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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�
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.
2014Des 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 où 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.
2014Accurate 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
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
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
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
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 ».
-