m i c " defects and thus intrinsic experimental limitations for data analysis. The quasi impossibility to grow single crystals have strongly limited the understanding of its microscopic properties. It has been suggested that, in this hexagonal lattice, the crystal field ground state is the highly anisotropic doublet Jz : ] _+3/2) (gll = 3 g j along the c axis, g± = 0 along the basal plane) [11,12]. The cases of C e C u 6 and CeRu2Si 2 developped in the next section will prove the importance of the anisotropy.

3. CeCu6: strong anisotropy in transport properties The orthorhombic compound CeCu 6 [13,14] shows heavy fermion properties with an enormous linear temperature term ~ = 1500 mJ mol ] K -2 in the specific heat [15-17]. Here, large single crystals can be obtained [14]. Extensive investigations of the properties of CeCu 6 have been already reported [18-20]. We have recently performed p and Q measurements [21] on a single crystal with current and heat flow along the three principal axes. [100], [010] and [001]. The temperature T~ of the resistivity maximum differs by a factor of nearly two between the [100] axis ( T ~ ~ = 8.5 K) and the [010] and [001] axis ( T ~ h = 14.7 K and T~l' = 15.4 K); this anisotropy also appears in the residual resistivity P0 (Og = 15.7 ~ 2 c m , P~ = 8.3 F ~ c m and P~ = 9.2 ~ 2 c m ) and in the coefficient B of the linear temperature term at low temperature (B ~ = 40, B b = 15 and B " = 19 ~ 2 c m K - ] ) (fig. 2). The interesting point is that B and P0 roughly scale with (T~a) ]. The mean value of B along the three directions is comparable to that observed in CeAl 3 [5]. The relation B - ( T ~ ) 1 found for each direction has already been found in high pressure study of CeCuzSi2

J. Flouquet et aL / Fermi liquid systems

[22]. This behavior shows that, in heavy fermion compounds, the linear temperature dependence of the resistivity is characteristic of the approach to the coherent regime. Measurements down to 26 m K show that, along [100], in agreement with results reported in ref. [15] for a polycrystal, the T 2 dependence of p holds only below 100 m K with a coefficient A = 112.3 ~f~cmK -2 three and half times larger than that reported for CeAI 3. This enhancement seems directly related to the range of validity of the T 2 law since this interval is about 3.5 times narrower in CeCu 6 than in CeAI 3. For CeCu6, the coherent regime may exist only below 100 mK. A comparison with recent data [19] obtained on single crystals having large residual resistivities ( A ' = 2 6 rL~cmK-2; pg = 100 rtf~cm) seems to indicate a sensitivity of the coefficients A and B to sample purity. However, it is difficult to conclude since the residual resistivity pg = 100 ~tf~cm, i.e. the effect of residual disorder reported in ref. [19], is larger than any thermal disorder observed in our sample (where the maximum resistivity p ~ --- 80 F~cm). On the other hand, for CeAI 3, variations of P0 from 0.7 to 5 pA2cm do not seem to affect the coefficient A [5].

100

,

,

,

,

,

CeCu6

Thermoelectric power (Q) experiments along each of the three directions show the existence of two positive anomalies: a large peak at 60 K and another one below 5 K well resolved along the [001] direction but marked only by a shoulder along the others. The connection of the anisotropy of p and Q with the anisotropy of the magnetization is not obvious since the easy magnetization axis is the c axis [18].

4. CeRu2Si2: anisotropy and polarized phase CeRu2Si 2 is a tetragonal material in which the cerium ions are close to the intermediate valence regime. This is suggested by the occurrence of a weak susceptibility maximum at T M = 10 K (far above the corresponding value (700 mK) for CeAl3) and by the magnitud~ of the ~ term ( = 380 mJ mol 1 K 2) [17,23,34]. The striking feature is that by many aspects ( 7 , 0 ) CeRu2Si 2 is similar to UPt 3 which has attracted considerable attention because of the occurrence of supercon-

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Fig. 2. Temperature dependence of the resistivity of C e C u 6 for the three main axis. The insert represents the low temperature variation.

50

100

H (kOe)

150

2o0

Fig. 3. Magnetization of a single crystal of CeRu2Si 2 at 4.2 K with H along the c axis (HIIc) and H in the (a, b) plane ( H i c) and magnetoresistivity of the same crystal with HIIc and 1 in the (a, b) plane. The insert represents the temperature dependence of the low field susceptibility of a second crystal for the two field directions.

88

J. Flouquet et al. / Fermi liquid 4vstems

ductivity at T~ ~ 0.5 K [25]; on the contrary, CeRu2Si 2 stays in its normal phase down to 40 mK [26]. the recent observation [24] of an inflexion point in the magnetization curve of a CeRu2Si 2 polycrystal gave us a strong stimulation for performing such experiments on single crystals [27]. A remarkable feature (fig. 3) is the huge anisotropy of the magnetic properties: The magnetization occurs mainly along the c axis ( X j X . ~- 15). Anomalies in x ( T ) and M ( H ) appear also only along the c axis. Magnetoresistivity experiments performed at 4.2 K give a maximum of p at the field H ~ = 7 5 kOe where M exhibits an inflexion point. The magnetic moment above H c and the low field anisotropy imply the Jz = I + 3 / 2 > doublet as crystal field ground state. Qualitatively, the variations of M and p at 4.2 K reproduce the scheme of the transition from non-magnetic (case of Ti) to magnetic (case of Mn) impurities dissolved in normal metals (Cu or Au) as discussed in a H a r t r e e - F o c k theory for 3d magnetic moments [28,29]. However, a maximum in X at T M = 10 K along the c axis is not predicted by a model of spin 1 / 2 impurities [30]; coherent effects must be considered. The magnetic field produces a polarized phase of heavy fermions by selecting only one magnetic component: an induced ferromagnet. Such a situation is reminiscent of that found for the IV compound TmSe where a metamagnetic transition occurs in a magnetic field from an antiferromagnetic to an induced ferromagnetic phase [31]. This produces a polarized state [32,33] and the strong scattering in the resistivity due to the paramagnetic doublet ground state drops drastically [34,35]. The rapid decrease in p and the fast increase of M at H---400 kOe in YbB12 [36] may belong to the same class of phenomena. CeRu2Si 2 exhibits an enhancement of the scattering at H~ as UPt 3 [37]. However, the magnetic anomaly has a bidimensional character in UPt 3 and an unidimensional character in CeRu2Si 2. For UPt 3, the study of the superconducting transition proves that the magnetic dimensionality plays an important role in the apparition of superconductivity [38]. Does the quasi one-dimensional magnetic properties of CeRu2S i2 prevent superconductivity?

describe the Kondo effect for an impurity [39]:

In 2

exp

m(~+H)dc

'

kBT

-S

where the partition function is given by a collection of free independent spin S of S: angular component m subjected to an internal field c with a Lorentzian distribution A P(£)

1

'/T ~2 q- A2 "

A simple extension is to introduce an additional characteristic energy E o which describes a symmetrical shift of the Lorentzian distribution:

1

A

p(,) = ~

1

+

a2+(~- E0)2 a2+(~+ E0)~ For a lattice, E 0 may reflect a drop in the density of states at the Fermi energy ( E 0 / A >_1), i.e. reproduce the idea of the opening of a pseudo gap by hybridization. The calculated magnetization and specific heat correctly reproduce the occurrence of a maximum in C / T at 350 mK and in X at 700 mK for CeAI 3 with the following choice of parameters: E 0 = 2.5 K and A = 1.5 K. Furthermore, it is obvious that, at low temperature, the model predicts an inflexion point in M ( H ) at H c = E 0 and an initial increase of the specific heat coefficient 3' in a magnetic field as observed experimentally [40]. The transition to a magnetic ordered state can be discussed if the intersite exchange coupling J,j is reduced to a molceular field H = J, jrn. Fig. 4 represents the temperature variation of C / T for a non-magnetic I

I

i

i

¢/T ( J mole -I K-2 ) 5

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T~(N M)

}

T~(M)

5. Phenomenologicalmodel 0

The aim of the following discussion is purely to find the simplest possible ingredients capable to reproduce the present results. The starting formula is the free energy F of the single resonant model developped to

+s

F= -kTf~p(c)

I

o

i

I

i

2

T (K)

4

Fig. 4. Variations of C / T as a function of T derived from a model of two splitted Lorentzian for a non-magnetic (NM) case (dashed line) and for a magnetic (M) case (full line).

J. Flouquet et al. / Fermi liquid systems

and a magnetic phase. The crossover into the coherent regime, located for the non-magnetic state at T = T* (where C / T is maximum) seems to persist in the magnetic phase. Such a situation could exist in CeB6 [41] or in TmS [42]. At the ordering temperature T N, only a part of the 4f moment is magnetically ordered, whereas, below T*, all f and d electrons are dynamically coupled. Finally, following a phenomenological model developped to describe the scattering of K o n d o impurities [43], the maximum in the magnetoresistivity can be reproduced with a charge difference Z = 1 and a level degeneracy l ~ 0. The phase shifts ~ ~ and ~ ~ for each + 1 / 2 component ( S = 1 / 2 ) are derived from the Friedel sum rule and from the magnetization: Z ~r + ~ 21+ 1 "~

M ~r - 8 ~ and 2 l + 1 -

Independent scatterings for spin up and spin down conductions electrons are assumed.

89

lattice. The systematic study of the apparent disorder in IV systems could lead to surprising results.

7. Conclusion Our aim was to report on new experiments on three different heavy fermion compounds. Further experiments must be performed in order to infirm or reinforce some of our statements. Parallely to the interest of unusual superconducting pairings, there is a large variety of basic experiments that can be realized in normal phases especially to look for collective excitations below T*.

Acknowledgements The authors thank Dr. F. Lapierre, J.M. Mignot, J. Souletie, Pr. B. Castaing, P. Nozieres, N.E. Phillips and J. Sierro for stimulating discussions.

6. Entering in an intermediate valent regime Up to now, a strongly interacting heavy fermion compound (like liquid 3He) behaves as a set of spin 1 / 2 particles. The condition S = 1 / 2 and the occurrence of a strong coupling anisotropy may play a major role in pairing possibilities. The expression " h e a v y fermion" must be regarded critically because a large part of the effective mass results from a renormalized effective band mass which is difficult to estimate when, as in CeA13, the energy associated with the band (A + Eo) is comparable to that (E0) describing correlations between particles. For CeAI3, the relevant parameters are the enhancement by a factor four of the ratio x / C T over the free electron band value [4] and also the " n o r m a l " value of the fermion Gri~neisen coefficient (Of --- - 2 0 0 ) after normalization by the effective mass enhancement ( m * / m = 400): I2~' -----0.5 [5]. As pointed out for an IV phase, the intersite coupling decreases by one order of magnitude in comparison with the energy of the ground state [44]. At first approximation, only renormalized band effects exist. The most serious difficulty in studying possible excitations above the ground state is that the scattering by impurities is not really independent from bare lattice effects (notably for A -- kBTK) and that it may be larger than the inelastic scattering between f and d particles. For IV compounds, the main question is whether new types of coupling through long range interactions exist, reflecting the possibility of finite solubility of one valence state in a homogeneous intermediate valence

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J. Flouquet et a L / Fermi liquid systems

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