Magnetic ground state of Ce Ni1-x Cux: A calorimetric investigation

Article abstract of DOI:10.1103/PhysRevB.71.134401

Ce Ni1-x Cux is a substitutional magnetic system where the interplay of the different magnetic interactions leads to the disappearance of the long-range magnetic order on the CeNi side. The existence of inhomogeneities (spin clusters or phase coexistence) has been previously detected by magnetization and muon spin relaxation (μSR) spectroscopy measurements. These inhomogeneities are always observed regardless of the different preparation methods and must, then, be considered as intrinsic. We present a detailed specific heat study in a large temperature range of 0.2 to 300 K. The analysis of these data, considering also previous neutron scattering, magnetic characterization, and μSR results, allows us to present a convenient description of the system as inhomogeneous on the nanometric scale. Two regimes are detected in the compositional range depending on the dominant Ruderman-Kittel-Kasuya-Yosida or Kondo interactions. We propose that the long-range magnetic order at low temperatures is achieved by a percolative process of magnetic clusters that become static below the freezing temperature Tf. In this scenario the existence of a quantum critical point at the magnetic-nonmagnetic crossover must be discarded. This situation should be considered as an example for other substitutional compounds with anomalous magnetic or superconducting properties.

Full text of DOI:10.1103/PhysRevB.71.134401

PHYSICAL REVIEW B 71, 134401 ͑2005͒
Magnetic ground state of CeNi1−xCux: A calorimetric investigation
N. Marcano, J. I. Espeso, J. C. Gómez Sal, and J. Rodríguez Fernández
Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Facultad de Ciencias, Universidad de Cantabria,
39005 Santander, Spain
J. Herrero-Albillos and F. Bartolomé
Instituto de Ciencia de Materiales de Aragón, CSIC Universidad de Zaragoza, 50009 Zaragoza, Spain
͑Received 8 November 2004; published 6 April 2005͒
CeNi1−xCux is a substitutional magnetic system where the interplay of the different magnetic interactions
leads to the disappearance of the long-range magnetic order on the CeNi side. The existence of inhomogene-
ities ͑spin clusters or phase coexistence͒ has been previously detected by magnetization and muon spin relax-
ation ͑␮SR͒ spectroscopy measurements. These inhomogeneities are always observed regardless of the differ-
ent preparation methods and must, then, be considered as intrinsic. We present a detailed specific heat study in
a large temperature range of 0.2 to 300 K. The analysis of these data, considering also previous neutron
scattering, magnetic characterization, and ␮SR results, allows us to present a convenient description of the
system as inhomogeneous on the nanometric scale. Two regimes are detected in the compositional range
depending on the dominant Ruderman-Kittel-Kasuya-Yosida or Kondo interactions. We propose that the long-
range magnetic order at low temperatures is achieved by a percolative process of magnetic clusters that become
static below the freezing temperature Tf. In this scenario the existence of a quantum critical point at the
magnetic-nonmagnetic crossover must be discarded. This situation should be considered as an example for
other substitutional compounds with anomalous magnetic or superconducting properties.
DOI: 10.1103/PhysRevB.71.134401
PACS number͑s͒: 75.40.Ϫs, 65.40.Ba, 75.30.Mb
I. INTRODUCTION
͑iii͒ At higher temperatures the shape of the thermal varia-
tion of the specific heat depends on the crystal electric field
splitting.
͑iv͒ The very-low-temperature CP/T behavior can show
departures from the predictions of the Fermi-liquid theory,2
which could serve as the sign of the different scenarios de-
voted to explaining the physics involved in these processes
͑spin fluctuations that exist close to a QCP,3–5 Griffiths phase
situation,6 disorder-driven mechanisms like “Kondo
disorder,”7,8 etc.͒.
Strongly correlated electron metals have been for a long
time now one of the most fruitful fields for the discovery of
new physical phenomena ͑heavy fermions, non-Fermi liquids
͑NFL’s͒, quantum critical points ͑QCP’s͒, or unconventional
superconductivity͒, some of them pointing toward a possible
new description of the quasiparticle ground state. At the
same time, the study of such magnetic compounds has pro-
vided additional insight into the understanding of the basic
magnetic interactions ͑Ruderman-Kittel-Kasuya-Yosida
RKKY, 4f-3d hybridization, crystal electric field, etc.͒ and
the interplay between them.
In many of these systems an additional problem appears
when we are dealing with atomic substitutions. This problem
is described in general as “disorder effects” and could consist
of compositional local inhomogeneities, phase coexistence,
and clusterization processes that in most cases are intrinsic to
the samples and cannot be avoided by means of different
preparation methods or supplementary thermal treatments.
Specific heat ͑CP͒ measurements provide a powerful
method for studying the behavior of these compounds1 for
the following reasons.
͑i͒ The magnetic contribution to the specific heat and the
corresponding magnetic entropy are related to the energy
levels of the magnetic ions ͑cerium in our case͒ and the
shape of the anomalies corresponding to the magnetic phase
transition gives a valuable indication as to the nature of the
transition.
͑ii͒ The electronic coefficient ␥ provides relevant infor-
mation concerning the conduction band density of states at
the Fermi level.
All these features can be used in the analysis of the
CeNi1−xCux system, which has been studied over the last few
years with different macroscopic and microscopic techniques
such as magnetization, resistivity, neutron diffraction, muon
spin relaxation ͑␮SR͒ spectroscopy, etc. The main features
of this system at the present stage can be summarized as
follows.
͑a͒ A complex magnetic behavior was determined9 with a
change from an antiferromagnetic ͑AFM͒ ͑CeCu͒ to ferro-
magnetic ͑FM͒ ground state well characterized by neutron
diffraction in CeNi0.4Cu0.6 ͑Ref. 10͒ evolving toward the eva-
nescence of the long-range magnetic order around
CeNi0.8Cu0.2
͑b͒ A “spin-glass-like” phase was surprisingly found at
temperatures above the ferromagnetic order state.11
.
A
“cluster-glass” scenario seems the most plausible hypothesis
according to ac and dc magnetization measurements.12
͑c͒ In fact, recent ␮SR studies confirm the presence of a
highly inhomogeneous magnetic state at temperatures ex-
tending from the long-range ordered ground state to the para-
magnetic regime. This inhomogeneous magnetic state con-
sists of long-range ordered and nonordered fractions, the
latter increasing with temperature.13
1098-0121/2005/71͑13͒/134401͑11͒/$23.00
134401-1
©2005 The American Physical Society

MARCANO et al.
PHYSICAL REVIEW B 71, 134401 ͑2005͒
TABLE I. Crystallographic data of the CeNi1−xCux compounds
obtained at 300 K from x-ray diffraction data.
͑d͒ The Ni-rich part of the diagram is very sensitive to
thermal treatments due to the proximity to the crystalline
structural change ͑FeB to CrB͒.14 There appears a marked
decrease in magnetization as a consequence of the strong
reduction of the magnetic Ce moments due to the increasing
hybridization effects. Some traces of hysteresis, however,
have been detected at very low temperatures in the
CeNi0.85Cu0.15 and CeNi0.9Cu0.1 compounds, which show
that the magnetic moments have not been completely ex-
hausted yet.15
Many questions still await answers in order for there to be
a full understanding of such complex behavior. The aim of
this paper is to show how the specific heat measurements
performed in these series between 0.2 and 300 K provide
valuable information that sheds light on the underlying phys-
ics in the system.
x
Structure
a ͑Å͒
b ͑Å͒
c ͑Å͒
V ͑Å3͒
0.1
0.15
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CrB
CrB
FeB
FeB
FeB
FeB
FeB
FeB
FeB
FeB
3.800͑4͒ 10.490͑1͒ 4.355͑7͒ 173.7͑3͒
3.814͑3͒ 10.565͑8͒ 4.390͑5͒ 176.9͑3͒
7.267͑4͒
7.280͑3͒
7.304͑2͒
7.324͑4͒
7.354͑1͒
7.369͑4͒
7.383͑4͒
7.410͑3͒
4.425͑2͒
4.428͑2͒
4.445͑1͒
4.471͑2͒
4.500͑5͒
4.506͑2͒
4.528͑2͒
4.543͑1͒
5.598͑3͒ 180.1͑1͒
5.602͑2͒ 180.6͑1͒
5.624͑2͒ 182.6͑1͒
5.646͑3͒ 184.9͑1͒
5.658͑5͒ 187.2͑1͒
5.674͑2͒ 188.4͑1͒
5.688͑3͒ 190.2͑2͒
5.682͑2͒ 191.3͑1͒
Starting in the first part of the article with a detailed pre-
sentation of the preparation methods, crystallography, and
quality control of the samples, we go on to present the spe-
cific heat measurements and their corresponding analysis.
Special care has been taken in order to elucidate the nature of
the anomalies found in these compounds. Particular attention
has been paid to the analysis of the low-temperature regime
at the compositional region near the magnetic-nonmagnetic
crossover.
arc-melted button between melts in order to improve homo-
geneity.
The crystalline structures of the CeNi1−xCux compounds
studied have been determined by x-ray diffraction at 300 K.
The alloys with xϾ0.15 crystallize in the FeB-type ortho-
rhombic structure ͑Pnma space group͒, whereas the alloys
with xഛ0.15 crystallize in the CrB-type orthorhombic struc-
ture ͑Cmcm space group͒.15 Both crystalline structures are
built from Ce trigonal prisms ͑with a transition metal in the
center͒. The two structures only differ in the relative dispo-
sition of the trigonal prisms, which is the reason why Ce-
Ce and Ce-͑Ni,Cu͒ distances evolve continuously from one
structure to the other.19
II. EXPERIMENTAL DETAILS
A. Instruments
The specific heat measurements in the range 0.2ϽT
Ͻ6 K were performed at the Instituto de Ciencia de Materi-
ales de Aragón on a fully automated quasiadiabatic
calorimeter16 refrigerated by adiabatic demagnetization of a
paramagnetic salt17,18 using the heat pulse technique and ger-
manium thermometry over the whole temperature range.
Small ingots of around 0.5 g weight were used for the mea-
surement, mixed with Apiezon N grease to achieve good
thermal contact between the sample and the calorimetric set
͑heater and thermometer͒ even at the lowest temperature.
The calorimeter is equipped with a mechanical switch that
allows temperatures down to 0.2 K to be reached. The abso-
lute accuracy of the instrument has been estimated to be
about 1%.
The calorimetric data between 2 and 300 K were obtained
in a commercial Quantum Design microcalorimeter at the
University of Cantabria using the relaxation technique. In
this case, thin slab-shape samples 0.5 mm thick and weigh-
ing around 8 mg were used. Matching between both sets of
data in the common temperature range is quite satisfactory.
The Rietveld analyses were performed considering Ni and
Cu atoms randomly distributed on the same 4c site while the
Ce lies on the other 4c site.
The narrowness of the x-ray peaks of the as-quenched
sample spectra guarantees the good crystallization of the
samples except for the Cu-rich ones, i.e., x=0.8 and 0.9,
which showed a broadening of the diffraction peaks. A high-
vacuum annealing at 425 °C ͑just below the melting point of
CeCu͒ for one week was carried out on the samples with x
Ͼ0.2. Noticeable narrowing of the peaks was induced only
for x=0.8 and 0.9 samples, whereas no significant modifica-
tions were observed for any of the other compositions.
The compounds with compositions close to the FeB
-CrB-type structure change ͑x=0.1, 0.15, and 0.2͒ present
signs of a mixture of FeB and CrB phases for the as-
quenched preparations. The stabilization of the proper struc-
ture in each case was reached with a high-vacuum annealing
at 440 °C for one week.
Table I summarizes the crystallographic data derived from
the Rietveld refinements at 300 K for the studied com-
pounds. The cell volume increases with the Cu content; the
20
evolution is quite similar to that reported for CeNiyPt1−y
both these sets of data are plotted in Fig. 1.
;
B. Samples
The samples in the present work are those with x=0.1,
0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. They are all
polycrystalline samples prepared by carefully melting to-
gether stoichiometric amounts of the appropriate high-purity
starting elements in an arc-melting furnace under inert Ar
atmosphere. Five melts were performed with flipping of the
In order to get a better characterization of the microstruc-
ture of the samples, a scanning electron microscopy ͑SEM͒
study has also been performed.
The intensity of backscattered electron images taken with
a SEM is proportional to the local atomic number of the
sample. Therefore, these backscattered electron images yield
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MAGNETIC GROUND STATE OF CeNi1−xCux: A…
PHYSICAL REVIEW B 71, 134401 ͑2005͒
FIG. 1. Cell volume derived from x-ray diffraction for
CeNi1−xCux together with that for CeNiyPt1−y. The drawn lines are
just a visual guideline.
semiquantitative information about composition homogene-
ity in a sample. Samples were prepared for SEM observa-
tions by using grinding paper disks of different grain sizes
and final polishing with diamond paste. The SEM backscat-
tered electron images were produced with a JEOL electron
microscope at the University of Cantabria. Magnifications
between 350 and 2000 were reached and quantitative elec-
tron probe microanalysis techniques were used to verify the
chemical composition of the phases. We used secondary
electron images to distinguish composition fluctuations from
physical voids in the sample by comparing them with the
backscattered electron images.
FIG. 2. ͑a͒ Temperature dependence of the specific heat for
CeNi1−xCux compounds with xϾ0.15 and YNi. The YNi data has
been fitted ͑solid line͒ considering a Debye and an electronic term
͑see text for details͒. ͑b͒ The specific heat for xഛ0.15 compounds
and LaNi. The CeNi and LaNi data were taken from Ref. 22. The
data corresponding to different compounds have been shifted for
clarity. The inset in Fig. 2͑b͒ shows the total specific heat in the
low-temperature regime.
The backscattered electron images revealed that most of
the sample volume consists of a matrix with the expected
nominal composition in each case. Electron probe mi-
croanalysis measurements revealed that this matrix is homo-
geneous in all the scanned regions ͑ϳ5 ␮m2͒. We observed
that as Cu concentration increases, samples tend to be very
brittle and so the tendency to present cracks in the surface
becomes more important.
A. Estimate of the electronic coefficient ␥
The value of the linear electronic coefficient of the spe-
cific heat ͑␥͒ has been taken as the extrapolation of the linear
part of the CP/T vs T2 curves at zero temperature, according
to the law CP=␥T+␤T3 followed at low temperatures, thus
overlooking any anomalous contribution. The values ob-
tained in this way are indicative of the electronic correlation
enhancement, i.e., how far the corresponding compound is
from the free-electron picture. Figure 3 illustrates this ex-
trapolation for some selected compositions of the series. The
III. RESULTS
The specific heat measurements performed in the tem-
perature range 0.2 to 300 K on the compositions with FeB-
type structure ͑xϾ0.15͒ and also the isostructural YNi are
shown in Fig. 2͑a͒. The temperature dependence of CP for
YNi can be accounted for if an electronic term and a lattice
one following the Debye function are taken into consider-
ation. The fit of these data yields the values
␥
=5 mJ/K2 mol and ␪D=250 K, which are in good agreement
with previous results.21 Figure 2͑b͒ displays the temperature
dependence of the specific heat for the compositions with
CrB-type structure ͑xഛ0.15͒ together with the already re-
ported data for the specific heat of CeNi and the isostructural
nonmagnetic LaNi compound.22 In that case, the temperature
dependence of CP for LaNi follows a Debye function with
␥=5 mJ/K2 mol and ␪D=190 K.22
The inset of Fig. 2͑b͒ shows the low-temperature region
͑from 0.2 to 6 K͒ of the total specific heat. The origin and
nature of the anomalies shown here will be explained later
on.
FIG. 3. Specific heat plotted as CP/T vs T2 showing the tem-
perature range with a linear behavior whose extrapolation to T
=0 K allows the estimation of the electronic specific heat coeffi-
cient ␥. Only some compositions are presented for clarity.
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MARCANO et al.
PHYSICAL REVIEW B 71, 134401 ͑2005͒
TABLE II. Magnetic characteristics of the compounds studied: Critical temperature ͑TC͒; reduced magnetic entropy at TC
͓Smag͑TC͒/R ln 2͔; linear electronic specific heat coefficient ͑␥͒; magnetic moment per Ce ion ͑␮͒ obtained from neutron diffraction analysis
͑Ref. 9͒; Kondo temperature ͑TK͒ estimated from ͑i͒ the magnetic entropy ͑Smag͒, ͑ii͒ the magnetic susceptibility ͑␪P/10͒ ͑Ref. 19͒, ͑iii͒
quasielastic neutron scattering ͑QENS͒ ͑Ref. 9͒, and ͑iv͒ the minimum in the electrical resistivity ͑T min/5͒ ͑Ref. 19͒.
␳
Smag͑TC͒
TC ͑K͒ ͑units of R ln 2͒
TK
Smag
TK
TK
TK
T
min/5 ͑mJ/K2 mol͒ ␮ ͑␮ ͒ field ⌬1, ⌬2 ͑K͒
␳
␥
Crystal electric
Compound
QENS ␪P/10
B
CeNi0.1Cu0.9
CeNi0.2Cu0.8
CeNi0.3Cu0.7
CeNi0.4Cu0.6
CeNi0.5Cu0.5
CeNi0.6Cu0.4
CeNi0.7Cu0.3
CeNi0.8Cu0.2
CeNi0.85Cu0.15
CeNi0.9Cu0.1
CeNi
2.9
3.0
3.1
2.7
2.5
2.6
2.2
1.8
0.67
0.58
0.45
0.46
0.58
0.60
0.48
0.19
4.3
0.7
1
0.4
0.1
0.8
1
131
117
1
44, 105
47, 128
50, 116
49, 120
55, 146
62, 135
50, 116
51, 115
46, 100
67, 159
4.8
1.9
6.1
2.4
2.8
3.7
4.7
114
6.0
2
112
0.6
4.6
1.4
1.7
151
0.51
4.7
154
4.8
6.6
176
4.1
6
7.2
4.5
206
8.2
217
13.8
150
140 ͑Ref. 22͒
65 ͑Ref. 22͒
differences over the series are clearly seen both in the size of
the CP/T values and in the slopes of the CP/T vs T2 data,
which correspond to the differences in the Sommerfeld coef-
Returning to the ␥ evolution for the CeNi1−xCux system
displayed in the same plot, we became aware of the coinci-
dence of the ␥ variation between the two series for Ni con-
tent larger than 0.5, whereas the compounds closer to CeCu
follow a different trend from the CeNiyPt1−y series. Thus,
that comparison allows dividing the behavior of the series
into two main composition regimes. Hence, the first regime
would range from CeCu to CeNi0.4Cu0.6, where the values of
␥ do not significantly change. This region corresponds to the
composition regime where the evolution from an AFM
͑CeCu͒ to an FM ͑CeNi0.4Cu0.6͒ ground state occurs, giving
rise to complex ordered spin structures as has been estab-
lished from neutron diffraction.9 This evolution is considered
to be caused by the enhancement of the ferromagnetic inter-
actions closely related to the change in the density of states
at the Fermi surface driven by the differences in d-electron
density between Cu and Ni,25 and it has been also found in
similar RNi1−xCux series with light26 and heavy rare-earth
elements.27
The evolution from CeNi0.4Cu0.6 to CeNi defines the sec-
ond regime. Once the FM ground state sets in, the competi-
tion between RKKY, predominantly ferromagnetic, and
Kondo interactions leads the system from a magnetic local-
ized to a delocalized Pauli paramagnetic state ͑CeNi͒, and
this competition defines the second regime. In that regime,
the coefficient ␥ is enhanced for compounds with increasing
hybridization favored by the decreasing cell volume, reach-
ing values of more than 200 mJ/K2 mol around the x=0.15
composition.
ficient ͑␥͒ and the Debye temperature ͑␪ ͒, respectively.
D
Table II reports the values of ␥ obtained for all the compo-
sitions.
According
to
the
reported
values
͑130–220 mJ/K2 mol͒ these samples can be considered as
moderate heavy fermions. The compositional dependence of
␥ is plotted in Fig. 4. We have shown in the same figure the
23
dependence of ␥ on composition for CeNiyPt1−y
,
another
Ce system typifying the evolution from a localized magnetic
state to a delocalized one according to the Doniach
diagram.24 In the latter case, the value of the ␥ coefficient
increases when the Ni content does ͑increasing hybridiza-
tion͒ and reaches
a maximum value of more than
200 mJ/K2 mol around the crossover where the change from
a magnetic localized regime to a delocalized one takes place.
Once the 4f delocalization is predominant, the ␥ value de-
creases with increasing Ni concentration.
Let us come back to the cell volume variations of both
series displayed in Fig. 1. In spite of the change in crystal-
lographic structure existing in CeNi1−xCux, the cell volume
evolves similarly for both series, both CePt and CeCu having
nearly the same value of the cell volume. They do, however,
have to be distinguished. While in CeNi1−xCux the random
substitution on the nonmagnetic site Ni-Cu not only changes
the cell volume but also modifies the electronic state ͑by
changing the number of conduction electrons͒, in the
CeNiyPt1−y case the Ni-Pt substitution modifies only the cell
FIG. 4. Electronic specific heat coefficient ͑␥͒ for CeNi1−xCux
series as a function of Ni composition ͑1−x͒ and for CeNiyPt1−y
series as a function of Ni ͑y͒. The solid line is drawn as a guide
only. The dashed lines set the crossover between the two composi-
tional regimes defined in the text.
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MAGNETIC GROUND STATE OF CeNi1−xCux: A…
PHYSICAL REVIEW B 71, 134401 ͑2005͒
volume. However, the evolution of ␥ is the same in both
series in the thus defined second regime in CeNi1−xCux. The
evolution of ␥ was explained in CeNiyPt1−y as driven by the
competition between RKKY and Kondo interactions under
an internal molecular field23 according to
TK
␥ ϰ
T2K + H2
where H is the Zeeman energy of the magnetic moment as-
sociated with a Ce3+ ion within the molecular field created by
the other ions.
Then it is deduced from the previous comparison that the
electronic effects do not play a predominant role in the bal-
ance between RKKY and 4f conduction band hybridization
of the second regime. It can thus be concluded that the driv-
ing parameters of the ␥ behavior in this region are the inter-
atomic distances rather than the electronic effects.
B. Estimate of Cmag
The magnetic contribution to the specific heat Cmag has
been estimated for all the compounds by taking as the lattice
contribution of each magnetic compound that of the isomor-
phous nonmagnetic one after taking into account the mass
corrections.28 Thus, the specific heat of LaNi ͑Ref. 21͒ has
been taken as the nonmagnetic reference for the CrB-type
structure compounds ͑x=0.1 and 0.15͒ while for the rest of
the compositions ͑FeB-type structure͒ the specific heat of
YNi was used. Using this procedure, Cmag reflects any effect
related to the magnetic order transition and/or the tempera-
ture dependence of the electronic term.
FIG. 5. Cmag versus temperature in the low-temperature regime
plotted separately for the two composition regions defined: ͑a͒ 0.9
ഛxഛ0.6 or region where the system evolves from AFM to FM
ground state; ͑b͒ xഛ0.6 or region where hybridization effects and
disorder are predominant ͑for details see text͒.
ϳ3TC, which becomes more important as Cu content in-
creases ͑i.e., as the exchange interactions became more sig-
nificant͒. All these features support the predominant role
played by RKKY interactions in setting the magnetic ground
state in this regime.
We note a perceptible change in Cmag between x=0.6 and
0.5. Instead of a hump, CeNi0.5Cu0.5 exhibits a clear ␭-type
anomaly, with a critical temperature TC=2.3 K. This
anomaly becomes even sharper for CeNi0.6Cu0.4 and shifts
slightly up to TC=2.5 K. For the following compounds, the
anomaly broadens and its maximum value decreases with
increasing Ni content. Thus, for compositions with xϽ0.2,
the anomaly in Cmag cannot be distinctly defined.
The temperature corresponding to the maximum of the
magnetic specific heat coincides with the freezing tempera-
ture ͑Tf͒ determined by ac magnetic susceptibility measure-
ments, which for the compounds in this second regime was
associated with a “spin-glass-like” state developing at tem-
peratures above the long-range ferromagnetic order.11,29 Mi-
croscopic studies performed with ␮SR in order to shed light
on the nature and origin of these transitions suggested an
intermediate inhomogeneous magnetic state above Tf evolv-
ing into a long-range ordered state at lower temperatures.
However, from the specific heat point of view, no other tran-
sitions were observed at temperatures below the maxima.
This feature agrees with the idea of a continuous evolution
͑better than a second-order phase transition͒ to the long-
range ordered state found from neutron and muon results at
low temperatures.9,13 Earlier calorimetric studies presented in
CeNi0.4Cu0.6 revealed a very narrow peak at 1 K.10 The ac-
tual measurement in this composition is identical to the al-
ready published one except for that feature. We should bear
in mind that Ref. 10 was an in-depth study on one composi-
1. Cmag in the low-temperature regime
Figure 5 displays Cmag as a function of the temperature in
the low-temperature range for both regimes as defined in the
previous section.
Thus, Fig. 5͑a͒ illustrates the region 1ϾxϾ0.6 where the
evolution from an AFM to a FM ground state occurs due to
the modifications of the positive and negative RKKY inter-
actions. Figure 5͑b͒ shows Cmag for compounds with x
Ͻ0.6, where the magnetic ground state evolves from ferro-
magnetism to a Pauli paramagnetic state ͑CeNi͒.
Starting from the Cu-rich side, CeNi0.1Cu0.9 shows an
anomaly at the critical temperature ͑TC͒ corresponding to the
antiferromagnetic transition ͑TN=2.5 K͒ defined as the in-
flection point above the maximum of the Cmag curve. The
anomaly becomes broader and its maximum value decreases
as Ni content increases; it is significantly reduced for the x
=0.6 compound, which shows a broad hump centered at
about 2 K. It is worth noting the smooth variation of Cmag
above the ordering temperature for these compounds. These
effects are related to the existence of a short-range magnetic
order well above TN. In fact, the competition between posi-
tive and negative interactions leads to complex ordered spin
structures, as was corroborated by neutron diffraction.9 In
addition, quasielastic neutron scattering ͑QENS͒ carried out
in the x=0.9, 0.8, and 0.6 compounds9 indicated a supple-
mentary inelastic Gaussian contribution persisting up to
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MARCANO et al.
PHYSICAL REVIEW B 71, 134401 ͑2005͒
FIG. 6. Cmag/T vs T representation of the specific heat for
CeNi1−xCux in the compositions studied. The inset shows the varia-
tion for the compounds near the magnetic-nonmagnetic crossover
x=0.1, 0.15, and 0.2.
FIG. 7. Magnetic entropy Smag for CeNi1−xCux versus reduced
temperature ͑T/TC͒ for some selected compounds for clarity.
studies by both macroscopic ͑ac and dc measurements14͒ and
microscopic ͑␮SR͒ techniques13 in the compositions around
x=0.2 revealed that the magnetic moments are not yet ex-
hausted. However, long-range magnetic order, if it exists,
cannot be detected by neutron diffraction ͑within the resolu-
tion of the available experiments͒ since we are dealing with
very reduced values of the magnetic moment. Both experi-
ments point toward some kind of short-range ordered states
and reveal evidence of spin-glass arrangements even for the
x=0.1 compound.15 Thus, on the basis of all these results we
have to discard an intrinsic electronic effect leading to a NFL
or quantum phase transition around the x=0.2 compound.
The divergence in Cmag/T could be related to the existence of
a small anomaly at lower temperatures as was found at very
low temperature in ac susceptibility measurements.14
tion of the CeNi1−xCux series and at that point when even the
magnetic phase diagram of the series was unknown, we were
not aware of the enormous relevance of a meticulous char-
acterization of the samples. In fact, the sample reported in
Ref. 10 was melted in an induction furnace and its degree of
inhomogeneity was not as well characterized as in the
present study.
Taking a look at the very-low-temperature side of the
Cmag/T vs T curves ͑Fig. 6͒, we can observe an almost con-
stant behavior for CeNi0.85Cu0.15 and CeNi0.9Cu0.1. This fact
indicates the Fermi-liquid character of these compounds. It
should be remarked that the Fermi-liquid theory should only
be applied at very low temperatures where it describes the
low-lying excitations of the quasiparticles ͑interacting fermi-
ons͒. However, it is usually assumed that this label can be
extended to compounds presenting a constant electronic spe-
cific heat and magnetic susceptibility coefficient and a T2
dependence of the electrical resistivity. This is the case of the
Cmag/T corresponding to the x=0.1 and 0.15 compounds.
In f-electron compounds, strong electronic correlations
can lead to anomalous low-temperature properties indicating
that the system changes from a nonmagnetic to a magnetic
ground state as a certain parameter such as concentration or
pressure is tuned. Typically, the NFL behavior is associated
with a diverging specific heat Cmag/T when approaching T
=0 K, while the Fermi-liquid theory predicts a constant
value.2
In CeNi1−xCux, the whole set of measurements traces a
crossover from magnetic order to Pauli paramagnetic behav-
ior, thereby crossing some critical concentration range where
long-range order is expected to vanish. Therefore, one might
ask if NFL behavior might arise in this system. In that sense,
what is remarkable is the divergence of the Cmag/T curve at
low temperatures for the x=0.2 compound.
We are tempted to attribute the origin of this divergence to
its being the critical concentration where long-range mag-
netic order is suppressed, then presenting a NFL behavior as
in other series exhibiting, in principle, a similar scenario.
However, the existence of a transition at lower temperature
cannot fully be ruled out. The low-temperature dependence
of Cmag/T must be carefully analyzed in this case. In fact, as
we mentioned in the Introduction, very-low-temperature
2. Low-temperature magnetic entropy
The magnetic entropies have been calculated from
T
C
Smag
=
magdT
T
͵
0
and are shown in Fig. 7 versus the reduced temperature
͑T/TC͒ for the most significant compounds.
The first point to be discussed in the entropy analysis is
the shape of these curves below TC. As has been pointed out
in other series such as RGa2 ͑Ref. 30͒ and RNi2Si2 ͑Ref. 31͒,
this shape is closely related to the magnetic structure and the
thermal demagnetization processes. The more simple the
structure is with well-defined anisotropy, the sharper the in-
crease of the entropy is, defining a clear change of slope at
TC. This is the case shown in Fig. 7 where for the AFM
complex structures ͑x=0.8 and 0.9͒ a slow and progressive
increase is observed, whereas for the FM one ͑x=0.4͒ the
increase is sharper and a knee is observed at TC. It must be
stressed, however, that in this CeNi1−xCux series these varia-
tions are quite smooth even in the case of FM compounds,
indicating a high degree of magnetic disorder.
In all the compounds the magnetic entropy at TC is lower
than R ln 2=5.76 J/K mol, the value expected from a dou-
blet ground state, which is the one presented by the Ce3+ ion
under an orthorhombic framework. The reasons for such a
reduction are at least two. The first one is the complexity of
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PHYSICAL REVIEW B 71, 134401 ͑2005͒
the magnetic order with the existence of short-range interac-
tions above TC; this should be the main effect in compounds
close to the CeCu limit. The value of the entropy at TC de-
creases as the structure become more complex ͑RKKY com-
peting interactions being the predominant effect͒. The second
is the Kondo effect, which reduces the value of the magnetic
moment that is the dominant effect once the ferromagnetic
order is established ͑xϽ0.6͒ and the Kondo interactions be-
come more important. This reduction of the magnetic mo-
ment has been confirmed by neutron and muon
experiments.10,13
From this last assumption a TK value can be estimated
considering the reduction of the magnetic entropy at TC from
the value of R ln 2.32 Using a simple two-level model with an
energy splitting of kBTK, we can calculate the reduced en-
tropy at TC as
⌬S
− TK
ͩ ͪ
TC
TK
exp͑− TK/TC͒
TC 1 + exp͑− TK/TC͒
= ln 1 + exp
+
.
ͫ
ͬ
ͫ
ͬ
R
This expression must be applied for transitions at TC to
long-range magnetic order which is not really the case in our
compounds with xഛ0.6. In this case the specific heat
anomaly at TC is related to the spin freezing temperature.
However, the frozen state in these samples is a cluster glass
with ferromagnetic correlations inside the clusters. So, even
if some fraction of the entropy is invested in the disorder
among the clusters, this approach can give us an estimate of
the Kondo temperatures in order to follow their composi-
tional evolution.
FIG. 8. Magnetic contribution to the specific heat as a function
of temperature for CeNi0.1Cu0.9 and CeNi0.6Cu0.4. Solid lines corre-
spond to the paramagnetic CEF contribution ͑see text͒. The arrows
indicate the main transitions in the CEF level scheme.
with the excited states separated from the ground state by
energy gaps ⌬1 and ⌬2. A Schottky-type anomaly is then
expected. The hump around 50 K appearing in all the studied
compounds of this series has been analyzed considering this
CEF scheme. The values ⌬1 and ⌬2 that correspond to the
best Schottky-type fit to Cmag are listed in Table II.
The values obtained for the energy gaps are similar for all
the compositions with FeB-type structure. Moreover, the
level scheme found for the CrB compositions ͑x=0.15 and
0.1͒ is close to that corresponding to the FeB-type ones. This
result indicates that the NiuCu dilution does not signifi-
cantly modify the CEF splitting over the series.
For the compounds with no anomaly in Cmag͑xϽ0.2͒, the
estimation of TK requires another kind of analysis. From the
Sacramento and Schlottmann calculations for a J=1/2
impurity,33 one could estimate TK as the value at which 45%
of the total R ln 2 entropy is recovered.
Taking into account the perturbation induced by the short-
range order correlations and the two different methods used
to estimate the TK values, these results must be looked at
with care.
Special attention must be paid to the CrB compositions.
On the one hand, the presence of Schottky-type anomalies
supports the fact that the magnetic moment in Ce3+ remains
localized still for those concentrations close to CeNi. On the
other hand, the values found for the samples with CrB- and
FeB-type structures reveal that the change in the structure
does not introduce significant changes in the CEF sensed by
the Ce3+ ions. This fact indicates the similarity between both
structures as was already pointed out in Sec. II B.
In contrast, it is worth noting that for the intermediate
valence compound CeNi, a broad contribution centered
around 140 K was observed in previous works,22 it being
related to spin fluctuations present in compounds close to the
onset of ferromagnetism.
Table II summarizes the values obtained for TK in each
composition. In the same table we also present the values of
TK estimated from susceptibility19 as ͉␪ ͉/10, QENS,9 and
P
resistivity as T␳min/5.19
It is clear that the right estimate of TK comes from
QENS,9 where the quasielastic signal is directly related to
the Kondo temperature. The values of TK obtained from the
magnetic entropy are clearly overestimated due to the en-
tropy associated with cluster-glass formation.
The most interesting feature in all these estimates of TK is
that, in spite of the different magnitudes due to the different
time scales of the techniques,23,34,35 the same relative varia-
tion of TK is found with increasing Ni content.
The magnetic contribution to the specific heat in the para-
magnetic range together with the best fit obtained for the
Schottky anomaly and the associated level scheme are illus-
trated in Fig. 8 as an example for x=0.9 and 0.4.
Finally, the magnetic entropy Smag calculated from the
experimental Cmag variation for the 0.2ϽTϽ300 K tempera-
ture range is shown in Fig. 9 for some selected compounds.
C. Cmag above TC
Apart from short-range order correlations that might be
present above the critical temperature, the contribution to
Cmag above TC arises from the crystal electric field ͑CEF͒.
Due to the low-symmetry site occupied by the Ce3+ ions in
this system, the J=5/2 multiplet splits into three doublets,
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MARCANO et al.
PHYSICAL REVIEW B 71, 134401 ͑2005͒
fects and then long-range magnetic order is hard to reach.
For compounds presenting higher Ni content, ␥ behaves as in
the nonordered heavy fermion compounds, ␥ϰ1/TK. In our
case, however, important magnetic disorder effects still pre-
vail. Furthermore, the magnetic moments are not completely
exhausted and they can arrange at very low temperatures.
The complex magnetic behavior and the evolution of the
interactions in this series can be condensed into the sketch
picture displayed in Fig. 10.
Previous bulk data suggested the existence of a spin-
glass-like phase ͑cluster glass in Fig. 10͒ appearing in a cer-
tain compositional range of this series at the freezing tem-
perature Tf.11,29 In addition, long-range ferromagnetic order
at very low temperatures has been confirmed by neutron dif-
fraction and ␮SR studies at least down to the composition
with x=0.2.13,36 The low-temperature specific heat for these
compounds exhibits marked anomalies at the freezing tem-
perature Tf detected by ac susceptibility. The shape of these
anomalies, however, differs from what is expected around
the freezing temperature in a canonical spin glass.
So far, a lot of experimental data on the magnetic heat
capacity behavior of “spin-glass” systems are available and
few analytical expressions have been used to reproduce the
data for the different samples.37,38 Furthermore, the “spin-
glass” label has been used to refer to many different situa-
tions ranging from the canonical spin glass ͑i.e., individual
spins͒ to magnetic ground states close to superparamag-
netism ͑i.e., noninteracting magnetic spin clusters͒. That is
the reason why some of the spin-glass properties do not seem
to be universal. In particular, it is usually assumed that the
magnetic contribution to the specific heat shows no sharp
anomaly at the freezing temperature, as found in other heavy
fermion intermetallic compounds with spin-glass-like behav-
ior: U2PdSi3 ͑Ref. 39͒, URh2Ge2 ͑Ref. 40͒, and more re-
cently CeNi2Sn2 ͑Ref. 41͒. Our data do not conform with this
common phenomenology although the magnetic macro-
scopic measurements clearly point to a spin-glass-like behav-
ior.
Furthermore, the magnetic entropy reached at the spin
freezing temperature is estimated to be 45–60 % of the the-
oretical value R ln 2 corresponding to the full degeneracy of
the magnetic ground state ͑see Table II͒. That percentage is
slightly higher than those reported for the crystalline dilute
spin-glass systems MnCu or AuFe ͑22–33 %͒,42 even taking
into account the entropy reduction due to the Kondo effect in
our samples. Compared with our data, a similar specific heat
and magnetic entropy behavior was also found in amorphous
Gd33Al67 ͑Ref. 43͒ and Er50Ni50 ͑Ref. 44͒ alloys. In addition,
for those compounds the magnetic specific heat showed a
T3/2 temperature dependence, indicating a collective excita-
FIG. 9. Magnetic entropy Smag versus temperature extended to
the whole temperature range studied for CeNi1−xCux.
For increasing temperature, Smag progressively increases due
to the thermal population of the excited CEF levels, reaching
saturation around 250 K. The saturation values are close to
the theoretical value R ln 6=14.89 J/K2 mol, as corresponds
to the magnetic entropy of a Ce3+ ion with a total angular
momentum J=5/2.
IV. DISCUSSION
The actual calorimetric study has provided useful infor-
mation about the complex magnetic ground state found in
this system. This section is devoted to analyzing the behavior
of the series, based on both the present and the previous
studies carried out so far.
First, we must remember the division of the series into
two regimes with respect to the heat capacity findings. Al-
though the competing interactions are present throughout the
series, the heat capacity reflects that the interplay of the in-
teractions and their magnitude are tuned by the composition.
Thus, we have defined a first regime where the competing
exchange interactions are predominant ͑0.9ഛxϽ0.6͒, giving
rise to a change from antiferromagnetism to ferromagnetism.
The second regime is characterized by the increasing Kondo
effect driven by the decreasing cell volume. As the Kondo
effect becomes more important, the Ce magnetic moments
become smaller and the effective magnetic interaction de-
creases.
In this second regime the general behavior is the same as
that expected for a Kondo lattice system and the electronic
coefficient ␥ evolves similarly as in the CeNiyPt1−y series,23
a
clear example of a Doniach-predicted behavior.
The maximum value of ␥ corresponds to the composition
where Kondo interactions overcome the molecular field ef-
FIG. 10. Schematic evolution of both the
magnetic behavior and structural characteristics
of CeNi1−xCu. The two different regimes deduced
from these calorimetric studies are indicated to-
gether with the predominant interaction.
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MAGNETIC GROUND STATE OF CeNi1−xCux: A…
PHYSICAL REVIEW B 71, 134401 ͑2005͒
developing in the system. The combination of all the tech-
niques, however, yields a complete dynamical picture of the
system since ␮SR bridges the gap between neutron scatter-
ing on the one side and bulk magnetic measurements on the
other.45
Considering our experimental results as a whole, we can
propose a convenient description of our system. Although
x-ray and neutron diffraction and scanning microscopy re-
flect a clear homogeneity of the samples at the micrometric
level, we can be sure that inhomogeneities in the random
distribution of Cu and Ni appear at the nanometric scale.
When temperature is lowered from the paramagnetic state,
zones or clusters of magnetic moments develop due to the
increasing short-range magnetic interactions. These clusters
freeze at Tf, losing their dynamical aspect. At lower tempera-
tures the magnetic clusters interact among one another and
percolate giving rise to a long-range ferromagnetic state at
very low temperatures, in agreement with the ferromagnetic
excitations detected by specific heat below Tf. This percola-
tive process scenario, reminiscent of that proposed by Dag-
otto in manganites,46 is also present in many substitutional
systems such as semiconductors, high-TC superconductors,
etc.47 The CeNi1−xCux is, thus, a clear example of a strongly
correlated electron system presenting a percolative scenario.
In the compound with x=0.2, the inhomogeneous mag-
netic state, where ordered and nonordered fractions coexist,
extends over a wider temperature range. A fully long-range
ordered state has been detected by ␮SR and ac susceptibility
only below Tϳ0.5 K.13,14 At this point we must remember
that this compound was reported to be on the crossover point
of the localized-nonlocalized magnetism.11 In fact, the spe-
cific heat measurements do not show any sharp transition as
in the other compounds. Furthermore, recent neutron diffrac-
tion studies do not detect any magnetic contribution within
the experimental limits.36 These results evidence how re-
duced the Ce magnetic moments are in this alloy. Being so
close to the disappearance of the magnetic moment and re-
garding the Cmag/T behavior presented in the previous sec-
tion, one could be tempted to look for a NFL behavior de-
fining a QCP in this composition. In that sense, Fig. 12
shows the corresponding fits of Cmag/T for this compound to
a log10 T law, as observed for the case of magnetic
fluctuations3,48,49 and a T−1+␭ law as proposed by Castro
Neto,6 who considered the existence of Griffiths phases con-
sisting of magnetic clusters embedded in a nonmagnetic ma-
trix.
FIG. 11. Temperature dependence of the magnetic contribution
to the specific heat Cmag for the FM compounds x=0.3, 0.4, 0.5, and
0.6 in the form Cmag versus T3/2. The data corresponding to differ-
ent compounds are shifted for clarity.
tion. These results indicate that ferromagneticlike spin waves
can be excited in these amorphous alloys, although the over-
all magnetic structure is not ferromagnetic but rather spin
glass. Figure 11 displays the temperature dependence of our
magnetic specific heat in the form Cmag versus T3/2 for the
ferromagnetic compounds with x=0.3, 0.4, 0.5, and 0.6. It is
clear that Cmag follows the T3/2 dependence up to about 0.7Tf
in a similar manner as observed in the amorphous com-
pounds cited above. The existence of these ferromagnetic
correlations yields a faster enhancement of the magnetic en-
tropy than that of a canonical spin glass ͑linear with tempera-
ture͒. Thus, the maximum value of the specific heat has to be
reached at a lower temperature than the 1.6Tf expected
value,37,38 as happens in our series.
Neutron diffraction experiments at very low temperature
have been conclusive in ascertaining the existence of long-
range ferromagnetic order in x=0.6, 0.5, and 0.4
compounds.9,36 Neutron scattering is primarily a probe of
long-range correlations. On the other hand, ␮SR spectros-
copy is a much more local probe than neutron scattering, and
needs no large coherence length. Consequently, it is particu-
larly sensitive to short-range order and other forms of disor-
dered magnetism.45 Our ␮SR results indicate the existence of
an inhomogeneous magnetic state at temperatures above the
freezing temperature Tf ͑seen by ac susceptibility͒ that runs
into the paramagnetic regime. The basic feature of this inter-
mediate state is the coexistence of ordered and nonordered
fractions. The latter decreases as the temperature is lowered
until Tf is reached. Below Tf, long-range order, from the
␮SR perspective,45 prevails. Nevertheless, it must be stressed
that this long-range ordered state detected by muons presents
strong local magnetic disorder as reflected by the wide field
distribution on the muon site arising from those analyses.
This distribution leads to the fast damping of the oscillatory
pattern observed in the series.13
Although the agreement factor of these fits is comparable
to other systems reported as non-Fermi liquids,50 in the
present case we have powerful reasons to discard this option.
On the one hand, ac susceptibility shows the presence of a
transition at very low temperatures as we pointed out above14
and, on the other hand, the hysteresis loops in x=0.15 and
0.1 compounds observed at very low temperatures15 suggest
the existence of a “cluster-glass” state even for those com-
pounds that macroscopically behave as Fermi liquids with
almost constant Cmag/T value.
It is worth remembering the difference in the time win-
dow for the study of spin dynamical processes when com-
paring the different experimental techniques used so far ͑i.e.,
ac susceptibility, specific heat, neutron diffraction, and ␮SR
spectroscopy͒. Each technique can access a different charac-
teristic range of the rate of spin fluctuations and hence pro-
vides different information concerning dynamical processes
In our opinion the scenario for this series has to be under-
stood without considering a QCP situation, but rather as only
a consequence of the increasing hybridization in an intrinsic
inhomogeneous state that leads to a cluster-glass situation.
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MARCANO et al.
PHYSICAL REVIEW B 71, 134401 ͑2005͒
close to the Cu-rich side, where the exchange interactions are
predominant and the magnetic behavior resembles our find-
ings in other RNi1−xCux series ͓i.e., NdNi1−xCux ͑Ref. 26͔͒
leading to changes from AFM to FM; the second one with
ferromagnetic behavior, but governed by Kondo interactions
tending to reduce the Ce magnetic moment that vanishes in
the CeNi compound in agreement with the Doniach model,
in a similar way as already found in CeNiyPt1−y.
23,24 We have
also carefully investigated the possible existence of a NFL
behavior for compositions close to the apparent vanishing of
the long-range order and its relation to a QCP. Considering
the analysis of CP alone, we can report an enhanced Fermi-
liquid behavior ͑␥ constant with temperature͒ for compounds
with xϽ0.2, and a divergence in the CP/T vs T plot for x
=0.2 that could be reasonably well reproduced by NFL mod-
els. However, supplementary information obtained by ac and
dc susceptibility measurements confirms the existence of
hysteresis loops at low temperature indicating that the mag-
netic moments are not fully exhausted in those compounds.
The QCP scenario is then not required to account for that
behavior in this series.
A special effort has been made to understand the magnetic
contribution of CP related to the spin-glass state or the short-
range magnetic ordered state. For this purpose we have con-
FIG. 12. ͑a͒ Cmag/T vs log10 T and ͑b͒ log10͑Cmag/T͒ vs log10
plots for the x=0.2 compound. The reported a, T0, and ␭ values are
T
sidered previous ␮SR, neutron diffraction, and ␹ results,
ac
the parameters obtained from the ͑a͒ Cmag/Tϰa log10͑T/T0͒ and ͑b͒
which provide different time scales of those dynamical pro-
cesses. The critical temperature TC represents the tempera-
ture where ferromagnetic correlations are established, al-
though full long-range order only occurs at lower
temperatures. Considering the overall results it has been pos-
sible to propose a convenient description of our system. Its
magnetic behavior can be fully described by the existence of
interacting magnetic clusters that percolate, giving rise to a
long-range magnetic state at very low temperatures.
Our results allow us to extend the large variety of ex-
amples of systems presenting intrinsic inhomogeneities that
reach long-range magnetic order by percolating processes
͑semiconductors, manganites, high-TC superconductors, etc.͒
to strongly correlated electron metals underlying those intrin-
sic disorder effects that cannot be avoided and are present in
mixed or substitutional systems.
C
mag/TϰT−1+␭ laws.
The more important the reduction of the magnetic moment
is, the lower is the stability of a long-range ordered state.
This interpretation resembles the situation described by
Coleman,4,5 where the magnetic moments still persist in the
NFL regime, as is the case in MnSi.51
The possibility of the scenario proposed here could be
extended to other substitutional systems such as UCu5−xPdx
͑Refs. 52 and 53͒ and more recently UPd2−xSnx ͑Ref. 54͒ and
CePtSi1−xGex ͑Ref. 55͒ where intrinsic disorder effects were
detected. This fact points out the importance of careful
analysis of such systems close to the evanescence of the
long-range magnetic order ͑the QCP͒. The quantum effects
due to new electronic ground states must be confirmed by
studies carried out in good single-crystal samples and avoid-
ing substitutional effects.
ACKNOWLEDGMENTS
V. CONCLUSIONS
This work was supported by the MAT2003-06815, the
FERLIN-ESF program, and the Fundación Ramón Areces.
N.M. and J.H.-A. acknowledge MEC for partial financial
support. Fruitful discussions with G. M. Kalvius are also
acknowledged. We are indebted to David Culebras for his
invaluable technical assistance.
CeNi1−xCux is a complex system that presented many
open questions concerning its magnetic behavior. The analy-
sis of the specific heat has shed light on most of those in-
triguing experimental facts. In this way, it has been possible
to distinguish two main concentration regimes: the first one
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