Spin-glass like dynamics of ferromagnetic clusters in La$_{0.75}$Ba$_{0.25}$CoO$_3$

1 MB·

by  Devendra Kumar

Spin-glass like dynamics of ferromagnetic clusters in La Ba CoO
0.75 0.25 3
∗
Devendra Kumar
UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore-452001,India
WereportthemagnetizationstudyofthecompoundLa0.75Ba0.25CoO3 whereBa2+ dopingisjust
above the critical limit for percolation of ferromagnetic clusters. The field cooled (FC) and zero
field cooled (ZFC) magnetization exhibit a thermomagnetic irreversibility and the ac susceptibility
show a frequency dependent peak at the ferromagnetic ordering temperature (TC≈203 K) of the
clusters. Thesefeaturesindicateaboutthepresenceofanon-equilibriumstatebelowTC. Inthenon-
equilibriumstate,thedynamicscalingoftheimaginarypartofacsusceptibilityandthestaticscaling
6 ofthenonlinearsusceptibilityclearlyestablishaspin-glasslikecooperativefreezingofferromagnetic
1 clusters at 200.9(2) K. The existence of spin-glass like freezing of ferromagnetic clusters is further
0 substantiated by the ZFC aging and memory experiments. We also observe certain dynamical
2 featureswhicharenotpresentinatypicalspin-glass,suchas,initialmagnetizationafterZFCaging
first increases and then decreases with the wait time and an imperfect recovery of relaxation in
n
negative temperature cycling experiments. This imperfect recovery transforms to perfect recovery
a
J onconcurrentfieldcycling. Ouranalysissuggeststhattheseadditionaldynamicalfeatureshavetheir
origin in inter-clusterexchangeinteraction andclustersize distribution. Theinter-clusterexchange
8
interaction above the magnetic percolation gives a superferromagnetic state in some granular thin

] filmsbutourresultsshowtheabsenceoftypicalsuperferromagneticlikestateinLa0.75Ba0.25CoO3.
l
e
PACSnumbers: 75.47.Lx,75.50.Lk,75.40.Gb
-
r Keywords: DisorderedCobaltites,spin-glass,cluster-glass,superferromagnetism
t
s
.
t
a I. INTRODUCTION ture, grow in number and size on lowering the temper-
m ature, and finally may undergo a spin-glass like freezing
onfurther decreasingthe temperature. The inter-cluster
-
Magnetically ordered materials exhibit interesting
d interaction increases on decreasing the temperature and
n physicalpropertieswhenthe spin-spincorrelationlength attains its peak value at the freezing temperature.9,10
o is limited to the range of few nanometers.1 The size of
Above the magnetic percolation of ferromagnetic clus-
c correlation length can be limited by limiting the physi-
[ ters, there is a finite possibility of exchange interac-
cal size of material by forming few nanometer size sin- tion between the neighbouring clusters in addition to

1 gle domain nanoparticles. Ensemble of such nanoparti-
the long range dipolar interaction. This additional ex-
v clesareobservedtoexhibitintriguingdynamicalfeatures,
change interaction, present above the magnetic percola-
9 such as, superparamagnetic like thermal blocking, spin-
tionofFenanoparticlesinFeAggranularthinfilmscause
0
glass like cooperative freezing, or ferromagnet like cor-
0 acrossoverfromspin-glasslike stateto asuperferromag-
2 relation between nanoparticle moments in the so called netic state.12
superferromagnetic state.2 The nature of physical state
0
. inthenanoparticleassemblyisdeterminedbythecompe- The presence of short range ferromagnetic clusters in
1
tition between the anisotropy energy, dipole interaction,
0 cobaltites with formula La1−xAxCoO3 (where A is di-
6 and exchange interaction. For dilutely packed nanopar- valent ion Ba2+ or Sr2+) have been observed in a num-
1 ticle system a superparmagnetic state is observed, but ber of reports.11,13–20 The density of ferromagnetic clus-
: fordenselypackednanoparticlesystemwithstrongdipo- ters increases on increasing the Ba2+ or Sr2+ doping,
v
lar interactions, a spin-glass like cooperative freezing is
Xi reported.2–4 The presence of additional exchange inter- and above a critical doping (xc), the ferromagnetic clus-
ters percolate. The percolation occurs at x =0.2 for
c
r action in the densely packed nanoparticle system with Ba2+ andx =0.18forSr2+.17–20 Theferromagneticclus-
a strong dipolar interactions can also result in a superfer- c
ters tend to retain their cluster nature and do not com-
romagnetic state.2,5–8
pletely agglomerate to form a continuous phase even
The size of correlationlength can also be limited even above the percolation.11,16,19,20 Below x both the Ba2+
c
without limiting the physical size of the system. In and Sr2+ doped systems exhibit a spin-glass like cooper-
number of materials, e.g. in phase separated mangan- ative freezing.17,20 Above x , ac susceptibility measure-
c
ites (La0.25Nd0.75)0.7Ca0.3MnO3, La0.7−xYxCa0.3MnO3 ments on La1−xSrxCoO3 show some characteristics of
(0 ≤ x ≤ 0.15), and cobaltite La1−xSrxCoO3 (0.18 ≤ cluster-glass dynamics while no such signature has been
x ≤ 5.0), this happens due to formation of short range detected in La1−xBaxCoO3 (x=0.2 and 0.3).11,19–21 Re-
ferromagneticclustersintheparamagneticmatrixatthe centlyadetailedstudyonLa Ba CoO hassuggested
0.5 0.5 3
ferromagnetic transition.9–11 The ferromagnetic clusters the presence of interacting superparamagnetic like dy-
in manganites are metallic and are associated with the namics in the ferromagnetic clusters.16 This makes the
temperature driven first-order insulator to metal transi- region between x and x=0.5 for Ba2+ doping interest-
c
tion. These clusters appear at the transition tempera- ing as at one end we have the spin-glass like dynamics,
2
while at the other end, we have the interacting super-
Experiment
paramagnetic like dynamics with surprising absence of FDiitfference
dynamical features in between. 4000 La0.75Ba0.25CoO3
nit)
In this manuscript we have performed a compre- b. u
hofenLsaiv0e.75inBvae0s.2ti5gCaotOio3n wofhitchhe lmieasgjnuestticaabllyovoerdtheeredcristtiactael ensity (ar 2000
doping for percolation. The ferromagnetic clusters in Int
La Ba CoO will experience the long range dipolar
0.75 0.25 3
interaction as well as the short range nearest neighbour 0
inter-cluster exchange interaction, and therefore, are a
20 40 (deg.) 60 80
good candidate for studying the interplay of these com-
Figure 1: (Color online) X-ray diffraction pattern of
peting interactions. Our resultsshow thatferromagnetic
clustersinLa Ba CoO undergoaspin-glasslikeco- La0.75Ba0.25CoO3atroomtemperature. Thesolidcirclesrep-
0.75 0.25 3 resents the experimental X-ray diffraction data, the red line
operativefreezing. IncontrasttoLa1−xSrxCoO3 (0.18≤ ontheexperimentaldataexhibitstheRietveldrefinementfor
x ≤ 5.0) and La0.7−xYxCa0.3MnO3 (0 ≤ x ≤ 0.15), rhombohedralR-3cstructurewithχ2=1.26,theshortvertical
the cluster-glass transition in La0.75Ba0.25CoO3 nearly linesshowtheBraggpeakpositions,andthebottomblueline
coincides with the ferromagnetic ordering and the inter- displays the difference between the experimental and calcu-
cluster interaction is found to be unaffected of temper- lated pattern.
ature. Concurrent to the spin-glass like dynamics, we
observe the signature of an additional dynamical mech-
anism which has been attributed to the exchange inter- The oxygencontent is determined by thermogravimetric
actionsbetweenthe ferromagneticclusters. Ouranalysis analysis which is close to its stoichiometric value of 3.0.
also show that unlike FeAg granular films, the perco-
lation of ferromagnetic clusters in La Ba CoO do
0.75 0.25 3
not establish a typical superferromagnetic like state in III. RESULTS AND DISCUSSIONS
the system.
A. Thermomagnetic irreversibility
II. EXPERIMENTAL DETAILS
Figure 2 shows the magnetization versus temperature
curvesatthefieldof5Oe,100Oe,500Oe,and10000Oe
Polycrystalline samples of La Ba CoO are pre- in field cooled (FC) and zero field cooled (ZFC) proto-
0.75 0.25 3
pared through pyrophoric method as described in our cols. In FC protocol the sample is cooled to 5 K in
earlier work.16 The stoichiometric ratio of high purity presence of applied field and data is recorded in heat-
(99.99%) La O , BaCoO , and Co(NO ) 6H O are dis- ing run without changing the field. In ZFC protocol the
2 3 3 3 2 2
solvedindilute nitricacidandtriethanolamine(TEA) is sample is cooled to 5 K in zero field, then field is ap-
addedtothefinalsolutionkeepingpHhighlyacidic. This plied, and data is recorded in the heating run. Both FC
◦
solutionisdriedat100 Cwhichfinallyburnsandyields and ZFC magnetization curves exhibit a paramagnetic
blackpowder. Theblackpowderispelletizedandheated to ferromagnetictransitiononcoolingandthe transition
◦
at 1125 C for 12 h in air. These samples are character- temperature (T ) is estimated to be around 203 K by
C
ized by XRD diffraction on a Bruker D8 Advance x-ray the temperature derivative of the 500 Oe FC magnetiza-
diffractometer using Cu Kα radiation. The magnetiza- tion curve (see inset of Fig. 2). On further cooling, the
tionmeasurementsareperformedon7TSQUIDMPMS- ZFC magnetization curve diverges from their respective
XL (Quantum Design) and 14T PPMS-VSM (Quantum FC curveata temperature T andexhibits a peak ata
irr
Design). The residual field in 7T SQUID MPMS-XL is temperature T , while the FC magnetization curve con-
p
set below 0.05 Oe by using the flux gate and compensa- tinuetogrow,albeitwithaslowerrate. Onincreasingthe
tioncoilsofultralowfieldattachmentbeforeperforming applied magnetic field, the ZFC peak flattens and T
irr
thezeroandlowfieldmagnetization,aging,andmemory and T shift to the lower temperatures. The existence
p
experiments. The data used in scaling analysis has been of thermomagnetic irreversibility is generally observed
corrected for demagnetization factor. in spin-glass,24–26 cluster-glass,27,28 superspin-glass,29,30
The x-ray data has been analyzed by the Rietveld re- superparamagnets,31 and anisotropic ferromagnets.32–37
finement method using FULLPROF software22 and the Theappearanceofferromagneticordering14atT negate
C
results show that the sample is single phase and crys- the possibility of an atomic spin-glass state. The ferro-
tallizesinrhombohedralstructurewithspacegroupR-3c magnetic state in these systems consists of small perco-
whichisinagrementwiththepreviousreport.23Figure1 lating ferromagnetic metallic clusters and the absence of
displays the room temperature x-ray diffraction pattern exchange bias effect rules out the possibility of existence
of La Ba CoO along with its Rietveld fit profile. of a spin-glass phase at the interface of ferromagnetic-
0.75 0.25 3
The goodness of fitting χ2 is 1.26and the lattice param- clusters.13–18,20,38 This suggests that the observed ther-
eters of the unit cell are a=5.4549(2) and c=13.3194(2). momagnetic irreversibilitymay have its originin the dy-
3
1.5 -00..0020 dM/dT 12 0000....01245HHHHzzzz14.5 mu/mole) 0.8
/f.u.)1B.0 -0.040 90 T (1K8)0 270 mole) 01..98HHzz ’ (e mole)
M ( ZFZFFCFCCC 1 15500 O0O0 eeOOee ’ (emu/6 14.0 201 T (K) 202 0.4" (emu/
0.5 ZFC 500 Oe
FC 500 Oe
ZFC 10000 Oe
FC 10000 Oe
0.0 0 0.0
0 100 T (K) 200 300
200 210 220 230
T (K)
Figure 2: (Color online) Magnetization versus temperature
underfieldcooled(FC)andzerofieldcooled(ZFC)protocols Figure3: (Coloronline)Temperaturedependenceofreal(χ′)
at various fields. The inset shows the dM/dT for 500 Oe FC andimaginary(χ′′)partofacsusceptibilitiesatvariousmea-
data. surement frequencies. The inset shows the expanded view of
′′
χ close to thepeak.
namics of ferromagnetic-clusters.
perature at the zero field dc limit, ǫ=(T −T )/T is the
g g
reduced temperature, and ν is the static critical expo-
nent. The relaxation time (τ) is related to correlation
B. AC susceptibility and dynamic scaling length (ξ) as τ ∝ ξz where z is the dynamic critical ex-
ponent. Thus for a cluster-glass transition, the relation
The results of ac susceptibility measurements at for relaxationtime τ (correspondingto a givenmeasure-
0.05 Hz, 0.1 Hz, 0.2 Hz, 0.4 Hz, 0.9 Hz, and 1.8 Hz at ment frequency f, τ=f−1) can be written as
3suOsceepatcibfiieliltdya(rχe′)diesxphlaiybeitdsian pFeiga.k3a.tTthheerfeearlropmaratgnofetaicc τ =τ0(T/Tg−1)−zν, (1)
transition,butasshownintheinsetofFig.3,incontrast where τ is the spin flipping time of fluctuating spin-
0
to long range ferromagnets, the peak temperature (T ) clusters. Foragivenmeasurementfrequencyf,the peak
f
increases on increasing the measurement frequency. The temperature of corresponding χ′(f), i.e. T , is associ-
f
existenceoffrequencydependenceinthepeakpositionof ated with the cluster-glass transition temperature. The
χ′ indicates that correlation length of the ferromagnetic imaginary part of ac susceptibility (χ′′(f)) also exhibits
orderdoesnotdivergeatT . Thisfrequencydependence a frequency dependent peak, and in this case, the inflec-
C
in the peak of χ′ is detectable only below 2 Hz. This is tion point in χ′′(f) is identified as the cluster-glasstran-
possibly due to strong contributions coming from within sition temperature (T ). Through T calculated from
f f
the ferromagneticregionsathigherfrequencies. Sazonov χ′(f) and χ′′(f) is expected to show qualitatively simi-
et al.21 failed to detect the frequency dependence in χ′ lar features, frequency dependence in χ′′(f) is more pro-
possibly due to higher measurement frequencies. Unlike nounced, and therefore estimation of T from inflection
f
La1−xSrxCoO3 (0.18 ≤ x ≤ 5.0),11,19 we do not observe point of χ′′(f) is relatively more accurate. For 0.05 Hz,
′′
any secondary peak or hump in χ . The frequency de- the error in estimation of T is large and so it has
f
′
pendenceinthepeakofχ generallymanifestsinthenon- been left out from the fitting process. Fig 4 (a) shows
equilibrium magnetic states, such as spin-glass, cluster- the fitting of equation 1 to τ versus T data following
f
glass, superspin-glass, and super-paramagnets, and the the procedure described in Ref. 16. The data fits well
parameterΦ=∆T /(T ∆log f)whichquantifythisde- with τ ∼ 10−38s, T =200.9(1)K, and zν=18(1). While
f f 10 0 g
pendence is observed to be around (0.02-0.005)for spin- the good fitting of equation 1 to the data suggests the
glass, cluster-glass, superspin-glass, and interacting su- possibility of spin-glass like phase transition, the value
perparamagnetsandaround(0.1-0.3)fornon-interacting of fitting parameter τ is unphysical and zν is outside
0
superparamagnets.39–43 The data shown in the inset of the range of canonical spin-glass(4-10)44,45 but is close
Fig. 3 gives Φ around 0.0005. Since the possibility of an to the values reported in cluster or superspin -glass(10-
atomic spin-glass state or a spin-glass phase at the in- 15).29,30,46–48 The anomaly in τ can be due to error in
0
terface of ferromagneticclusters have been already ruled estimationoffittingparametercausedbylimitedspanof
out, the observed frequency dependence can be because frequency range.
of spin-glass like freezing or superparamagnet like ther- Incriticalslowingdowndescriptionofspin-glassphase
mal blocking of interacting ferromagnetic clusters. transition, χ′′(ω,T) should behave according to the dy-
If the observed frequency dependence of T is due to namic scaling equation proposed by Geschwind et al.49
f
critical slowing down of fluctuating clusters, as in the Tχ′′(ω,T)ǫ−β =g(ωǫ−zν), (2)
case of spin-glass transition, the spin-cluster correlation
length (ξ) should diverge as ξ ∝ ǫ−ν on approaching T where ω =2πf, β is the critical exponent corresponding
g
from T>T . Here T is the cluster-glass transition tem- to the order parameter, and g is a universal function of
g g
4
1000 60
1000Oe
2 (a) (b) 200Oe
=0.22(1) 50Oe
zv=18(2) 10Oe
ln()1 1"-00 T Tg= 02.0095.H9(z2) 2/(+)3/Hnl0
10 0.1Hz
0.2Hz
0 0.4Hz
0.9Hz
1 1.8Hz
0
-4.74 -4.68 -4.62 1E39 1E49 1E59 1E69 1E79 1E89 0.00 0.03 2/( ) 0.06
ln(T/Tg-1) -zv /H
Figure5: (Color online) Scalingplot of thenonlinearsuscep-
Figure4: (Coloronline)(a)FrequencydependenceofT plot-
ted as lnτ versus lnǫ. The straight line is the least sfquare tibility above Tg based on theequation 6.
′′
fittingof equation 1. (b) Dynamicscaling of χ (ω,T)on the
basis of equation 2.
where ǫ=(T-T )/T is the reduced temperature, β is
g g
the critical exponent for spin-glass order parameter, γ
′′ is the critical exponent for spin-glass susceptibility, and
itsargument. Ifthefrequencydependenceinχ isindeed
′′ F(x) and G(x) are the scaling functions. The scaling is
due to spin-glass like phase transition, then χ curves of
achievedbyplottingχ /H2β/(β+γ) versusH2/ǫβ+γ for
different frequency should collapse on a single universal NL
χ at different fields and varying the parameters T ,
curve (g) for the proper value of critical exponents β, zν NL g
β and γ such that all data collapse on a master curve.
and transition temperature T . As shown in Fig. 4 (b),
g In the limit of ǫ →0 the abscissa and ordinate have a
a nearly perfect collapse of the data over two decades
span of many decades, and therefore, are plotted on log
of frequency is obtained for T =209.9(2) K, β=0.22(1)
g
scales. The log scale plotting gives equal weightage to
and zν=18(2) which confirms the presence of spin-glass
alldatapointsirrespectiveoftheiraccuracywhichsome-
like phase transition in the system. The value of the
times hides the departure from the good scaling.24,54 To
parameters zν and T are in agrement with the values
g
test the scaling equation in a better way, Geschwind et
obtained from equation 1.
al.54 have rewritten the scaling equation such that the
argument of scaling function is linear in ǫ
C. Static scaling of nonlinear susceptibility
χ =H2β/(β+γ)G˜(ǫ/H2/(β+γ)). (6)
NL
Themagnetization(M)attheuniformappliedfieldH The contribution of nonlinear susceptibilities to magne-
can be expanded in terms of nonlinear susceptibilities as tization diminishes as H approaches to zero, and in this
limit, the magnetizationaboveT gives a reasonableap-
g
M(H)=M0+χ1H +χ2H2+χ3H3+...., (3) proximation of χ1. We have used the magnetization at
0.5 Oe as the approximate value of χ . Figure 5 shows
1
whereM isthespontaneousmagnetization,χ isthelin-
0 1 the scaling plot obtained using equation 6 for the data
earsusceptibility,andχ ,χ .. arenonlinearsusceptibili-
2 3 at 10 Oe, 50 Oe, 200 Oe, and 1000 Oe. The data taken
ties. For anatomic spin-glass,M and the coefficients of
0 at four different fields collapse best on a master curve
evenpowerofH i.e χ ,χ .. arezerowhile coefficientsof
2 4 forβ=0.22,γ=40,andT =200.9K.Thereasonablescal-
g
theoddpowerofH i.e χ ,χ .. divergesasT approaches
3 5 ing of the χ by equation 6 supports the occurrence
T inthecriticalregime.50–52Forcluster-glass,ifthenon- NL
g of a spin-glass like phase transition at T . The values
g
linearresponseofisolatedferromagneticclustersissmall,
of parameter β and T are in agrement with that of the
g
coefficientsoftheoddpowerofH willalsodivergeinthe
dynamic scaling. The parameter β lies in the range of
criticalregimesimilartothatofatomicspin-glass.53 The
cluster or superspin -glass while parameter γ is large
overallnonlinearsusceptibilityχ whichdivergesinthe
NL in comparison to typical values observed in spin-glass,
criticaltemperatureregimeinaspin-glasssystemcanbe cluster-glass,or superspin-glass.9,45,53,55
written as
D. Aging and memory experiments
χ =χ −M/H =χ H2+χ H4+..... (4)
NL 1 3 5
The existence of a spin-glass like phase transition in
The phenomenological theory of spin-glass by Suziki
La Ba CoO is also investigatedthrough aging and
predicts that the χ should follow the static scaling 0.75 0.25 3
NL
relation50 memory experiments. In FC (ZFC) aging experiments,
the system is cooled from 250 K to temperature of ag-
χNL =ǫβF(H2/ǫβ+γ) (5a) ing (Ta) in presence of field Ha (zero filed), and at Ta
after isothermal waiting of t s, the field is switched off
or χ =H2β/(β+γ)G(H2/ǫβ+γ), (5b) w
NL (field H is applied) and the magnetization is recorded
a
5
as a function of time. The results of field cooled aging
esowcexufanpritttehiernedtirmismmiunepoenFretedrsigmpea.paat6renan8(emd0ane)taKngamcnneaeadntgwsdinthwes5itci0iithnzhasKehtdtiaoifsosrnterrsbeiHpbexeeuhacnt=itibio5vrinte0espl0oyotf.OhratTeeenhdeiaseffroteedtocreptocorpaoceyy--f 3.40 3 M (10 emu/mole) ttww==03(as0)00s 33..568031M (10 emu/mole)00 1t 0(s0)0 tttwww===10310s000000000s0s 567432 M (emu/mole) 456M(0)(emu/mole)8640(b)20t0w0(s)40 015000 KK44454680 M(0)(emu/mole) tttt====0625s000000s00ss
energy barriers as well as in systems with a spin-glass 3.36 tw=10000s
like transition.4 The value of initial magnetization ob- t (s) t (s)
100 1000 10000 100 1000 10000
served after switching off the field at 80 K increases on
(c) (d)
increasing the wait time similar to that of superparam- 120 e)
a5g0nKet.sTohresapginin-gglaexsspebruimt eenxthsibpietsrfoarmcoemdpilnexthbeeZhaFvCioprraot- 4 S(t)10 t=0s 0 emu/mol
90 t=600s M (
tocol show a distinct behavior in superparamagnet and t=2000s -2
t=5000s
spin-glass systems. In superparamagnets the wait time
dependence inZFCagingiseitherabsentorsignificantly 60 t(s) -4 T (K)
1000 10000 0 100 200

small (when several competing sources of anisotropy are
present) in comparison to the wait time dependence of
Figure 6: (Color online) (a) Wait time dependence of decay
FCaging,whileinspin-glasssystems,astrongwaittime
of thermoremanent magnetization (TRM) at 80 K. The in-
dependenceisobservedbothincaseofFCaswellasZFC set shows the wait time dependence of TRM at 50 K. (b)
aging.4,26 Figure 6 (b) exhibits the results of ZFC aging Wait time dependence of ZFC magnetization at 50 K. The
experiments at T =50 K and H =100 Oe. The presence solid line is the fit of equation 7 to data. The inset shows
a a
of strong wait time dependence in ZFC aging as in the the wait time dependence of M(0) at 50 K and 10 K. (c)
caseofFCagingreconfirmsthepresenceofspin-glasslike S(t)=(1/H)(dM(t)/lnt) for different wait times at 50 K. (d)
transition in the system. The observedtime dependence Thetemperaturedependenceof∆M=MZmFemC-MZreFfC. MZmFemC
in magnetization is fitted with stretched exponential de- hasanintermediatestopof104 sat130Kinthecoolingrun.
cay
tw (s) M0 Mg τ×103 β χ2/DOF R2
M(t)=M −M exp(−(t/τ)β), (7)
0 g 0 57.73(8) 7.0(1) 4.3(2) 0.502(8) 0.00142 0.99920
where M is the contribution of intrinsic ferromagnetic 6×102 74.77(9) 7.2(1) 4.8(2) 0.488(8) 0.00139 0.99921
0
component, M is the initial magnetizationof the glassy 2×103 61.26(9) 7.3(1) 5.2(2) 0.488(7) 0.00103 0.99941
g
component, τ is the time constant, and β represents 5×103 54.5(1) 7.4(1) 6.1(3) 0.476(7) 0.00027 0.99982
the distribution of energy barriers with 0 ≤ β ≤ 1 for
spin-glass. The equation 7 fits well to the time depen- TableI:Fitparametersobtainedfromthefittingofequation7
dent magnetization data of Fig. 6 (b) and the fitting to themagnetization data of Fig. 6 (b).
parameters are shown in Table I. The parameters M
g
and τ increase while β decreases on increasing the wait
time. Inspin-glasssystems,themagnetizationatt=0i.e. laxation times and peaks around tw.56,59,60 Figure 6 (c)
M(0) is observedto decrease on increasing the wait time show the S(t) versus lnt curves for various tw. As ex-
(t ),4,26,56 butinLa Ba CoO ,itfirstincreasesand pected in a spin-glass system, we get a peak in S(t)
w 0.75 0.25 3
then decreases. The ZFC aging experiments performed at tewff which shifts to higher t on increasing tw but
at 10 K (not shown here) also give similar results. A the order of shift is relatively small in comparison to
comparison of wait time dependence of M(0) at 50 K atomic spin-glass. In atomic spin-glass tewff≈tw, while
and 10 K is shown in inset of Fig. 6 (b). A similar in our case, tewff>tw and show a weak tw dependence.
behavior in M(0) is also observed in ZFC aging of fer- tewff varies from 4760 s to 6144 s on changing tw from
romagnetic La0.8Ca0.2MnO3 (18 nm) and antiferromag- 600 s to 5000 s. The observed tw dependence of tewff
netic La0.2Ca0.8MnO3 (15 nm) nanoparticles.57,58 The resembles with that reported for La0.7−xYxCa0.3MnO3
initial increase in M(0) with t suggests the presence (0 ≤ x ≤ 0.15) manganites which exhibit a cluster-
w
of an additional dynamical mechanism along with the glass behavior and La0.8Ca0.5MnO3 (18 nm) nanopar-
spin-glass like freezing. In La Ca MnO nanoparti- ticles which exhibit a superspin-glass like behavior.10,57
0.8 0.2 3
cles, the additional dynamic features have been claimed The resultsofZFC memoryexperimentsaredisplayed
to be coming from the development of a superferromag- in Fig. 6 (d). In ZFC memory experiment, the sys-
netic (SFM) like state i.e. development of ferromagnetic tem is cooled in zero field from 300 K to 10 K with
like correlation among the superspins.57 an intermediate stop of 104 s at 130 K, and at 10 K,
For spin-glass, Lundgren et. al. have suggested that 100 Oe field is applied and the magnetization (MZmFemC)
the quantity S(t)=(1/H)(dM(t)/lnt) is proportional to is recorded in the heating run. The reference ZFC mag-
thespectraldensityofrelaxationtimes(g(τ)),andthere- netization (Mref ) is also recorded under same proto-
ZFC
fore,S(t)versustplotsgiveanestimateofg (τ =t)for col but without an intermediate stop. The difference in
tw
givent . Ast increases,g (τ)shiftstowardslongerre- magnetization i.e. ∆M=Mmem-Mref versus temper-
w w tw ZFC ZFC
6
ature exhibits a broad dip around 140 K which signals
tstloihotfkocaeZpotFlctdoiChsuoerpomisnenyrelgysamttteoiohvmbreeysercfrieronvemoeelFedzimniiniggnbg.,ers6ruyasnsn(tid.dte)smTtcahshogeeunirnnfiemgfdromeearmrtesg,otothtrihhnyeeegiinenaoxtbesiZssrpeFtmiernCvne-adcgpteilaiarotoosnes-f 5547 T=5M (emu/mole)0 K, tH1=100 Oe T=554047 K, tH2=M (emu/mole)100 OeT=50 K, Ht=3100 Oe 4584 T=5t01M (emu/mole) K, H=100 Oe T55=4470 K,t H2M (emu/mole)=0 Oe tT(s=)50 K, H=t3100 Oe
a spin-glasslike state in La0.75Ba0.25CoO3.4,26,61,62 Here (a) t (s0) 400t 0(s)8000 42 (b) t (0s) 5000 10000
we note that in contrast to atomic spin-glass, the ob- 0 6000 12000 0 6000 12000
served dip in ZFC memory is broad and does not ex- T=50 K, H=0 Oe T=40 K, H=0 Oe T=50 K, H=0 Oe T=50 K, H=0 Oe T=40 K, H=100 Oe T=50 K, H=0 Oe
hetehrnibaetliilttmyieaiscorbc(oossseumcrpovpepelreidsctpeiflinnirpsesp)juuipinvsegermnstapiumticnihoe-ngol.lofanstSghsueeorcrhfltuhcaclautnbusteatehthriaa-nvgtgiloaomrsfsaaisgwtnoghemeetnriicce- 22118942 M (emu/mole) t1 222111788814t20M (emu/mole) 5t 0(s0)0 t130000 22129005 M (emu/mole)(td1) 222111888246t20M (emu/mole) 50t 0(s0) t130000
spin-glass, and therefore, the observation time in units (c)
t (s) t (s)
of microscopic flipping time is relatively shorter in com- 0 7000 14000 0 7000 14000
parison of atomic spin-glass.57,58,63,64 Because of this,
the length scales probed during the experimental time Figure7: (Coloronline)Magnetizationrelaxationwithanin-
scale are shorter and the condition for rejuvenation, i.e. termediatenegativetemperaturecycling (a) in ZFCprotocol
the length scales probed during the experimental time without field change, (b) in ZFC protocol with field change,
scale are larger than the so called overlap length, is not (c)inFCprotocolwithoutfieldchange,and(d)inFCproto-
satisfied.63 TheabsenceofcompleterejuvenationinZFC col with field change. Insetsshow therelaxation att1 and t3
memory may also occur if the probing field is strong only (after adjusting the time scale for t2) for the respective
measurement protocols.
enough to perturb the intrinsic non-equilibrium dynam-
ics of the spin-glass system.
the same functional form as t but starts with a lower
1
initial value. A concurrent field cycling along with tem-
E. Temperature-field cycling and relaxation peraturecycling(0Oe,50K,t s,-100Oe,40K,t s,-0Oe,
1 2
50K, t s) nearly removes this shift, and in this case, re-
3
The relaxation experiments discussed in previous sec- laxation during t3 is a continuation of relaxation dur-
tion are complemented with an intermediate negative ing t1 similar to that observed in superspin or cluster
thermal cycling with and without field change. Fig- -glass. The results ofnegative temperaturecycling with-
ure. 7 (a) shows the result of temperature cycling with- out the field change in ZFC and FC protocols support
out field changeunder ZFC protocol. Here the sample is the finding of ZFC aging experiments (discussed in sub-
cooled to 50 K in zero field, then 100 Oe field is applied section IIID) that there exists an additional relaxation
and the magnetization is recorded as a function of time mechanismapartfromthe usualcluster-glassrelaxation.
for t s. Then keeping the field constant, the temper- This additional relaxation mechanism seems to enhance
1
ature is changed to 40 K and magnetization is recorded the effect of cluster-glass relaxation, i.e. it contributes
fort s. Thereaftertemperatureisagainbroughtto50K positivelytomagnetizationwhenspin-clustersaligndur-
2
(without changing the field) and magnetization is taken ing the relaxation process (ZFC aging) but contributes
for t s. According to Sun et al., for superspin or cluster negatively to magnetization when spin-clusters random-
3
-glass, the relaxation during t should be the continua- ize during the relaxation (FC aging). The field cycling
3
tionof the relaxationat t .26,61,65 The inset ofFig.7 (a) probably blocks the continuous contribution of this ad-
1
shows the relaxation at t and t . The relaxation dur- ditional relaxation mechanism to magnetization during
1 3
ing t has the same functional form as during t but is negative temperature cycling, and thus, leave us with a
3 1
shifted upward. Figure.7 (b) exhibits the relaxationun- normal cluster-glass like relaxation behavior.
der similar protocolbut with a zero field between t and
1
t . As clear from the inset of Fig. 7 (b), the relaxation
3
att is the continuationoft withoutany apparentshift IV. FURTHER DISCUSSIONS
3 1
in magnetization as expected from a typical superspin
or cluster -glass. This indicates that a concurrent field Theexperimentalresultsdiscussedintheprevioussec-
cycling destroys the mechanism responsible for the ob- tions clearly establish the existence of a spin-glass like
served additional upward shift in magnetization during phase transitionalongwith anadditionalconcurrentdy-
the negative thermal cycling. The results of similar set namics in the so called cluster-ferromagnetic state of
of negative thermal cycling experiments performed with La Ba CoO . Thespin-glasslikebehaviormayarise
0.75 0.25 3
and without field cycling under FC protocol are shown duetocooperativefreezingofferromagneticclustersorit
in Fig. 7 (c) and (d) respectively. The insets show the can be due to coexistence of a spin-glass phase with the
relaxation during t and t . When field remains zero ferromagnetic-clusters. The absence of exchange bias ef-
1 3
during the temperature cycling, the relaxation at t has fectsuggeststhe lackofferromagnetspin-glassinterface.
3
7
superferromagnets.3,70,71 These theoretical predictions
175K
150K have been substantiated by relaxation measurements on
125K
granular multilayers, magnetic clusters, and nanoparti-
1E-6 cle assemblies.9,70–76 Experimentally n ≈ 1 (n ≯ 1) is
W(t) observedforclusterorsuperspin-glasswhilen>1forsu-
perferromagnets. Mao et al. have performed mean field
calculations including both the dipolar and exchange in-
teractionsamongthenanoparticlesandtheirresultsshow
1E-7 that n ≈ 1 (with superspin-glass behavior) for small ex-
1000 t(s) 10000
change interactions which smoothly transforms to n>1
Figure 8: (Color online) Relaxation rate versus time on log (withsuperferromagneticbehavior)ongraduallyenhanc-
scale at 175 K,150 K,and 125 K.Thestraight lineshow the ing the strength of exchange interaction keeping dipolar
fittingof equation 8 to respective data. interaction intact.8
Figure 8 shows the W(t) versus time on the log-log
scale at 175 K, 150 K, and 125 K. Here the system is
This along with the values of critical exponents zv and
cooledfrom 300 K to respective relaxationtemperatures
β, the wait time dependence of S(t), and the lack of
under a field of 100 Oe, and after temperature stabiliza-
complete rejuvenation in ZFC memory indicate that the
tion, the field is switched off and the thermoremanent
spin-glass like behavior is coming from the cooperative
magnetization is recorded as a function of time. The fit-
freezingofferromagnetic-clustersinsteadofatomicspins.
ting of equation 8 to the W(t) versus time data gives
Theexistenceofspin-glasslikedynamicsinassemblyof
n=0.99(7) at 175 K, n=0.89(4) at 150 K and n=0.93(3)
ferromagneticclusters require(a)dense packingofferro-
at 125 K. The n values are close to one (from lower
magnetic clusters with random orientationof anisotropy
side) supporting the cluster-glass dynamics. We have
axes,and(b)stronginter-clusterdipolarinteractions. In
alsofittedthe time dependence ofthermoremanentmag-
La Ba CoO , the ferromagnetic clusters with ran-
0.75 0.25 3
netization with stretched exponential, power law, and
domly oriented anisotropyaxes percolates,13–15,17,18 and
power law with finite remanence and only the power
so, there is also a reasonable possibility of exchange in-
law function fits with reasonable parameters and error
teractionbetweentheneighboringferromagneticclusters.
values. The power law with finite remanence fit gives
Theexchangeinteractionbetweentheferromagneticclus-
huge error in fitting parameters. This along with n<1
ters can be because of exchange bridges between the
rulesoutthe possibilityofsuperferromagnetlikedynam-
surface atoms of neighbouring ferromagnetic clusters or
icsandsuggeststhatinter-clusterexchangeinteractionin
can be due to tunneling exchange coupling between the
neighbouringmetallicferromagneticclusters.5,6,66,67 The La0.75Ba0.25CoO3,ifpresent,arerelativelyweakincom-
parisonto the dipolar interaction. We alsonote that the
competingdipolarandinter-clusterexchangeinteraction
value of exponent n remains nearly constant on chang-
in a disordered random anisotropy system can lead to a
superferromagnetic state.2,7,12,66,68 The superferromag- ing the relaxation temperature from 125 K (0.6255Tg)
to175K(0.875T )whichindicatesthattheinter-cluster
neticstateexhibitsdynamicfeaturesasinthecluster(or g
interaction remains nearly constant on approaching T .
superspin)-glass, but in contrast to cluster-glass which g
This behaviour is quite different from other phase sep-
has a zero thermoremanence magnetization in the limit
of t→∞, superferromagnet has a finite remanence.69–71 arated mangnaites and cobaltites where n increases on
approaching T due to increase in inter-cluster dipolar
The decay of relaxation rate (W(t)) of thermorema- g
interactions which has been attributed to enhancement
nentmagnetization(m(t))canbeusedtodistinguishthe
in cluster density.9,73
cluster(orsuperspin)-glassdynamicsfromthatofsuper-
The presence of additional dynamic features in the
ferromagnet. According to Monte Carlo simulations of
cluster-glass state can be understood by taking into ac-
Ulrich et al., the W(t) of an assembly of nano-particles
countthecompetitionbetweenthedipolarandexchange
with dipole interaction decays as a universal power law
interactions. The energy E of a cluster i in an ensemble
after some crossovertime t 3 i
0
of ferromagnetic clusters can be written as
d
W(t)= lnm(t)=At−n, (8) (µ .µ )−3(µ .rˆ )(µ .rˆ )
dt Ei =−KVi(µˆi.nˆi)2+X i j r3i ij j ij
where exponent n depends on the packing density of j ij
nano-particles and A is a temperature dependent con- − X Jijµi.µj −µi.H, (9)
stant. Depending on the value of exponent n, the m(t)
decaysasstretchedexponentialm(t)=m exp(−(t/τ)1−n) j(nn)
0
(for n<1), power law m(t)=m t−A (for n=1), or whereK istheanisotropyconstant,nˆ isthe unitvector
1 i
power law with finite remanence m(t)=m∞ + m1t1−n along the easy axis, Vi is the volume, µi is the moment
(for n>1). The decay with exponent n<1 is associ- of ith cluster, r is the distance between the ith and
ij
ated with dilute systems, n=1 is associated with clus- jth cluster and H is the appliedfield. J representsthe
ij
ter or superspin -glass, and n>1 is associated with exchange coupling between the nearest neighbour i and
8
j. The anisotropy energy favours the alignment of clus- clusters in the above process (i) and (ii) adds up.
ter moment along the easy axis nˆ which may vary ran- This gives an additional upward shift (downward
i
domly from cluster to cluster. The first term of dipolar shift) in magnetization for ZFC protocol (for FC
energy favours antiferromagnetic coupling while the sec- protocol)when the temperatureis broughtbackto
ond term attempt to align these clusters randomly. The T. Switching off (on) the field during relaxation in
anisotropyenergyandthedipolarinteractioninadensely ZFC protocol (FC protocol) at T-∆T weakens (ii)
packed cluster system give a spin-glass like cooperative because now Zeemanenergy does not support (op-
dynamics.3,8 The exchangeenergyfavoursthe alignment pose) the alignment effort of exchange energy. Ad-
of nearest neighbours while Zeeman energy favours the ditionally,thecontributionfromalignmentofsmall
alignment of all the cluster moments along the field di- clusterin(i)willbeoppositeof(ii). Becauseofthe
rection. The anisotropy energy gives two equal energy oppositesignincontributionfrom(i)and(ii),they
minima, one along and the other opposite to nˆ . The tendstocanceleachotherandthereforethereisno
i
dipolarenergy,the exchangeenergy,andtheZeemanen- significantshiftinmagnetizationwhenthetemper-
ergymaylowertheenergyofoneoftheseminimaandthe ature is raised back to T.
clustermomentwilltransformtothelowenergystateby
thermal activation. For weak exchange interaction and
small field, the dominant anisotropy and dipolar energy
V. CONCLUSIONS
termscauseacluster-glasslikedynamics.8 The ensemble
offerromagneticclustersinoursystemwillhaveacluster
We have performed a detailed investigation of
sizedistribution. Thesmallclustershavealargersurface
La Ba CoO which lies just above the critical dop-
to volume ratio, and therefore, the exchange interaction 0.75 0.25 3
ing for percolationof ferromagneticclusters. Our results
may be more significant for small clusters embedded be-
show an irreversibility in FC-ZFC magnetization and a
tween the larger clusters. The dynamics of these small
frequency dependent peak in ac susceptibility which co-
clusters will strongly depend on the alignment of their
neighboring moments. The additional dynamical behav- incides with the ferromagnetic ordering (TC ≈ 203 K)
of the clusters. The contribution from magnetic order-
ior observed in the aging and negative temperature cy-
ingwithintheclustersmaskthefrequencydependenceof
clingexperimentscanbepossiblyunderstoodonthebasis
ac susceptibility above2 Hz. The FC-ZFCirreversibility
of the above picture as follows:
and frequency dependence in the peak of ac susceptibil-
1. AfterazerofieldquenchfromaboveT ,duringzero
g ityindicateabouttheexistenceofanon-equilibriumstate
field aging, the small embedded clusters will try to
below T . The dynamicscaling ofthe imaginarypartof
C
align according to their big neighbors because of
ac susceptibility, the static scaling of the nonlinear sus-
significant exchange interaction while the big ones
ceptibility, and the ZFC aging and memory experiments
will relax according to the spin-glass model. The
giveconclusiveevidenceofthespin-glasslikecooperative
initial enhancement in magnetization is possibly
freezingofferromagneticclusters(T =200.9(2)K)inthe
g
a result of thermally activated alignment of small
non-equilibrium state.
clusters which is slowly overcome by the dominant
The results of ZFC aging experiments in the non-
spin-glass like relaxationof larger clusters.
equilibriumstateindicateabouttheexistenceofanaddi-
2. The applied field lowers the energy of easy axis di- tionaldynamicalmechanismapartfromthetypicalspin-
rectionpointingalongthefield. Ifthisnowbecomes glass dynamics. The presence of this additional dynami-
the low energy state, the activated alignment of calmechanismis further substantiatedby relaxationun-
cluster moments along the field direction will en- dernegativetemperaturecycling. Ouranalysisshowthat
hance the magnetization with time. Switching off this additional dynamical mechanism possibly have its
thefieldfavorstherestoringofspin-glasslikeorder- origin in the inter-cluster exchange interaction and clus-
ingduetoprevalentdipoleinteraction. Thiscauses ter size distribution. The inter-cluster exchange inter-
magnetization to decay with time. Because of the actions can create a superferromagnetic state in the en-
hierarchical nature of dynamics in spin-glass, the sembleofdenselypackedferromagneticclusters. Thede-
setofenergybarriersrelaxedattemperature Tare cay of relaxation rate of thermoremanent magnetization
differentfromthatofT-∆T.Whenthetemperature and the decay of thermoremanent magnetization sug-
isloweredtoT-∆TafterarelaxationatT,thecon- gest the absence of typical superferromagnetic state in
tributionto relaxationatT-∆Tcomesfrom(i)the La0.75Ba0.25CoO3.
spin-glass like relaxation of active clusters (at T-
∆T)andtheexchangeinducedactivatedalignment
of their small neighbors and (ii) exchange induced A. Acknowledgement
activated alignment of small clusters by the frozen
neighbours which were active at T.
We are thankful to A. Banerjee for support and dis-
3. If the field is not changed during thermal cycling cussions. We also acknowledge M. Gupta for XRD mea-
thenexchangeinducedactivatedalignmentofsmall surements.
9
∗
Electronic address: [email protected], [email protected] 016003 (2010).
1 X.BatlleandA.Labarta,J.Phys.D:Appl.Phys.35,R15 27 I. G. Deac, J. F. Mitchell, and P. Schiffer, Phys. Rev. B
(2002). 63, 172408 (2001).
2 S.BedantaandW.Kleemann,J.Phys.D:Appl.Phys.42, 28 X. H. Huang, J. F. Ding, Z. L. Jiang, Y. W. Yin, Q. X.
013001 (2009). Yu,and X. G. Lia, J. Appl.Phys. 106, 083904 (2009).
3 M.Ulrich,J.García-Otero, J.Rivas,andA.Bunde,Phys. 29 D. Parker, V. Dupuis, F. Ladieu, J.-P. Bouchaud, E.
Rev.B 67, 024416 (2003). Dubois, R. Perzynski, and E. Vincent, Phys. Rev. B 77,
4 M. Sasaki, P. E. Jönsson, H. Takayama, and H. Mamiya, 104428 (2008).
Phys. Rev.B 71, 104405 (2005). 30 K. Hiroi, K. Komatsu, and T. Sato, Phys. Rev. B 83,
5 C. Frandsen, C. W. Ostenfeld, M. Xu, C. S. Jacobsen, 224423 (2011).
L. Keller, K. Lefmann, and S. Mørup, Phys. Rev. B 70, 31 M.Knobel,W.C.Nunes,L.M.Socolovsky,E.DeBiasi,J.
134416 (2004). M. Vargas, and J. C. Denardin,J. Nanosci. Nanotechnol.
6 C. Frandsen and S. Mørup, Phys. Rev. Lett. 94, 027202 8, 2836 (2008).
(2005). 32 D.Kumar,K.P.Rajeev,J.A.Alonso,andM.J.Martínez-
7 S. Bedanta, T. Eimüller, W. Kleemann, J. Rhensius, F. Lope, Phys. Rev.B 88, 014410 (2013).
Stromberg, E. Amaladass, S. Cardoso, and P. P. Freitas, 33 P. S. Anil Kumar, P. A. Joy, and S. K. Date, J. Phys. :
Phys. Rev.Lett. 98, 176601 (2007). Condens. Matter 10, L487 (1998).
8 Z. Mao and Xi Chen, J. Phys.: Condens. Matter 23, 34 P.S.AnilKumar,P. A.Joy, andS.K.Date, Bull. Mater.
226005 (2011). Sci. 23, 97 (2000).
9 F. Rivadulla, M. A. Lop´ez-Quintela, and J. Rivas, Phys. 35 P. A. Joy, P. S. Anil Kumar, and S. K. Date, J. Phys. :
Rev.Lett. 93, 167206 (2004). Condens. Matter 10, 11049 (1998).
10 R. S. Freitas, L. Ghivelder, F. Damay, F. Dias, and L. F. 36 T. Song, R. M. Roshko, and E. D. Dahlberg, J. Phys. :
Cohen, Phys. Rev.B 64, 144404 (2001). Condens. Matter 13, 3443 (2001).
11 D.N.H.Nam,K.Jonason,P.Nordblad,N.V.Khiem,and 37 R.M. Roshkoand L. Xi, J. Appl.Phys. 93, 6653 (2003).
N. X.Phuc, Phys.Rev.B 59, 4189 (1999). 38 W.LuoandF.Wang,Appl.Phys.Lett.93,176102(2008),
12 J. Alonso, M. L. Fdez-Gubieda, J. M. Barandiarán, A. J. Gesheva, Appl. Phys. Lett. 93, 176101 (2008), W. Luo
Svalov,L.FernándezBarquín,D.AlbaVenero,andI.Orue, and F. Wang, Appl.Phys. Lett. 90, 162515 (2007).
Phys. Rev.B 82, 054406 (2010). 39 G. F. Goya and V. Sagredo, Phys. Rev. B 64, 235208
13 D.Phelan,D.Louca,S.Rosenkranz,S.-H.Lee,Y.Qiu,P. (2001).
J. Chupas, R. Osborn, H. Zheng, J. F. Mitchell, J. R. D. 40 D.Y.Cong,S.Roth,J.Liu,Q.Luo,M.Pötschke,C.Hür-
Copley, J. L. Sarrao, and Y. Moritomo, Phys. Rev. Lett. rich, and L. Schultz,Appl.Phys. Lett. 96, 112504 (2010).
96, 027201 (2006). 41 M. Thakur, M. Patra, S.Majumdar, and S. Giri, J. Appl.
14 P.Tong,J.Yu,Q.Huang,K.Yamada,andD.Louca,Phys. Phys.105, 073905 (2009).
Rev.Lett. 106, 156407 (2011). 42 J. A. De Toro, M. A. López de la Torre, M. A. Arranz, J.
15 R.X.Smith,M.J.R.Hoch,W.G.Moulton,P.L.Kuhns, M.Riveiro,J.L.Martínez,P.Palade,andG.Filoti,Phys.
A.P.Reyes,G.S.Boebinger,H.Zheng,andJ.F.Mitchell, Rev.B 64, 094438 (2001).
Phys. Rev.B 86, 054428 (2012). 43 J. L. Dormann, L. Bessais, and D. Fiorani, J. Phys. C:
16 D.KumarandA.Banerjee,J.Phys.: Condens.Matter25, Solid StatePhys. 21, 2015 (1988).
216005 (2013). 44 J. Souletie and J. L. Tholence, Phys. Rev. B 32, 516
17 P. Mandal, P. Choudhury, S. K. Biswas, and B. Ghosh, (1985).
Phys. Rev.B 70, 104407 (2004). 45 K.H.FischerandJ.Hertz,SpinGlasses (CambridgeUni-
18 M. Kriener, C. Zobel, A. Reichl, J. Baier, M. Cwik, K. versity Press, Cambridge, 1991).
Berggold, H. Kierspel, O. Zabara, A. Freimuth, and T. 46 E. Svanidze and E. Morosan, Phys. Rev. B 88, 064412
Lorenz, Phys. Rev.B 69, 094417 (2004). (2013).
19 J. Wuand C. Leighton, Phys. Rev.B 67, 174408 (2003). 47 K. Vijayanandhini, Ch. Simon, V. Pralong, V. Caignaert,
20 D.SamalandP.S.AnilKumar,J.Phys.: Condens.Matter and B. Raveau,Phys.Rev.B 79, 224407 (2009).
23, 016001 (2011). 48 V.K. Anand,D.T. Adroja, and A.D.Hillier, Phys. Rev.
21 A. P. Sazonov, I. O. Troyanchuk, H. Gamari-Seale, V. V. B 85, 014418 (2012).
Sikolenko,K.L.Stefanopoulos,G.K.Nicolaides,andY.K. 49 S. Geschwind, D. A. Huse, and G. E. Devlin, Phys. Rev.
Atanassova,J.Phys.: Condens.Matter21,156004(2009). B 41, 4854 (1990).
22 J. Rodríguez-Carvajal, Phys. B (Amsterdam) 192, 55 50 M. Suzuki,Prog. Theor. Phys.58, 1151 (1977).
(1993); Program FULLPROF, LLB-JRC, Laboratoire 51 K. Wada and H. Takayama, Prog. Theor. Phys. 64, 327
Léon Brillouin, CEA-Saclay, France, 1996. (1980).
23 I. Fita, R. Szymczak, R. Puzniak, A.Wisniewski, I. O. 52 S. Fujiki and S. Katsura, Prog. Theor. Phys. 65, 1130
Troyanchuk,D.V.Karpinsky,V.Markovich,andH.Szym- (1981).
czak, Phys. Rev.B 83, 064414 (2011). 53 T. Jonsson, P. Svedlindh,M. F. Hansen, Phys. Rev. Lett.
24 S. D. Tiwari and K. P. Rajeev, Phys. Rev. B 72, 104433 81, 3976 (1998).
(2005). 54 S. Geschwind, D. A. Huse, and G. E. Devlin, Phys. Rev.
25 J. A. Mydosh, Spin Glasses: An Experperimental Intro- B 41, 2650 (1990).
duction (Taylor and Francis London 1993). 55 R. Mathieu, J. A. De Toro, D. Salazar, S. S. Lee, J. L.
26 V.BishtandK.P.Rajeev,J.Phys.: Condens.Matter22, Cheong, P.Nordblad, EPL 102, 67002 (2013).
10
56 L.Lundgren,P.Svedlindh,P.Nordblad,andO.Beckman, 67 V. N. Kondratyev and H. O. Lutz, Phys. Rev. Lett. 81,
Phys. Rev.Lett. 51, 911 (1983). 4508 (1998).
57 V. Markovich, I. Fita, A. Wisniewski, G. Jung, D. 68 D.KechrakosandK.N.Trohidou,J.Magn.Magn.Mater.
Mogilyansky, R. Puzniak, L. Titelman, and G. Gorodet- 262, 107 (2003).
sky,Phys. Rev.B 81, 134440 (2010). 69 R.V.Chamberlin,J.Hemberger,A.Loidl,K.D.Humfeld,
58 V. Markovich, R. Puzniak, Y. Skourski, A. Wisniewski, D.Farrell,S.Yamamuro,Y.Ijiri,andS.A.Majetich,Phys.
D. Mogilyanski, G. Jung, and G. Gorodetsky, J. Phys.: Rev.B 66, 172403 (2002).
Condens. Matter 24, 266001 (2012). 70 Xi Chen, W. Kleemann, O. Petracic, O. Sichelschmidt,
59 L. Lundgren, P. Svedlindh, and O. Beckman, Phys. Rev. S. Cardoso, and P. P. Freitas, Phys. Rev. B 68, 054433
B 26, 3990 (1982). (2003).
60 Eric Vincent, Lecture Notes in Physics Vol.716, 7 (2007), 71 Xi Chen, S. Sahoo, W. Kleemann, S. Cardoso, and P. P.
arXiv:cond-mat/0603583. Freitas, Phys.Rev.B 70, 172411 (2004).
61 M. Bandyopadhyay and S. Dattagupta, Phys. Rev. B 74, 72 M. Patra, S. Majumdar and S. Giri, J. Phys.: Condens.
214410 (2006). Matter 22, 116001 (2010).
62 R.Mathieu,P.E.Jönsson,P.Nordblad,H.A.Katori,and 73 Y. K. Tang, Y. Sun, and Z. H. Cheng, Phys. Rev. B 73,
A. Ito, Phys.Rev.B 65, 012411 (2001). 012409 (2006).
63 P.E.Jönsson, H.Yoshino,H.Mamiya,and H.Takayama, 74 D. De, A. Karmakar, M. K. Bhunia, A. Bhaumik, S. Ma-
Phys. Rev.B 71, 104404 (2005). jumdar, and S. Giri, J. Appl.Phys. 111, 033919 (2012).
64 C. Raj Sankar, S. Vijayanand, S. Verma, and P.A. Joy, 75 X. Wang, Y. Sui, Y. Li, M. Xu, W. Wang, P. Liu, and J.
Solid State Commun. 141, 307 (2007). Tang, J. Appl.Phys.109, 07C322 (2011).
65 Y. Sun, M. B. Salamon, K. Garnier, and R. S. Averback, 76 S. Karmakar, B. K. Chaudhuri, C. L. Chan, and H. D.
Phys. Rev.Lett. 91, 167206 (2003). Yang, J. Appl.Phys. 108, 113916 (2010).
66 M. F. Hansen, C. B. Koch, and S. Mørup, Phys. Rev. B
62, 1124 (2000).