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On the pressure-temperature phase diagram of the Kondo compound CeAl 2

B. Barbara, J. Beille, B. Cheaito, J.M. Laurant, M.F. Rossignol, A. Waintal, S. Zemirli

To cite this version:

B. Barbara, J. Beille, B. Cheaito, J.M. Laurant, M.F. Rossignol, et al.. On the pressure-temperature phase diagram of the Kondo compound CeAl 2. Journal de Physique, 1987, 48 (4), pp.635-640.

�10.1051/jphys:01987004804063500�. �jpa-00210479�

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On the pressure-temperature phase diagram

of the Kondo compound CeAl2

B. Barbara, J. Beille, B. Cheaito, J. M. Laurant (*), M. F. Rossignol, A. Waintal and S. Zemirli Laboratoire Louis-Néel, C.N.R.S., (*) C.R.T.B.T., C.N.R.S., 166X, 38042 Grenoble Cedex, France

(Requ le 20 novembre 1985, révisé le 24 novembre 1986, accepté le 28 novembre 1986)

Résumé.

2014

Des mesures de diffraction des rayons X sous pression, effectuées sur un monocristal de

CeAl2 réduit en poudre, ne montrent aucune anomalie de volume jusqu’a 150 kbar. Cependant des expériences de résistance électrique sous fortes pressions effectuées sur un monocristal provenant de la même origine montrent une anomalie faible au voisinage de 80 kbar, à la température ambiante, i.e. près de la pression pour laquelle Croft et Jayaraman [2] ont observé une très importante anomalie de volume.

L’anomalie de résistance a été suivie jusqu’à 2 K, et conduit à une nouvelle ligne dans le diagramme

P-T de CeAl2. Cette ligne, interprétée en termes d’un changement entre deux régimes Kondo, coupe la ligne critique un ordre magnétique apparaît. Sur la base de ces résultats obtenus sur CeAl2, nous proposons un

diagramme de phase détaillé pour les systèmes Kondo magnétiquement ordonnés.

Abstract.

2014

Accurate high pressure X-ray diffraction experiments performed on a powdered single crystal of CeAl2 do not show any sizeable volume anomaly up to 150 kbar. However, high pressure electrical resistance

experiments performed on a single crystal of the same batch show a continuous anomaly near 80 kbar at room temperature, i.e. close to the pressure for which Croft and Jayaraman [2] observed a strong volume anomaly.

The resistance anomaly has been followed down to 2 K leading to a new line in the P-T diagram of CeAl2. This line, interpreted in terms of a crossover between two different Kondo regimes, crosses the critical line where magnetic order takes place. On the basis of our results in CeAl2, a detailed phase diagram for magnetically ordered Kondo compounds is proposed and discussed.

Classification

Physics Abstracts

72.15

1. Introduction.

The a - y transition of Ce-metal has been discovered 60 years ago [1] and since, it has been the object of

an increasing interest on both experimental and

theoretical sides. This transition is also observed on

several Ce-based intermetallics such as CeAl2 [2, 3], CexThl-x [4], CeNi [5] and even on metallic praseodymium [6] at extremely high pressures. The

a - y transition was first thought to correspond to a

f-d promotion [7]. Johansson suggested that it was

instead a Mott transition from localized f electrons

to an f-band [8]. Subsequently various experiments

have supported the idea that some sort of f delocali- zation is responsible for the a - y transition. More

recently this transition has been attributed to a

volume collapse of a compressible Kondo model [9, 10]. This last interpretation is coherent with the 7 %

softening observed on the bulk modulus of CeAl2 compared to other RAl2 compounds [11]. However,

it does not exclude the possibility of slight deviations

from an integer valency (say by a few per cent)

or/and of electrons having a f angular symmetry and

a sufficiently large radial extension so that they

contribute to the diffuse part of the form factor [12]

(these two aspects being however not relevant in

driving the a - y transition).

The apparition of a dense Kondo regime can be

associated either with a first order transition (a - y )

or with a simple regular variation of the volume (no transition), depending on the counterbalence be- tween’ Kondo and elastic volume energies. Ce metal [1, 13], CexTh1-x [4] for x 0.27, Pr metal [6] and

CeNi [5] show a first-order transition whereas

CexThl - x [4] for x > 0.27 shows a simple regular

variation of the volume. Concerning CeAl2, this question was not clear up to now. After X-ray experiments under pressure [4] it has been widely admitted that CeAl2 presents a a - y transition.

However, the nature of this transition has been

questioned after other lattice parameter experiments

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004804063500

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636

performed on the alloys CeAl2 _ eCue with

0 -- E -- 0.05 [14]. On the other hand sample depend-

ences of the bulk modulus have been observed in different alloys of CeAl2 by Bartholin et al. [3].

In this paper we first describe accurate lattice parameter and electrical resistance experiments per- formed on a single crystal of CeAl2 (prepared on a

W crucible, which has been shown to provide best samples [11]) under a quasi-hydrostatic pressure

reaching 150 kbar. Lattice parameter experiments performed at room temperature do not show any

sign of valence anomaly ; however, one cannot exclude the possibility of very smooth anomaly

between 50 and 100 kbar, considering the error bars.

Resistance experiments clearly show a smooth ano- maly near 80 kbar at room temperature. This anoma- ly evoluates rather strongly between 300 K and 2 K

leading to a new line in the P-T diagram of CeAl2. We interpret this line as a crossover from a

weak to a strong Kondo behaviour. Then we estab- lished that this line intercepts the critical line where

CeAl2 orders magnetically [15, 18]. The possible

influence of a crossover, from weak to strong Kondo effect, on the nature of the magnetic order is

discussed for the first time. This point as well as

other features of the P-T diagram of CeAl2 can be

understood on the simple basis of a volume depen-

dent Kondo temperature (see Ref. [10] and refer-

ences therein) already used in the study of the susceptibility of CeAl2 under pressure [12, 16, 19].

2. Pressure. Volume behaviour.

The set up [17] used for determining the pressure- volume behaviour of CeAl2 is built around a rotating molybdenum anode generator Marconi-Elliott type GX 21. The X-ray beam is made quasi-mono-

chromatic and focussed using a set of two metallic

mirrors. The diffraction pattern is recorded by a

curved gaseous detector INEL CPS 120 (with a

radius of curvature of 25 cm and an angular aperture of 120°), transmitted to and analysed by a microcom-

puter. The sample (carefully powdered single crystal)

was enclosed within a diamond anvil cell [20]. At

each pressure, at least three reflections have been

simultaneously observed in CeAl2. The pressure was determined with an accuracy of ± 2 kbar from the reflection lines (at least two) of NaCI powder mixed

with the CeAl2 one.

The pressure-volume behaviour obtained in our

sample of CeAl2 is given figure 1. First of all the absence of any well defined transition is clearly

observed in contradiction with the result of reference

[2]. A reasonable fit to the Birch-Murnagham equation [21], performed up to 140 kbar leads to values for the isothermal bulk modulus B

=

-dP/d(V/Vo) and its pressure derivative Bó in

very good agreement with those previously obtained

at lower pressures (see Fig. 1). Therefore, if CeAl2

Fig. 1.

-

Volume-pressure evolution of CeAl2 at room temperature and fit to the Birch-Mumagham [20]

equation. The values obtained for the bulk modulus

Bo

=

68.7 kbar and its first derivative Bo’

=

3.0 are in good agreement with other determinations [3, 11].

exhibits an anomaly in its P-V behaviour below 150 kbar, this anomaly must be very smooth : the conventional a - y transition does not exist in our

sample CeAl2. This. result is very important since it

was, up to now, very well admitted that the archetyp-

ical Kondo system CeAl2 presented a typical cerium-

like a - y transition. It has been carefully checked by doing several runs in pressure. On the other hand another group performed the same kind of exper- iment on another sample of CeAl2 [22] ; their sample, as ours, does not show any sharp feature in

the P-V behaviour.

3. High pressure resistivity.

Let us now consider our electrical resistance meas- urements of CeAl2 under pressure. They have been

undertaken on a small single crystal of CeAl2 (0.05 x 0.2 x 1.6 mm3) from the same batch that the

sample used for X-ray experiments. A Bridgman

anvil technique [23] was used in attaining quasi- hydrostatic pressures P 140 kbar in a higher

pressure cell developed in our laboratory. A Pb

pressure manometer was sandwiched with our

CeAl2 crystal between steatite disks into a pyrophyl-

lite gasket. The pressure within the cell was deduced from a calibration of the superconductive transition

of Pb versus pressure [24]. We also used the Pb (I-II)

transformation fixed point at 130 kbar.

The pressure evolution of the resistance of our

single crystal of CeAl2 is given figure 2. First of all

we must mention that isobars all saturate to

-

35 mn, but the curve A. This lack of saturation is

just due to the bad quality of electrical contacts at

low pressure in this type of cell. In the curve B a

pressure of 16.2 kbar is large enough to compact the system and therefore to considerably improve electri-

cal contacts. General features of these curves are

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Fig. 2.

-

Electrical resistance-temperature plot for a CeAl2 single crystal at different hydrostatic pressures. The

high temperature portion of the curve A shows an excess

of resistance which is due to bad electrical contacts at low pressures. Results for LaAl2 at normal pressure are also

given.

similar to those of Nicolas-Francillon et al. [25] and

Probst and Wittig [26]. However, one cannot extract any well defined anomaly from these data due to

their limitation in a very restricted pressure range

(P -- 16 kbar) in the first case and to an important

pressure dependent background in the second case (experiments have been performed on powders and

therefore compacting effects modify the resistivity).

On the contrary, if plotted in a resistance-pressure

system of coordinates, data of figure 2 clearly shows

a transition from a resistive to a less resistive state when the pressure increases (Fig. 3). When lattice contributions are subtracted (normalized resistance

of LaA12) it appears that the spin disorder resistivity

vanishes in the high pressure phase. It is here

assumed that lattice contributions are not affected

by a pressure ; this is generally the case for very stable materials such as CeAl2 or LaAl2. This

Fig. 3.

-

Isotherms deduced from results of figure 2.

Notice that, in the high pressure side, the resistance of

CeAl2 is nearly equal to that of LaAl2 (arrows) suggesting

a vanishing of the spin disorder resistivity.

hypothesis is furthermore corroborated by our X-ray experiments which clearly show that the MgCu2

structure of CeAl2 is preserved up to 150 kbar.

The continuous vanishing of the spin disorder resistivity of CeAl2 under pressure (beyond crystal

field effects [27] which always lead to a finite magnetic resistivity at low temperature) must be attributed, in the paramagnetic phase, to strong reductions in the time averaged moments (m2) T’

associated with a larger Kondo temperature ; in the high pressure/low temperature region, the Kondo temperature, TK is larger than that, TK, of the low

pressure/high temperature one. A continuous vol-

ume anomaly could be associated with this modifi- cation in the TK of CeAl2, as suggested by the

following arguments :

i) the evolution of the Neel temperature of CeAl2 with pressure has been in the past well interpreted in terms of a volume dependent Kondo temperature where TK (pJ) is the usual exponential

function and where the functional dependence pJ(P ) has been determined from resistivity exper- iment [16, 12] (nearly linear dependence).

ii) the bulk modulus of well characterized crystals

of CeAl2 is by 7 % softer than the one of the other RA12 (an interpolation using the phenomenological

relation B - (valency/volume) was used [11]).

iii) The surprising coincidence of the pressures at which our resistance anomaly and the transition of Croft and Jayaraman [4] occurs in another sample of CeAl2,

iv) the alloys CexThl - x, which are for x > 0.27 similar to CeAl2 under pressure, show a linear

resistivity/volume relationship when the temperature

evoluates [4].

It is worth noticing that in CeAl2 the resistance anomaly is continuous at any temperature between 300 and 2 K, showing that the transition a - y does

not exist in CeAl2 in this temperature range, for P :!S: 140 kbar ; instead we are in the presence of a continuous passage betwen two Kondo regimes (weak to strong) of the same Kondo phase. If the

arguments given above are relevant, then a smooth

volume anomaly would be associated with this

crossover and the line of figure 4 would represent the inflection point of isotherms above the critical

point (Pc7 Tc) given in the Kondo collapse model [9, 10]. In CeAl2 it is not clear whether the upper critical

point lies at very low temperature or both critical

points collapse at finite temperature. This last possi- bility seems more probable since, in the Gibbs energy, the volume dependent term of Kondo origin

is relatively small, compared to the normal one, as this can be estimated from the softening [11] of 7 %

of the bulk modulus. Accurate P- V experiments are

in progress which should allow to study in more

details the P- V diagram of CeAl2. At the present

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638

time we shall only consider that, at each tempera- ture, there is a pressure P above which local

paramagnetic moments vanish in CeAl2 due to a large Kondo temperature. The corresponding P-T

line is shown figure 4. It has been defined from the inflection point of each isotherm of figure 3 (another definition, in which the characteristic pressure is taken on each isotherm at Rmax/2, where Rmax is the

resistance maximum of crystal field origin, leads to

the same curve). This curve characterizes the cross- over from weak to strong Kondo regime in which the

time averaged paramagnetic moment vanishes prog-

ressively. It is very different from a a - y transition line. In particular its curvature is positive instead of negative. The existence of a continuous passage from a magnetic to a non magnetic state in CeAl2 had been in fact suggested many years ago from a linear extrapolation of susceptibility measure-

ments performed between 0 and 18 kbar at 1.5 K [19], i.e. in the magnetically ordered phase. This point brings us to the question of possible connec-

tions between a crossover from weak to strong

Kondo behaviour occurring in the vicinity of a magnetically ordered phase.

Fig. 4.

-

Line of crossover between weak and strong Kondo regimes in CeAl2. Note the positive curvature.

4. P-T Phase diagramme.

In CeAl2 the crossover line intercepts the critical line where the modulated magnetic order takes place at

about 2.5 K and 23 kbar (Fig. 5). It is interesting to

note that a phase transition from this modulated structure to a simple antiferromagnetic structure of

type II has been observed near this point [15]. More precisely, below 15 kbar, CeAl2 orders according to

Fig. 5.

-

Temperature-pressure phase diagram of CeAl2

for T (K ) 10 and 10 =s= T (K ) 20 (inset). Three diffe- rent magnetic phases are present : the paramagnetic phase (regions II and III), the modulated phase (regions IV), the type II phase (region VI). Magnetically ordered CeAl2

shows also a coexistence between modulated and type II phases, in region V [15]. Expected lines separating re- gions V with regions IV or VI are indicated in dashed. In the paramagnetic region, two different crossover lines are given. One of them is not yet very well determined

(triangles and dashed). It is interpreted as due to the

crossover between thermal (II) and weak Kondo behaviour

(III). The second one determined from our pressure

experiments corresponds to the crossover between weak and strong Kondo behaviour (see also Ref. [32]).

a modulated structure and at lower temperature it shows a coexistence of the modulated and type II

structure. Between 15 and = 20 kbar these two types of magnetic order coexist at any temperature and

above = 23 kbar only the type II structure persists [31]. In particular the amplitude of the components of the propagation vector characterizing the mod-

ulated structure vanishes at 1.7 K and 22 kbar. It is remarkable that this point (0) falls exactly on the extrapolation, in the ordered phase, of the crossover

like determined in the paramagnetic phase (this extrapolated portion of line coincides, in fact, with

the curve determined, in the ordered phase, using

the same criterion). This suggests that the change

from the modulated to the type II antiferromagnetic

structure is a consequence of the increase of the Kondo temperature from TK to T H It is tempting to

consider that the crossover line, associated with the

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progressive change from weak to strong Kondo behaviour in the paramagnetic region, becomes

critical in the magnetically ordered region (transition

between the modulated and type II structure). In

such a case another critical line, characterizing the

type II to paramagnetic transition in the high

pressure region, should be observed (region VI).

However, this line might be rather steep due to a strong pressure dependence of the Neel temperature, reducing the extention of the phase VI and therefore making difficult to observe it, unless very careful

investigations are performed between 20 and 40 kbar.

It might be interesting to note that the weak

(TK/T 1 ) or strong (TK/T > 1 ) Kondo regimes correspond to the weak or strong coupling limits in a

Kondo inpurity. The resistivity should vary - Ln T in the first case and - T2 (Fermi liquid) in the second

case. These different aspects will be discussed in

forthcoming papers.

Another aspect of the observation of a high

pressure magnetic structure in CeAl2 concerns the

nature of the magnetic ground state of the f.c.c.

Kondo lattice. In CeAl2, inhomogeneous moment

reductions (modulated structure [18]) have been

attributed to the competition between « normal » ferromagnetic interactions (PrAl2, as well as most

other rare earth-Al2 are ferromagnetic) and negative

interactions [18] resulting from the Kondo effect in a compound [12] (Kondo lattice). This interpretation

is still consistent with the conclusions of this paper,

since, in the high pressure region, negative interac-

tions of Kondo origin should become relevant lead-

ing to a simple antiferromagnetic structure. Within this interpretation the type II antiferromagnetic

structure would characterize the ground state of the

f.c.c. Kondo lattice. It would be interesting to see

whether such a conclusion could be obtained theoret-

ically on the basis of coherence effects in the f.c.c.

Kondo lattice. However, one must mention that

Jarlborg et al. [30] found that nesting properties of LaAl2 correspond reasonably well to the type II

antiferromagnetic structure of CeAl2. This indicates

that, even in the absence of Kondo lattice effects,

the magnetic structure of CeAl2 would be essentially unchanged.

Let us now consider some other features of t1¡e

P-T phase diagram of CeAl2 given in figure 5. The high temperature-low pressure region (weak Kondo coupling) is divided in two portions :

-

in region II, the Kondo temperature is negli- gible compared to the thermal energy kT and therefore the paramagnetic moment M2> T is not

reduced (in other words the Kondo enhanced quan-

tum fluctuation time TK - hlkTK is much larger

than the characteristic time of thermal fluctuations TT - 1 /k7J . The behaviour of CeAl2 tends here to a

normal « thermal behaviour ».

-

in region III, TK : TT, and cerium paramagne- tic moments are reduced (paramagnetic scattering of CeAl2 led to J (M2) - 0.6 Jl-B at 8 K). The be-

haviour of CeAl2 is here of weak Kondo Type.

The hatched line separating these two regions

indicates the crossover from thermal to weak Kondo behaviour. This line can be located from e.g. specific

heat experiments and/or resistivity experiments.

Specific heat results at P

=

0 [29] showed that the full entropy R In 2 of the crystal field ground state of CeAl2 is reached at 15 K ; this temperature has been interpreted as characteristic of the crossover above mentioned between Kondo and thermal fluctuations at P

=

0 [12]. Two other data points can be obtained

from the resistance experiments of figure 1 where,

above the resistivity cusp of the Neel temperature, another cusp is clearly observed which must be

connected with the evolution of TK/T. We believe

that the cusp at the right end of the resistance

plateau of figure 1 (curves A and B) can be used to

locate the crossover between thermal and Kondo behaviour at P $= 0 (Fig. 5 inset). It is interesting to

note that the resistance plateau (i.e. both cusps)

vanishes between 16.2 and 21.7 kbar : this is consis- tent with a progressive increasing of the Kondo

temperature with pressure which tends to suppress at the same time the magnetic order and the

crossover to a weak Kondo regime.

Finally the recent work of Schefzyk et al. [31]

shows the existence of a specific heat anomaly at a temperature T, slightly larger than TN, with

d Ln TcldP > 0. Our resistivity experiment under

pressure also shows an anomaly in the paramagnetic phase. We have followed this anomaly between 5

and 25 kbar ; it can very well be estrapolated when

P > 0 to the anomaly at Schefzyk et al. In our

interpretation the line Tc (P ) characterizes the oc- currence of a magnetic short range order in a Kondo system (magnetic coherence).

5. Conclusion.

To conclude, we have shown that our sample of CeAl2 does not present any sharp anomaly in its

pressure behaviour. Instead we evidenced a cros- sover line characterizing the passage from weak to strong Kondo regimes. The intersection of this line with the critical line along which magnetic order

takes place is located where the magnetic structure changes from modulated to type II. This suggests the

possibility of a new type of transition within the

magnetically ordered phase between different ordered phases of low and large TK. Such a phase

transition would be the analog, below TN, of the

weak Kondo coupling to Fermi liquid crossover

observed above TN. On the other hand a crossover

line between regions dominated by thermal fluctua-

tions or where single ion Kondo fluctuations start to

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640

be observed has been identified. The existence of such crossover lines should be general and their

identification in CeAl2 should help in understanding

anomalies observed in other Kondo compounds.

Acknowledgments.

It is a pleasure to thank A. Draperi for valuable technical assistance and C. Lacroix for interesting

discussions.

References

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mun. 44, 8 (1982) 1199.

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Appl. Phys. 50, 3, 11 (1979) 2300.

[13] See e.g. JAYARAMAN, A., Phys. Rev. B 137, 1A (1975) 179 and references therein.

[14] WAINTAL, A., ROSSIGNOL, M. F. and BARBARA, B., internal report, C.N.R.S., not published.

[15] BARBARA, B., BOUCHERLE, J. X., ROSSIGNOL, M. F. and VETTIER, C., Phys. Rev. Lett. 45, 11 (1980) 938.

[16] BARBARA, B., CYROT, M., LACROIX-LYON-CAEN,

C. and ROSSIGNOL, M. F., J. de Physique Col- loq. 40 (1979) C5-340.

[17] DRAPERI, A., LAURANT, J. M., ROSSIGNOL, M. F.

and WAINTAL, A., to be published.

[18] BARBARA, B., BOUCHERLE, J. X., BUEVOZ, J. L., ROSSIGNOL, M. F. and SCHWEIZER, J., Solid State Commun. 24 (1977) 48.

[19] BARBARA, B., BARTHOLIN, H., FLORENCE, H., ROSSIGNOL, M. F. and WALKER, E., Physica

86-88B+C (1977) 177.

[20] PAUREAU, J. J., WAINTAL, A., FRUCHART, R. and SENATEUR, J. P., European High Pressure Re- search Group, 16th Annual Conf., Reading,

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[21] BIRCH, F., J. Geophys. Res. 57 (1952) 227.

[22] VEDEL, I., REDON, A. M., LEGER, J. M., MIGNOT,

J. M. and FLOUQUET, J., J. Magn. Magn. Mat.

54-57 (1986) 361.

[23] EICHLER, A. and WITTIG, J., Z. Ang. Phys. 25 (1968) 319.

[24] EILING, A. and SCHILLING, J. S:, J. Phys. F: Metal.

Phys. 11 (1981) 623.

[25] NICOLAS-FRANCILLON, A., PERCHERON, A., ACHARD, J. C., GOROCHOV, O., CORNUT, B., JEROME, D. and COQBLIN, B., Solid State Com-

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[31] The study of different magnetic phases of CeAl2 under pressure is given in reference [15].

However, similar approaches based on the effect

of very small amounts of impurities have been

also performed, see e.g. BARBARA, B., BOUCHERLE, J. X., BUEVOZ, J. L., ROSSIGNOL,

M. F. and SCHWEIZER, J., J. Magn. Magn. Mat.

14 (1979) 221 ;

SCHEFZYK, R., LIEBE, W., STEGLICH, F., GOTO, T.

and LUTHI, B., J. Magn. Magn. Mat. 45 (1984)

229.

[32] BARBARA, B., BEILLE, J., CHEAITO, B., LAURANT,

J. M., ROSSIGNOL, M. F., WAINTAIL, A. and

ZEMIRLI, S., Phys. Lett. 113A (1986) 381.

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