DSpace at VNU: Critical behavior of La0.7Ca0.3Mn1 x NixO3manganites exhibiting the crossover offirst- and second-order phase transitions

Solid State Communications 184 (2014) 40–46 Contents lists available at ScienceDirect Solid State Communications journal homepage: www.elsevier.com/locate/ssc Critical behavior of La0.7Ca0.3Mn1 À xNixO3 manganites exhibiting the crossover of first- and second-order phase transitions The-Long Phan a, Q.T Tran b, P.Q Thanh c, P.D.H Yen d, T.D Thanh a,e, S.C Yu a,n a Department of Physics, Chungbuk National University, Cheongju 361-763, South Korea Center for Science and Technology Communication, Ministry of Science and Technology, 113 Tran Duy Hung, Hanoi, Vietnam c Faculty of Physics, Hanoi University of Science, Vietnam National University, Hanoi, Vietnam d Faculty of Engineering Physics and Nanotechnology, VNU – University of Engineering and Technology, Xuan Thuy, Cau Giay, Hanoi, Vietnam e Institute of Materials Science, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam b art ic l e i nf o a b s t r a c t Article history: Received 29 October 2013 Received in revised form 28 December 2013 Accepted 29 December 2013 by F Peeters Available online January 2014 We used Banerjee0 s criteria, modified Arrott plots, and the scaling hypothesis to analyze magneticfield dependences of magnetization near the ferromagnetic–paramagnetic (FM–PM) phasetransition temperature (T C) of perovskite-type manganites La0.7Ca 0.3Mn1 À xNixO (x ¼0.09, 0.12 and 0.15) In the FM region, experimental results for the critical exponent β ( ¼0.171 and 0.262 for x ¼ 0.09 and 0.12, respectively) reveal two first samples exhibiting tricriticality associated with the crossover of first- and second-order phase transitions Increasing Ni-doping content leads to the shift of the β value ( ¼0.320 for x ¼ 0.15) towards that expected for the 3D Ising model (β ¼0.325) This is due to the fact that the substitution of Ni ions into the Mn site changes structural parameters and dilutes the FM phase, which act as fluctuations and influence the FM-interaction strength of double-exchange Mn3 ỵ Mn4 ỵ pairs, and the phase-transition type For the critical exponent γ ( ¼0.976–0.990), the stability in its value demonstrates the PM behavior above T C of the samples Particularly, around T C of La0.7Ca 0.3Mn1 À xNixO compounds, magnetic-field dependences of the maximum magnetic-entropy change can be described by a power law of |Δ Smax| p H n, where values n ¼ 0.55–0.77 are quite far from those (n ¼ 0.33–0.48) calculated from the theoretical relation n ẳ1 ỵ ( 1)/( ỵ ) This difference is related to the use of the mean-field theory for the samples exhibiting the magnetic inhomogeneity & 2014 Elsevier Ltd All rights reserved Keywords: A Perovskite manganites D Critical behavior D Magnetic entropy change Introduction It is known that hole-doped lanthanum manganites of La1 À x (Ca, Sr, Ba)xMnO with x ẳ0.3 (corresponding to Mn3 ỵ / Mn4 ỵ ẳ 7/3) usually exhibit colossal magnetoresistance (MR) and magnetocaloric (MC) effects around their the ferromagnetic–paramagnetic (FM–PM) phase-transition temperature (the Curie temperature, T C) [1] With this doping content, double-exchange (DE) FM interactions between Mn3 ỵ and Mn4 ỵ are dominant as comparing with super-exchange antiFM interactions of Mn3 þ –Mn3 þ and Mn4 þ –Mn4 þ pairs The strength of magnetic interactions thus depends on the average bond length 〈Mn–O〉, and bond angle 〈Mn–O–Mn〉 of the perosvkite structure Different compounds have different bond parameters, which are related to Jahn–Teller lattice distortions due to strong electron–phonon coupling [2] In reference to the n Corresponding author Tel.: ỵ 82 43 261 2269; fax: ỵ 82 43 275 6416 E-mail address: scyu@chungbuk.ac.kr (S.C Yu) 0038-1098/$ - see front matter & 2014 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.ssc.2013.12.032 symmetry of MnO octahedra, it has been noted that cooperative Jahn–Teller distortions are present in an orthorhombic structure rather than in the rhombohedral one [3] Among hole-doped manganites, orthorhombic La0.7 Ca 0.3MnO3 is known as a typical material exhibiting MR and MC effects much greater than those obtained from the other compounds Particularly, depending on bulk or nanostructured sample types, its T C in the range of 222–265 K [3–9] can be tuned towards room temperature by doping Sr, Ba or Pb [10– 14] Meanwhile, the transition-metal doping (such as Co, Fe, Ni and so forth) lowers TC [15–17] Additionally, its discontinuous FM–PM transition at T C is followed up with structural changes, and is known as a first-order magnetic phase transition (FOMT) [8,9,12] This discontinuous phase transition can be rounded to a continuous one of a second-order magnetic phase transition (SOMT) upon the doping, and reduced dimensionality (i.e., finite-size effects), and external fields [5,7,11,12,16,18,19] The assessment of a continuous SOMT can base on the success in determining the critical exponents β , γ , and δ associated with temperature dependences of the spontaneous magnetization, Experimental details Three polycrystalline perovskite-type manganites La0.7 Ca0.3 Mn1 À x Nix O3 with x ¼0.09, 0.12 and 0.15 were prepared by solidstate reaction as using purity commercial powders La2 O3, CaO, NiO, and MnCO (99.9%) as precursors These powders combined with stoichiometrical quantities were well mixed and ground, and then pre-annealed at 900 1C for 24 h After preannealing, three mixtures were re-ground and pressed into pellets, and annealed at 1300 1C for 72 h in air For reference, the parent compound La0.7 Ca 0.3MnO3 was also prepared with the same conditions as described X-ray diffraction (XRD) patterns of the final products checked by an X-ray diffractometer (Bruker AXS, D8 Discover) revealed the single phase in an orthorhombic structure (the space group Pbnm) of La 0.7Ca0.3 Mn1 À xNixO3 samples, see Fig 1(a) Basing on the XRD data, we calculated the lattice parameters (a, b, and c) and unit cell (V), as shown in Table The variation of these parameters indicates the substitution of Ni ions (could be Ni ỵ , Ni þ , and/or Ni4 þ ) for Mn in the perovskite structure Magnetic measurements were performed on a superconducting quantum interference device magnetometer (SQUID) The TC values obtained from the flexion points in temperature dependences of magnetization, M(T), with the applied field H ¼ 100 Oe, Fig 1(b) are about 200, 185 and 170 K for x ¼0.09, 0.12 and 0.15, respectively, which are lower than the value TC E 260 K of the parent compound Results and discussion Fig shows M–H data and inverse Arrott plots (H/M versus M2 ) at different temperatures around the FM–PM phase transition of La 0.7Ca0.3 Mn1 À xNixO It appears from the M–H data that there is no saturation magnetization value in spite of the H variation up to 40 kOe This is assigned to the existence of the magnetic inhomogeneity or short-range FM order At a given temperature, higher Ni-doping content reduces the magnetization With increasing temperature, nonlinear M–H curves in the FM region become linear because the samples enter the PM state Different from the parent compound [5,7–9,12], there is (004)/(242) (123)/(321) (202)/(040) (022)/(220) Intensity (arb units) x = 0.15 x = 0.12 x = 0.09 x=0 30 40 50 60 70 2θ (degree) 1.2 H = 100 Oe 0.9 Normalized M Ms(T), inverse initial susceptibility, χ –1 (T), and critical isotherm at TC, respectively [20,21] Distinguishing the FOMT from the SOMT can be based on the criteria proposed by Banerjee [22], who performed H/M versus M2 curves (where H is the field, and M is the magnetization) in the vicinity of T C, and then suggested that their positive or negative slopes are indication of a secondor first-order phase transition, respectively Reviewing previous studies, one can see that many works focused on La0.7Ca0.3 MnO3-based materials showing the FOMT and/or SOMT However, the crossover region from first-order to second-order phase transitions, and some related physical properties, such as the magnetic entropy change versus T and H, ΔS m(T, H), have not been widely studied Furthermore, there is no much attention given to the assessment of a magnetic ordering parameter (n) determined from the relations n ¼ þ (β À1)/(β þ γ ) [23], and from a power law |Δ Smax(H)| p Hn [24] (where |Δ Smax| is the maximum magnetic entropy change around TC) To get more insight into the above problems, we prepared La0.7Ca0.3 Mn1 À xNi xO3 compounds, and have studied their critical behaviors upon Banerjee0 s criteria [22], modified Arrott plots and the scaling hypothesis [20,21] The determined critical values are then discussed together with the magnetic ordering parameter n 41 (121) T.-L Phan et al / Solid State Communications 184 (2014) 40–46 x=0 x = 0.12 x = 0.09 x = 0.15 0.6 0.3 0.0 50 100 150 200 250 300 T (K) Fig (Color online) (a) Room-temperature XRD patterns, and (b) normalized M(T) curves with the applied field of H¼100 Oe for La0.7Ca0.3Mn1 À xNixO3 (x ¼0, 0.09, 0.12, and 0.15) Table Values of the lattice parameters and unit cell calculated from XRD analyses of La0.7Ca0.3Mn1 À xNixO3 with x¼ 0.09, 0.12 and 0.15 Sample, x a (Å) b (Å) c (Å) V (Å3) 0.09 0.12 0.15 5.473 5.467 5.461 5.474 4.461 5.451 5.450 5.450 7.711 7.707 7.719 7.713 230.46 229.70 229.73 230.10 no S-like shape in the M–H curves, and negative slopes in the H/ M versus M2 curves, see Fig These tokens demonstrate our Ni-doped samples undergoing the FOMT [22,25] According to the mean-field theory (MFT) proposed for a ferromagnet exhibiting the SOMT and long-range FM interactions [26], the free energy GL is expanded in even powers of M: GL ẳaM2 ỵbM4 ỵ HM, where a and b are temperaturedependent parameters Minimizing GL as GL/M ẳ0 results in the relation H/M ẳ2a ỵ 4bM2 It means that if magnetic interactions of the FM system exactly obey the MFT, M2 versus H/M curves in the vicinity of TC are parallel straight lines At TC, the M2 and H/M line passes through the origin [27,28] However, these features are absent from the Arrott performance shown in Fig 2(b, d, and e) It means that magnetic interactions in the samples could not be the long-range type The critical exponents β ¼0.5 and γ ¼1.0 (in the normal Arrott plots [20,27]) based on the MFT are thus not suitable to describe magnetic interactions taking place in our samples Within the framework 42 T.-L Phan et al / Solid State Communications 184 (2014) 40–46 182 K 182 K ΔT = K 80 60 220 K 220 K 40 20 10 20 30 50 168 K 168 K 60 8 ΔT = K M (emu/g) 80 40 206 K 206 K 40 20 0 10 20 30 40 50 ΔT = K 60 152 K 152 K 190 K 190 K 40 20 H/M (102, Oe.g/emu) 10 20 30 H (kOe) 40 50 M2 (103, emu/g)2 Fig (Color online) M–H data and inverse Arrott plots for La0.7Ca0.3Mn1 À xNixO3 with (a, b) x ¼0.09, (c, d) x¼ 0.12, and (e, f) x¼ 0.15 of the MFT, we need to find other sets of the critical-exponent values reflecting more frankly the magnetic properties of the samples This work is based on the modified Arrott plot (MAP) method [20], which is generalized by the scaling equation of γ β state, (H/M)1/ ¼c1 ε þ c2M1/ , where c1 and c2 are temperaturedependent parameters, and ε ¼(T À TC)/TC is the reduced temperature β and γ values can be obtained from the asymptotic relations [18,20] M s Tị ẳ M ị ; Tị ẳ h0 =M0 ị ; M ¼ DH 1=δ ; ε ¼ 0; ε o 0; ε 0; ð1Þ ð2Þ ð3Þ where M0, h0 , and D are the critical amplitudes Additionally, according to the static-scaling hypothesis [21], M is a function of ε and H, MH; ị ẳ jj f H=jj ỵ γ Þ This equation reflects β that, with determined β and values, plotting M/ versus H/ ỵ makes all data points falling on the f À and f ỵ branches for T o TC and T T C, respectively Here, determining the critical parameters is based on the MAP method, and started from the scaling equation of state Correct β and γ values make M–H data points falling on a set of parallel straight lines in the β γ β performance of M1/ versus (H/M)1/ Moreover, the M1/ versus γ (H/M)1/ line passes through the origin at T C Similar to the MFT case, our analyses indicated that the exponent values β ¼0.365 and γ ¼1.336 expected for the 3D Heisenberg model [21] not match with the descriptions of the MAP method Only β ¼0.25 and γ ¼ 1.0 expected for the tricritical MFT model (T-MFT), and β ¼0.325 and γ ¼1.241 expected for the 3D Ising model [12,29] can be used as initially trial values to find optimal exponent values for the samples with x ¼0.09 and 0.12, and for x ¼0.15, respectively With these trial values, Ms(T) and χ 0(T) data would be obtained from the linear extrapolation in the high-field β region for the isotherms to the co-ordinate axes of M1/ and 1/ γ 1/ γ (1/χ ) ¼(H/M) , respectively The Ms(T) and χ 0(T) data obtained from the linear extrapolation are then fitted to Eqs (1) and (2), respectively, to achieve better β , γ and TC values, as can be seen from Fig These new values of β , γ , and TC are continuously used for next MAP processes until their optimal values are achieved Notably, the T C values of the samples obtained from M–T measurements were also used as reference in the fitting With such the careful comparison, only the sets of critical parameters with T C E 199.4 K, β ¼ 0.171 0.006 and γ ¼ 0.976 0.012 for x ¼0.09; TC E 184.4 K, β ¼ 0.262 0.005 and γ ¼ 0.979 0.012 for x ¼0.12; and TC E 170 K, β ¼0.320 0.009 T.-L Phan et al / Solid State Communications 184 (2014) 40–46 64 56 12 72 TC = 199.4 ± 0.3 γ = 0.976 ± 0.012 185 190 195 200 205 210 186 K x = 0.09 β = 0.171 γ = 0.976 TC = 199.5 ± 0.1 β = 0.171 ± 0.006 43 214 K 215 40 30 TC = 184.3 ± 0.1 γ = 0.979 ± 0.012 TC = 184.5 ± 0.1 β = 0.262 ± 0.005 170 175 180 185 190 195 200 50 40 β = 0.320 ± 0.009 160 165 200 400 170 175 180 600 174 K x = 0.12 30 β = 0.262 γ = 0.978 20 10 200 K 0 0.6 185 0.4 200 400 600 152 K TC = 169.6 ± 0.5 K γ = 0.990 ± 0.082 30 TC = 170.0 ± 0.1 K 155 M1/βx106 (emu/g)1/β 50 -1 Ms (emu/g) 60 χ0 (x102, Oe.g/emu) 800 x = 0.15 β = 0.320 γ = 0.990 T (K) Fig (Color online) Ms(T) and χ 0À ðTÞ data fitted to Eqs (1) and (2) for La0.7Ca0.3Mn1 À xNixO3 with (a) x¼ 0.09, (b) x ¼0.12, and (c) x¼ 0.15 and γ ¼0.990 0.082 for x ¼0.15 are in good agreement with the MAP descriptions, see Fig With the obtained critical β β γ exponents, the scaling performance of M/| | versus H/ ỵ curves, see Fig and their inset, reveals the M–H data points at high-magnetic elds falling into two f and f ỵ universal branches for T o TC and T TC, respectively These results prove the reliability in value of the critical values obtained from our work It should be noticed that the MAP method only works well for the fields (HL ) higher than 28, 24 and 12 kOe for x ¼ 0.09, 0.12 and 0.15, respectively At the fields lower than HL , there may be rearrangement of magnetic domains, the effect due to the uncertainty in the calculation of demagnetization factor, and/or the persistence of the FOMT (particularly for the samples with x ¼0.09 and 0.12) [12,30] Unexpected errors for critical values can thus be occurred, leading to the scattering of the M–H data points (at the fields lower than H L) from the universal curves [5,12], as can be seen in Fig For the exponent δ , its value can be obtained from fitting the isotherms at T ¼T C to Eq (3) Basically, the δ values determined from Eq (3) would be equal to those calculated from the Widom relation ẳ ỵ / [21] In our work, δ values are about 6.7, 4.7, and 4.1 for x ¼0.09, 0.12 and 0.15, respectively Clearly, with increasing Ni concentration in La 0.7Ca0.3 Mn1 À xNixO 3, there is a shifting tendency of the exponent values (β , γ and δ ) towards those of the MFT (with β ¼0.5, γ ¼ and δ ¼3) This is tightly related to the FOMT–SOMT transformation The better applicability of the MAP method has been found for the samples with high-enough Ni concentrations as x 0.12 We believe that the substitution 180 K 0.2 0.0 200 400 (H/M) 1/γ 800 600 (Oe.g/emu) 1000 1/γ Fig (Color online) MAPs of M1/β versus (H/M)1/γ with the critical exponents obtained for La0.7Ca0.3Mn1 À xNixO3 with (a) x ¼0.09, (b) x ¼0.12 and (c) x¼ 0.15 of Ni ions into the Mn site not only changes structural parameters of 〈Mn–O〉 and 〈Mn–O–Mn〉, but also leads to the additional presence of anti-FM interactions related to Ni ions (for example, super-exchange pairs of Ni ỵ Ni2 ỵ , Ni3 ỵ Ni3 ỵ , Ni4 ỵ Ni ỵ , and/or Ni2 ỵ ,3 ỵ ,4 ỵ Mn3 ỵ ,4 ỵ ) besides pre-existing anti-FM interaction pairs of Mn3 ỵ Mn3 þ and Mn4 þ –Mn4 þ These factors act as uctuations, reduce the strength of Mn3 ỵ Mn4 ỵ FM interactions (which thus reduce the magnetization and TC values), and influence the phase transition as well Comparing with the theoretical models [21,29], one can see that the values of γ ( ¼0.976–0.990) are quite stable, demonstrating the complete PM state in the samples at temperatures above TC For the FM region, however, β ¼0.262 for x ¼0.12 is close to that expected for the T-MFT (β ¼ 0.25) This sample thus exhibits tricriticality associated with the crossover of first- and second-order phase transitions Similar results were also found in some manganites [9,12,29] A smaller value of β ¼ 0.171 for x ¼0.09 reveals this sample lying in the region close to the crossover, where the FOMT is still persistent It is also known that the MAP application for the materials with the presence 44 T.-L Phan et al / Solid State Communications 184 (2014) 40–46 T < TC 102 T < TC M/|ε|β 150 101 100 T > TC 50 T > TC 100 104 105 106 107 T < TC 102 200 T < TC 150 M/|ε|β M/|ε|β (emu/g) 2.0x103 4.0x103 H/|ε|β+γ 0.0 101 100 T > TC 50 T > TC 0.0 100 104 105 2.0x106 4.0x106 H/|ε|β+γ 106 107 T < TC 102 200 T < TC 150 M/|ε|β 101 T > TC 100 T > TC 50 100 0.0 104 105 2.0x106 4.0x106 H/|ε|β+γ 106 H/|ε|β+γ belong to other universality classes (such as the T-MFT and 3D Ising models) if 1/2 o s o 2, which can be the case taking place in our samples Together with assessing the critical behaviors of La0.7Ca0.3Mn1À xNixO3 samples, we have also considered the magnetic-entropy change (ΔSm) and its field dependence, as shown in Fig At a given temperature for each sample, À ΔSm increases with increasing H Around TC, À ΔSm(T) curves reach the maxima, |ΔSmax| The |ΔSmax| values determined for x¼0.09, 0.12, and 0.15 in the field H¼ 40 kOe are about 7.1, 5.2, and 3.4 J kg À K À 1, respectively, which are smaller than those obtained from the parent compound [7] Though the Ni doping reduces the |Δ Smax| value, the linewidth of the À Δ Sm(T) curves become broadened due to the FOMT– SOMT transformation, enhancing the refrigerant capacity (RC) Particularly, at TC the H dependences of |Δ Smax| can be well described by a power law of |Δ Smax| p Hn [24], where values n ¼0.55, 0.68, and 0.77 for x ¼0.09, 0.12, and 0.15, respectively These values are different from those (n ¼0.33, 0.41, and 0.48 for x ¼0.09, 0.12, and 0.15, respectively) calculated from the relation n ẳ1 ỵ ( 1)/( ỵ ) [23] As shown in Ref [34], n is known as a function of T, H and |Δ Sm|, which can also be obtained from the relation n ¼dln|ΔS m|/dlnH Depending on the variation of these parameters, n would be different It reaches the minimum at temperatures in the vicinity of T C [23] We believe that a large deviation of the n values obtained from two routes is because the exponent values β and γ determined from the MAP method are much different from those expected for the MFT In other words, La0.7Ca0.3 Mn1 À xNi xO3 samples are not conventional ferromagnets There are the magnetic inhomogeneity, and the existence of FOMT and/or SOMT properties (particularly for two samples with x ¼0.09 and 0.12 lying in the crossover region) For conventional ferromagnets obeyed the MFT, n is equal to 2/3 However, experimental results based on the framework of the SOMT (MFT) theory for inhomogeneous ferromagnets, like the present cases, introduce the values n different from 2/3 [23,24] 107 (Oe) Fig (Color online) Scaling performance of M/|| versus H/|| ỵ γ in the log scale at temperatures T o TC and T TC for La0.7 Ca0.3Mn1 À xNixO3 with (a) x ¼ 0.09, (b) x ¼ 0.12 and (c) x ¼ 0.15 The insets plot the same data in the linear scale of the FOMT makes of their exponent values different from those expected for the theoretical models, such as the cases of La0.7Ca0.3 MnO3 and La0.9 Te0.1MnO [5,31] With a higher Nidoping content of x ¼0.15, one can see that its β value ( ¼0.320) is close to that expected for the 3D Ising model (β ¼0.325), indicating the existence of short-range FM order associated with the magnetic inhomogeneity, and FM/anti-FM mixed phase It comes to our attention that the β value tends to shift towards the values of the Heisenberg model and MFT if Ni content (x) in La 0.7Ca0.3 Mn1 À x NixO is higher than 0.15 For inhomogeneous ferromagnets, the critical values usually depend on the magnetic field ranges employed for MAP analyses because of a significant field-induced change in the nature and range of the FM interaction [9,32] Performing a renormalization group analysis of exchange-interaction systems, Fisher et al found the exponent values depending on the range of exchange interaction characterized by J(r) ẳ 1/r d ỵ s (where d, and s are the dimension of the system, and the interaction range, respectively) [33] The MFT exponents are valid for s o 1/ while the Heisenberg ones are valid for s The exponents Conclusions We studied the critical behavior and related physical properties of manganites La 0.7 Ca 0.3 Mn1 À x Ni x O (x ¼ 0.09, 0.12 and 0.15) around their T C values Detailed analyses of the M–H–T data based on the MAP method revealed the stability in value of γ E 1, demonstrating the real PM behavior above T C in the samples However, in the FM region, experimental results revealed the sample x ¼0.15 undergoing the SOMT Its β exponent is close to that expected for the 3D Ising model For the samples with lower Ni-doping contents, their exponents β ( ¼ 0.171 and 0.262 for x ¼ 0.09 and 0.12, respectively) indicate the samples exhibiting tricriticality associated with the FOMT– SOMT transformation; in which, the FOMT is dominant at the fields lower than H L Short-range FM interactions are thus found in all the samples Interestingly, around T C , field dependences of |Δ S max | can be described by a power law |Δ S max | p H n The n values ( ¼0.55–0.77) obtained from the power-law fitting are higher than those (n ¼ 0.33–0.48) calculated from the relation n ẳ ỵ( 1)/( þ γ ) We believe that the deviation of the n values obtained from two ways is related to the using of the approximate MFT (the MAP method) for unconventional ferromagnets (with the existence of the magnetic inhomogeneity, and FOMT and/or SOMT properties), where the exponent values β and γ determined are much different from those expected for the MFT T.-L Phan et al / Solid State Communications 184 (2014) 40–46 45 8 6 4 40 kOe |ΔSmax| α Hn (with n = 0.55) 30 kOe 2 20 kOe 200 210 220 230 10 20 30 40 6 4 40 kOe |ΔSmax| α Hn (with n = 0.68) 30 kOe 20 kOe 170 10 kOe 180 190 200 210 |ΔSmax| (Jkg-1K-1) 190 -ΔSm (J.kg-1.K-1) 10 kOe 0 10 20 30 40 3 40 kOe 2 30 kOe |ΔSmax| α Hn (with n = 0.77) 20 kOe 1 10 kOe 150 160 170 180 190 T (K) 10 20 30 40 H (kO e) Fig (Color online) À ΔSm(T) curves with the fields H¼ 10, 20, 30 and 40 kOe, and field dependences of |ΔSmax| at TC fitted to a power law |ΔSmax| p Hn for La0.7Ca0.3Mn1 À xNixO3 with (a, b) x¼ 0.09, (c, d) x¼ 0.12 and (e, f) x ¼ 0.15 Acknowledgment This research was supported by the Converging Research Center Program through the Ministry of Science, ICT and Future Planning, Korea (2013K000405), and by NAFOSTED of Vietnam (103.02-2010.38) References [1] A.P Ramirez, J Phys.: Condens Matter (1997) 8171 [2] A.J Millis, B.I Shraiman, R Mueller, Phys Rev Lett 77 (1996) 175 [3] J Mira, J Rivas, L.E Hueso, F Rivadulla, M.A Lopez Quintela, J Appl Phys 91 (2002) 8903 [4] A Berger, G Campillo, P Vivas, J.E Pearson, S.D Bader, E Baca, P Prieto, J Appl Phys 91 (2002) 8393 [5] H.S Shin, J.E Lee, Y.S Nam, H.L Ju, C.W Park, Solid State Commun 118 (2001) 377 [6] S Taran, B.K Chaudhuri, S Chatterjee, H.D Yang, S Neeleshwar, Y.Y 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À xNixO3 with (a) x ¼ 0.09, (b) x ¼ 0.12 and (c) x ¼ 0.15 The insets plot the same data in the linear scale of the FOMT makes of their exponent values different from those expected for the theoretical... 0.7Ca0.3 Mn1 À xNixO It appears from the M–H data that there is no saturation magnetization value in spite of the H variation up to 40 kOe This is assigned to the existence of the magnetic inhomogeneity... online) M–H data and inverse Arrott plots for La0.7Ca0.3Mn1 À xNixO3 with (a, b) x ¼0.09, (c, d) x 0.12, and (e, f) x 0.15 of the MFT, we need to find other sets of the critical- exponent values