Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 Contents lists available at ScienceDirect Solid State Communications journal homepage: www.elsevier.com/locate/ssc Criticalbehaviorof La0.7Ca0.3Mn1 À xNixO3 manganitesexhibitingthecrossoverof first- andsecond-orderphasetransitions Q1 Q2 Q3 The-Long Phan a,b,1, Q.T Tran b,e, P.Q Thanh b,c, P.D.H Yen b,d, T.D Thanh b,e, S.C Yu a,b 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 We used Banerjee0 s criteria, modified Arrott plots, andthe scaling hypothesis to analyze magnetic-field dependences of magnetization near the ferromagnetic–paramagnetic (FM–PM) phase-transition temperature (TC) of perovskite-type manganites La0.7Ca0.3Mn1 À xNixO3 (x ¼0.09, 0.12 and 0.15) In the FM region, experimental results for thecritical exponent β ( ¼0.171 and 0.262 for x ¼0.09 and 0.12, respectively) reveal two first samples exhibiting tricriticality associated with thecrossoverof first- andsecond-orderphasetransitions Increasing Ni-doping content leads to the shift ofthe β value ( ¼0.320 for x ¼ 0.15) towards that expected for the 3D Ising model (β ¼ 0.325) 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, andthe phase-transition type For thecritical exponent γ ( ¼0.976–0.990), the stability in its value demonstrates the PM behavior above TC ofthe samples Particularly, around TC of La0.7Ca0.3Mn1 À xNixO3 compounds, magnetic-field dependences ofthe maximum magnetic-entropy change can be described by a power law of |ΔSmax| p Hn, where values n ¼0.55–0.77 are quite far from those (n ¼0.33–0.48) calculated from the theoretical relation n ẳ1 ỵ ( 1)/( þ γ) This difference is related to the use ofthe mean-field theory for the samples exhibitingthe magnetic inhomogeneity & 2014 Published by Elsevier Ltd Keywords: A Perovskite manganites D Criticalbehavior D Magnetic entropy change Introduction It is known that hole-doped lanthanum manganitesof La1 À x(Ca, Sr, Ba)xMnO3 with xẳ 0.3 (corresponding to Mn3 ỵ / Mn4 ỵ ẳ7/3) usually exhibit colossal magnetoresistance (MR) and magnetocaloric (MC) effects around their the ferromagnetic– paramagnetic (FM–PM) phase-transition temperature (the Curie temperature, TC) [1] With this doping content, double-exchange (DE) FM interactions between Mn3 ỵ and Mn4 ỵ are dominant as comparing with super-exchange anti-FM interactions of Mn3 ỵ Mn3 ỵ and Mn4 ỵ Mn4 ỵ pairs The strength of magnetic interactions thus depends on the average bond length 〈Mn–O〉, and bond angle 〈Mn–O–Mn〉 ofthe 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 symmetry of MnO6 octahedra, it has been noted that cooperative Jahn–Teller distortions are E-mail address: ptlong2512@yahoo.com (T.-L Phan) Tel.: ỵ82 43 261 2269 present in an orthorhombic structure rather than in the rhombohedral one [3] Among hole-doped manganites, orthorhombic La0.7Ca0.3MnO3 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 TC 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 TC is followed up with structural changes, and is known as a firstorder 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 thecritical exponents β, γ, and δ associated with temperature dependences ofthe spontaneous magnetization, Ms(T), inverse initial susceptibility, χ0–1(T), andcritical isotherm at TC, respectively [20,21] Distinguishing the FOMT from the SOMT can be based on the criteria proposed by 0038-1098/$ - see front matter & 2014 Published by Elsevier Ltd http://dx.doi.org/10.1016/j.ssc.2013.12.032 Please cite this article as: T.-L Phan, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2013.12.032i 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 Three polycrystalline perovskite-type manganites La0.7Ca0.3 Mn1 À xNixO3 with x ¼0.09, 0.12 and 0.15 were prepared by solidstate reaction as using purity commercial powders La2O3, CaO, NiO, and MnCO3 (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 pre-annealing, three mixtures were re-ground and pressed into pellets, and annealed at 1300 1C for 72 h in air For reference, the parent compound La0.7Ca0.3MnO3 was also prepared with the same conditions as described X-ray diffraction (XRD) patterns ofthe 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 La0.7Ca0.3Mn1 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 Ni2 ỵ , Ni3 ỵ , and/or Ni4 ỵ ) 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 ¼100 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 ofthe 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 La0.7Ca0.3Mn1 À xNixO3 It appears from the M–H data that there is no saturation magnetization value in spite ofthe H variation up to 40 kOe This is assigned to the existence ofthe 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 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] (004)/(242) (123)/(321) 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 Experimental details (202)/(040) (022)/(220) Intensity (arb units) (121) Banerjee [22], who performed H/M versus M2 curves (where H is the field, and M is the magnetization) in the vicinity of TC, and then suggested that their positive or negative slopes are indication of a second- or first-order phase transition, respectively Reviewing previous studies, one can see that many works focused on La0.7Ca0.3MnO3-based materials showing the FOMT and/or SOMT However, thecrossover region from first-order to second-orderphase transitions, and some related physical properties, such as the magnetic entropy change versus T and H, ΔSm(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 ẳ1 ỵ( 1)/( ỵ ) [23], and from a power law |ΔSmax(H)| p Hn [24] (where |ΔSmax| is the maximum magnetic entropy change around TC) To get more insight into the above problems, we prepared La0.7Ca0.3Mn1 À xNixO3 compounds, and have studied their critical behaviors upon Banerjee0 s criteria [22], modified Arrott plots andthe scaling hypothesis [20,21] The determined critical values are then discussed together with the magnetic ordering parameter n Normalized M 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 T.-L Phan et al / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 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 ofthe 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 According to the mean-field theory (MFT) proposed for a ferromagnet exhibitingthe SOMT and long-range FM interactions [26], the free energy GL is expanded in even powers of M: GL ẳ aM2 ỵ bM4 ỵ HM, where a and b are temperature-dependent parameters Minimizing GL as ∂GL/∂M¼0 results in the relation H/M ẳ 2a ỵ4bM2 It means that if magnetic interactions ofthe 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 longrange type Thecritical exponents β ¼0.5 and γ ¼ 1.0 (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 ofthe MFT, we need to find other sets ofthe critical-exponent values reflecting more frankly the magnetic properties ofthe samples This work is based on the modified Arrott plot (MAP) method [20], which is generalized by the γ β scaling equation of state, (H/M)1/ ẳc1 ỵc2M1/ , where c1 and c2 Please cite this article as: T.-L Phan, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2013.12.032i 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 T.-L Phan et al / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 182 K 80 182 K ΔT = K 60 220 K 220 K 40 20 10 20 30 M (emu/g) 80 40 50 168 K 168 K 60 8 206 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) ΔT = K 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 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 ¼ 0.09, (c, d) x¼ 0.12, and (e, f) x ¼0.15 are temperature-dependent 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 ẳ DH 1=δ ; ε ¼ 0; ε o 0; ε 40; ð1Þ ð2Þ ð3Þ where M0, h0, and D are thecritical amplitudes Additionally, according to the static-scaling hypothesis [21], M is a function of ε and H, MðH; εÞ ẳ jj f H=jj ỵ ị This equation reflects that, with β β γ determined β and γ values, plotting M/ versus H/ ỵ makes all data points falling on the f and f ỵ branches for T oTC and T 4TC, respectively Here, determining thecritical 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 β γ parallel straight lines in the performance of M1/ versus (H/M)1/ 1/β 1/γ Moreover, the M versus (H/M) line passes through the origin at TC Similar to the MFT case, our analyses indicated that the exponent values β ¼0.365 and γ ¼ 1.336 expected for the 3D Heisenberg model [21] not match with the descriptions ofthe MAP method Only β ¼ 0.25 and γ ¼ 1.0 expected for the tricritical MFT model (T-MFT), and β ¼ 0.325 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 x¼ 0.09 and 0.12, and for x¼ 0.15, respectively With these trial values, Ms(T) and χ0(T) data would be obtained from the linear extrapolation in the high-field region for the isotherms to the co-ordinate axes of β γ γ M1/ and (1/χ0)1/ ¼(H/M)1/ , respectively The Ms(T) and χ0(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 TC values ofthe samples obtained from M–T measurements were also used as reference in the fitting With such the careful comparison, only the sets ofcritical parameters with TC E199.4 K, β ¼0.171 70.006 and γ ¼0.9767 0.012 for x ¼ 0.09; TC E184.4 K, β ¼0.262 70.005 and γ ¼0.979 70.012 for x¼ 0.12; and TC E170 K, β ¼0.320 70.009 and γ ¼ 0.9907 0.082 for x¼ 0.15 are in good agreement with the MAP descriptions, see Fig With the obtained critical exponents, the scaling perforβ β γ mance 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 oTC and T 4TC, respectively These results prove the reliability in value ofthecritical 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 Please cite this article as: T.-L Phan, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2013.12.032i 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 TC = 199.5 ± 0.1 β = 0.171 ± 0.006 185 190 195 200 205 210 40 30 170 215 TC = 184.3 ± 0.1 γ = 0.979 ± 0.012 TC = 184.5 ± 0.1 β = 0.262 ± 0.005 175 180 190 185 195 200 -1 TC = 199.4 ± 0.3 γ = 0.976 ± 0.012 50 χ0 (x102, Oe.g/emu) 64 Ms (emu/g) 72 56 60 50 40 30 TC = 169.6 ± 0.5 K γ = 0.990 ± 0.082 TC = 170.0 ± 0.1 K β = 0.320 ± 0.009 155 160 165 170 175 180 185 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) x ¼0.09, (b) x¼ 0.12, and (c) x ¼0.15 x104 12 186 K x = 0.09 β = 0.171 γ = 0.976 214 K 0 200 400 M1/β (x106, emu/g)1/β 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 T.-L Phan et al / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 174 K 0.6 β = 0.262 γ = 0.978 20 10 600 200 K 200 400 600 800 152 K x = 0.15 β = 0.320 γ = 0.990 0.4 180 K 0.2 0.0 x = 0.12 30 200 400 600 800 1000 (H/M)1/γ (Oe.g/emu)1/γ Fig (Color online) MAPs of M1/β versus (H/M)1/γ with thecritical exponents obtained for La0.7Ca0.3Mn1 À xNixO3 with (a) x ¼0.09, (b) x¼ 0.12 and (c) x ¼ 0.15 12 kOe for x ¼0.09, 0.12 and 0.15, respectively Atthe 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 ofthe 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 ofthe M–H data points (at the fields lower than HL) 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 ¼TC to Eq (3) Basically, the δ values determined from Eq (3) would be equal to those calculated from the Widom relation δ ẳ1 ỵ / [21] In our work, 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 La0.7Ca0.3Mn1 À xNixO3, there is a shifting tendency ofthe exponent values (β, γ and δ) towards those ofthe MFT (with β ¼0.5, γ ¼1 and δ ¼ 3) This is tightly related to the FOMT–SOMT transformation The better applicability ofthe MAP method has been found for the samples with high-enough Ni concentrations as x 40.12 We believe that the substitution 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 Ni2 ỵ Ni2 ỵ , Ni3 ỵ Ni3 ỵ , Ni4 ỵ Ni4 ỵ , and/or Ni2 ỵ ,3 ỵ ,4 ỵ Mn3 ỵ ,4 ỵ ) beside pre-existing anti-FM interaction pairs of Mn3 þ –Mn3 þ and Please cite this article as: T.-L Phan, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2013.12.032i 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 T.-L Phan et al / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 102 102 M/|ε|β (emu/g) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 101 100 104 101 100 105 106 107 104 102 101 100 104 105 106 107 H/|ε|β+γ (Oe) Fig (Color online) Scaling performance of M/|| versus H/|| ỵ in the log scale at temperatures To TC and T 4TC for La0.7Ca0.3Mn1 À xNixO3 with (a) x¼ 0.09, (b) x¼ 0.12 and (c) x ¼ 0.15 The insets plot the same data in the linear scale Mn4 ỵ Mn4 ỵ These factors act as uctuations, reduce the strength of Mn3 ỵ Mn4 þ FM interactions (which thus reduce the magnetization and TC values), and influence thephase 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 ¼ 0.262 for x ¼0.12 is close to that expected for the T-MFT (β ¼ 0.25) This sample thus exhibits tricriticality associated with thecrossoverof first- and secondorder phasetransitions Similar results were also found in some manganites [9,12,29] A smaller value of β ¼ 0.171 for x¼0.09 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 ofthe FOMT makes of their exponent values different from those expected for the theoretical models, such as the cases of La0.7Ca0.3MnO3 and La0.9Te0.1MnO3 [5,31] With a higher Ni-doping content of x ¼0.15, one can see that its β value (¼ 0.320) is close to that expected for the 3D Ising model (β ¼0.325), indicating the existence of shortrange 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 ofthe Heisenberg model and MFT if Ni content (x) in La0.7Ca0.3Mn1 À xNixO3 is higher than 0.15 For inhomogeneous ferromagnets, thecritical values usually depend on the magnetic field ranges employed for MAP analyses because of a significant field-induced change in the nature and range ofthe 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/rd ỵ s (where d, and s are the dimension ofthe system, andthe interaction range, respectively) [33] The MFT exponents are valid for s o 1/2 while the Heisenberg ones are valid for s The exponents belong to other universality classes (such as the T-MFT and 3D Ising models) if 1/2 o s o2, which can be the case taking place in our samples Together with assessing thecritical 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 |ΔSmax| value, the linewidth ofthe À ΔSm(T) curves become broadened due to the FOMT–SOMT transformation, enhancing the refrigerant capacity (RC) Particularly, at TC the H dependences of |ΔSmax| can be well described by a power law of |ΔSmax|pHn [24], where values n¼0.55, 0.68, and 0.77 for x¼0.09, 0.12, and 0.15, respectively These values are different from those (n¼0.33, 0.41, and 0.48 for x¼ 0.09, 0.12, and 0.15, respectively) calculated from the relation nẳ1ỵ( 1)/( ỵ ) [23] As shown in Ref [34], n is known as a function of T, H and |ΔSm|, which can also be obtained from the relation n¼d ln|ΔSm|/d ln H Depending on the variation of these parameters, n would be different It reaches the minimum at temperatures in the vicinity of TC [23] We believe that a large deviation ofthe 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, La0.7Ca0.3Mn1À xNixO3 samples are not conventional ferromagnets There are the magnetic inhomogeneity, andthe existence of FOMT and/or SOMT properties (particularly for two samples with x¼0.09 and 0.12 lying in thecrossover region) For conventional ferromagnets obeyed the MFT, n is equal to 2/3 However, experimental results based on the framework ofthe SOMT (MFT) theory for inhomogeneous ferromagnets, like the present cases, introduce the values n different from 2/3 [23,24] Conclusions We studied thecriticalbehaviorand related physical properties ofmanganites La0.7Ca0.3Mn1 À xNixO3 (x ¼0.09, 0.12 and 0.15) around their TC values Detailed analyses ofthe M–H–T data based on the MAP method revealed the stability in value of γ E1, demonstrating the real PM behavior above TC in the samples However, in the FM region, experimental results revealed the Please cite this article as: T.-L Phan, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2013.12.032i 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 8 6 4 40 kOe |ΔSmax| α Hn (with n = 0.55) 30 kOe 2 20 kOe 10 kOe 190 200 210 220 230 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) -ΔSm (J.kg-1.K-1) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 T.-L Phan et al / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 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 10 T (K) 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| pHn for La0.7Ca0.3Mn1 À xNixO3 with (a, b) x¼ 0.09, (c, d) x¼ 0.12 and (e, f) x ¼ 0.15 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 ( ¼0.171 and 0.262 for x ¼0.09 and 0.12, respectively) indicate the samples exhibiting tricriticality associated with the FOMT–SOMT transformation; in which, the FOMT is dominant atthe fields lower than HL Shortrange FM interactions are thus found in all the samples Interestingly, around TC, field dependences of |ΔSmax| can be described by a power law |ΔSmax| p Hn The n values (¼ 0.55–0.77) obtained from the power-law fitting are higher than those (n¼ 0.33–0.48) calculated from the relation n ¼1 þ(β À1)/(β þ γ) We believe that the deviation ofthe n values obtained from two ways is related to the using ofthe approximate MFT (the MAP method) for unconventional ferromagnets (with the existence ofthe 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 Acknowledgment This research was supported by the Converging Research Center Program through the Ministry of Science, ICT and Future Planning, Korea (2013K000405) 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 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T.-L Phan, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2013.12.032i ... 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. .. curves (where H is the field, and M is the magnetization) in the vicinity of TC, and then suggested that their positive or negative slopes are indication of a second- or first-order phase transition,... (nẳ 0.330.48) calculated from the relation n ẳ1 ỵ( 1)/( ỵ ) 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