Universal behavior for magnetic entropy change in magnetocaloric materials:
An analysis on the nature of phase transitions
Claudia Marcela Bonilla
*
Instituto de Ciencia de Materiales de Aragón, Departamento de Física de la Materia Condensada, CSIC–Universidad de Zaragoza,
Pedro Cerbuna 12, 50009 Zaragoza, Spain
Julia Herrero-Albillos
Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Albert-Einstein-Str. 15, 12489 Berlin, Germany
Fernando Bartolomé, Luis Miguel García, and María Parra-Borderías
Instituto de Ciencia de Materiales de Aragón, Depto. de Física de la Materia Condensada, CSIC–Universidad de Zaragoza,
Pedro Cerbuna 12, 50009 Zaragoza, Spain
Victorino Franco
Departamento de Física de la Materia Condensada, ICMSE-CSIC, Universidad de Sevilla, P.O. Box 1065, 41080 Sevilla, Spain
共Received 29 September 2009; revised manuscript received 21 April 2010; published 16 June 2010
兲
A universal curve for the change in the magnetic entropy has been recently proposed for materials with
second-order phase transitions. In this work we have studied the universal behavior of the magnetocaloric
effect in the family of cobalt Laves phases, RCo
2
, and mixed manganites, La
2/3
共Ca
x
Sr
共1−x兲
兲
1/3
MnO
3
, which
exhibit first- and second-order phase transitions. The rescaled magnetic entropy change curves for different
applied fields collapse onto a single curve for materials with second-order phase transition as opposed to the
first-order phase transition compounds, for which this collapse does not hold. This result suggests that the
universal curve may be used as a further criterion to distinguish the order of the phase transition.
DOI: 10.1103/PhysRevB.81.224424 PACS number共s兲: 75.30.Sg, 77.80.B⫺, 71.20.Eh
I. INTRODUCTION
Magnetic refrigerators are a new environmentally friendly
option to the conventional devices based on compression ex-
pansion of gases. These devices also show higher energy
efficiency than those based on ozone-depleting gases. The
physical basis behind the operation of this kind of equipment
is the magnetocaloric effect 共MCE兲. Currently, the develop-
ment of this technology is tied to the research in materials
presenting optimal magnetocaloric properties 共namely, large
magnetic entropy change, ⌬S
M
, and large refrigerant capac-
ity, RC兲 near room temperature.
The RC of a given refrigerant is defined as the area below
the ⌬S
M
共T兲 curve between the temperatures of the cold and
hot reservoirs 共T
cold
and T
hot
, respectively兲.
1
T
cold
and T
hot
are usually taken as those temperatures where ⌬S
M
equals
⌬S
M
peak
/ 2. Therefore, in order to get high RC values both the
height and the width of the ⌬S
M
peak have to be considered.
Materials presenting first-order magnetostructural phase
transitions frequently show giant magnetocaloric effect
共GMCE兲,
2
i.e., very large values of ⌬S
peak
. However, first-
order phase transitions have two important drawbacks,
namely, the narrowness of the ⌬S
M
curve and the presence of
hysteresis, which leads to low operation frequencies and
cooling power.
3
To overcome these problems, compounds
undergoing second-order phase transitions may be used. Al-
though these compounds do present smaller 兩⌬ S
M
peak
兩 than
GMCE materials, they do not show thermal hysteresis and
their ⌬S
M
共T兲 is extended through a wider temperature range.
The compromise between an optimal RC and the lack of
hysteresis makes compounds with second-order phase tran-
sitions better candidates for the development of magnetic
cooling devices at the present moment.
The MCE is frequently characterized by measuring mag-
netization M共H兲 curves at different temperatures, allowing
⌬S
M
共T兲 to be obtained by means of the Maxwell relations.
Recently, V. Franco et al.
4–6
have described the universal
behavior for the ⌬S
M
共T兲 in compounds with second-order
phase transition. In addition to the intrinsic beauty of a uni-
versal behavior, this curve allows the prediction of the field
dependence of ⌬S
M
共T兲 even in those materials that do not
follow a mean-field approach; and it can be used to make
extrapolations in temperature or field close to the entropy
change peak. From a theoretical point of view, the universal
curve can be derived from the equation of state and the criti-
cal exponents of the system; from a practical point of view,
the phenomenological approach allows to construct the uni-
versal curve without knowing the critical exponents or the
equation of state for the material under study.
The universal behavior of ⌬S
M
has been confirmed in
several second-order transition compounds, including Fe-
based amorphous alloys such as FeMoCuB,
7
FeCrMoBCu,
8,9
and FeZrBCu,
10
in Gd and the intermetallic Er
1−x
Dy
x
Al
2
,
11
and in TbCo
2
,
12
among others.
It is interesting to note that the collapse of these curves is
observed not only in the near vicinity of the transition but in
a wide temperature range. This raises the question as to
whether the collapse of the ⌬S
M
共T兲 curves is a manifestation
of a universal behavior or not. A study on first-order phase
transitions should shed light on the subject: a breakdown of
the universal curve is expected for first-order phase transi-
tions if the underlying cause is universality associated to
critical phenomena and intrinsic to second-order phase tran-
sitions. Otherwise, ⌬S共T兲 curves may collapse in the same
way for first order as for second-order transitions.
PHYSICAL REVIEW B 81, 224424 共2010兲
1098-0121/2010/81共22兲/224424共7兲 ©2010 The American Physical Society224424-1
The aim of this work is to systematically study the behav-
ior of this universal curve for the magnetic entropy change
for two families of compounds, which present both first- and
second-order phase transitions. Indeed, we aim at showing
whether or not a breakdown of the universal behavior of
⌬S
M
occurs in first-order phase transitions.
We have chosen the cobalt Laves phases family due to its
rich phenomenology: first, compounds formed with light
rare-earth ions are ferromagnets while those formed with
heavy rare earths are ferrimagnets. Second, the magnetic or-
der is established through a second-order phase transition in
all of them except in ErCo
2
, HoCo
2
, and DyCo
2
, where the
magnetic ordering is coupled to a structural change, leading
to a first-order magnetostructural transition. Additionally the
ferrimagnetic HoCo
2
and ferromagnetic NdCo
2
undergo a
first-order spin reorientation transition 共SRT兲 below their
magnetic ordering transitions. The structural, electronic, and
magnetic properties of this family of compounds have been
thoroughly studied.
13,14
Moreover, the large entropy change
showed by ErCo
2
has led to studies on the MCE properties
of the pure and pseudobinary Co Laves phases.
2,15–22
We have also selected the ferromagnetic manganites,
La
2/3
共Ca
x
Sr
共1−x兲
兲
1/3
MnO
3
with x=0, 0.5, and 1, in order
to give more generality to our results. The physical
properties of these materials have been reported in
literature.
23–27
A magnetic transition with first-order charac-
ter has been determined for La
2/3
Ca
1/3
MnO
3
at 260 K while
La
2/3
共Ca
0.5
Sr
0.5
兲
1/3
MnO
3
and La
2/3
Sr
1/3
MnO
3
show a second-
order transition at 340 K and 370 K, respectively.
23,24
II. EXPERIMENTAL DETAILS AND DATA ANALYSIS
Intermetallic samples of RCo
2
with R = Tb, Pr, Nd, Dy,
and Ho were prepared by melting the pure metallic precur-
sors in an induction furnace under Ar atmosphere. The alloys
were later annealed under Ar atmosphere at 850 ° C for 8–12
days depending on the sample. The policrystalline mangan-
ites La
2/3
共Ca,Sr兲
1/3
MnO
3
were obtained from La
2
O
3
, CaCo
3
,
Mn
2
O
3
, and Sr
2
Co
3
as precursors. The starting powders were
ground, pelleted, and sintered following a standard ceramic
method.
24
A highly pure single phase was found in all the
samples as checked by x-ray diffraction.
Field dependence of magnetization measurements were
performed in a Quantum Design MPMS-5S superconducting
quantum interference device magnetometer. M共H兲 isotherms
were obtained by varying the field between 0 to 5 T for all
samples. Between 30 and 70 M共H兲 curves were measured in
a range from 6 to 400 K, depending on the sample ordering
temperature, T
c
, and the presence of SRT.
The magnetic ordering at zero field occurs at 40 K, 78 K,
98 K, 138 K, and 231 K for PrCo
2
, HoCo
2
, NdCo
2
, DyCo
2
,
and TbCo
2
, respectively. The SRT temperature T
SRT
for
NdCo
2
and HoCo
2
are 42 K and 16 K, respectively. The
magnetic ordering temperatures for La
2/3
Ca
1/3
MnO
3
,
La
2/3
Sr
1/3
MnO
3
, and La
2/3
共Ca
0.5
Sr
0.5
兲
1/3
MnO
3
were identified
as 260 K, 340 K, and 370 K, respectively. These data are
fully consistent with those previously reported.
23,24,28
The magnetic entropy change ⌬S
M
共T兲 can be obtained
from the M共H兲 curves by applying a numerical approxima-
tion to the equation
⌬S
M
=
冕
0
H
冉
M
T
冊
H
dH 共1兲
replacing the partial derivative for finite differences and nu-
merically solving the integrals for each value of H.
In Fig. 1 panel a we can observe the M共H兲 curves mea-
sured for HoCo
2
for 58 values of temperature between 6 and
225 K. Correspondingly panel b shows the ⌬S
M
共T兲 curves
normalized to their maximum value ⌬S
peak
for 25 field val-
ues.
The construction of the phenomenological universal curve
is based on the collapse of the ⌬S
M
共T , H兲 points correspond-
ing to equivalent states of the system into one single point in
the new curve. Those equivalent states have the same height,
h,inthe⌬S
M
/ ⌬S
M
peak
curves. For each value of the applied
field and any arbitrary value of h, two reference temperatures
共T
r1
⬍T
c
and T
r2
⬎T
c
兲 are found so that ⌬S
M
共T
r1
兲/ ⌬S
M
peak
=⌬S
M
共T
r2
兲/ ⌬S
M
peak
=h. The collapse of the normalized en-
tropy change curves can be then obtained by defining a new
variable for the temperature axis,
, given by the expression
FIG. 1. 共Color online兲共a兲 Magnetization measurements as func-
tion of field for different temperatures for HoCo
2
. The values of
applied field during the measurement were 0, 0.1, 0.2, 0.3, 0.4, 0.5,
0.6, 0.7, 0.9, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.5,
4.0, 4.5, and 5.0 T. 共b兲 Normalized entropy change versus tempera-
ture for different applied fields for HoCo
2
.
BONILLA et al. PHYSICAL REVIEW B 81, 224424 共2010兲
224424-2
=
再
− 共T − T
c
兲/共T
r
1
− T
c
兲
T ⱕ T
c
共T − T
c
兲/共T
r
2
− T
c
兲
T ⬎ T
c
.
冎
共2兲
In this work we have identified T
c
as the temperature of the
maximum entropy change
29
and we have selected h=0.5
when constructing the universal curve for each sample. By
construction, the temperature axis is rescaled in a different
way below and above T
c
imposing the constraint that the
reference points in the new curve correspond to ⌬S
M
共
= ⫾1兲/ ⌬S
M
peak
=h.
The existence of the universal curve for second-order
phase transitions has been already theoretically grounded.
30
The assumption that different physical magnitudes 共such as
magnetization兲 scale, in the vicinity of a second-order tran-
sition, is well supported both theoretically and
experimentally.
31
Based on this statement we can consider
the scaling equation for a magnetic system given by
32
H
M
␦
= h
冉
t
M
1/

冊
, 共3兲
where M is the magnetization, H is the applied field, t is the
reduced temperature,
␦
and

are critical exponents for the
critical isotherm 共t=0兲 and the magnetization behavior along
coexistence 共H =0, t ⬍ 0兲, respectively, and h共x兲 is a scaling
function. This h共x兲 is the same for systems belonging to the
same universality class provided that the magnetization and
magnetic field units are such that h共0兲=1 and h共−1兲=0. The
Eq. 共3兲 can be written as
M
兩t兩

= m
⫾
冉
H
兩t兩
⌬
冊
共4兲
the product

␦
=⌬, determines the gap exponent and the ⫾
sign is related to t⬎0 and t ⬍ 0, respectively.
Combining Eqs. 共2兲 and 共4兲 and after some algebra
30
the
entropy change can be expressed as
⌬S
M
/a
M
= ⫾ 兩t 兩
1−
␣
冕
0
H/兩t兩
⌬
dx关

m
⫾
共x兲 − ⌬xm
⫾
⬘
共x兲兴
= 兩t兩
1−
␣
s
˜
共t/H
1/⌬
兲 = H
1−
␣
/⌬
s共t/H
1/⌬
兲, 共5兲
where a
M
=T
c
−1
A
␦
+1
B, with A and B the critical amplitudes at
coexistence 关M =A共−t兲

兴 and along the critical isotherm 共H
=BM
␦
兲, respectively. Here s共x兲 is the scaling function. If the
reduced temperature t is rescaled by a factor proportional to
H
1/⌬
, and the magnetic entropy change by a
M
H
共1−
␣
兲/⌬
the
expression 共5兲 shows that the experimental data would col-
lapse onto the same curve. This demonstration proves that
the MCE data of different alloys belonging to the same uni-
versality class should collapse in a common universal curve.
In this way the universal curve can also be constructed
analytically if the equation of state and the critical exponents
of a material are known. However, from a practical point of
view, the phenomenological approach allows the use of the
universal curve for practical purposes without knowing those
details about the material.
A single reference temperature can be used to collapse all
the curves.
30
However, the use of two reference temperatures
has been necessary in some special cases to obtain a satis-
factory universal curve. In particular to correct the presence
of a minority magnetic phase in LaFe
10.8
Si
2.2
共Ref. 12兲 or the
influence of the demagnetization factor.
33
In the present
work, we have used two references instead of one in order to
assure that if a breakdown of the universal behavior of ⌬S
M
should occur, it could not be ascribed to any of the previ-
ously mentioned artifactual causes.
III. RESULTS AND DISCUSSION
The normalized entropy change as a function of the res-
caled temperature
for the magnetic ordering transitions of
the RCo
2
and the mixed manganites compounds are shown in
Figs. 2 and 3, respectively.
Panels a and b of Fig. 2 show our results for the second-
order phase transitions of TbCo
2
and PrCo
2
. The results for
the second-order transitions in the mixed manganites family,
La
2/3
Sr
1/3
MnO
3
and La
2/3
共Ca
0.5
Sr
0.5
兲
1/3
MnO
3
, are presented,
respectively, in panel a and b of Fig. 3. The collapse of all
these data into a unique curve—in a very wide temperature
range—for the RCo
2
compounds and the manganites is a
further confirmation of the general validity of the treatment
in second-order phase transition compounds. Indeed, the uni-
versal behavior of ⌬S
M
had been independently demon-
strated for another TbCo
2
sample.
12
Panel d of Fig. 2 shows the result for DyCo
2
. From mere
inspection of the graph, it is evident that—for temperatures
below T
c
—the curves do not overlap, pointing out that this
alloy does not follow a universal curve for magnetic entropy
change. In the case of ferromagnetic first-order phase transi-
tion of La
2/3
Ca
1/3
MnO
3
共see panel c in Fig. 3兲 a breakdown
of the universal behavior for the normalized entropy change
can be observed.
The collapse for
⬎0 is due to the paramagnetic behav-
ior. Magnetization scales with 共
H/ kT兲 and therefore it is
possible to collapse ⌬S for every compound in the paramag-
netic region. For values −1 ⬍
⬍0 the deviation from col-
lapse cannot be very large, as the curves coincide by con-
struction. The reference points are such that ⌬S
M
共
= ⫾1兲/ ⌬S
M
peak
=h, where h is arbitrary 共0 ⬍ h ⬍ 1兲, in conse-
quence the collapse is broken only below
=−1. Within the
range −1⬍
⬍0 the collapse is real in second-order transi-
tions and only apparent in first-order transitions. Therefore,
the effect of the order of the transition is decisive only below
=−1, in this phenomenological approach. In principle, the
presence of a minority magnetic phase in the sample, or the
demagnetizing factor could be responsible of an apparent
breakdown of the universal curve.
12,33
However, as was
pointed out previously, two reference temperatures have been
used throughout this work and therefore the effect of those
phenomena have been excluded. For this reason, we ascribe
the breakdown of the universal curve in DyCo
2
and
La
2/3
Ca
1/3
MnO
3
to the first-order nature of their phase tran-
sitions.
Furthermore, the observed behavior in first-order phase
transitions suggests that the collapse of the ⌬S
M
curves is
related to the universality intrinsic to second order phase
transitions.
31
The breakdown of the universal behavior can
be quantified from the vertical spread of the points for values
UNIVERSAL BEHAVIOR FOR MAGNETIC ENTROPY… PHYSICAL REVIEW B 81, 224424 共2010兲
224424-3
below
=−1. We have calculated the width W of the vertical
spreading of each scaled entropy change curve relative to its
mean value at an arbitrarily chosen
⬍−1. The dispersion is
then given by
dispersion = 100 ⫻
W共
=−5兲
⌬S
M
/⌬S
M
peak
共
=−5兲
. 共6兲
In Table I we list the values obtained for the dispersion in the
studied compounds. Clearly, for compounds with first-order
phase transition the dispersion always remain superior to
100%, i.e., the width of the vertical spreading is larger than
the mean value of ⌬ S
M
/ ⌬S
M
peak
for that value of
. For com-
pounds with second-order phase transition the dispersion is
never larger than 30%, which may be due to the experimen-
tal uncertainty. From our results, we expect this behavior
holds regardless the family of compounds.
The compounds NdCo
2
and HoCo
2
are selected to high-
light the influence of the thermodynamical order of the tran-
sition on whether or not the rescaled entropy curves collapse
into a universal behavior. NdCo
2
shows a typical ferromag-
netic second-order phase transition at T
c
=95 K and a SRT of
first order at lower temperature 关T
SRT
⬃42 K 共Ref. 20兲兴.As
is shown in panel c of Fig. 2—where
=0 corresponds to
T
c
—the expected collapse for a second-order phase transition
is observed except in the vicinity of the SRT. The observed
shift of the SRT peaks is due to: first, the usual dependence
of the critical temperature on the applied field in first-order
phase transitions
20
and second—and more significantly—due
to the scaling around T
c
.
The situation is similar for HoCo
2
where a ferrimagnetic
transition occurs at T
c
=86 K and the SRT takes place at
T
SRT
⬃16 K 共Ref. 20兲 although for this system both are first-
order phase transitions. Panel e of Fig. 2—where again
=0 corresponds to T
c
—shows the breakdown of the universal
curve for the T
c
of HoCo
2
. Below T
c
, the rescaled ⌬S
M
curves show a behavior very similar to that observed in
DyCo
2
. Further splitting of the ⌬S
M
共
兲 curves at lower tem-
peratures 共
⬍−6兲 comes from the SRT contribution, as in
NdCo
2
.
It is now interesting to compare the result of scaling the
⌬S
M
curves for the different compounds. On one hand, as it
is shown in panels a–c of Fig. 4, there is a common collapse
within the scaled entropy change curves for second-order
phase transition compounds. For TbCo
2
, PrCo
2
, and NdCo
2
,
showed in panel a, the collapse to a common curve is satis-
fied except at lower
. The existence of a universality for the
⌬S
M
curves relies on the scaling with temperature of the
magnetization 共and, consequently, of the magnetic entropy兲
near a second-order phase transition.
30,31
Therefore, every
system from the same universality class, i.e., with the same
critical exponents, will collapse into a common curve. Due to
the fact that TbCo
2
is a ferrimagnet while PrCo
2
and NdCo
2
are ferromagnets the common collapse is not satisfied. On
the other side panels b and c of Fig. 4 show the comparison
between ferromagnets: within the manganite family and for
La
2/3
共Ca
0.5
Sr
0.5
兲
1/3
MnO
3
and PrCo
2
systems, respectively.
The result is fully consistent with our previous statement.
Second-order paramagnetic-ferromagnetic phase transition
of double exchange materials close to half-filling 关which is
the case of manganites La
2/3
共Ca
x
Sr
1−x
兲
1/3
MnO
3
兴 belongs to a
Heisenberg three-dimensional universality class
34,35
as well
FIG. 2. 共Color online兲 Normalized entropy change as a function of the rescaled temperature
for the cobalt Laves phases studied in this
work. A universal curve for the second-order phase transitions of TbCo
2
共panel a兲, PrCo
2
共panel b兲, and NdCo
2
共panel c兲 is demonstrated
while a breakdown of the universal curve for the first order phase transitions of DyCo
2
共panel d兲 and HoCo
2
共panel e兲 can be observed. The
panel f shows a comparison of the rescaled curves for PrCo
2
and DyCo
2
共vertical axis in logarithmic scale兲.
BONILLA et al. PHYSICAL REVIEW B 81, 224424 共2010兲
224424-4
as second-order ferromagnets RCo
2
.
36
In consequence the
scaled entropy change curves for La
2/3
Sr
1/3
MnO
3
and
La
2/3
共Ca
0.5
Sr
0.5
兲
1/3
MnO
3
do collapse in good approximation
to a common behavior; as well as the curves for PrCo
2
and
La
2/3
共Ca
0.5
Sr
0.5
兲
1/3
MnO
3
regardless both systems belong to
different families of compounds.
Panel f of Fig. 2 and panel d of Fig. 3 shows the scaled
entropy changes in semilogarithmic axis for PrCo
2
and
DyCo
2
and La
2/3
Ca
1/3
MnO
3
and La
2/3
共Sr
0.5
Ca
0.5
兲
1/3
MnO
3
,
respectively. Both figures allow direct comparison between
the results for a first- and a second-order phase transition in
each family.
The abruptness of the changes in physical magnitudes at
the first-order transition of DyCo
2
are small, making its as-
cription as first order very difficult by inspection of experi-
mental results alone. Usually the order of the transition can
be distinguished from different experimental techniques such
as specific heat, differential scanning calorimetry or resistiv-
ity, among others. However, these experiments usually in-
volve long and careful measurements very near T
c
. More-
over, conventional calorimetric measurements for samples of
nanoscopic sizes are not sensitive enough to follow the rapid
changes in temperature for such sample sizes.
37
Even for
bulk DyCo
2
samples the establishment on the order of its
transition is not straightforward. First, the jump in resistivity
at the transition is small and not very abrupt,
2
and the tem-
perature range of metastability is very narrow.
22
Additionally,
an applied pressure well below 1GPa 共Refs. 38 and 39兲 or a
chemical dilution of 20% with Tb, for example,
40
are enough
to destabilize the first-order character of the transition. All
these results suggest that DyCo
2
is a first order case near the
critical point, i.e., on the border of second-order phase tran-
sition.
A criterion from purely magnetic measurements can be
proposed as an alternative to calorimetric techniques. Usu-
ally the Banerjee criterion
41
has been employed to establish
the magnetic phase transition character. By studying the
presence of a negative slope region on the isothermal plots of
H/ M versus M
2
first-order phase transitions can be identi-
fied. In Fig. 5 we show that the criterion is clear for all the
RCo
2
but DyCo
2
. Indeed, neither magnetization nor Arrot
plots allow clear determination of the order of that phase
FIG. 3. 共Color online兲 Normalized entropy change as a function of the rescaled temperature
for the manganites studied in this work.
A universal curve for the second-order phase transitions of La
2/3
Sr
1/3
MnO
3
共panel a兲 and La
2/3
共Ca
0.5
Sr
0.5
兲
1/3
MnO
3
共panel b兲, is demonstrated,
while a breakdown of the universal curve for the first order phase transitions of La
2/3
Ca
1/3
MnO
3
共panel c兲. Panel d shows a comparison of the
rescaled curves for La
2/3
Ca
1/3
MnO
3
and La
2/3
共Ca
0.5
Sr
0.5
兲
1/3
MnO
3
共vertical axis in logarithmic scale兲.
TABLE I. Dispersion for scaled entropy change values at
=
−5.
Order Compound Dispersion 共%兲
First order DyCo
2
116.41
HoCo
2
131.31
La
2/3
Ca
1/3
MnO
3
105.55
Second order TbCo
2
17.17
PrCo
2
26.74
La
2/3
共Sr
0.5
Ca
0.5
兲
1/3
MnO
3
9.09
La
2/3
Sr
1/3
MnO
3
14.00
UNIVERSAL BEHAVIOR FOR MAGNETIC ENTROPY… PHYSICAL REVIEW B 81, 224424 共2010兲
224424-5
