arXiv:cond-mat/0703399v1 [cond-mat.str-el] 15 Mar 2007
Mesoscopic Magnetic States in Metallic Alloys with Strong Electronic Correlations: A
Percolative Scenario for CeNi1−xCux
N. Marcano,1,2J.C. G´ omez Sal,1J.I. Espeso,1J.M. De Teresa,3P.A. Algarabel,3C. Paulsen,4and J.R. Iglesias5
1Dpto. CITIMAC, Universidad de Cantabria, 39005 Santander, Spain
2Cavendish Laboratory, University of Cambridge, Cambridge CB3 OHE, United Kingdom
3ICMA, CSIC - Universidad de Zaragoza, 50009 Zaragoza, Spain
4Centre de Recherche sur les Tr` es Basses Temp´ eratures, CNRS, 38042 Grenoble, France
5Instituto de F´ ısica, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazil
(Dated: February 4, 2008)
We present evidence for the existence of magnetic clusters of approximately 20˚ A in the strongly
correlated alloy system CeNi1−xCux (0.7 ≤ x ≤ 0.2) based on small angle neutron scattering exper-
iments as well as the occurrence of staircase-like hysteresis cycles during very low temperature (100
mK) magnetization measurements. An unusual feature is the observation of long-range ferromag-
netic order below the cluster-glass transition without any indication of a sharp transition at a Curie
temperature. These observations strongly support a phenomenological model where a percolative
process connects the cluster-glass state observed at high temperatures with the long-range ferromag-
netic order observed by neutron diffraction experiments at very low temperatures. The model can
account for all the puzzling macroscopic and microscopic data previously obtained in this system,
providing a new perspective with regard to the magnetic ground state of other alloyed compounds
with small magnetic moments or weak ferromagnetism with intrinsic disorder effects.
PACS numbers: 71.27.+a, 75.25.+z, 75.60.Ej, 61.12.Ex, 75.30.Kz
In order to investigate the physical and, in particular,
magnetic properties of materials it is often common and
useful to substitute different species of ions into metals
and alloys.Ability to tune to the properties of mat-
ter in this manner allows one to distinguish between
the applicability and limitations of different theoretical
models to describe the emergent phenomena. For ex-
ample, one of the first experimental demonstration of
the Doniach’s phase diagram was from the systematic
study of Pt substitutions in the CeNi1−xPtxseries [1, 2].
Doniach’s phenomenology explains the competition be-
tween the RKKY interaction and the conduction band
hybridization in Cerium and Uranium compounds some
of them exhibiting Non-Fermi Liquid (NFL) behaviour
in the proximity of a Quantum Critical Point (QCP) .
In some important examples, substitution or doping can
improve a desired property, as is the case for materials
such as High-TC and other unconventional supercon-
ductors [5, 6] as well colossal magneto-resistance (CMR)
compounds [7, 8].
It is clear that strong electron correlations are the ori-
gin of the properties for all the examples mentioned above
and as a consequence, many of these properties are sen-
sitive to disorder [9, 10]. One is thus left with the puzzle
of how to discern whether the observed novel phenomena
or change in property is due intrinsic inhomogeneities or
simply a consequence of the poor quality of sample. It is
important to note that regardless of the amount of substi-
tution one can not guarantee that the distribution of the
dopant to be uniform across the volume of the sample.
It is indeed possible to achieve crystallographic homo-
geneity in the doped compounds, however one has to be
attentive to the implications at the electronic level. The
average effect of such substitutions is clearly observed
in macroscopic measurements such as X-ray diffraction,
magnetization, specific heat etc, but the heterogeneities
only become apparent when microscopic techniques, par-
ticularly muon spin relaxation (µSR) and transmission
electron microscopy are used. In this sense, even appro-
priate annealing which brings about perfect homogenisa-
tion is unable to prevent intrinsic inhomogeneities from
forming in alloyed and doped compounds. A special ef-
fort is thus required in the structural, electrical and mag-
netic characterization by combining macroscopic and mi-
croscopic techniques in order to obtain a complete picture
of the magnetic behaviour of such complex systems.
Accordingly, a complete study of the CeNi1−xCuxse-
ries has been developed in recent years using policrys-
talline samples [11, 12, 13, 14]. The main objective of
this study was the search for a NFL behaviour close to
the QCP expected around x = 0.2 (CeCu is antiferromag-
netic (AFM) and evolves to ferromagnetism (FM) for x
< 0.8, while CeNi is an intermediate valence compound).
However, our study revealed puzzling and seemingly con-
tradictory physical results. In particular: (i) Long-range
magnetic order has been obtained from neutron diffrac-
tion  at low temperatures (AFM for compounds 1 <
x ≤ 0.7 and FM for compounds with 0.6 ≤ x ≤ 0.3). In
addition, a cluster-glass state was determined in the FM
compositions by ac-susceptibility (χac) at higher temper-
atures. The freezing temperature, Tf, coincides with the
maxima observed in specific heat measurements ; (ii)
No indication of any Curie temperature, TC, associated
to the onset of long-range ferromagnetic order was ob-
0.1 K-5 K
Intensity (Arb. Units)
H = 50 Oe
Relaxation Rate (µs-1)
FIG. 1: Summary of the macroscopic and microscopic mea-
surements for CeNi0.4Cu0.6: Top: The real (χ′
ac field of 1 Oe at various selected frequencies, and the mag-
netic contribution to the specific heat Cmag in the low tem-
perature regime.The inset shows the magnetic neutron-
diffraction pattern (difference between 0.1 and 5 K). Points
indicate the experimental data and the line is the calculated
pattern. Vertical ticks mark the magnetic reflections for |? q|
= 0. Bottom: Field Cooled and Zero Field Cooled magneti-
zation in a field of 50 Oe. The inset shows the temperature
dependence of the µSR relaxation rates λtrans and λlong, and
the paramagnetic volume fraction, FPM, in the intermediate
ac) and imagi-
ac) components of the ac-susceptibility in an applied
served by macroscopic measurements (χac, CP). µSR
results indicate the formation of dynamic spin clusters
below a characteristic temperature T∗, above Tf ;
(iii) considering the whole analysis of the results, the ex-
istence of a QCP was not required to describe the evolu-
tion of the magnetism along the series, although the low
temperature CP for x = 0.2 follows a NFL-like temper-
ature dependence. In this letter we present new Small
Angle Neutron Scattering (SANS) and very low temper-
ature (100 mK) magnetization data that lead us to pro-
pose a new phenomenological model that encompasses
all the previous macroscopic and microscopic measure-
ments obtained on this system. This picture is based
on a cluster-glass state percolating into long-range ferro-
(ʜ~ 20Å )
(ʜ≥ 1000Å )
FIG. 2: Schematic illustration of the magnetic state according
to the proposed magnetic cluster model in different temper-
ature regions. Tf denotes the cluster-glass freezing tempera-
ture and T∗the establishment of the intermediate magnetic
inhomogeneous state (see text); the latter is determined by
µSR ZF-measurements, and the former from the cusp in ac-
susceptibility. Note that these two temperatures evolve with
the composition along the series. ξ corresponds to the mag-
netic correlation length.
magnetic order at low temperature.
Figure 1 displays the experimental evidence of the
above mentioned behaviour for one characteristic com-
pound (x = 0.6) as a representative sample presenting
long-range ferromagnetic order at a very low tempera-
ture (see inset of Figure 1 top). The magnetic structure
is collinear FM with the Ce magnetic moments strongly
reduced (0.6 µB) and lying along the b-axis. Both χ′
accurves (displayed in Figure 1 top) show a pro-
nounced maximum at Tf, which shifts with the frequency
of the ac applied field, and is associated to a cluster-glass
state [15, 16]. The specific heat, also plotted in the same
figure, exhibits a broad anomaly near Tf. Such cluster-
glass state is further supported by the irreversibility of
the field cooled and zero field cooled magnetic susceptibil-
ity (see Figure 1 bottom), although it starts at T∗> Tf.
This last observation implies the presence of short range
correlations well above Tf.
The µSR experiments have been essential in order to
interpret these results. The characteristic parameters ob-
tained from such a study are summarized in the inset of
Figure 1 bottom. The most significant result is the exis-
tence of an intermediate state developing at Tf< T < T∗
and consisting of long-range ordered and non-ordered
phases, the former increasing as the temperature is grad-
ually decreased. This situation, described as the Grif-
fiths phase  was also detected by µSR in the NFL
CeCoGe1.8Si1.2compound . Below Tf, muons detect
long-range order with a broad molecular field distribution
on the muon site, which indicates the presence of strong
local spin disorder in this state. It has to be stressed
that no magnetic contribution (within the experimental
resolution limits) is observed in the neutron diffraction
pattern obtained just below Tf. In fact, µSR is particu-
larly sensitive to short-range order correlations  and
therefor, a shorter coherence length (as compared to neu-
T= 1.57 K
ξ= 27 (1)Å
FIG. 3: Magnetic SANS intensity as a function of the neutron
momentum transfer, q, for CeNi0.4Cu0.6 compound at 1.57 K,
just below Tf. The solid lines are a fit to a Lorentzian-squared
q dependence (see text). ξ corresponds to the estimated cor-
relation length obtained from the fit. The inset shows the
same fit in a log-log scale.
tron diffraction) suffices to define the long-range ordered
state from µSR. The overview of Figure 1 also reveals the
absence of a transition temperature below Tf. Thus, the
main question arising from this puzzling phenomenology
is the mechanism that leads the system from a cluster-
glass into a long-range ordered state. In order to answer
this crucial question, and bearing in mind all the previ-
ously described results, we propose a phenomenological
picture that takes into account all the experimental evi-
Such a picture is shown in Figure 2 and illustrates
the magnetic state of the system in different tempera-
ture regimes. As the temperature is lowered from the
paramagnetic state, regions where the magnetic moments
fluctuate together, or clusters, develop due to the rise in
short-range magnetic interactions. The temperature that
corresponds to the cluster-formation (T∗) corresponds
to the upper boundary of the intermediate inhomoge-
neous state detected by µSR. The volume fraction of
these dynamic entities increases when the temperature
decreases and they freeze at Tf, as seen by χac, CP and
dc-magnetization. At this temperature, and just below,
magnetic correlations are large enough to be considered
as long-range order by µSR but not by neutron diffrac-
tion.Very low temperatures are required (T ≪ Tf)
in order to detect magnetic contribution in the neutron
diffraction pattern.Taking into account this body of
evidence, we propose a percolative process in order to
describe the emergence of the long-range ferromagnetic
state from the cluster-glass one below Tf. According to
such a mechanism, the size of the magnetic clusters would
increase below Tf, leading to a domain-like ferromagnetic
state at very low temperature.
The proposed percolative scenario is strongly sup-
ported by two recent experiments. Their results are de-
The most direct evidence one can gain of the existence
of magnetic clusters is obtained by SANS. For this reason,
we have carried out SANS measurements for several sam-
ples of the series using the D16 instrument located at the
Institute Laue-Langevin (ILL) in Grenoble. The SANS
magnetic signal was obtained at temperatures below the
Tf detected by macroscopic measurements, and the in-
coherent nuclear scattering was removed by subtracting
the SANS signal recorded in the paramagnetic regime
(T > T∗). The results obtained for the x= 0.6 sample
are presented in Figure 3 as an example. When analyz-
ing the small angle magnetic scattering cross section of a
sample exhibiting inhomogeneous states, such as cluster-
glass behaviour, one would expect to have two different
contributions: a Lorentzian term [I = A/(q2+1/ξ2)] rep-
resenting the short-range ferromagnetic order associated
to the fluctuations of the spin system and characteris-
tic of temperatures around the Curie temperature, and
a Lorentzian-squared one [I = B/(q2+ 1/ξ2)2] arising
from scattering from static regions of local spin ordering
[20, 21]. However, in the present case, only the latter is
needed to account for the magnetic SANS signal. The
conclusions that can be obtained from these results are:
(i) The direct observation of clusters of magnetic origin
and the clear evidence that long-range magnetic order
still has not been fully established at the lowest mea-
sured SANS temperature (1.57 K for x = 0.6); (ii) We
have obtained a correlation length of 27˚ A, which is re-
lated with the average size of the magnetic clusters. It
is worth noting that the results obtained for the x= 0.2
sample give much smaller values of the magnetic SANS
intensity and correlation length, as is expected due to the
highly reduced magnetic moment in this composition as
a consequence of the enhancement of the Kondo effect
and weaker magnetic interactions.
Magnetization was measured down to 100 mK (T ≪
Tf) using a SQUID magnetometer equipped with a
miniature dilution refrigerator at the CRTBT (CNRS,
Grenoble). Two representative compounds with x = 0.4
and 0.5 were studied, both of which display long-range
ferromagnetic order at this low temperature as detected
by neutron diffraction measurements. Figure 4 shows two
different hysteresis loops obtained for CeNi0.6Cu0.4at T
= 100 mK. The magnetization changes in a series of large
discrete jumps giving rise to a multistep pattern. When
repeating the hysteresis loop measurement, the steps ap-
pear at different field values. This unexpected staircase-
like behaviour only appears in the very low tempera-
ture loops (well below Tf), vanishing for slightly higher
temperatures (300 mK). Similar results have been ob-
tained in CeNi0.5Cu0.5. The observed features in these
cycles, clearly related to avalanches of domain flips, are
the mesoscopic analogue of the Barkhausen noise [22, 23].
The magnetic domains appear in conventional ferro-
magnets in order to minimize the magnetostatic energy.
In the present case, they are a consequence of the ther-
-10 -505 10
FIG. 4: Hysteresis loops obtained for CeNi0.6Cu0.4 at 100
mK. The inset shows the numerical simulation of a hysteresis
loop (see details in the text). The magnetization is in relative
units and the magnetic field in units of the exchange constant,
mally activated percolative process of static ferromag-
netic clusters reaching a minimum energy state. This
mechanism increases the magnetic correlation length up
to values that can be detected by neutron diffraction
(∼ 103˚ A). The process is driven by the increasing im-
portance of the RKKY interaction as the temperature de-
creases. This interaction, then, competes with the local
anisotropy, giving rise to a structure of magnetic domains
that displays an ”asperomagnetic” mesoscopic state such
has been reported by Coey  in the case of amorphous
systems. The present situation is clearly reminiscent of
that case, but occurring in crystalline samples.
Recent simulations based on disorder, anisotropy and
competing magnetic interactions reproduce satisfactorily
the experimental situation. We have used an Ising-like
Hamiltonian with a positive exchange interaction and an
anisotropy term that has been fixed at random for each
one of the random size clusters in which the lattice has
been split. Furthermore, in order to simulate the dis-
order in the interactions, we have “isolated” some spins
(typically 2.5%), by eliminating the ferromagnetic link
with their neighbours. Within this model, we have per-
formed a Monte Carlo simulation on a 3-dimensional lat-
tice at zero temperature. One typical hysteresis cycle
so obtained is shown in the inset of Figure 4, exhibiting
remarkable resemblance with the experimental results.
In conclusion, the use of many different techniques has
allowed us to propose a mesoscopic cluster-glass state
and to extend the percolative scenario, already described
in manganites , magnetic semiconductors  and di-
luted magnets , to strongly correlated electron metal-
lic systems and, in particular, to Ce and U based inter-
metallic compounds. This model has been confirmed by
SANS and the staircase hysteresis loops at low tempera-
tures and represents, in our opinion, a more general sit-
uation than the particular case of CeNi1−xCuxand cer-
tainly shed light on the nature of magnetic ground state
of other compounds with small magnetic moments, weak
ferromagnetism and intrinsic disorder effects [28, 29, 30].
This work is supported by the MAT2003-06815 project
and the ECOM COST Action P16. N. Marcano acknowl-
edges the Spanish MEC for financial support. We thank
G.M. Kalvius, L. Fernandez Barquin, J. Rodriguez Fer-
nandez, B. Coqblin, S. Magalhaes, V. Sechovsky, M.B.
Maple, G.G. Lonzarich, P. Haen and S.S. Saxena for use-
ful discussions. ILL staff is acknowledged for support
with the SANS measurements.
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