M. Dhahri*a,
J. Dhahria and
E. K. Hlilb
aLaboratoire de la matière condensée et des nanosciences, Univercité de Monastir, 5019, Tunisia. E-mail: mmay988205@gmail.com
bInstitut Neel, CNRS et Univercité Joseph Fourrier, B. P. 166, 38042 Grenoble, France
First published on 31st January 2018
A detailed study of the structural, magnetic, magnetocaloric and electrical effect properties in polycrystalline manganite La0.5Sm0.1Sr0.4Mn0.975In0.025O3 is presented. The X-ray diffraction pattern is consistent with a rhombohedral structure with Rc space group. Experimental results revealed that our compound prepared via a sol–gel method exhibits a continuous (second-order) ferromagnetic (FM) to paramagnetic (PM) phase transition around the Curie temperature (TC = 300 K). In addition, the magnetic entropy change was found to reach 5.25 J kg−1 K−1 under an applied magnetic field of 5 T, corresponding to a relative cooling power (RCP) of 236 J kg−1. We have fitted the experimental data of resistivity using a typical numerical method (Gauss function). The simulation values such as maximum resistivity (ρmax) and metal–semiconductor transition temperature (TM–Sc), calculated from this function, showed a perfect agreement with the experimental data. The shifts of these parameters as a function of magnetic field for our sample have been interpreted. The obtained values of β and γ, determined by analyzing the Arrott plots, are found to be TC = 298.66 ± 0.64 K, β = 0.325 ± 0.001 and γ = 1.25 ± 0.01. The critical isotherm M (TC, μ0H) gives δ = 4.81 ± 0.01. These critical exponent values are found to be consistent and comparable to those predicted by the three-dimensional Ising model with short-range interaction. Thus, the Widom scaling law is fulfilled.
Also, manganese oxides especially, the La–Sr based manganite (La0.7Sr0.3MnO3) exhibit a metal–semiconductor transition (MSc) accompanied by a FM–PM transition near TC. In a recent publication4,5 there have been few models that can explain the transport mechanism in manganites. Among them, we point out the small polaron hopping (SPH) model and the 3D Mott's variable range hopping (VRH) in the semiconducting region, the adiabatic small polaron hopping mechanism6 and electron–electron, electron–phonon processes in the metallic region. These laws are very important in manganite research because they very well describe the observed high-temperature variation (T > TM–Sc) in the conduction mechanism.
However, there is still no clear conclusion of whether or not the resistivity ρ(T) can be continuously predicted by temperature from the metal phase to the semiconductor phase for individual manganite with only one equation.
Unfortunately, few studies have been done on the mathematical model which can describe the carrier transport behavior of manganite as a function of temperature around the metal semiconducting transition and the relation between magnetic and electrical properties. In this contribution, we have determined the correlation between electrical and magnetic properties and we have developed a mathematical model to quantitatively analyze the temperature-dependent resistivity.
Many previous reports were dedicated that the most accepted interpretations for the origin of these properties are the double exchange model and Jahn–Teller effect7 which they are used to identify the magnetic phase transitions (FM–PM).
Therefore, to understand better the relation between CMR effect and the semiconductor–metal transition, two important questions about FM–PM transition should be clarified: one is the order of phase transition; the other is the common universality class. A most useful approach is the consideration of the critical exponents. These describing the thermodynamic properties near the phase transition can be used to elucidate interactions mechanisms near TC. In earlier theoretical works,8,9 the critical behavior related to the FM–PM transition in manganites within the DE model was first described with long-range mean-field theory. Later, depending on the computational technology for the CMR of manganites, Motome and Furukawa suggested that the FM–PM transition should belong to Heisenberg's universality Class.10,11 However, some research have predicted that the critical exponents in manganites are in agreement with a short range exchange interaction model with the estimated critical exponent values related to either 3D-Heisenberg or 3D-Ising model.
Critical exponents for manganites show wide variation that almost covers all universality classes and different experimental tools are used for their determination. Ghosh et al.12 reported that the calculated values of the critical exponent β is equal to 0.37 for the manganite La0.7Sr0.3MnO3. However, La0.8Sr0.2MnO3 is in good agreement with that in mean-field model13 with a relative high value of β (=0.5). While a very low critical exponent of β = 0.14 identified in the single crystal La0.7Ca0.3MnO3 suggested that the FM–PM transition in this system is of a first rather than second order type.14 With this variety in mind, it is worthwhile to study the critical behavior in the same perovskite manganite.
The present work aims basically at investigating the structural, magnetic, magnetocaloric, electrical and the critical behavior for La0.5Sm0.1Sr0.4Mn0.975In0.025O3 (LSSMIO) manganite. We used four kinds of different theoretical models, which are mean field, 3D-Heisenberg, 3D-Ising, and tricritical mean field to explain the critical behavior in the manganite.
In this method, the stoichiometric amounts of high purity nitrate Sr(NO3)2·6H2O; La(NO3)3·6H2O; Mn(NO3)2·4H2O; Sm(NO3)3·6H2O and In(NO3)3·xH2O precursors were taken as starting materials in appropriate stoichiometric ratio powder and pH was adjusted between 6.5 and 7. In the first step, the precursor solution was prepared by dissolving the constituents (precursors/starting materials) with desired composition in deionized water. In the next step, the homogeneous precursor solution was heated to 90 °C under constant stirring to eliminate the excess water and get a dry fluffy porous mass. Subsequently, the obtained sol was cooled before the addition of ethylene glycol (EG) (1:1; EG:CA) and citric acid (CA) (CA:metal ion molar ratios of 1, 2 or 3) which they were used as polymerization/complexation (PC) agents. The process was heated first at 340–380 K with a vigorous stirring to evaporate water, accelerate the poly-esterification reaction between CA and EG and increase viscosity. Then the temperature was raised up to 450 K forming a dark viscous gel which slowly turned into a dark resin. This resin was easily powdered in an agate mortar and was calcined at 600 K for 7 h in oxygen atmosphere to eliminate the other organic compounds the carbons gases and give a fine powder. Finally, the resulting powder was uniaxial pressed at 105 Pa into pellets with a thickness of 2 mm and diameter of 8 mm. The obtained black pellets were sintered in air at 900 °C for 15 h.
Indeed, the microstructure was observed by a scanning electron microscope (SEM) using a Philips XL30 and semi-quantitative analysis was performed at a 20 kV accelerating voltage using energy dispersive X-ray analyses (EDAX). To extract the critical exponent of the sample accurately, the magnetic measurements were performed in the range of 0–5T, near the FM to PM phase transition using a BS1 and BS2 magnetometer developed in Louis Neel Laboratory, Grenoble. In fact, the isothermals are corrected by a demagnetization factor Da that has been determined by a standard procedure from the low-field linear-response regime at a low temperature (μ0Happl − DaM).
Parameters | La0.5Sm0.1Sr0.4Mn0.975In0.025O3 |
Structure type | Rhombohedral |
Space group | Rc |
Lattice parameter | |
a (Å) | 5.478(2) |
c (Å) | 13.379(5) |
Vunit cell (Å3) | 347.69 |
(O)Biso (Å2) | 1.530 |
(O)x | 0.4460(5) |
(La, Sm, Sr)Biso (Å2) | 0.786(3) |
(Mn, In)Biso (Å2) | 0.543(4) |
Discrepancy factors (%) | |
Bragg R-factor | 2.14 |
Rp | 6.7 |
Rwp | 9.1 |
RF-factor | 5.8 |
Goodness of fit χ2 | 1.68 |
The average crystallite size values have been estimated from the full width at half maximum of X-ray diffraction peaks and calculated using Scherer's equation given as:16
(1) |
βSch = βobs2 − βinstr2 | (2) |
(β2)instr = Utan(θ)2 + Vtan(θ) + W | (3) |
The DS value is found to be 87 nm. The SEM micrograph of our sample is shown in Fig. 2(b). Moreover, the average grain size was calculated using the average size linear intercept method from the micrograph. The value is DSEM = 210 nm. We note that the grain size obtained by SEM (DSEM) is much larger than that calculated by Scherrer's formula which can be explained by the fact that each particle observed by SEM is formed by several crystallized grains.18
In order to check the existence of all elements in these compounds, energy dispersive X-ray analysis (EDAX) was carried out at room temperature. EDAX spectrum represented in Fig. 2(a) reveals the presence of La, Sm, Sr, Mn, In and O elements, which confirms that there is no loss of any integrated elements during the sintering within experimental errors. The typical cationic composition for the sample is represented in Table 2. EDAX analysis shows that the chemical composition of the sample is close to the nominal one within the experimental uncertainties.
Chemical species | Nominal composition | ||||
---|---|---|---|---|---|
La | Sm | Sr | Mn | In | |
0.499 | 0.101 | 0.389 | 0.974 | 0.026 | La0.5Sm0.1Sr0.4Mn0.975In0.025O3 |
M(T) = M0 + M3/2T3/2 + M2T2 | (4) |
Fig. 3 Temperature dependence of magnetization M(T) measured at 0.05T for LSSMIO sample and the solid line (black color) is the nonlinear curve fit following eqn (4). The (a) inset shows the plot of dM/dT as a function of temperature at μ0H = 0.05T and the (b)inset shows the temperature dependence of the inverse magnetic susceptibility. |
In order to better understand the magnetic behavior of our sample, we have fitted the inverse of the susceptibility as a function of temperature χ0−1(T) defined as (M = χH) (inset. b Fig. 3), using the following Curie–Weiss (CW) law:
(5) |
(6) |
Generally, the difference between (θCW = 304 K) value and (TC = 300 K) value depends on the substance and is associated with the presence of short-range ordered slightly above TC, which may be related to the presence of a magnetic inhomogeneity. From the determined Curie constant C, we have deduced the experimental effective moment μexpeff using the following relation:23
(7) |
μeff(S) = gμB[S(S + 1)]1/2 | (8) |
Then, we have calculated the effective paramagnetic moment per formula unit which can be written as:
(9) |
The μexpeff and μtheoeff values are 4.85 and 4.47, respectively. The experimental value of the effective paramagnetic moment is higher than the theoretical one. It is the signature of Mn4+ and Mn3+ clusters,24 which can be explained by the presence of a short-range magnetic order in the paramagnetic phase. So that in this phase, the magnetic spins do not exist as individuals, they rather exist in small groups.
Magnetization versus magnetic field (M − μ0H) curve of LSSMIO compound at 5 K is plotted in Inset. a Fig. 4. The sample closely reach a constant value of magnetization under an applied field μ0H = 1.5T. The estimated magnetic moments from magnetization data at 5 K is 3.45μB per formula unit. A rough estimation of the expected magnetic moment can be made, based on the chemical formula La0.53+Sm0.13+Sr0.42+(Mn0.6 − x3+Mn0.44+)Inx3+O32− leading to a magnetic moment: Msp = (4 × (0.6 − x) + 3 × 0.4)μB = (3.6 − 4x)μB, as Mn3+ and Mn4+ ions have magnetic moments of 4μB and 3μB respectively. The calculated value of magnetic moment per formula unit is 3.5μB for our sample.
The magnetocaloric effect MCE which is an intrinsic property of all magnetic materials, is the tendency of the material to heat up or cool down during the application or removal of a magnetic field. The applied importance of the MCE is easily appreciated from the fact that for many years it has been used successfully to reach ultra-low temperatures in a research environment.25,26 Besides, it is maximized when the material is near its magnetic ordering temperature (Curie temperature TC). In order to examine this property we have carried out the isothermal M(μ0H) measurements at different temperatures (with temperature interval δT = 4 K) in the FM–PM transition region for the sample (Fig. 4). To guarantee that magnetization data were determined in isothermal conditions, the sweep rat of the magnetic field was set slowly enough. The magnetization curve of LSSMIO manganite at temperatures below TC exhibit sharp increase of magnetization at low fields and then a gradual saturation at high fields reflecting a paramagnetic behavior.27 This magnetization is all the smaller as the temperature is high, which means that the thermal agitation is important.
In order to enquire the efficiency of our sample in the magnetic refrigeration systems, the magnetic entropy change ΔSM(T, μ0H) due to the application of a magnetic field μ0H can be calculated from a family of isothermal M–μ0H curves, using the following formula:
(10) |
From Maxwell relation (−ΔSM) was induced by changing the magnetic field from zero to (μ0H):
(11) |
(12) |
Using eqn (12), we have calculated the magnetic entropy change under different field changes for LSSMIO manganite as seen in Fig. 5. The change of magnetic entropy of a magnetic material has the largest value near a phase transition, where the magnetization changes rapidly with temperature.28,29
The peak magnitude increases with the increase in the applied magnetic field μ0H but the peak position is closely unaffected because of the second order nature of the ferromagnetic transition in this compound. It should be noted that for each magnetic applied field (−ΔSmaxM) reaches the maxima value at the Curie temperature (TC = 300 K). The values of (−ΔSmaxM), which increases with increasing the applied magnetic field, are 5.25 and 2.11 kg−1 K−1 upon a magnetic field change of 5T and 2T, respectively.
On the other hand, magnetic refrigerants are desired to have not only a large (−ΔSM) but also a large refrigerant relative cooling power (RCP). This factor corresponds to the amount of heat per kilogram that can be transferred between the cold and hot tanks during an ideal refrigeration cycle and defined as:30
RCP = −ΔSmaxM × δTFWHM | (13) |
The RCP values as well as the maximum values of the magnetic entropy change under a magnetic applied field of 5T are summarized in Table 3. We remarque that the RCP factor undergo a moderate increase with the amplification of the magnetic field. The significant value of the RCP would confirms the transport of a greater amount of heat in an ideal refrigeration cycle. We can see that these results are interesting compared with other compounds31–38 reported in the literature (Table 3), so we can estimate that our compound is a potential candidate to be used in the magnetic refrigeration.
Material | TC (K) | μ0H (T) | ΔSmaxM (J kg−1 K−1) | RCP (J kg−1) | Ref. |
---|---|---|---|---|---|
La0.5Sm0.1Sr0.4Mn0.975In0.025O3 | 300 | 5 | 5.25 | 236 | This work |
Gd | 293 | 5 | 9.5 | 410 | 31 |
Gd5Si2Ge2 | 275 | 5 | 18.5 | 535 | 32 |
La0.7Sr0.3MnO3 | 370 | 5 | 5.15 | 252 | 33 |
La0.7Ca0.3MnO3 | 264 | 5 | 7.7 | — | 34 |
La0.7Ca0.2Sr0.1MnO3 | 308 | 5 | 7.5 | 374 | 35 |
La0.57Y0.1Ba0.23Ca0.1MnO3 | 300 | 5 | 4.34 | 349 | 36 |
La0.67Sr0.33Mn0.9Cr0.1O3 | 328 | 5 | 5 | — | 37 |
La0.7Ca0.1Pb0.2Mn0.9Al0.05Sn0.05O3 | 295 | 5 | 2.3 | 135 | 38 |
La0.7Ca0.1Pb0.2Mn0.85Al0.075Sn0.075O3 | 290 | 5 | 2 | 176 | 38 |
Franco et al.39 proposed that the phenomenological universal curve is made by normalizing all the magnetic entropy change (ΔS(T, μ0H)/ΔSmaxM). Here, ΔSmaxM presents the peak maximum of the magnetic entropy change at different magnetic fields (shown in Inset. Fig. 5) and by rescaling the temperature axis, namely (θ), below and above TC, as noted:
(14) |
Knowing that Tr1 and Tr2 present the temperatures of two reference points corresponding to ΔSM(Tr1,2) = 1/2ΔSmaxM. It's clear from this figure that the experimental points distribute on one universal curve. The existence of the universal curve of (−ΔSM) around TC confirms the second nature order phase transition.
(15) |
If the Gauss function (14) is available for predicting resistivity at temperatures across the measurement range for our compound, the minimum value of ρ is given by ρ(T|T→∞). Thus, the physical significance of parameter ρ(Tu) is the resistivity of manganite materials at high temperatures. The maximum value of ρ is given by the parameters ρ(Tu), A, and w in the following form:
(16) |
So, the Gauss function (14) will be rewrite as:
(17) |
We analyses the resistivity curves of LSSMIO compound at different magnetic field using the above approach (17). Optimized parameters employed to simulate the component spectra are also listed in Table 4 and the fitted curves are shown in Inset. a Fig. 6.
The correlation coefficient R2 (Table 4) which is close to 1, showed a satisfactory agreement between experimental and the modeled data which implies that the metal–insulator transition temperature, TM–Sc can be confirmed more precisely by the Gauss function simulation for our compound. The comparison between the peaks of the best-fitted value of Td, determined from Gauss function, and the experimental data demonstrated that parameter Td corresponds to the metal–semiconducting transition temperature, TM–Sc.
Thus, the transition temperature TM–Sc can be confirmed more precisely for other magnetic fields using the appropriate Gauss function simulation.
The Fig. 7 illustrates the dependency of ρmax on the applied magnetic field for LSSMIO sample. The best fit of this curve show that the successful logistics equation could properly give a quantitative relationship between ρmax and magnetic field μ0H via nonlinear curve fitting, and the logistic function is given by:
(18) |
Fig. 7 Experimental and simulated ρmax as a function of magnetic field. The inset shows TM–Sc vs. μ0H for LSSMIO compound. |
The optimized parameters A, B, C and P will be determined from the fitting of the experimental data (Table 5).
A | B | C | P | S | I | |
---|---|---|---|---|---|---|
La0.5Sm0.1Sr0.4Mn0.975In0.025O3 | 0.426 | −0.265 | 1.231 | 2.632 | 3.628 | 296.095 |
The plots of ρmax versus magnetic field for LSSMIO compound have been fitted with the logistic model, which successfully describes the experimental behavior of the maximum resistivity ρmax of our sample. According to this model, it's clear that ρmax decreases with an increased magnetic field μ0H, which implies a logistic increase in the carrier density. For this reason, it is appropriate to use the logistic model for predicting the maximum resistivity ρmax before one magnetic field is applied.
It's clear from Fig. 6 that TM–Sc shifts toward higher temperatures when μ0H increases, which confirm that there is a relationship between TM–Sc and the applied magnetic field. Inset. Fig. 7 shows the construct of the TM–Sc of LSSMIO compound as a function of applied magnetic field. We can see that there is a linear relationship between the two variables.
Therefore, the functional form between TM–Sc and magnetic field is expressed by a line eqn (19), and it is given by:
TM–Sc(μ0H) = S × μ0H + I | (19) |
The obtained constants are illustrated in Table 5. Notably, it can be observed that the theoretical results of TM–Sc derived by eqn (19) are consistent with the experimental data in Table 4. Therefore, it may be stated that the magnetic field is correlative with TM–Sc. In addition, it is appropriate to use eqn (19) in forecasting TM–Sc before one magnetic field is applied.
The shifts of the TM–Sc to the high-temperature range with the increase of the applied magnetic field can be explained by the reduction in the charge carriers delocalization uniformly caused by the applied magnetic field, which in turn might result in reducing the resistivity and also cause local ordering of the magnetic spins in the same way. Due to this linear ordering, the FM metallic state may suppress the PM insulating regime. Therefore, it may be stated that the conduction electrons (eg1) are completely polarized inside the magnetic domains hence, the peak temperature (TM–Sc) shifts to the high temperature side with applications of the magnetic field.
(20) |
Inset b Fig. 6 presents the variation of MR as a function of the temperature at different applied magnetic fields (2–5T). It is very interesting to note from this figure that MR presents a pic around TC then it gradually decreases at high temperatures. The maximum magnetoresistance values of our compound at the metal–semiconducting (MSc) transition temperature are found 52%, 58%, 63% and 69% under applied magnetic field of 2, 3, 4 and 5T, respectively.
The reason for the higher MR% observed at high temperature is attributed to the involved spin polarized tunneling between grains or spin dependent scattering of polarized electron at grain boundaries.41,42
Ms(T) = M0|ε|−β, ε < 0, T < TC | (21) |
χ0−1(T) = (h0/M0)εγ, ε > 0, T > TC | (22) |
M = DH1/δ, ε = 0, T = TC | (23) |
In our work, we have used different methods to investigate the critical behavior of the LSSMIO sample, namely the modified Arrott plots (MAP) method, the Kouvel–Fisher method (KF) and critical isotherm analysis (CI).44,45 The first method used to calculate the critical exponents is the MAP method (also called Arrott–Noakes plots). In this technique, the M = f (μ0H) data is converted into series of isothermal (M1/β vs. (μ0H/M)1/γ) depending on the following relation:44
(μ0H/M)1/γ = (T − TC)/T1 + (M/M1)1/β | (24) |
Fig. 8(a–c) shows the modified Arrott plots (MAP) based on the Arrott–Noakes equation of state eqn (24), at different temperatures by using the 3D Ising model (β = 0.325, γ = 1.240), the 3D Heisenberg model (β = 0.365, γ = 1.336) and the tricritical mean field (β = 0.25, γ = 1), respectively for LSSMIO compound. In order to select the best model which describes this system, we calculated their relative slopes (RS) which are defined as: RS = S(T)/S(TC) (where S(T) is the slope of the quasi-straight line in the high-field region at T). In the most ideal case, all RSs should be equal to 1 because the modified Arrott plot is a series of parallel straight lines.
Fig. 8(d) shows the RS vs. T curve for LSSMIO sample for the four models, mean field model, 3D-Heisenberg, 3D-Ising and tricritical mean field model. The RS of 3D-Heisenberg and tricritical mean-field models obviously deviates from the straight line of RS = 1 but the RS of 3D-Ising model is around to this line. Therefore, the first Arrott plot gives the best results among these three models, indicating the critical properties of LSSMIO compound can be described with 3D-Ising model.
Based on these isotherms, the spontaneous magnetization MS(T, 0) and the inverse susceptibility χ0−1(T) data are extracted from the linear extrapolation from the high-field region to the intercepts with the axes M1/β and (μ0H)1/γ, respectively. In Fig. 9(a) we have plotted the temperature dependence of MS(T, 0) as green squares and χ0−1(T) as blue squares with their fitting curves using eqn (21) and (22), respectively. We find that the final fitted curves reproduce the experimental data perfectly and give two sets of critical exponents (Table 6). It can be seen that TC obtained from the critical analysis of the modified plot agrees well with that obtained from the M(T) curves in Fig. 3, and the obtained critical exponents are very close to those in the 3D Ising model.
Material | Method | TC (K) | β | γ | δ | Ref. |
---|---|---|---|---|---|---|
Mean-field model | Theory | 0.5 | 1.0 | 3.0 | 49 | |
3D-Heisenberg model | Theory | 0.365 ± 0.003 | 1.336 ± 0.004 | 4.80 ± 0.04 | 49 | |
3D-Ising model | Theory | 0.325 ± 0.002 | 1.241 ± 0.002 | 4.82 ± 0.02 | 49 | |
Tricritical mean-field model | Theory | 0.25 | 1 | 5 | 50 | |
La0.5Sm0.1Sr0.4Mn0.975 In0.025O3 | MAP | 298.66 ± 0.64 | 0.325 ± 0.001 | 1.25 ± 0.01 | This work | |
KF | 301.83 ± 0.43 | 0.327 ± 0.002 | 1.259 ± 0.001 | This work | ||
CI (cal) | — | — | — | 4.84 | This work | |
CI (exp) | — | — | — | 4.81 ± 0.01 | ||
La0.7Ca0.2Sr0.1Mn0.85Cr0.15O3 | 234.54 ± 0.6 | 0.322 ± 0.03 | 1.2 ± 0.17 | 4.752 | 51 | |
La0.6Sr0.4Mn0.9V0.1O3 | 356.407 | 0.316 | 1.243 | 4.947 | 52 | |
La0.6Ca0.2Sr0.2MnO3 | 344.456 | 0.498 | 1.053 | 2.992 | 53 | |
La0.57Nd0.1Sr0.33MnO3 | 352.23 | 0.368 | 1.191 | 4.236 | 54 | |
La0.7Ca0.2Sr0.1MnO3 | 284 | 0.394 | 0.925 | 3.34 | 55 | |
La0.75Sr0.25MnO3 | — | 0.40 ± 0.02 | 1.27 ± 0.06 | 4.12 ± 0.33 | 56 | |
La0.7Sr0.3MnO3 | 360.2 | 0.377 ± 0.004 | 1.168 ± 0.006 | 4.10 ± 0.01 | 57 | |
La0.7Sr0.3Mn0.95Co0.05O3 | 320.4 | 0.403 ± 0.005 | 1.159 ± 0.007 | 3.88 ± 0.01 | 57 | |
La0.7Sr0.3Mn0.9Co0.1O3 | 281.6 | 0.457 ± 0.007 | 1.114 ± 0.005 | 3.44 ± 0.01 | 57 | |
La0.7Sr0.3Mn0.85Co0.15O3 | 273.9 | 0.418 ± 0.004 | 1.187 ± 0.006 | 3.84 ± 0.01 | 57 | |
La0.5Sm0.1Sr0.4MnO3 | 313.36 ± 0.36 | 0.324 ± 0.01 | 1.240 ± 0.13 | 4.83 ± 0.01 | 58 |
As the next step in the scaling analysis, we have followed the Kouvel Fisher method to determine more accurately β, γ and TC.47
MS(T)[dMS(T)/dT]−1 = (T − TC)/β | (25) |
χ0−1(T)[dχ0−1(T)/dT]−1 = (T − TC)/γ | (26) |
After this method both MS[dMS/dT]−1 and χ0−1[dχ0−1/dT]−1 have a linear behavior with respect to T with slopes 1/β and 1/γ, respectively, as shown in Fig. 9(b).
One of the advantages of this method is that the value of the critical temperature is not introduced a priori but extracted from the intercept of the straight fitted lines on the temperature axis. The values of β, γ and TC obtained by KF method are also summarized in Table 6.
It is worth remarking how MAPs and the Kouvel Fisher method give close values of all critical parameters, confirming the robustness of the results. After eqn (23), the critical exponent δ can be extracted from the fitting of the critical isotherm to be compared with the values obtained from the scaling law (eqn (27)). Inset Fig. 10 shows the critical isotherm at T = 300 K in log–log scale as this should render a straight line (as it happens), whose slope is δ. We obtained δ = 4.81 ± 0.01 for LSSMIO compound (Table 6).
Fig. 10 Isothermal M vs. μ0H plot of LSSMIO sample at TC = 300 K; the inset shows the same plot in log–log scale and the solid line (red color) is the linear fit following eqn (19). |
Furthermore, according to the statistical theory, these three critical exponents have to obey the Widom scaling relation:48
(27) |
Using this scaling relation, the value of δ is equal to 4.84 for β and γ obtained from the MAP method. Thus, the critical exponents found in this study obey the Widom scaling relation remarkably well, implying that the obtained β and γ values are reliable.
To put our obtained results in the context of previous works, we summarize in the Table 6 the values of the critical exponents obtained for our sample, those expected from theoretical models49,50 and the previous reports on Sr-doped manganites. It is found that the value of (β = 0.325 ± 0.001) for our compound is quite close to that expected from 3D Ising model (β = 0.325 ± 0.002). Similar results have been found for other compounds such as La0.7Ca0.2Sr0.1Mn0.85Cr0.15O3 (ref. 51) and La0.6Sr0.4Mn0.9V0.1O3 (ref. 52) with β = 0.322 ± 0.03 and β = 0.316, repectively. However, few other Sr-doped compounds, listed in Table 6, have β values close to those of the mean field and 3D Heisenberg models, such as La0.6Ca0.2Sr0.2MnO3 with (β = 0.498 and γ = 1.053),53 La0.57Nd0.1Sr0.33MnO3 with (β = 0.368 and γ = 1.191)54 and La0.7Ca0.2Sr0.1MnO3 with (β = 0.394 and γ = 0.925).55 Besides, the critical exponent values of La0.75Sr0.25MnO3 compound56 are all between mean-field values and three-dimensional-(3D)-Ising-model values (β = 0.40 ± 0.02, γ = 1.27 ± 0.06), but those determined for La0.7Sr0.3Mn1−xCoxO3 sample57 do not belong to any universality class (β = 0.403 ± 0.005; γ = 1.159 ± 0.007 for x = 0.05) and (β = 0.457 ± 0.007; γ = 1.114 ± 0.005 for x = 0.1). To further understand the magnetic interactions in In-doped perovskite manganite, in earlier studies we have carefully investigated the critical behavior of the reference compound La0.5Sm0.1Sr0.4MnO3 (ref. 58) (β = 0.324 ± 0.01; γ = 1.240 ± 0.13). It belong to the same universality class (3D Ising model) and it show that short-range ferromagnetic order is present in the reference compound around the critical temperature.
Performing a renormalization group analysis of exchange interaction systems, Fisher et al.59 have found that the values of the critical parameters depend on the range of exchange interactions with the form J(r) = 1/rd+σ (d and σ are the dimension of the system and the interaction range, respectively). It has been argued that, if σ is greater than 2, the Heisenberg framework is valid for a 3D-isotropic ferromagnet, However, if σ is less than 3/2, it is the mean field framework, which is valid. In the intermediate range of 3/2 ≤ σ ≤ 2 the FM behavior belongs to varied classes such as (3D Ising and tricritical mean field model) which depend on σ.
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