arX

iv:1

211.

4936

v1 [

cond

-mat

.str

-el]

21

Nov

201

2

A critical examination of magnetic states of La0.5Ba0.5CoO3: non-Griffiths phase and

interacting ferromagnetic-clusters

Devendra Kumar∗ and A. BanerjeeUGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore-452001,India

We report detailed dc magnetization, linear and non-linear ac susceptibility measurements onthe hole doped disordered cobaltite La0.5Ba0.5CoO3. Our results show that the magnetically or-dered state of the system consists of coexisting non-ferromagnetic phases along with percolatingferromagnetic-clusters. The percolating ferromagnetic-clusters possibly undergo a 3D Hisenberglike magnetic ordering at the Curie temperature of 202(3) K. In between 202 and 220 K, the lin-ear and non-linear ac susceptibility measurements show the presence of magnetic correlations evenwhen the spontaneous magnetization is zero which indicates the presence of preformed short rangemagnetic-clusters. The characteristics of these short range magnetic-clusters that exist above Curietemperature are quite distinct than that of Griffiths phase e.g the inverse dc susceptibility exhibitsan field independent upward deviation, and the second harmonic of ac susceptibility is non-negative.Below Curie temperature the system exhibit spin-glass like features such as irreversibility in the fieldcooled and zero field cooled magnetization and frequency dependence in the peak of ac susceptibility.The presence of a spin or cluster -glass like state is ruled out by the absence of field divergence inthird harmonic of ac susceptibility and zero field cooled memory. This indicates that the observedspin-glass like features are possibility due to progressive thermal blocking of ferromagnetic-clusterswhich is further confirmed by the Wohlfarth’s model of superparamagnetism. The frequency depen-dence of the peak of ac susceptibility obeys the Vogel-Fulcher law with τ0 ≈ 10−9s. This togetherwith the existence of an AT line in H-T space indicates the existence of significant inter-clusterinteraction among these ferromagnetic-clusters.

PACS numbers: 75.47.Lx, 75.30.Kz , 75.30.Cr

I. INTRODUCTION

The transition metal oxides e.g. manganites, cuprates,and cobaltites exhibit complex phase diagram includ-ing the microscopically inhomogeneous electronic statesdue to interplay of various competitive electronic en-ergies such as electron kinetic energy, electron-electroncoulomb repulsion, spin-spin, spin-orbit, and crystal fieldinteractions.1,2 Of these oxides, the cobaltite LaCoO3 ex-hibits a unique property of temperature and doping de-pendent spin state transition.3,4 The Co3+ ion in LaCoO3

can exist in low spin (LS) state with configuration t62ge0g

(S=0), intermediate spin (IS) state with configurationt52ge

1g (S=1), and high spin (HS) state with configuration

t42ge2g (S=2). The LaCoO3 have a charge transfer insu-

lator type non-magnetic ground state with Co3+ ion inthe LS state; it starts showing magnetic moment above30 K and exhibits a paramagnetic like behavior above100 K.5,6 This change in magnetic moment and behav-ior is attributed to thermally driven spin state transi-tion of Co3+ ion, but the nature of transition whetherit is a LS-IS transition or LS-HS transition is still notcompletely settled.4–11 The hole doping of LaCoO3 byreplacing the trivalent La3+ with divalent Sr2+ or Ba2+

generates Co4+, and each of these Co4+ transforms theirsix nearest Co3+ neighbors into the IS state by form-ing octahedrally shaped spin-state polarons.12–14 In thesepolarons, eg electrons of Co3+ are delocalized and areshared by Co3+ and Co4+ ions of the polaron, whilet2g electrons of both the ions are localized and coupleferromagnetically via double exchange interaction. For

small hole doping these isolated spin state polarons arestable within the nonferromagnetic matrix. Additionalhole doping enhances the number density of spin statepolarons, and above a critical doping of x=0.04, the en-hanced polaron density causes a decay of polaronic statedue to ferromagnetic (FM) interaction between the intra-polaronic Co3+ ions at the cost of the antiferromagnetic(AFM) intra-polaronic interaction.15 This, in turn, re-sults in the formation of hole rich ferromagnetic spinclsuters embedded in non-ferromagnetic insulating ma-trix. On further enhancing the hole doping, at a criticalconcentration (x=0.18 for Sr2+ and x=0.2 for Ba2+), theferromagnetic metallic clusters eventually percolates giv-ing rise to long range ferromagnetic ordering and metallicconductivity.16

For La1−xSrxCoO3, a spin-glass like state is ob-served below the critical doping concentration forpercolation of ferromagnetic metallic regions andabove this a ferromagnetic or a ferromagnetic-clusterstate is reported.16–18,20,48 But a recent report onLa0.5Sr0.5CoO3 show the presence of glassy dynamicseven in the so called ferromagnetic or ferromagnetic-cluster state which has been attributed to coexistenceof spin or cluster -glass like phase along with percolatingferromagnetic-clusters.21 The absence of exchange biaseffect in La0.5Sr0.5CoO3 clearly indicates that the spinor cluster -glass like phase is not present at the interfaceof ferromagnetic-clusters and non-ferromagnetic matrix,but instead, it probably coexist as small patches alongwith the percolating backbone of ferromagnetic-clusters.The nature of magnetic state in La1−xBaxCoO3 with

2

Ba2+ having a larger ionic radii than Sr2+ (ionic radiiof La3+=1.216Å, Ba2+=1.47Å, Sr2+=1.31Å) is relativ-ity less studied, and early reports indicate the presence offerromagnetic-metallic ground state for x > 0.2.16,17 Thehigher ionic radii of Ba2+ (a) enhances the local random-ness due to larger size mismatch between the Ba2+ andLa3+ ions, and (b) reduces the overall distortion fromideal pervoskite structure and so the tolerance factor tapproaches to 1. This enhancement in tolerance factorstraightens the Co-O-Co bonds which in turn increasesthe ferromagnetic coupling due to double exchange inter-action between Co3+ and Co4+ ions. Furthermore, theBa2+ doping enhances the concentration of Jahn-Teller(J-T) active IS state because of lattice expansion and theformation of J-T magnetopolaron is found to be mostpreferable for Ba doped cobaltites.13,22

In this paper we present a detailed study ofLa0.5Ba0.5CoO3 with an aim to understand its differentmagnetic states. We have compared our results with thatof La0.5Sr0.5CoO3 to bring out possible effect of differ-ences in ionic radii of Ba2+ and La2+. Our results showthat the magnetically ordered state of La0.5Ba0.5CoO3

consist of non-ferromagnetic phases coexisting along withpercolating backbone of ferromagnetic-clusters. Theseferromagnetic-clusters have critical exponent γ=1.29(1)indicating the possibility of 3D Hisenberg like spin order-ing at the Curie temperature (TC) of 202(3) K. Above TC ,short range magnetic-clusters with characteristics quitedifferent from Griffiths phase exists up to around 220 K.Below TC the magnetic state of the system exhibits spin-glass like behavior, but in contrast to La0.5Sr0.5CoO3,this behavior does not originate from coexistence ofspin or cluster -glass like phases along with percolatingferromagnetic-clusters. Furthermore, our analysis showthat the spin-glass like dynamics in La0.5Ba0.5CoO3 isdue to superparamagnetic like thermal blocking of thedynamics of interacting ferromagnetic-clusters.

II. EXPERIMENTAL DETAILS

Polycrystalline La0.5Ba0.5CoO3 samples are preparedby pyrophoric method24 using high purity (99.99%)La2O3, BaCoO3, and Co(NO3)26H2O. The stichometricratio of La2O3, BaCoO3, and Co(NO3)26H2O are sepa-rately dissolved in dilute nitric acid and then these solu-tionis are mixed with the triethanolamine (TEA) keep-ing the pH highly acidic. The final solution is dried at100 ◦C, which burns and yields a black powder that is pal-letized and subsequently annealed at 1100 ◦C for 12 hour.These samples are characterized by X-Ray diffraction ona Bruker D8 Advance X-ray diffractometer using Cu-Kα radiation. The dc magnetization measurements areperformed on a 14 T Quantum Design physical prop-erty measurement system-vibrating sample magnetome-ter and the low field ac susceptibility measurements arecarried out on a ac-susceptibility setup which is describedin Reference 25.

20 40 60 80

0

5000

10000

Inte

nsity

(arb

. uni

t)

(deg.)

Experiment Fit Difference La0.5Ba0.5CoO3

Figure 1: Room temperature X-ray diffraction pattern ofLa0.5Ba0.5CoO3. The solid circles show the experimental X-ray diffraction data, the red line on the experimental datashow the Rietveld refinement for simple cubic Pm-3m struc-ture with χ2=1.34, the short vertical lines give the Braggpeak positions, and the bottom blue line gives the differencebetween the experimental and calculated pattern.

The X-ray diffraction data of La0.5Ba0.5CoO3 is col-lected at room temperature and analyzed with Rietveldstructural refinement using FULLPROF software.26 Fig-ure 1 show the XRD data, the Rietveld fit profile, theBragg positions, and the difference in experimental andmodel results. The Rietveld refinement show that thesample is single phase and crystallizes in simple cubicPm-3m structure with lattice constant a=3.8726(2)Å andunit cell volume V =58.078(4)Å3. The unit cell volume indisordered cobaltites depends on the oxygen stoichiome-try, and the comparison of our result with that of Ref-erence 27 suggests that the oxygen non-stoichiometry ismuch less than 0.05. The average crystallite size of thesample is estimated from XRD data using the Scherrerformula which comes around 85 nm.

III. RESULTS AND DISCUSSIONS

A. DC Magnetization

1. Thermomagnetic irreversibility

Figure 2 show the temperature variation of magneti-zation in field cooled (FC) and zero field cooled (ZFC)protocol. In the FC protocol, the sample is cooled to5 K in presence of measuring field and the magnetizationis recorded in heating run keeping the field constant. InZFC protocol the sample is cooled to 5 K in zero field andthen the measuring field is applied and magnetization isrecorded as a function of temperature in the heating run.On lowering the temperature, around 200 K, both theFC and ZFC magnetization curves show a rapid increasein magnetization which is indicative of paramagnetic to

3

0 100 200 3000

2

4

6

8

ZFC 0.05T FC 0.05T ZFC 0.08T FC 0.08T ZFC 0.07T FC 0.07T ZFC 1.00T FC 1.00T

M (1

03 em

u/m

ole)

T (K)

Figure 2: Temperature dependence of magnetization underFC and ZFC protocol at various measuring fields.

ferromagnetic transition. On further lowering the tem-perature, the FC curve keeps evolving while the ZFCcurve bifurcates with that of FC at the temperature Tirr

and exhibits a broad peak at a temperature Tp. On in-creasing the measuring field Tirr and Tp decreases withan enhancement in broadening of ZFC peak. At 1 T, theFC and ZFC curves almost coincide. The bifurcation inFC-ZFC magnetization along with a peak in ZFC mag-netization indicates about the presence of a spin-glass,28

cluster-glass,29,30 super-paramagnetic,31,32 or ferromag-netic state.33 At low fields, Tp < Tirr, and below Tp

the FC magnetization is not constant with temperature.This observation is not in agreement with that of canon-ical spin-glasses and suggests that the system is possiblyin a cluster-glass, super-paramagnetic, or ferromagneticstate. Similar observations has been made on the otherrelatively well studied half doped disordered cobaltiteLa0.5Sr0.5CoO3.18,20,21

2. Non-Griffiths phase

Figure 3 displays the temperature variation of in-verse dc susceptibility (H/M) at 500, 700, 800, and1000 Oe taken under the FC protocol. The H/M fitswell with the Curie-Weiss law at high temperatures(T>250 K) with Curie constant C=1.59(1) emu-K/mole-Oe and Weiss constant θ=210(2) K. The Curie constantgives an effective value of paramagnetic moment µeff =3.566(3) µB/f.u. and the positive value of Weiss constantindicates the dominance of ferromagnetic correlations inthe ordered state. The inverse dc susceptibility exhibits aupward deviation from the Curie-Weiss law at a tempera-ture T ∗(≈ 250 K) which is greater than TC , and this devi-ation is found to be field independent. This upward devi-ation of the inverse dc susceptibility from Curie-Weiss lawin La0.5Ba0.5CoO3 is quite distinct from that of Griffithsphase seen in many randomly doped transition metal ox-ides, where the presence of short range ferromagnetic

180 240 3000

20

40

60 500 Oe 700 Oe 800 Oe 1000 Oe

H/M

(mol

e-O

e/em

u)

T (K)

Figure 3: H/M versus temperature at 500, 700, 800, and1000 Oe. The solid line show the fitting of Curie-Weiss equa-tion.

correlations above TC gives a field dependent downwarddeviation in the inverse dc susceptibility.36–38 A simi-lar non-Griffiths like behavior in the inverse dc suscepti-bility is also observed in La1−xSrxCoO3.34,35 In case ofLa1−xSrxCoO3, the small angle neutron diffraction mea-surements exhibit a sharp onset in the spin correlation atthe temperature T ∗ (where the inverse susceptibility de-viates from Curie Weiss law) which has been interpretedas the emergence of short range ferromagnetic clusters.35

Above TC , the deviation from Curie Weiss law in inversedc susceptibility can be (a) because of composition fluc-tuation and so having regions with Curie temperaturehigher than the bulk of the sample or (b) because ofthe presence of short range ferromagnetic correlationswith zero spontaneous magnetization as is the case forGriffiths phase. The possible mechanism for upward de-viation in inverse dc susceptibility, whether it is due tocomposition fluctuation or due to existence of short rangeferromagnetic correlations, will be probed by the Arrotplot and ac susceptibility measurements in section III A 3and III B 2 respectively.

3. Coexistence of ferromagnetic and non-ferromagnetic

phases

In figure 4 (a) we show the magnetization versus fieldplot at 10 K, 40 K, 180 K, 190 K, 200 K, 210 K, and220 K. At low temperatures, for example at 10 K themagnetization exhibits a saturation like behavior at highmagnetic fields which is typical of a ferromagnet, buta careful observation of the data indicates the presenceof a non-saturating magnetization along with the satu-rating ferromagnetic component. The presence of thisnon-saturating component prohibits the magnetizationfrom saturating even at high magnetic fields. The exis-tence of a non-ferromagnetic component along with theferromagnetic component is in agreement with the clus-

4

ter model of other disordered cobaltite La1−xSrxCoO3

where a number of studies have shown the presenceof non-ferromagnetic Co3+ matrix with antiferromag-netic interactions that coexist along with ferromagnetic-clusters.18,20 The ferromagnetic component can be ex-tracted from the total magnetization by assuming thatthe total magnetization can be written as Mtot=MF +χAFH where MF is the saturation value of ferromag-netic component and χAF is the slope of M vs. H curveat high fields. Using this to fit the magnetization ver-sus field curve above 12 T at 10 K, we estimate thesaturation magnetization of ferromagnetic component as1.855(1) µB/Co. The value of saturation magnetizationof ferromagnetic-clusters is smaller than that expectedfrom the spin only value (Ms = gSµB = 2.5µB) whenboth the Co3+ (S=1) and Co4+ (S=3/2) are in IS state.It is to be noted that the similar results about the differ-ence in experimental and expected saturation magneti-zation has also been reported on La1−xSrxCoO3.18,20 Onthe basis of the band structure calculations, Ravindraet al.39 have shown that the hole doping in these materi-als reduces the ionicity, enhances the Co-O hybridization,and stabilizes the IS state. Due to enhanced Co-O hy-bridization the expected average Co moment is reducedcompared to the prediction of simple ionic model.

In figure 4 (b), the M-H isotherms around 200 K areplotted as M2 versus H/M which is known as Arrotplot.40 In these plots, the intercept of the linear fittingof high field data on the X and Y axis gives inverse sus-ceptibility and spontaneous magnetization respectivelyand the one passing through origin gives the ferromag-netic transition temperature TC . At 201 K, the value ofspontaneous magnetization is 0.058 µB/Co which showsthe presence of ferromagnetic interactions, and the linepassing through origin will correspond to M2 versus H/Mcurve lying in between 201-202 K indicating that the TC

lies in between. Above TC , e.g. at 204 K and 205 K, thespontaneous magnetization (MS) is zero indicating theabsence of long range ferromagnetic ordering.

B. AC Susceptibility

In order to get a better understanding of the magnet-ically ordered state, we have performed ac susceptibilitymeasurements at low fields which probe the dynamics ofthe system at the time scales decided by the measuringfrequency range. The magnetization (M) of a system canbe expressed in terms of the applied field (H) as:

M(H) = M0 + χ1H + χ2H2 + χ3H

3 + ... (1)

where M0 is the spontaneous magnetization, χ1 is thelinear susceptibility and χ2, χ3,.. are the nonlinear sus-ceptibilities which can be identified with the Taylor se-ries expansion of M(H) = M0 +(1/1!)(dM/dH)H=0H +(1/2!)(d2M/dH2)H=0H

2 + .. .

0 10 200

20

40 (b)

M2 (1

03 em

u/m

ole)

2

205 K

205 K 204 K 202 K 201 K 199 K 197 K

H/M (Oe-mole/emu)

197 K

0 5 100

5

10 (a)

220 K210 K200 K190 K180 K

40 K

H (T)

M(1

03 em

u/m

ole)

10 K

Figure 4: (a) Magnetization versus field at 10 K, 40 K, 180 K,190 K, 200 K, 210 K, and 220 K (b) Arrot plot (M2 vs. H/M)of the magnetization isotherms at 197 K, 199 K, 201 K, 202 K,204 K, and 205 K. The solid black lines are straight line fitto M2 vs. H/M curve at high field which are extrapolated toH=0.

1. Nature of magnetically ordered state

Figure 5 show the real part of linear ac susceptibil-ity (χ

′

1) measured in the ac field of 2.21 Oe and fre-quency 1131, 333, 131, 11, and 1 Hz. The χ

′

1 exhibitsa broad peak similar to that of ZFC magnetization andthe peak position (TB) in χ

′

1 decreases on increasing themeasuring frequency (see inset (b) of figure 5) which isa common feature of spin-glass, cluster-glass, and super-paramagnetic systems; and the presence of frequency de-pendence in TB clearly rules out the possibility of nor-mal long range ferromagnetic state. The frequency de-pendence in χ

′

1 is quantified as Φ = ∆TB/(TB∆log10f)and the estimated value of Φ is 0.0023 which is inagreement with typical values seen in canonical spin-glasses, cluster-glasses, superspin-glass, or interactingsuper-paramagnets (0.02-0.005),41–43 and two order ofmagnitude lower than that observed in noninteractingsuper-paramagnets (0.1-0.3).44,57

In absence of the time inversion symmetry breakingfield, M(H)=-M(−H), all the even terms in equation2 i.e. χ2, χ4 are zero. The χ2 is observed in presenceof a superimposed external dc field or an internal fieldwhich originates from magnetically correlated spins. For

5

100 200 300

0

1

2

3

(c)

160 200 2400.00

0.03

0.06

" (em

u/m

ole)

T (K)

1131 Hz 333 Hz 11 Hz

200 2100

6

T (K)

-1(d

-1/d

T)

(b)

163 173

2.4

2.6

1' (em

u/m

ole)

T (K)

1131 Hz 333 Hz 111 Hz 11 Hz 1 Hz

' 1 (em

u/m

ole)

T (K)

(a)

Figure 5: Temperature dependence of the real part of linearac susceptibility at various frequencies at the field of 2.21 Oe.The inset (a) show the χ−1dχ−1/dT versus T at 131 Hz at acfield of 0.57 Oe, the inset (b) show the expanded view of thepeak in ac susceptibility and the inset (c) show the imaginarypart of linear ac susceptibility.

canonical spin-glass the coefficient of even powers of H inequation 2 are zero. The real part of nonlinear suscepti-bility χ2 is plotted in figure 6. χ2 is zero in paramagneticphase, has a small positive peak at 202 K, then a largenegative peak around 167 K, and thereafter it slowly ap-proaches to zero. Both the positive and negative peaksin χ2 diminishes on increasing the ac field. Below TC ,the negative value of χ2 clearly show the presence of fer-romagnetic ordering which also rules out the presenceof canonical spin-glass state, but the possibility of co-existence of spin-glass phase along with ferromagnetic-clusters remains open. Moreover, Kouvel-Fisher (K-F)analysis of the ac susceptibility data is performed to getan accurate measure of TC and an idea about the natureof ferromagnetic transition. According to K-F, the zerofield susceptibility in the vicinity of TC varies as46

Y =

[

d

dT(lnχ−1

0 )

]

−1

=(T − TC)

γ(2)

In the inset (a) of figure 5 we have plotted the Y ver-sus temperature for 131 Hz at the AC field of 0.57 Oein the vicinity of TC . The temperature range of straightline fitting is selected in such a way that fitting of anysubset of the data gives the same parameters (withinthe error bar). The inverse of the slope and the inter-cept on T axis gives γ=1.29(1) and TC=202(3) K re-spectively. The TC value is in agreement with that ob-tained from Arrot plot in section III A 3. The γ valuelies between the 3D Ising model (1.241) and the 3DHisenberg model (1.386) and are in close agreement withthe values reported for La0.67Sr0.33CoO3 (γ=1.310(1)),La0.5Sr0.5CoO3 (γ=1.27(2)) and conventional ferromag-net Ni (γ=1.34(1)).47–49 The La0.67Sr0.33CoO3 is be-lieved to order as 3D Hisenberg model and therefore asimilar ordering is also expected for La0.5Ba0.5CoO3

100 200 300

-0.8

-0.4

0.0

195 210

0.00

0.07

0.14

2' (1

0-3 O

e-1 e

mu/

mol

e)

T (K)

2' (1

0-3 O

e-1 e

mu/

mol

e)

8.47 Oe 4.71 Oe 2.83 Oe 1.89 Oe 0.94 Oe

T (K)

Figure 6: Temperature dependence of the real part of secondharmonic of AC susceptibility at various fields at 131 Hz. Theinset show the expanded view of the small peak at 202 K.

2. Further evidence of non-Griffiths phase

From figure 6 we note that the χ2 starts to have a nonzero value below 220 K, a temperature which is higherthan the TC obtained from Kouvel-Fisher (K-F) analysis.Also, the imaginary part of the first harmonic of ac sus-ceptibility (χ

′′

) starts showing a non zero value below ≈220 K (see inset (c) of figure 5). These two observationsclearly show the presence of magnetic correlations muchabove TC where the spontaneous magnetization is zero(see figure 4 (b)). The absence of spontaneous magneti-zation establish un-ambiguously that the non zero valueof χ2 and χ

′′

are not because of some ferromagnetic re-gions with TC higher than the bulk of the system due tocomposition fluctuation. This suggests that the presenceof magnetic correlation above TC is due to the preforma-tion of short range magnetic-clusters at a temperatureT ∗ much above TC and so the observed upward devia-tion from the Curie Weiss law in inverse dc susceptibilitypossibly owes its origin to the existence of short rangemagnetic-clusters. We note that while in small angle neu-tron diffraction measurements of La1−xSrxCoO3 the fer-romagnetic correlations appears at the T ∗,35 the non-zerovalue of χ2 and χ

′′

1 in ac susceptibility of La0.5Ba0.5CoO3

is detectable only below 220 K which is lower than theT ∗(≈250 K). In the preformed magnetic-cluster state, weget a positive value of χ2 which changes sign on loweringthe temperature below TC (see figure 6). This is quitedifferent from Griffiths phase where a negative value ofχ2 is observed at all T below the Griffiths temperature.38

In La1−xSrxCoO3, the upward deviation of inverse dcsusceptibility in presence of short range ferromagnetic-clusters (TC<T<T ∗) is argued to be possibly dueto antiferromagnetic interactions which may favor an-tiparallel alignment of these clusters resulting in sup-pression of susceptibility.35 The existence of shortrange ferromagnetic-clusters above TC and how theseferromagnetic-clusters give a field independent upwarddeviation in inverse dc susceptibility is not yet prop-

6

100 200 300-3

-2

-1

0

0 3 6 9

1

2

|' 3(

max

)|(10

-3 O

e-2- e

mu/

mol

e)H(Oe)

8.47 Oe 4.71 Oe 2.83 Oe 1.89 Oe 0.76 Oe

3' (10-3

Oe-2

em

u/m

ole)

T (K)

Figure 7: Temperature dependence of the real part of thirdharmonic of ac susceptibility at various fields at 131 Hz. Theinset show the ac field dependence of the peak value of χ3

which is χ′

3(max)).

erly understood. Here we show the existence of non-Griffiths like phase in La0.5Ba0.5CoO3 which in combina-tion with the results of La1−xSrxCoO3 suggest that thenon-Griffiths like phase may also be present in other holedoped disordered cobaltites. Our results gives a deeperinsight about the characteristics of such non-Griffithsphase which would be important in developing a defini-tive understanding of them.

3. Absence of spin or cluster -glass like transition

After discarding the possibility of canonical spin-glassstate on the basis of non zero value of χ2 we needto identify the origin of frequency dependence in TB

from remaining possibilities, which are the cluster-glass,the super-paramagnetism, and the coexistence of non-ferromagnetic glassy (e.g. spin or cluster-glass type)phases along with the ferromagnetic clusters. In the firsttwo cases, relaxing entities are the superspins i.e. themoment of a single magnetic domain (cluster) instead ofatomic spins and the maximum feasible size of the do-main (cluster) is that of a crystallite; while for the lastcase, as reported for La0.5Sr0.5CoO3, the relaxing enti-ties are not well understood and the atomic-spins, spin-clusters are among the various possibilities. It is quitedifficult to distinguish whether the slowing down in spindynamics is due to progressive thermal blocking or due tospin-glass like cooperative freezing of the fluctuating en-tities. To determine the nature of spin dynamics, we havemeasured the third harmonic of ac susceptibility (χ3)which is proportional (and opposite in sign) to spin-glasssusceptibility (χSG). The negative divergence of χ3 at Tg

in the limit of H → 0 gives the direct evidence of spin-glass like critical slowing down of the fluctuating entitiesand hence confirms unambiguously the presence of spinor cluster -glass phase.50,51 The temperature dependenceof the real part of the third harmonic of ac susceptibility

5.00 5.25 5.50 5.75

0.8

1.6

1.5 1.8 2.1

-1.8

-1.2

-0.6

3 (10

-3 O

e-2 e

mu/

mol

e)

T-3 (10-7 K-3)

1 (em

u/m

ole)

T-1 (10-3 K-1)

Figure 8: χ′

1 versus T−1 above the blocking temperature. The

inset show the χ′

3 versus T−3 above the blocking temperature.The straight lines are the fitting of equation 4 and 5 to thedata.

at 131 Hz and at different ac fields is plotted in figure 7.The magnitude of the peak in χ3 (χ

′

3(max))) dependson the ac field and the field dependence of χ

′

3(max)) isplotted in the inset of figure 7. The χ

′

3(max)) does notdiverge as H → 0 which clearly shows that fluctuatingentities does not freeze in a spin or cluster -glass state.We do not observe any ZFC memory effect which furthersupports the absence of spin or cluster -glass like freezingin the system. These results suggest that the observedfrequency dependence in χ1 is possibility due to progres-sive thermal blocking of fluctuating entities.

4. Superparamagnetic behavior of ferromagnetic clusters

The existence of super-paramagnetism behavior, i.e.progressively thermal blocking of single domain magneticclusters is further substantiated by Wohlfarth’s model ofsuperparamagnets which shows that the magnetizationof an ensemble of magnetic clusters is given as52,53

M = n〈µ〉L(〈µ〉H/kBT ) (3)

where n is the number of clusters per unit volume, 〈µ〉 isthe average magnetic moment of the clusters, kB is theBoltzman constant, and L(x) is the Langevin function.Above the blocking temperature (TB), the expansion ofLangevin function in powers of H gives

χ1 = n〈µ〉2/3kBT = P1/T (4)

and

χ3 = (n〈µ〉/45)(〈µ〉/kBT )3 = P3/T (5)

The equation 4 and 5 show that for superparamagneticclusters, above TB, χ1 and χ3 varies as a linear func-tion of T−1 and T−3 respectively. The figure 8 showthe χ

′

1 versus T−1 which is linear above TB. Similarly,

7

167.4 168.0 168.6

1E-3

0.01

0.1

1

Scaling law Vogel-Fulcher Law

5.94 5.96 5.98

-6

-3

0

ln

TP-1 (10-3 K-1) (s

)

TB(K)

Figure 9: Variation of relaxation time (τ ) with blocking tem-perature (TB). The solid line represents the fitting of Vogel-Fulcher law while the dashed line represents the fitting ofscaling law. The inset show the lnτ versus T−1

Band the solid

line is the fitting of Néel-Arrhennius law.

the inset of figure 8 display the linear variation of χ′

3

with T−3 above TB. The ratio of the fitting parameterP3 and P1 is used to estimate 〈µ〉, which comes around1.83 × 105 µB where µB is the effective Bohr magne-ton. Such a large value of 〈µ〉 is generally observed insuperparamagnet clusters. This is because the clusterconsists of a large number of atomic spins each havingthe magnetic moment of few µB (while normal paramag-net only have the atomic spins). Since 〈µ〉=MSV whereMS is the saturation magnetization and V is the vol-ume of cluster, assuming the clusters to be spherical,the average size of the clusters comes around 15 nm.The size of the magnetic clusters is much smaller thanthe crystallite size (≈ 85 nm calculated from the X-raydiffraction) which indicates that each crystallite containsa number ferromagnetic-clusters. Combining this withthe results of low temperature isothermal magnetizationof section III A 3, we infer that each crystallite of the sys-tem consists of percolating ferromagnetic-clusters coex-isting along with the non-ferromagnetic hole poor phases.

5. Inter-cluster interaction

The ferromagnetic-clusters coexisting with the non-ferromagnetic phases may interact with each other di-rectly through dipole-dipole interaction or via non-ferromagnetic matrix through exchange interactions.54

The degree of inter-cluster interaction and their effecton fluctuation dynamics is studied by fitting the fre-quency dependence of TB with Néel-Arrhennius, Vogel-Fulcher, and scaling law.55 The results of these fittingsare shown in figure 9 and its inset. For an ensemble ofnon-interacting superparamagnets, the relaxation time τfollows the Néel-Arrhennius law55

τ = τ0exp(

Ea

kBT

)

(6)

where Ea is the average anisotropy energy barrier, τ0is the times constant corresponding to characteristic at-tempt frequency, and kB is the Boltzman constant. Theexperimentally observed τ0 values for non-interactingsuper-paramagnets are in the range of 10−8 − 10−13 s.57

The inset of figure 9 show the fitting of equation 6 toT−1

B versus lnτ data which gives τ0 ≈ 10−402 s andEa=13.31 eV. The fitting of Néel-Arrhennius law yieldsun-physical values which rule out the possibility of non-interacting dynamics and hint the presence of cooperativedynamics due to inter-cluster interaction. The dynam-ics of the interacting superparamagnets is described byVogel-Fulcher law55

τ = τ0exp(

Ea

kB(T − T0)

)

(7)

where the temperature T0 which has a value between zeroand TB is often related to the strength of inter-cluster in-teraction. The fitting of equation 7 to the data of figure9 gives τ0 ∼10−9 s, Ea/kB=63(9) K and T0=164.3 K.The τ0 value obtained from the Vogel-Fulcher fitting isorders of magnitude larger than the spin-flip time ofatomic magnetic moments (∼ 10−13 s). This stronglysupports that the fluctuating entities are spin-clusterswith a significant inter-cluster interaction among them.Strong inter-cluster interactions can give rise to spin-glass like cooperative freezing, and in this case, the fre-quency dependence of peak in χ

′

1 is expected to followthe power law divergence of the standard critical slowingdown given by dynamic scaling theory28,55,56

τ = τ0(T/Tg − 1)−zv (8)

where τ is the dynamical fluctuation time scale corre-sponding to measurement frequency at the peak temper-ature of χ

′

1, τ0 is the spin flipping time of the relaxingentities, Tg is the cluster-glass (or spin-glass) transitiontemperature in the limit of zero frequency, z is the dy-namic scaling exponent, and v is the critical exponent.In the vicinity of cluster-glass transition, the spin clustercorrelation length ξ diverges as ξ ∝ (T/Tg − 1)−v andthe dynamic scaling hypothesis relates τ to ξ as τ ∼ ξz.The fitting of scaling law to the frequency dependence ofTB (see figure 9) gives τ0 ∼ 10−25 s, Tg=165.4 K, andzv=12.6. The value of exponent zv is higher than thatobserved in case of spin-glasses (2-10) and τ0 is orders ofmagnitude smaller than the values reported for cluster-glass (10−9-10−6 s) and spin-glass (10−11-10−13 s). Thevalue of τ0 is even smaller than the spin-flip time of asingle atom (∼ 10−13 s), which is un-physical, and thisindicates that the spin dynamics in the system does notexhibit the critical slowing down on approaching Tg asexpected from the dynamic scaling. Thus, it can be in-ferred that the inter-cluster interactions present in thesystem are significant, but not strong enough to cause aspin-glass like transition.

8

0 30 60 90

120

150

180

T B (K)

H2/3 (Oe2/3)

Figure 10: Field dependence of the peak in ZFC magnetiza-tion. The straight line show the fitting of AT line to data.

C. Further discussions

The presence of significant inter-cluster interaction canalso be reaffirmed from the field dependence of the peaktemperature (Tp) in ZFC magnetization curves. ForIsing spin-glass, mean field theory of spin-glass predictsa critical de Almeida-Thouless (AT) line in H-T spacewhich marks the spin-glass phase transition.59 AboveAT line the large field destroys the frozen spin state.The spin-glass transition temperature corresponds to thepeak in ZFC magnetization (Tp) and the AT line pre-dicts that Tp ∝ H2/3. The AT line like field depen-dence of Tp is not unique to spin or cluster -glass transi-tion but it has been also observed in some of interactingsuper-paramagnets which otherwise undergo a progres-sive thermal blocking.32,60 In figure 10 we have plottedthe field dependence of Tp which fits well with the ATline giving zero field spin-glass transition temperature(Tg) of 172(1) K. Since our ac susceptibility measure-ments have ruled out the possibility of a spin or clus-ter -glass like freezing, the existence of AT line in H-Tspace clearly indicates the presence of a significant inter-cluster interaction in the system. The inter-cluster in-teractions can originate from different type of magneticinteractions and the strength of these interactions gen-erally depends on the packing density of ferromagnetic-clusters. These magnetic interaction includes the longrange dipole-dipole interaction among the ferromagnetic-clusters along with the possibilities of exchange, tunnel-ing exchange and superexchange interactions.54

The absence of spin-glass like phase in La0.5Ba0.5CoO3

is in contrast with La0.5Sr0.5CoO3 where spin or clus-ter -glass like phase coexists along with the percolatingferromagnetic-clusters.21 The doping at A site of LaCoO3

with 50% Ba or Sr gives same hole concentration, andtherefore, the observed discrepancy in magnetically or-dered state of La0.5Ba0.5CoO3 and La0.5Sr0.5CoO3 canbe only due to difference in tolerance factor and local lat-tice distortions caused by the difference in ionic radii ofBa and Sr. A comparative study using the microscopic

probe e.g neutron scattering is required to understandhow this difference gives rise to non-spin-glass and spin-glass like behavior in these two systems.

IV. CONCLUSIONS

In conclusion, we have performed a comprehensive setof dc magnetization, linear and non-linear ac suscepti-bility measurements to understand the magnetic stateof the hole doped disordered cobaltite La0.5Ba0.5CoO3.The results of isothermal magnetization suggest that themagnetically ordered state of the system consists of per-colating ferromagnetic-clusters along with the coexistingnon-ferromagnetic phases. The Kouvel-Fisher analysis ofthe ac susceptibility gives TC=202(3) K and γ=1.29(1)indicating the possibility of 3D Hisenberg like magneticordering in the ferromagnetic-clusters. Above TC , thereexist a temperature range (TC <T <T ∗) where inverse dcsusceptibility exhibits a field independent upward devia-tion from Curie-Weiss law which is different from the fielddependent downward deviation seen in Griffiths phase.In this region, the spontaneous magnetization is zero, butthe real part of second harmonic of ac susceptibility andthe imaginary part of linear ac susceptibility are non-zero which indicate the presence magnetic correlationsthat do not originate from the variation in local TC dueto composition fluctuation. This suggests that possiblythese magnetic correlation comes from the preformationof short range magnetic-clusters.

Below TC the system exhibits thermomagnetic irre-versibility and frequency dependence in the peak ofac susceptibility which suggest the presence of spin-glass, cluster-glass, or superparamagnetic phases. Theabsence of field divergence in the peak of third har-monic of ac susceptibility and absence of ZFC mem-ory rule out the existence of spin or cluster -glass likephases and suggest that the observed spin-dynamics ispossibly due to super-paramagnetic like thermal block-ing of ferromagnetic-clusters. This is in sharp con-trast to La0.5Sr0.5CoO3 where the spin or cluster -glasslike phase coexist with the ferromagnetic-clusters. Thesuper-paramagnetic behavior of ferromagnetic-clustersin La0.5Ba0.5CoO3 is further confirmed by Wohlfarth’smodel of super-paramagnetism. The analysis of fre-quency dependence in the peak of ac susceptibility byNéel-Arrhennius, Vogel-Fulcher, and scaling law sug-gest the existence of significant inter-cluster interactionamong the ferromagnetic-clusters which is further con-firmed by the existence of AT line in the H-T space.

V. ACKNOWLEDGEMENT

We are thankful to P. Chaddah for useful discussions.We also acknowledge M. Gupta for XRD measurements.

9

∗ Electronic address: deven@csr.res.in1 E. Dagotto, Science 309, 257 (2005).2 M. Rini, R. Tobey, N. Dean, J. Itatani, Y. Tomioka, Y.

Tokura, R. W. Schoenlein, and A. Cavalleri, Nature 449,72 (2007).

3 D. Louca, J. L. Sarrao, J. D. Thompson, H. Roder, and G.H. Kwei, Phys. Rev. B 60, 10378 (1999).

4 A. Podlesnyak, S. Streule, J. Mesot, M. Medarde, E. Pom-jakushina, K. Conder, A. Tanaka, M. W. Haverkort, andD. I. Khomskii, Phys. Rev. Lett. 97, 247208 (2006).

5 P. M. Raccah and J. B. Goodenough, Phys. Rev. B 155,932 (1967).

6 M. Kriener, M. Braden, H. Kierspel, D. Senff, O. Zabara,C. Zobel, and T. Lorenz, Phys. Rev. B 79, 224104 (2009).

7 C. Zobel, M. Kriener, D. Bruns, J. Baier, M. Grüninger,T. Lorenz, P. Reutler, and A. Revcolevschi, Phys. Rev. B66, 020402(R) (2002).

8 S. Noguchi, S. Kawamata, K. Okuda, H. Nojiri, and M.Motokawa, Phys. Rev. B 66, 094404 (2002).

9 M. W. Haverkort, Z. Hu, J. C. Cezar, T. Burnus, H. Hart-mann, M. Reuther, C. Zobel, T. Lorenz, A. Tanaka, N. B.Brookes, H. H. Hsieh, H.-J. Lin, C. T. Chen, and L. H.Tjeng, Phys. Rev. Lett. 97, 176405 (2006).

10 P. G. Radaelli and S.-W. Cheong, Phys. Rev. B 66, 094408(2002).

11 M. A. Korotin, S. Yu. Ezhov, I. V. Solovyev, V. I. Anisi-mov, D. I. Khomskii, and G. A. Sawatzky, Phys. Rev. B54, 5309 (1996).

12 D. Louca and J. L. Sarrao, Phys. Rev. Lett. 91, 155501(2003).

13 D. Phelan, J. Yu, and D. Louca, Phys. Rev. B 78, 094108(2008).

14 A. Podlesnyak, M. Russina, A. Furrer, A. Alfonsov, E.Vavilova, V. Kataev, B. Büchner, Th. Strassle, E. Pom-jakushina, K. Conder, and D. I. Khomskii, Phys. Rev. Lett.101, 247603 (2008).

15 A. Podlesnyak, G. Ehlers, M. Frontzek, A. S. Sefat, A. Fur-rer, Th. Strassle, E. Pomjakushina, K. Conder, F. Demmel,and D. I. Khomskii, Phys. Rev. B 83, 134430 (2011).

16 M. Kriener, C. Zobel, A. Reichl, J. Baier, M. Cwik, K.Berggold, H. Kierspel, O. Zabara, A. Freimuth, and T.Lorenz, Phys. Rev. B 69, 094417 (2004).

17 P. Mandal, P. Choudhury, S. K. Biswas, and B. Ghosh,Phys. Rev. B 70, 104407 (2004).

18 J. Wu and C. Leighton, Phys. Rev. B 67, 174408 (2003).19 S. Mukherjee, R. Ranganathan, P. S. Anilkumar, and P.

A. Joy, Phys. Rev. B 54, 9267 (1996).20 D. Samal and P. S. Anil Kumar, J. Phys.: Condens. Matter

23, 016001 (2011).21 D. Samal and P. S. Anil Kumar, J. Appl. Phys. 111,

043902 (2012).22 D. Phelan, D. Louca, K. Kamazawa, M. F. Hundley, and

K. Yamada, Phys. Rev. B 76, 104111 (2007).23 J. Yu, D. Louca, D. Phelan, K. Tomiyasu, K. Horigane,

and K. Yamada, Phys. Rev. B 80, 052402 (2009).24 R. K. Pati, J. C. Ray, and P. Pramanik, J. Am. Ceram.

Soc. 84, 2849 (2001).25 A. Bajpai and A. Banerjee, Rev. Sci. Instrum. 68, 4075

(1997).26 J. Rodríguez-Carvajal, Phys. B (Amsterdam) 192, 55

(1993); Program FULLPROF, LLB-JRC, Laboratoire

Léon Brillouin, CEA-Saclay, France, 1996.27 I. O. Troyanchuk, N. V. Kasper, D. D. Khalyavin, A. N.

Chobot, G. M. Chobot, and H. Szymczak, J. Phys.: Con-dens. Matter 10, 6381 (1998).

28 J. A. Mydosh, Spin Glasses: An Experperimental Intro-duction (Taylor and Francis London 1993).

29 I. G. Deac, J. F. Mitchell, and P. Schiffer, Phys. Rev. B63, 172408 (2001).

30 X. H. Huang, J. F. Ding, Z. L. Jiang, Y. W. Yin, Q. X.Yu, and X. G. Lia, J. Appl. Phys. 106, 083904 (2009).

31 M. Knobel, W. C. Nunes, L. M. Socolovsky, E. De Biasi, J.M. Varg as, and J. C. Denardin, J. Nanosci. Nanotechnol.8, 2836 (2008).

32 A. K. Pramanik and A. Banerjee, Phys. Rev. B 82, 094402(2010).

33 P. S. Anil Kumar, P. A. Joy, and S. K. Date, J. Phys. :Condens. Matter 10, L487 (1998).

34 R. Caciuffo, J. Mira, J. Rivas, M. A. Senaris-Rodriguez,P. G. Radaelli, F. Carsughi, D. Fiorani, and J. B. Goode-nough, Europhys. Lett. 45, 399 (1999).

35 C. He, M. A. Torija, J. Wu, J. W. Lynn, H. Zheng, J. F.Mitchell, and C. Leighton, Phys. Rev. B 76, 014401 (2007).

36 R. B. Griffiths, Phys. Rev. Lett. 23, 17 (1969).37 C. Magen, P. A. Algarabel, L. Morellon, J. P. Araújo, C.

Ritter, M. R. Ibarra, A. M. Pereira, and J. B. Sousa, Phys.Rev. Lett. 96, 167201 (2006).

38 A. K. Pramanik and A. Banerjee, Phys. Rev. B 81, 024431(2010).

39 P. Ravindran, H. Fjellvag, A. Kjekshus, P. Blaha, K.Schwarz, and J. Luitz, J. Appl. Phys. 91, 291 (2002).

40 A. Arrott, Phys. Rev. 108, 1394 (1957).41 G. F. Goya and V. Sagredo, Phys. Rev. B 64, 235208

(2001).42 D. Y. Cong, S. Roth, J. Liu, Q. Luo, M. Pötschke, C. Hür-

rich, and L. Schultz, Appl. Phys. Lett. 96, 112504 (2010).43 M. Thakur, M. Patra, S. Majumdar, and S. Giri, J. Appl.

Phys. 105, 073905 (2009).44 J. A. De Toro, M. A. López de la Torre, M. A. Arranz, J.

M. Riveiro, J. L. Martínez, P. Palade, and G. Filoti, Phys.Rev. B 64, 094438 (2001).

45 J. L. Dormann, L. Bessais, and D. Fiorani, J. Phys. C 21,2015 (1988).

46 J. S. Kouvel and M. E. Fischer, Phys. Rev. 136, A1626(1964).

47 N. Menyuk, P. M. Raccah, and K. Dwight, Phys. Rev. 166,510 (1968).

48 S. Mukherjee, P. Raychaudhuri, and A. K. Nigam, Phys.Rev. B 61, 8651 (2000).

49 N. Khan, A. Midya, K. Mydeen, P. Mandal, A. Loidl, andD. Prabhakaran, Phys. Rev. B 82, 064422 (2010).

50 M. Suzuki, Prog. Theor. Phys. 58, 1151 (1977).51 A. Bajpai and A. Banerjee, Phys. Rev. B 55, 12439 (1997).52 E. P. Wohlfarth, J. Appl. Phys. 29, 595 (1958).53 T. Bitoh, K. Ohba, M. Takamatsu, T. Shirane, and S.

Chikazawa, J. Magn. Magn. Mater. 154, 59 (1996).54 S. Bedanta and W. Kleemann, J. Phys. D: Appl. Phys. 42,

013001 (2009).55 K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801

(1986).56 P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49,

435 (1977).

10

57 J. L. Dormann, D. Fiorani, and E. Tronc, J. Magn. Magn.Mater. 202, 251 (1999).

58 C. Djurberg, P. Svedlindh, P. Nordblad, M. F. Hansen, F.Bodker, and S. Morup, Phys. Rev. Lett. 79, 5154 (1997).

59 J. R. L. de Almeida and D. J. Thouless, J. Phys. A 11,

983 (1978).60 L. E. Wenger and J. A. Mydosh, Phys. Rev. B 29, 4156

(1984).