Magnetic phase diagram for a non-extensive system: Experimental connection with manganites

21 Pages·0.56 MB·English

by  M. S. Reis

Magneti
phase diagram for a non-extensive system: Experimental
onne
tion with
manganites
∗
M.S. Reis and V.S. Amaral
Departamento de Físi
a, Universidade de Aveiro, 3810-193 Aveiro, Portugal
3
0
0 J.P. Araújo
2 IFIMUP, Departamento de Físi
a, Universidade do Porto, 4150 Porto, Portugal

n
a I.S. Oliveira
J Centro Brasileiro de Pesquisas Físi
as, Rua Dr. Xavier Sigaud 150 Ur
a, 22290-180 Rio de Janeiro-RJ, Brazil
6 (Dated: February 2, 2008)
1
Inthepresentpaperwemakeathoroughanalysisofa
lassi
alspinsystem,withintheframework

ofTsallis nonextensivestatisti
s. FromtheanalysisofthegeneralizedGibbsfreeenergy,withinthe
]
h mean-(cid:28)eld approximation, a para-ferromagneti
phase diagram, whi
h exhibits (cid:28)rst and se
ond
c order phase transitions, is built. The feaqtures of the generalized, and
lassi
al, magneti
moment
e are mainly determined by the values of , the non-extensive parameter. The model is su

essfully
m applied to the
ase of La0.60Y0.07Ca0.33MnO3 manganite. The temperature and magneti
(cid:28)eld
- dependen
e of the experimental magnetization on this manganite are faithfully reprodu
ed. The
t agreement between rather "exoti
" magneti
properties of manganites and the predi
tions of the
a q
-statisti
s,
omes to support our early
laim that these materials are magneti
ally nonextensive
t
s obje
ts.
.
t
a
PACSnumbers:
m
-
d I. INTRODUCTION systems and tight-binding-like Hamiltonians29,30,31,
n 32 33
metalli
and super
ondu
tor systems, et
. The (cid:28)rst
o
c In the literature of manganites, various models have eviden
es that the magneti
properties of manganites
[ appeared as di(cid:27)erent attempts to reprodu
e the ele
tri

ould be des
ribed within the framework of Tsallis
and magneti
properties of these systems. Krivoru
hko statisti
s were presented by Reis et al.34, followed by an
2 et al.1, Nunez-Regueiro et al.2 and Dionne3 are in- analysis35 of the unusual paramagneti
sus
eptibility of
v 0.67 0.33 3 36
9 teresting examples of multiparameter models, but La Ca MnO , measured by Amaral et al. .
3 whi
h failed to a
hieve full agreement to experimental Maximizationqof Eq.1 subje
ted to the
onstraint of
2 data. Ravindranath et al.4
ompare resistivity data in tˆhe normalized -expe
tation value of the Hamiltonian
1 La0.6Y0.1Ca0.3MnO3 to di(cid:27)erent two-parameter models, H14,15:
0 whi
h do not agreeto ea
hother in the low-temperature Tr ˆρˆq
3 U = {H }
0 rwaonrgkeo.fORitvhaesreitnatel.r5e,sHtinugesoatettemal.p6t,sH
earnembaenfsoeutnadl.7in,Pthael q Tr{ρˆq} (2)
/
t etal.8,Philipetal.9,Szew
zyketal.10,Viretetal.11 and
a
12 and the usual normalization of the density matrix
m Tka
huketal. . Noneoftheseobtainedplainagreement Tr ρˆ =1
{ } ,yieldsthefollowingexpressionforthedensity
between experiment and theory, irrespe
t their number ρˆ
- matrix :
d of adjusting parameters and approa
h.
n On another hand, Tsallis generalized ρˆ= 1 [1 (1 q)β˜(ˆ U )]1/(1−q)
o statisti
s13,14,15,16, has been su

essfully applied to Zq − − H− q (3)
c 17
an impressive number of areas . The formalism rests
:
v on the de(cid:28)nition of generalized entropy13:
where
i 1 pq
rX Sq =k −qP1i i (1) Zq =Tr[1−(1−q)β˜(Hˆ−Uq)]1/(1−q) (4)
−
a
q pi β˜ = β/Tr ρˆq β
where ips t=he1entropki
index, are probabilities satis- is the partition fun
tion and { }. Here, is
fying i i and is a positive
onstant. The above theLagrangeparameterasso
iatedtotheinternalenergy.
P
formula
onverges to the usual Maxwell-Boltzmann def- Themagnetizationofaspe
imenis,a

ordingly,given
inition of entropy, and to thqe usu1al derived thermody- by14,15,16,34,35:
13,14,15,16,17
nami
fun
tions, in the limit → . Tr µˆρˆq
In what
on
erns Condensed Matter prob1le8,m19s,2,0a,2p1p,2l2i- Mq = Tr{ ρˆq } (5)

ations of Eq.1 in
lude: Ising ferromagnets , { }
23,24,25
mole
ular (cid:28)eld approximation , per
olation
µˆ
26 27,28
problems , Landau diamagnetism , ele
tron-phonon where is the magneti
moment operator.
2
However, Eq.3
an be wβr∗itten in a morβe∗
=onvβ˜e/n[i1en+t II. CLASSICAL MODEL
14,15,34,35
(fo1rmq)β˜Uq] in terms of , de(cid:28)ned as µ~
− . In parti
ular, to analy1z/e(tkhβe∗)physi
al sys- Considera
lHa~ssi
alspin submittedtoahomogeneous
tem des
ribed here, the quantity will repre- magneti
(cid:28)eld . The hamiltonian H is given by
34,35
sentthephysi
altemperatures
ale, asin referen
es .
= µHcosθ
Dis
ussion about the
on
ept of temperature and La- H − (6)
grange param1e4t,1e5r,s34i,n35,T37s,a38ll,3is9 statisti
s
an be found in θ ~µ H~
tqhe literature . In the pre0sentqwor1k, the where is the angle between and . Following the
parameter is restri
ted to the interval ≤ ≤ , pre- usual Tsallis formalism14,15,16,34,35, the magnetization
serving the entropy
on
avity35,40. Mq
an be determined from Eq.5, yielding:
In this paper we pursue the idea, based on novel ex- 1 1 1, x> 1
pmeargimneetni
taalllyanndonth-eexotreentis
iavleroebsujelt
st,s.thTathimsapnrgoapneirtteys aapre- Mµq =Lq(x)= (2−q)(cid:26)cothq−(x)x− x1, x< 11−−1qq (7)
pears in systems where long-range intera
tions and/or
x = µH/kT coth
fra
tality exist, and su
h features have been invoked in where , and q is the generalized q-
re
entmodelsofmanganites,aswellasintheinterpreta- hyperboli

o-tangent52. The above two bran
hes fun
-
tion of experimental results. They appear, for instan
e, tion results from the Tsallis
ut-o(cid:27)15,16,53,54. It is inter-
41
in the work of Dagotto and
o-workers , who empha- estingtonotethesimilaritybetweentheaboveresultand
size the role of the
ompetition between di(cid:27)erent phases the traditional Langevin fun
tion. In what follows, Eq.7
to the physi
al properties of these materials. Various will be
alled Generalized Langevin Fun
tion, and this
authors have
onsidered the formation of mi
ro-
lusters result
analsobederivedfrom theGeneralized Brillouin
of
ompeting phases, with fra
tal shapes, randomly dis- Fun
tion, introdu
ed in Ref.35, taking the limit of large
S
42,43
tributed in the material , and the role of long-range spin values ( → ∞). The result derived above is valid
44,45 q
intera
tions to phase segregation . Important exper- only for 0≤ ≤1.
χ
imental results in this dire
tion have also been reported The generalized magneti
sus
eptibility q shows the
46 47
by Marithew et al. , and Fiebig et al. . Parti
ularly usual dependen
e on the inverse of the absolute temper-
48
insightful is the work of Satou and Yamanada , who ature
derivedaCantor spe
traforthedouble-ex
hangehamil- ∂ qµ2
q
χ = lim M = =qχ
tonian, basis of theoreti
al models of manganites. q
H→0(cid:20) ∂H (cid:21) 3kT 1 (8)
A major di(cid:30)
ulty with Tsallis formulation
on
erns
q
thephysi
almeaningoftheentropi
parameter . Inthis
dire
tion, Be
k and Cohen49 have re
ently shown that and is proportχional to the usual paramagneti
Langevin
q sus
eptibility 1. A similar result was dedu
ed for the
the value of gives a dire
t measure of the internal dis- 35
generalized Brillouin fun
tion .
tribution of temperatures in an inhomogeneous system.
Although their results are not dire
tly applied to mag-
neti
systems,ithasbeenknownforsometimethatman-
III. MEAN-FIELD APPROXIMATION
ganitesaremagneti
allyinhomogeneoussystems(see,for
41,50
instan
e, Ref. and referen
es therein), and this fa
t
51 A. Gibbs Free Energy
hasbeenexploredveryre
entlybySalamonetal. , who
applied the idea of distribution of the inverse sus
epti-
bility (whi
h turns out to be equivalent to a distribu- TheGibbsfreeenergyis,withinthemean(cid:28)eldapprox-
tion of temperatures), to the analysis of the magneti
imation,
sus
eptibility and the e(cid:27)e
tive paramagneti
moment of kT Mq λ
0.7 0.3 3 G= −1( ′)d ′ H 2
La Ca MnO . µ Z Mq Mq Mq − Mq− 2Mq (9)
0
In whatfollows,wepresentamagneti
model for
las-
si
alspins(
luster),usingforthat,theTsallisgeneralized −1
wherethe(cid:28)rsttermistheentropy,withMq theinverse
statisti
. In the model, we
onsider that non-extensivity
fun
tionofthegeneralizedLangevinfun
tion(Eq.7); the
exists in the intra-
luster intera
tion, whereas the inter-
se
ond is the Zeeman term; and the third, the ex
hange

luster intera
tion remains extensive. This is important λ
energy,with asthemean (cid:28)eld parameter. The equilib-
to maintain the total magnetization proportional to the
rium magnetization
an be found from the minimization
number of
lusters. Following, the Gibbs free energy is q
of the Gibbs free energy and, depending on the value,
analyzed,withinthemean-(cid:28)eldapproximation,andase-
(cid:28)rst or se
ond order transition features emerge. At zero
ries of interesting magneti
features appears, as a
on- >
magneti
(cid:28)eld and for q 0.5the free energy presents, at
sequen
e of the intra-
luster non-extensivity. Finally, a
any temperature, only one minimum, for positive values

onne
tionbetweenthe modelproposedhereandtheex-
perimental data obtained from magneti
measurements of magnetization. Correspondingly, a se
Ton(qd)o=rdqeTr(p1)ara-
c c
performed on manganites support our thesis that these ferromagneti
phasetransitionTo(1
)
u=rsµa2tλ/3k , as
c
obje
ts are magneti
ally non-extensive. sket
hedin (cid:28)gure1(a). Here, istheCurie
3
temperatureforthestandardLangevinmodel. Fromnow as the GtenetraliMzed(HCu=ri0e,FTu)n=
ti0on. In the paramagnCet(qi
)
c
on,weintrodtu=
eTt/hTec(d1)imensihon=leµssHt/ekmTpc(e1r)atureand(cid:28)eld phasθep(,q) =≥λC,(q) and the fun
tions
parameters and . and are
onstants:
For q≤0.5 the nature of the phase transition is more q
C(q) =

omplex, presenting a typi
al behavior of (cid:28)rst order λ (14)
55
phase transition . As it is illustrated in (cid:28)gure 1(b),
t > t
SH
for su(cid:30)
iently high temperatures ( ), only one
= 0
q
minimumtat=Mt is observed. De
reasing temper- θ(q) =q
ature, at SH, a se
ond minimum appears with (cid:28)- p (15)
nite magnetization and further lowering temperature to
t=t χ
c q
, thisminimumbe
omesdegenerate,
orresponding Thetemperaturedependen
eof ,anditsinverse,are
= 0 t < t q < 0.5 q > 0.5
q c
to M . For the free energy global minimum quite distin
t, depending whether or , as
q >0.5
o

urs for (cid:28)nite magnetization, determining its equilib- displayedin(cid:28)gure3(a)and(b), respe
tively. For
t
c
riumvalue. However,this equilibrium stateis not ne
es- the sus
eptibility diverges at , the same temperaθtu(qr)e
sarily the one observed in (cid:28)nite times, sin
e there is an where its inverseinter
epts the temperature axis at p .
q < 0.5
energy barrier between the two minima, whi
h prevent For the sus
eptibility is always (cid:28)nite and peaks
t
thewholesystemtorea
htheglobalminimumofenergy. dis
ontinuouslyat c. Correspondingly, its inverseshows
The minimum temperature that
an sutsta=intSzCero mag- a dis
ontinuity with (cid:28)nite valuθe(sq)and the Curie-Weiss
netization is the one
orresponding to , where p
linear behavior extrapolates to , whi
h is lower than
t
the energy barrier goes to zero and only one minimum c
t .
SH
exists at (cid:28)nite magnetization. In a similar way,
or-
responds to the maximum temperature that
an sustain
a (cid:28)nite magnetization. These phenomena are the well-
C. In(cid:29)uen
e of Magneti
Field on the Phase
known `superheating' and `super
ooling'
y
les and are Transition
responsible for the thermal hysteresis normally observed
t t
55 SH SC
in (cid:28)rst orderphase transitions . The and tem-
FromtheanalysisoftheGibbsfreeenergy,we
on
lude
peratures
anbeanalyti
allyderivedfromthe
onditions q < 0.5 h = 0
q that for and , the equilibrium magnetiza-
des
ribed above,and arevalid onlyfor ≤0.5,yielding:
tion has a (cid:28)rst order phase transition. Su(cid:30)
iently high
h h
3 1 0q
t = magneti
(cid:28)eld ≥ , is able to removethe energybar-
SH
4(2 q) (10) rier between the two minima in the Gibbs free energy
−
and,
onsequently, the dis
ontinuity in the equilibrium
magnetization
urve. Thise(cid:27)e
tisillustratedin(cid:28)gure4.
h
0q
t =q Theexpressionfor
anbederivedfromthe
ondition
SC
(11) des
ribed above, yielding:
t 3(1 2q)
On the
ontrary, a
losed expression for c
annot be h0q = −
2 q (16)
derived, sin
e it involves trans
endental equations. The −
temperaturedependen
eoftheredu
edequilibriummag-
(2 q)/µ h/
q q
netization M − is sensitive to the features de- Thequantity M isparti
ularlyimportantfromthe
s
ribed above, as it
an be seen in (cid:28)gure 2 and inset experimental point of view, and is sket
hed in (cid:28)gure 5,
h=3h
0q
therein. for q<0.5 and .
q < 0.5
Another interesting point of this model, for ,
is the existen
e of a ferromagneti
phase transition in-
B. Magneti
Sus
eptibility du
ed by the magneti
(cid:28)eld,tin the paramagneti
phase.
c
Infa
t, su(cid:30)
iently
loseto , thetwominimahavesim-
χq q ilar free energy values and,
onsequently, the magneti
The generalized magneti
sus
eptibiliy , for any
(cid:28)eldenergyplaysade
isiverole,beingabletoswit
hbe-
value,
an be derived from Eq.9,
tween the low and high magnetization values, indu
ing
C(q) the (cid:28)rst orderphase transition. However, for su(cid:30)
iently
χq = t θ(q) (12) high temperatures t ≥ t0q, only one minimum exists for
p
− any value of magneti
(cid:28)eld, and the phase transition be-

omesofse
ondorder
hara
ter. Thise(cid:27)e
tisillustrated
t
where, in analogy with the usual Curie-Weiss law, we in (cid:28)gure 6. The temperature 0q, whi
h determines the
de(cid:28)ne
ontinuousor dis
ontinuous nature of the magnetization
−1 versus (cid:28)eld
urve is given by:
µ ∂ −1
C(q) = kTC(1)  ∂MMqq (cid:12)(cid:12)(cid:12)(cid:12)Mq(H=0,T) (13) t0q = 3(21−−qq)2 (17)
(cid:12)
4
t
0q
Above ,themagnetizationasafun
tionofmagneti
orderregion. As the magneti
(cid:28)eld in
reases, this points
(cid:28)eld
urves are
ontinuous, presenting, however, a
har- travels in the dashed
urve a

ording to the parametri
h
c
a
teristi

hangeofslopeatsu
hmagneti
(cid:28)eld ,whi
h equations:
varies linearly with temperature, and is given by 3 2h
q = −
h =α(t t′) 0h 6 h (21)
c − (18) −
1(3+h)2
where t =
0h
3q(1 q) 3 6 h (22)
t′ = − −
(2 q) (19)
− Inananalogousway,(cid:28)gure7(b)presentstheproje
tion
h q
ofthephasediagraminthe − planeforseveraltemper-
t>t
and c
atures . Ea
hlinedividestheplaneintheferromag-
1
α= neti
region (above the
urve) aqnd<in0.t5he paramagneti
1 q (20) region (below the
urve). For and su(cid:30)
iently
− tc

lose to , a (cid:28)eld indu
ed phase transition o

urs dis-

ontinuously,withthedashedareas
orrespondingtothe
regions of phase
oexisten
e. The magnitude of the dis-
D. Magneti
Phase Diagrams ∆
q

ontinuity M de
reases with in
reasing temperature,
t t
0q
disappearingfor ≥ (Eq.17), abovewhi
hthe transi-
At this point it is
onvenient to summarize the main
tionisalwaysse
ondorder(see(cid:28)gure6). Thedashedline
properties found so far: the behavior of the generalized (q0t,h0t)

uts the transition
urves at the point , whi
h
magnetization and the generalized magneti
sus
eptibil-
similarly as in the previous diagram, divides the
urves
itywithtemperatureand(cid:28)elddependgreatlyontheval-
q in regions of (cid:28)rst and se
ond order phase transition. As
ues assumed for the entropi
parameter . It
an be the
the temperature in
reases, the point movesup along the
usual se
ond order para-ferromagneti
phase transition

urvea

ordingtotheparametri
equationsgivenbelow,
but it
an be
ome (cid:28)rst order, exhibiting the properties
with
orresponding in
rease of the se
ond order region:
normally asso
iated to this type of transitions.
t q 6 t √t2+12t
Figure7(a)presentsthe − phasediagramforseveral q = − −
h 0t
6 (23)
values. The solid lines divide the plane in ferro (be-
h = 0
low) and paramagneti
(above) regions. For and
q >0.5
thepara-ferromagnqe<ti
0p.5hasetransitionisalways 6(t 3+√t2+12t)
se
ondorder,whereasfor thetransitionbe
omes h0t = −
t+6+√t2+12t (24)
(cid:28)rst order. The dotted lines limit the stability regions
of the fully ordered and disordered phases. In this way, h t
Theproje
tionofthephasediagraminthe − planeis
theshadedareabetweensu
hdottedlinesrepresentsthe q
presentedin (cid:28)gure7(
), only for =0.1,forsakeof
lear-
ferro-paramagneti
phase
oexisten
e,whi
h
anexistfor
tSC(q) t tSH(q) ness. Again,thedottedlineslimitthestabilityregionsof
temperatures in the interval ≤ ≤ .
h the para and ferromagneti
phases and the shaded area
Stri
tly, in the presen
e of the magneti
(cid:28)eld the
represents the values of (cid:28)eld and temperature where the
ferro-paramagneti
phase transition does not exist for
anytemperatureq, s<in
0e.5Mq isalwaysdi(cid:27)erentfromzero. atwndo tpehmapseesra
taunre
ho0eqxaisntd. t0Aq,btohveet
rearntsaiitnionvablue
eosmoefs(cid:28)seel
d-
In any
ase, for , if the magneti
(cid:28)eld is not too hc
ond order and the
hara
teristi
(cid:28)eld varies linearly
high, the Gibbs free energy presents two minima
lose
totc andwe
analwayssmdiasltlinguish twophases, onewith w(ti0tqh,ht0eqm)perature, as given by Eq.18. The open
ir
le
smallmagnetizationMq andotherwithlargemagne- divides the
urve into the (cid:28)rst and se
ond or-
large derphasetransitionregions,andtravelsalongthedashed
tization Mq . If the magneti
(cid:28)eld is su(cid:30)
iently high
h h
urve a

ording to Eqs.16 and 17.
0q
≥ ,theenergybarrierdisappearsandthetransition
be
omes
ontinuous. However, in this
ase, the transi-
tion o

urs with a
hara
teristi
slope
hange in magne- E. Generalized Landau Coe(cid:30)
ients
tizationversustemperature
urves,whi
hdoesnoto

ur
q > 0.5 q < 0.5
for . For and up to a
riti
al applied
In this se
tion we derive the generalized
oe(cid:30)
ients of
magneti
(cid:28)eld, the phase transition o

urs dis
ontinu- 55
ously, with the shaded areas
orresponding to the values Landau theory of phase tran=sitGio/nksT(.1)Let us assume a
q redu
ed Gibbs free energy G C , and magnetiza-
of temperature and where phase
oexisten
e o

urs. m= /µ
∆ q tion Mq . Thus, we are able to expand, for small
Themagnitudeofthedis
ontinuity M de
reaseswith
h h
0q values of magnetization, the previously presented Gibbs
in
reasing (cid:28)eld, disappearing for ≥ (Eq.16), above
free energy (Eq.9), yielding:
whi
h the transition is always se
ond order (see (cid:28)gure
4q). T,the dashedline
uts the transitionlinesat the point = Aqm2+ Bqm4+ Cqm6 hm
( 0h 0h), whi
h divides ea
h line into a (cid:28)rst and se
ond G 2 4 6 − (25)
5
where,
hara
ter, with addit<iona∗l hysteresis for (cid:28)elds below
3 than a
riti
al (cid:28)eld H Hc and tempera∗ture ranges be-
Aq = q(t−q) (26) tween TC and a
riti
al temperature TC. For the last
manganite
ited above, for instan
e, M(H) presents an
C
upward in(cid:29)e
tion point from T =150 K up to 220
c
9(8q 3 4q2)t K, with a
hara
teristi
(cid:28)eld H (T), for the in(cid:29)ex-
Bq = −5q3− (27) ion point, presenting an almost liner temperature de-
penden
e. In addition, a large thermal hysteresis is
C

learly delimited for temp∗eratures ranging from T up
27(54 318q+623q2 464q3+116q4)t to the bran
hing point TC ∼170 K. Analogous behav-
Cq = − 175q5− (28) ior0a.6r5e f0o.u35nd in3L64a0.67Ca0.33M0.5nO30.5and0L.9a50.80M.0n5O336625,
Sm Sr MnO and Pr Ca Mn Cr O ,
among others.
WithintheLandautheory,negativevaluesofthe
oef-
(cid:28)
ientBmean(cid:28)rstordertransitions. Inthisdire
tion,an
obvious
orrelationbetweenthemodelhereproposedand
q < Another interesting feature on the magnetization be-
the usual Landau theory is obtained, sin
e for 1/2,
haviorof some manganites
on
erns the H/M vs T mea-
q
the generalized Landau
oe(cid:30)
ient B also assume nega-
q < surement, that presents a strong downturn for temper-
tive values, assket
hed in (cid:28)gure 8. Note that for 1/2 C
atures nearly above T . It makes a deviation from
our model also predi
ts (cid:28)rst order transition. Addition- C
the simple Curie-Weiss law, and, at T , the mag-
ally,stillwithintheLandautheory,theparameterAusu-
=a(T T ) neti
state is qui
kly swit
hed to a ferromagneti
one.
0
ally takes the form A − (Curie Law), and this
Su
h feature is frequently found in the literature of
is exa
tly the relation found for the generalized Landau 0.60 0.07 0.33 3 0.8 363
manganites: La Y Ca MnO and La MnO ,
q

oe(cid:30)
ient A (Eq.26 and sket
hed on the inset of (cid:28)gure 0.55 0.45 366,67 0.825 0.175 0.86 0.14 368
Sm Sr MnO , La Sr Mn Cu O ,
8), that represents the inverse of the generalized sus
ep- 1−x x 3 x 69
Ca Pr MnO with ≤0.1 , among others.
tibility (Eq.12).
56,57,58
It is well know that, using the
lassi
al formu-
lation of the Landauhth/emory, omr s2imilar, a negative slope However, even with the enormous quantity of experi-
of the isothermplots vs. (Arrot Plot) would in-
mental measurements on manganites (see, for example,
di
ate a (cid:28)rst order phase transition.dT/hdums,=de0riving the the impressivereview by Dagottoand
o-workers41), the
minimum of the Gibbs free energy ( G ), we
an
h/m nature of the phase transition in ferromagneti
mangan-
express the quantity as:
itesisstilla
ontroversialissue. Inthisdire
tion,Amaral
h 62,63
= + m2+ (m2)2 etal.
laimfornewmodelstoexplainthe(cid:28)rst-se
ond
q q q
m A B C (29) order
hara
terofthetransitioninmanganites,evenwith
70
thetheoreti
alworksdevelopedbyJaimeetal. ,Alonso
q <
71 60
Asexpe
ted,for 1/2theGeneralizedArrotPlot has et al. and Novák et al. .
anegativeslope,indi
ating(cid:28)rstordertransition,whereas
q >
for 1/2, these plots are straight lines,
hara
teristi
of a ferromagneti
se
ond order phase transition.qT>hese Toinquireabout theorderof thephase transitionand
featuqre<s are presented in (cid:28)gure 9(a) and (b), for 1/2 des
ribe theoreti
ally the behavior of the relevant mag-
and 1/2, respe
tively. neti
quantities, Amaral and
o-workers62,63 used the
ma
ros
opi
Landau theory of phase transition, expand-
ing the free energy up to sixth power of th4e magnetiza-
IV. CONNECTIONS TO EXPERIMENTAL tion. IftheparameterB(withrespe
ttoM )isnegative,
RESULTS
the transition
an be (cid:28)rst-order like. In this
ase, the
magnetizationwill presentlarge(cid:28)eld
y
lingir∗reversib∗il-
A. A Brief Survey ity only for (cid:28)elds and temperatures below Hc and TC,
respe
tively. Further analysis of Landau theory, even
<
The experimental (cid:28)eld and temperature dependen- for B ∗ 0 and hig∗her temperatures and magneti
(cid:28)elds

ies of some manganites present interesting aspe
ts. (H>Hc and T>TC), show that the magnetization has a
58,59
Mira et al. analyzed the
hara
ter of the phase pe
uliarin(cid:29)exionpoint at a
hara
teristi
magneti
(cid:28)eld
2/3 1−y y 1/3 3 c c
transition in La (Ca Sr ) MnO and
on
luded H ,thatin
reaseslinearlywithtemperature,H (T)∝(T-
0 58,59
that for y=0 the magneti
transition is of (cid:28)rst or- T ). Miraetal. havealsoappliedtheLandautheory
der, whereas it is se
ond order for y=1. Other to manganites. However, the results of the Landau the-
works,in
luding those using nu
lear magneti
resonan
e ory are not su(cid:30)
ient to reprodu
e all pe
uliar magneti
60,61 36,62,63
(NMR), support this result . Amaral et al. properties of the manganites, su
h as the anomalousbe-
0.67 0.33 3 0.8 3
emphasized that La Ca MnO , La MnO and havior of the H/M vs. T quantity, presented on (cid:28)gure 5
0.60 0.07 0.33 3
La Y Ca MnO exhibit (cid:28)rst order transition and (cid:28)gure 12.
6
B. Experimental and theoreti
al results for In the present paper we have extended our analysis and
0.60 0.07 0.33 3
La Y Ca MnO presented new
ompelling eviden
es in this dire
tion by
dedu
ing a magneti
phase diagram whi
h mat
hes ob-
A ferromagneti

erami
La0.60Y0.07Ca0.33MnO3 was served experimental results on La0.60Y0.07Ca0.33MnO3,
prepared by standard solid-state methods63, and the along with some other magneti
properties of this
omq-
magnetization was measured using a Quantum Design pound. The interpretation of the entropi
parameter ,
49
SQUID magnetometer (55 kOe) and an Oxford Instru- givenbyBe
kandCohen ,intermsoftheratiobetween
ments VSM (120 kOe). the mean and width of the temperature distribution in
In this se
tion we will apply the general results ob- the system,
omes to support our proposal, sin
e man-
tained in the previous se
tions to a quantitative analy- ganiteshavelongbeenre
ognizedasobje
tswhoseprop-
sis of experimental data for La0.60Y0.07Ca0.33MnO3. As erties aredominated by intrinsi
inhomogeneities41,50,66.
stated in se
tion III, we work within the mean-(cid:28)eld ap- Su
hadistribution
anbetranslatedasadistributionof
x
proximation, for whi
h assume the expression: themagneti
sus
eptibility,aspointedoutbySalqamonet
51
µ(H +λ ) al. , from whi
h a temperature dependen
e of
an be
q
x= M expe
ted. Therefore,we
on
ludethattheTsallisnonex-
kT (30)
tensive statisti
s is a handy tool to study,
lassify and
λ predi
t magneti
and thermal properties of manganites.
where is the mean-(cid:28)eld parameter. Figure 10 presents
themeasuredandtheoreti
almagneti
momentasafun
-
C
tionofmagneti
(cid:28)eld,forseveraltemperaturesaboveT .
VI. ACKNOWLEDGEMENTS
The ex
ellent experimental-theoreti
al agreement is due
totheuseoftheTsallisstatisti
s,whi
hparametrizesthe
41 The authors thanks FAPERJ/Brasil, FCT/Portugal
system inhomogeneity .
q µ λ N
(
ontra
t POCTI/CTM/35462/99) and ICCTI/CAPES
Here, we use , , and , the number of
lusters in
(Portugal-Brasilbilateral
ooperation),for(cid:28)nan
ialsup-
the sample, as free parameters. The magneti
moment
µ
port. WearealsothankfultoA.P.Guimarães,E.K.Lenzi
of the
lusters follows the usual temperature tenden
y
q
and R.S. Mendes, for their helpful suggestions.
of a para-ferromagneti
transition, whereas in
reases
towards unity with in
reasing temperature ((cid:28)gure 11).
q
The temperaturedependen
e ofthe parameterwasex-
VII. FIGURE CAPTIONS
pe
ted,sin
eitshouldrea
htheunityforsu(cid:30)
ientlyhigh
temperatures.
From these results, we were able to
ompare the Figure 1. Gibbshfr=ee energy(Eq.9), within the mean (cid:28)eld
approximation and 0, as a fun
tion of the redu
ed mag-
anomalous downturn in H/M vs. T
urves, just with q >
q µ netization. (a) For 0.5 only one minimum is observed,
the temperature dependen
e oλf anNd , and the ap- for any value of temperature (t>tc or t<tc), representing a
proximately
onstant values of and . The
al
ulated
q qse<
ond order=phase transition for the magnetization. (b) For
H/M vs. T
urveis displayedin (cid:28)gure12,with its
or- c
0.5 and t t two degenerated minima are observed, rep-
responding experimental value. One should stress that resenting a (cid:28)rst order phase transition. The tSH and tSC
the solid line in this plot does not in
lude any other (cid:28)t- temperatures limit the 'superheating' and 'super
ooling'
y-
ting parameter. The
urvewas
al
ulatedusingonly the
le, responsible for the thermal hysteresis normally observed
(cid:28)tted parameters obtained from Figure 10. in (cid:28)rst order transitions.
Additionally, Amaral and
o-workerspointed out that Figure 2. Temperature dependen
e of tqhe redhu
=ed0equi-
La0.60Y0.07Ca0.33MnO362 presents two di(cid:27)erent features qlib<rium magnetization for several values of and . For
H M H
c 1/2 the transition shows a (cid:28)rst order
hara
ter, whereas
on the in(cid:29)e
tion point, at , of its vs.
urves: q>
for 1/2thetransitionisofse
ondordertype. Inset: Ther-
t(cid:28)(eie)mldapleHirncae,taufroretresmTin>petThr∗Caet,ruaarnengddee(TpiieC)nad<neTno
<beTsoe∗Cfrv.tehFdeig
huhyrasertea1r
3etsesirhsiosfwtoi
rs tmioanlsh,ywstitehretshisen'sourpmearlhlyeaotibnsge'rvaenddi'nsu(cid:28)presrt
ooordlienrg'ph
ya
sleest.ransi-
H T Figure 3. Temperature dependen
e of the generalized
theplotof c vs. obtainedfromexperimentaldata. It magneti
sus
eptibilityχq,anditsinverse,for(a)q<0.5and
q>
is striking the similarity between this plot and the
urve (b) 0.5.
shown in Fig. 7(
). Figure4. Temperaturedependen
eofhtheredqu=
edequilib-
62,63
anFdinMailrlya,ethteaAl.5r8r,o59tPalroetsveprryesseinmteildarbytoAtmhaorsaelperteasle.nted r
iiuenmtlymhagignhetmizaagtinoentif
or(cid:28)esledvehra≥l vha0lqu(eEsqo.f16,),ainsdablet0o.1r.eSmuo(cid:30)ve-
on (cid:28)gure 9(a)(b). thedis
ontinuityin the equilibriummagnetization
urveM.
q
Fqig=ure5. Tehm=pe3rha0tquredependen
eofthequantityh/ ,
for 0.2 and .
Figure 6. Magneti
(cid:28)elddependen
eoftheredu
edequi-
V. CONCLUSION
lqib=rium magnetization for several values of temperature, and
0.1. It represents a genuine ferromagneti
phase transi-
34,35
denIn
estwthoatptrheveimouasgnpeutbi
lip
arotipoenrstiesofwmeanpgraensietnetse
danevbie- Ftioornsuin(cid:30)d
ui
eendtlybyhimghagtnemetpi
er(cid:28)aetludr,esint≥thet0qpa(rEaqm.1a7g)n,etthi
etprhaanssei-.
suitably des
ribed in the framework of Tsallis statisti
s. tion be
omes
ontinuous.
7
q− 2
Figure 7. magneti
phase diagram summarizing the Figure 9. The generalized Arrot Plot (h/m vs. m
q> q<
main properties ftou−nqd in the model. (a) Proje
tion of the
urves), for (a) 0.5 and (b) 0.5.
phasediagramin plane. Thesolidlinesdividetheplane Figure 10. Measured(open
ir
les) andtheoreti
al (solid
in ferro (below) and paramagneti
(above) phases, with a re- lines -Eqs.7 and30) magneti
momentas a fun
tionof mag-
C
gion of phase
oexisten
e between the dotted lines (shaded neti
(cid:28)eld, for several values of temperatures above T =150
area). Ontheleftsideofthedashedline,thetransitionhasa K.
(cid:28)rst order
hara
ter, whereas a se
ond order
hara
ter ahri−seqs Figqure11µ. Temperaturedependen
eofthe(cid:28)ttingparam-
fortherightside. (b)Proje
tionofthephasediagramin eters, and (see text).
>
c
plane, for several temperatures t t . The
urves in this
ase Figure 12. Measured(open
ir
les) andtheoreti
al (solid
have a
omplete analogy to the previous one, however, it is lines)valuesofthequantityH/Mvs. T.Thesolidlineinthis
ferromagneti
above thetransition lines, sin
e itrepresentsa plot does not in
lude any (cid:28)tting parameters, and was
al
u-
ferromagneti
transition indu
ed by mh−agnteti
(cid:28)eld. (q
)=Pro- lated using only the (cid:28)tted parameters obtained from (cid:28)gure
je
tion of the phase diagram in the plane, for 0.1. 10.
h0q
Abovt0eq
ertain values of (cid:28)eld and temperatures (Eq.16), >Fig∗Cure 13. The linear temcperature dependen
e, for
and (Eq.17), thetransition be
omesse
ond oBrder. T T ,ofthe
hara
teristi
(cid:28)eldH ,whi
h
orrespondstothe
q
Figure8. ThegeneralizedLanqdau
oe(cid:30)q
<ient Baqsafun
- in(cid:29)ex0io.6n0p0o.i0n7tof0.t3h3eexpe3rimentaClM<vs<. H∗C
urves,measured
tion of the entropi
parameter . For 0.5, assume in La Y Ca MnO . For T T T the hysteresis is
negative values,indi
ating(cid:28)rstorderphasAe transition. Inset: indi
atedbytheshadedarea. It is striking thesimilarity be-
q
temperaturedependen
eoftheparameter ,thatrepresents tween this experimental plot and the theoreti
al one, shown
theinverse of thegeneralized sus
eptibility. in (cid:28)gure 7(
).
∗
16
Present address: Centro Brasileiro de Pesquisas Físi
as, C. Tsallis, Braz. J. Phys. 29, 1 (1999).
17
RuaDr.XavierSigaud150Ur
a,22290-180RiodeJaneiro- For a
ompleteand updatedlistof referen
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RJ, Brasil; Ele
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D. I. Uzinov, Introdu
tion to the Theory of Criti
al Phe-
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