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Critical scaling and percolation model in La0.57Gd0.1Sr0.33Mn0.9In0.1O3 manganite
Journal of Alloys and Compounds 688 (2016) 1251e1259
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Critical scaling and percolation model in La0.57Gd0.1Sr0.33Mn0.9In0.1O3 manganite Mounira Abassi a, *, Asma Zaidi a, J. Dhahri a, E.K. Hlil b a b
Laboratoire de la Mati ere Condens ee et des Nanosciences, Universit e de Monastir, 5019, Tunisia Institut N eel, CNRS et Universit e Joseph Fourier, B.P. 166, 38042, Grenoble, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 April 2016 Received in revised form 29 June 2016 Accepted 30 June 2016 Available online 2 July 2016
We investigated the influence of indium (In) doping on the structural, magnetic and electrical properties of polycrystalline sample La0.57Gd0.1Sr0.33Mn0.9In0.1O3, prepared by solegel method. The crystallographic study showed that the sample crystallizes in the rhombohedral system with R3c space group. The relationship between the electrical and magnetic properties of manganites was traced through analyzing the dependence of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 resistivity on temperature. The measurement of resistivity versus the temperature r(T) of the sample showed that a ferromagnetic (FM) to a paramagnetic (PM) transition at Curie temperature (TC) was usually accompanied by a simultaneous metal-insulator (M-I) transition. Resistivity decreased with increasing the applied magnetic field. The values of TMSc were found to move towards a high temperature side with increasing the magnetic field. The estimated critical exponents are (b ¼ 0.272 ± 0.004; g ¼ 1.067 ± 0.002 and a ¼ 0.453). The values of these exponents obtained from resistivity were very close to those predicted by the tricritical mean-field model. These results were in good agreement with the analysis of the critical exponents from magnetization measurements. The electrical resistivity was fitted with the phenomenological percolation model, which is based on the phase segregation of the ferromagnetic metallic regions and the paramagnetic semiconductor regions. So, we found that the estimated results were in accordance with the experimental data. © 2016 Elsevier B.V. All rights reserved.
Keywords: Manganites Metal-insulator transition Resistivity Critical exponents Percolation model
1. Introduction Rare-earth perovskite manganites have been extensively studied thanks to their interesting electrical and magnetic properties [1,2]. Mixed manganites R1xAxMnO3 (R being trivalent rare earth ions, A ¼ Ca, Ba, Sr) have attracted much interest because these materials exhibit a rich phase diagram as a function of temperature [3], magnetic field [4], magnetocaloric effect (MCE) and illustrate the metal-insulator as well as the phenomenon of colossal magnetoresistance (CMR) [5]. The physical properties of perovskite manganites for technological applications such as magnetic refrigeration, electrical resistivity and very high-density magnetic recording (read heads for hard disks or magnetic memories (MRAM) non-volatile) have been studied in the framework of double exchange (DE) interaction for the whole doped manganite [6]. An important part of the work on manganites aimed to present
* Corresponding author. Tel.: þ216 25 012 603; fax: þ216 73 500280. E-mail address: [email protected] (M. Abassi). http://dx.doi.org/10.1016/j.jallcom.2016.06.293 0925-8388/© 2016 Elsevier B.V. All rights reserved.
the effect of CMR. However, the study of the anomalies of the various thermo physical properties, like the specific heat in the vicinity of the magnetic phase transition, with investigating the values of the universal critical parameters, has not been given the sufficient importance in research projects done on manganites. As far as manganites are concerned, there have been a lot of experiments to estimate the critical exponents of ferromagnetic manganites. Systems showing a second order metal insulator phase transition obey one of the common universality classes. By contrast, the experimental estimates of the critical exponents are still controversial including those for short-range Heisenberg interaction [7,8], the mean-field values [9,10] and those which cannot be classified into any universality class ever known [11]. Although a lot of work has been conducted to study the Mn-site doping effects, the critical exponents as very important parameters for magnetic transition are less available for the Mn-site doping. In this study, we carried out a systematic and detailed analysis of the magnetization measurement of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 polycrystalline sample near the ferromagneticeparamagnetic phase transition temperature. We have also determined the critical
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temperature TC and the critical parameters b and g. Then, we tried to investigate the critical behavior in LGSMIO at its semiconductoremetallic transition via the measurements of resistivity. The values of these exponents obtained from resistivity are very close to those predicted by the tricritical mean-field model. Finally, we intend to study the dependence of the electrical resistivity properties as a function of the LGSMIO, volume fraction, f ; which is analyzed in the framework of the percolation theory. A percolation model based on the mixed phase consisting of itinerant electrons and localized magnetic polarons has been proposed to explain the observed results. 2. Experimental details
Scherer formula:
DS ¼
kl bhkl cos q
(1)
where b is the breadth of the observed diffraction line at its half intensity maximum, K is the so-called shape factor (¼0.9) and l is the X-ray wavelength used. The value of (DS) is 35 nm. The average size of the particle of this sample is determined using the Scanning Electron Microscope (SEM) images by finding the minimum and maximum dimensions of the large number of particles (seen in the inset of Fig. 2). The value is 102 nm. The grain sizes observed by SEM were larger than those calculated by Scherrer’s formula. This can be explained by the fact that each particle observed by SEM is formed by several crystallized grains. In order to check the existence of all elements in this sample, energy dispersive X-ray analysis (EDAX) was carried out at room temperature. EDAX spectrum represented in Fig. 2 reveals the presence of La, Gd, 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 this sample is represented in Table 1. This measurement confirmed the cationic composition of this sample and the absence of secondary phases in the X-rays.
The microstructure of ceramic materials in general and CMR materials in particular was highly affected by the preparation routes and heat treatments. Sol-gel routes are known to produce very high quality, homogeneous and fine particle materials [12]. Bulk polycrystalline sample with a compositional formula La0.57Gd0.1Sr0.33Mn0.9In0.1O3 was prepared by the sol-gel process. In this method, the metal nitrates taken in stoichiometric ratios were dissolved in an aqueous solution. Citric acid was added in 1:1 ratio and the pH was adjusted to a value between 6.5 and 7.0 by adding ammonia. After getting a sol-gel on slow evaporation, a gelating reagent-ethylene glycol- was added and heated between 160 and 180 C to get a gel. On further heating, this solution yielded a dry fluffy porous mass (precursor), which was calcined at 700 C for 6 h. Then, the powder was pressed into circular pellets. These pellets were sintered at 900 C in air for 12 h. Phase purity, homogeneity and cell dimensions were determined by powder XRD. The phase analysis was carried out using FULLPROF program based on Rietveld method. 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 (EDX). Magnetizations (M) vs. temperature (T) were measured using a BS1 magnetometer. The isotherm magnetization vs. magnetic field (m0H) was recorded by the SQUID-VSM system. This latter was measured in the range of 0e5 T and with a temperature interval of 2 K in the vicinity of Curie temperature (TC). To obtain the metal-insulator transition temperature (TM-Sc) and to study the influence of the magnetic field on resistivity, the electrical resistivity and magnetoresistance measurements were done by standard dc four-probe technique using a closed cycle helium refrigerator cryostat in applied fields of 0 T, 2 T and 5 T.
Fig. 3(a) shows the temperature dependence of magnetizations using an applied field of 0.05 T. This curve reveals that the La0.57Gd0.1Sr0.33Mn0.9In0.1O3 oxide exhibits a magnetic transition from a ferromagnetic (FM) to a paramagnetic (PM) state when increasing temperature. This transition occurs at the Curie temperature (TC) which is obtained from the peak of the dM/dT curves. The Curie temperature TC is found to be 336 K, which is close to room temperature. In order to clarify the nature of the FMePM phase transition, we measured the magnetic-field dependence of magnetization at different temperatures (MeHeT), around Curie temperature TC. Fig. 3(b) shows the evolution of magnetization versus the applied magnetic field obtained at different temperatures (isothermal magnetization). Obviously, the M(H) curves were found to increase with decreasing temperature in the selected temperature range, where thermal fluctuation of spins decreased with decreasing temperature. Moreover, from the Arrot-plot (inset of Fig. 3(b)) in which all curves M2 vs. H/M have positive slopes, the magnetic phase transition was found to be of a second order according to Banerjee’s criterion [13].
3. Results and discussion
4. Critical parameters determined from magnetization data
3.1. Structural properties
We determined the values of the critical exponents of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 sample from the magnetization data versus temperature and magnetic field M (H, T), to understand their magnetic and magnetocaloric properties. Experimental results revealed that this sample exhibits a continuous (second-order) paramagnetic (PM) to ferromagnetic (FM) phase transition. In mean field theory at near TC, M2 vs H=M at various T should form a progression of parallel straight lines. The line for T ¼ TC passes through the origin on this plot. Moreover, the intersections of these curves for T_TC with the H=M axis gave the values of 1=c0 ðTÞ at H ¼ 0. In the present case the curves in the Arrott plot are non linear, indicating that the mean-field theory (b ¼ 0.5, g ¼ 1 and d ¼ 3) Fig. 4(a) characteristics of systems with long-range interactions is invalid for the present phase transition. Therefore, the values of MS ðTÞ and c1 0 ðTÞ were determined using a modified Arrott plot (Also called modified Arrott plots
The X-ray diffraction (XRD) pattern of the La0.57Gd0.1Sr0.33Mn0.9In0.1O3 oxide recorded at room temperature is shown in Fig. 1. Such a diffraction indicates that the sample crystallized in a rhombohedral structure with an R3c space group, in which the La/ Gd/Sr atoms are at 6a (0, 0, 1/4) position, Mn/In at 6b (0, 0, 0) position and O at 18 e (x, 0, 1/4) position. The cell parameters are found to be a ¼ b ¼ 5.41 Å and c ¼ 13.13 Å. The quality of the agreement is evaluated through the adequacy of the fit indicator c2 ¼ 1.5. 3.2. Morphological properties From the reflection of 2q values of XRD profile, we can also calculate the average grain size (DS) from the XRD peaks using
3.3. Magnetic properties
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Fig. 1. X-ray diffraction pattern (solid curve) and Rietveld refinement result (open symbols) for La0.57Gd0.1Sr0.33Mn0.9In0.1O3 at room temperature. The difference between the data and the calculation is shown at the bottom.
Fig. 2. EDAX spectrum for La0.57Gd0.1Sr0.33Mn0.9In0.1O3. The insets shows the typical SEM.
Table 1 Results of EDAX analysis of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 sample. Element
Typical cationic composition from EDX
Wt%
At.%
Nominal composition
La Gd Sr Mn In
0.568 0.101 0.332 0.901 0.102
33.876 6.886 12.452 21.205 4.979
11.361 2.021 6.640 18.021 2.042
La0.57Gd0.1Sr0.33Mn0.9In0.1O3
(MAP)). In this technique, the M ¼ f ðHÞ data are converted into series of isotherms ðM1=b ¼ f ðH=MÞ1=g Þ depending on the following relationship [14]:
ðH=MÞ1=g ¼
ðT TC Þ þ ðM=M1 Þ1=b T1
(2)
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the estimated critical exponents are (b¼ 0.282 ± 0.004; g ¼ 0.975 ± 0.002 at TC ¼ 332.57 ± 0.11). The deduced values of the critical exponents were close to the theoretical prediction of tricritical mean-field model. Here, the obtained critical exponents values can be verified with the prediction of the scaling theory using the MðH; εÞ ¼ εb f ±ðH=εbþg Þ equation, where ε is the reduced C temperature ε ¼ TT TC and fþ and f are regular analytical functions above and below TC [15]. According to this scaling law, the plots of M=jεjb as a function of H=jεjbþg fall into two universal curves; one for temperatures above TC and the other for temperatures below TC. Fig. 6(d) shows the scaling plot on a linear scale and the inset shows the same data in a logarithmic scale. Here in, we can clearly see that all points follow two curves, one for T < TC and the other for T > TC. As a consequence, the obtained values of the critical exponents are reliable and in agreement with the scaling hypothesis. 5. Electrical properties 5.1. Magnetoresistance property The temperature dependence of the electrical resistivity under zero and 5 T magnetic field is shown in Fig. 7(a). We noticed an electric transition from a metallic (dr/dT > 0) to an insulator (dr/ dT < 0) state. This figure reveals a ferromagnetic-metallic to paramagnetic semiconductor transition near TMSc (the temperatures’ dr values obtained from the inflexion point of dT plots). Moreover, we are interested in the magnetoresistance property in the LGSMIO compound which is defined by the reduction of the electrical resistivity of the material via an applied magnetic field. The normalized resistivity decrease (MR %) is calculated using the following formula:
MRð%Þ ¼ Fig. 3. (a) Magnetic isotherms measured at different temperatures for La0.57Gd0.1Sr0.33Mn0.9In0.1O3 sample close to the Curie temperature (TC ¼ 336 K). Inset shows the plot of dM/dT vs T. (b). Isothermal magnetization in the temperature range under a magnetic field up to 5 T. Inset shows the Arrot plot (M2 vs. H/M) of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 compound.
The MAP is constructed using the critical exponents of 3DHeisenberg ðb ¼ 0:365; g ¼ 1:336 and d ¼ 4:80Þ Fig. 4(b), tricritical mean field theories ðb ¼ 0:250; g ¼ 1 and d ¼ 5Þ Fig. 4(c) and 3D-Ising ðb ¼ 0:325; g ¼ 1:241 and d ¼ 4:82Þ Fig. 4(d). Based on these curves, all models present quasi straight and nearly parallel lines in the high field region. Thus, it is somehow difficult to distinguish which one of them is the best for the determination of the critical exponents. In order to compare these results and select the best model which describes this system, we calculated their relative slopes (RS) which are defined as RS ¼ SðTÞ=SðTC Þ. If the modified Arrott plots show a series of absolute parallel lines, the relative slope should be kept to 1 irrespective of temperature [14]. As shown in Fig. 5, the RS of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 using mean-field, 3D-Heisneberg and 3D-Ising model clearly deviates from RS ¼ 1, but the RS of tricritical mean-field is close to it. The critical properties near the ferromagnetice paramagnetic phase transition temperature were analyzed from the data of static magnetization measurements for the sample, using various techniques such as modified Arrott plot (MAP) Fig. 6(a), KouveleFisher (KF) Fig. 6(b) method and critical isotherm (CI) analysis Fig. 6(c). The details of calculation are reported in Ref. [14]. The exponents b, g and d, are, respectively, associated with the spontaneous magnetization (MS), the inverse of the initial susceptibility (c1 0 ) and magnetization isotherm (M H at TC) [14]. In the present work,
rðH ¼ 0Þ rðHÞ 100 rðH ¼ 0Þ
(3)
where r (H ¼ 0) is the zero field resistivity and r (H) is the resistivity under an external magnetic field of 5 T. The MR dependence of the temperature is shown in Fig. 7(b). This curve displays the presence of peaks around the semiconductor-metal transition temperature (324 K) with a value of MR% around 67%, which is higher than that found in other manganite oxides [16,17]. Fig. 8(a) depicts the temperature derivative of resistivity normalized with respect to its value at Curie temperature. It is plotted as versus temperature from 290 to 360 K for the LGSMIO compound. The results illustrate qualitative features (the divergence near TC) relevant to transport theory. 5.2. Critical parameters determined from electrical resistivity For a better understanding of the nature of phase transition, we tried to determine the values of the critical exponents close to second order metal-semiconductor transition and to assign one of these models to second order systems. Depending on Fisher-Langer theory [18], specific heat at a constant pressure (Cp) and at the phase transition temperature is proportional to the temperature derivative of the resistivity at T ¼ TMSc. In ferromagnetic systems and other materials, the derivative of the resistivity presents a divergence of the type:
CP f
dr dT
¼
dr dh
¼ ha
(4)
where a is the specific heat critical exponent, Cp is the specific heat and h ¼ ½T TMSc =TMSc is the reduced temperature. The two power law forms of equation (4) below and above TM-SC given by
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Fig. 4. Modified Arrott plots: isotherms of M1/bvs.(m0H/M)1/g with (b) 3D-Heisenberg model, (c) tricritical mean-field model and (d) 3D-Ising model for La0.57Gd0.1Sr0.33Mn0.9In0.1O3 compound.
CðTÞ ¼
i 1 dr Aþ h ¼ ð hÞa 1 þ Bþ for T < TMSc rðTC Þ dT a (6)
Fig. 5. Normalized slope RS [defined as RS ¼ S(T)/S(TC)] as a function of temperature using several methods.
Geldert et al. [19] are:
CðTÞ ¼
i 1 dr Aþ h ð hÞa 1 þ Bþ for T < TMSc ¼ rðTC Þ dT a (5)
where A and B are constants, a and a’ are specific heat critical exponents below and above TMSc .The temperature derivative of resistivity normalized with respect to its value at TMSc ½ð1=rðTC Þðdr=dTÞ againsth is shown in Fig. 8(b) for LGSMIO sample and the equations (5) and (6) are fitted below and above TMSc at the same figure. The solid lines passing through the data are the best fits in the two regions. The values of constants A, B and the specific heat critical exponents below and above TMSc (a anda’ ) are obtained from the fitting analysis for the sample (present work) and the values of other manganites are listed in Table 2 for comparison. It is clear from Table 2 that the specific heat critical exponents for LGSMIO sample (a and a’ ) are found to be 0.453 which agrees well with that obtained using tri-critical mean-field model [20]. The specific heat critical exponents for Co doped sample (a anda’ ) are found to be 0.115 and this value is consistent with that obtained using 3D Heisenberg model. For the Cr doped samples, the plots below and above TMSc are agreement with the values of 0.001 close to the mean field theory, suggesting long range ferromagnetic order [21]. In addition, other critical exponents b and g are calculated from the Suezaki-Mori model [22] which relates the temperature derivative of the electrical resistivity to the reduced temperature (h) magnetic ordering as follows:
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Fig. 6. (a) The spontaneous magnetization MS(T, 0) and the inverse initial susceptibility 1=c0 as a function of a temperature, respectively for the LGSMIO sample. (b) KouveleFisher plots for the spontaneous magnetization MS(T, 0) and the inverse initial susceptibility 1=c0 ðTÞ. The solid lines are the linear fits. (c) Isothermal M vs m0H plot of LGSMIO at TC ¼ 336 K; the inset shows the same plot in log-log scale and the solid line is the linear fit. (d) Scaling plots M|ε|bvs. m0H|ε|(bþg), indicating two universal curves below and above TC for LGSMIO sample. Inset shows the same plots on a logelog scale.
dr ¼ Bþ hðaþg1Þ dT
for
T > TMSc
(7)
dr ¼ Bg hðaþlÞ=2 þ B hðaþg1Þ dT
for
T < TMSc
(8)
where the constants Bþ and B- incorporate the term involving the zone boundary energy gap Bg and ½ðT TMSc Þ=TMSc is the reduced temperature, as already denoted. Taking natural logarithm on both sides, equation (7) can be rewritten as,
Ln
dr ¼ ða þ g 1ÞLnðhÞ dT
forT > TMSc
(9)
The slope of ln(dr/dT) versus ln(hÞ plot gives the value of (a þ g 1). As a is obtained from the fit with Fisher-Langer method, we can obtain the value of g. Fig. 9(a) shows ln(dr/dT) vs ln(hÞ above TMSc and gives the value of a þ g 1 for LGSMIO sample. The slope is estimated to be 0.520± 0.02. Substituting the value of a (derived from the figure in this relation), the critical exponent g was obtained with a value equal to 1.067 ± 0.02. The first term in equation (8) involving Bg will be dominant at temperatures less than TM-SC because the scaling law gives ða þ gÞ=2 ða þ g 1Þ ¼ b. As done earlier, the equation (8) can also be rewritten by taking natural logarithm on both sides as:
Ln
dr ¼ dT
aþg 2
ða þ g 1Þ LnðhÞ
forT < TMSc
(10)
Using the scaling relation, (ð½a þ g=2Þ ða þ g 1Þ) is equal to the value of the critical exponent b. Hence, the direct slope of the plot ln(dr/dT) versus ln(h) below TMSc will give b value. Fig. 9(b) presents ln(dr/dT) versus ln(h) below TMSc for LGSMIO sample and the value of slope is obtained as b ¼ 0.272 ± 0.004. The critical exponents are determined for LGSMIO (present work) and theoretical models are listed in Table 3. Finally, one can note that Rushbrooke scaling relation a þ 2b þ g ¼ 2 (theory) which gives 2.064 for LGSMIO sample is also satisfied [23]. Generally, for LGSMIO, the exponents’ values obtained from the resistivity are very close to the ones predicted from the tri-critical mean-field model. These results are in agreement with the analysis of critical exponents from the magnetization measurements. This study shows a descriptive report of the correlation between magnetic and electrical transport properties. In light of this qualitative agreement, a strong correlation between electrical and magnetic properties in manganite of LGSMIO type near the phase transition temperature is proved. In order to understand the transport mechanism in our sample, we will use the theoretical model describing it. Taking LGSMIO sample into consideration, the resistivity rises when the temperature decreases from 324 K and reaches a maximum at TM-Sc. This region (T>TMSc ) is characterized by an insulator behavior which is described by the adiabatic small polaron hopping
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Fig. 8. (a): The thermal derivative of resistivity normalized with respect to resistivity value at TC for LGSMIO. (b) The thermal derivative of resistivity normalized with respect to resistivity value at TM-Sc, as a function of reduced temperature [h¼ (T-TM-Sc)/ TM-Sc] LGSMIO.
Fig. 7. (a): Variation of resistivity as function of temperature of the LGSMIO under 0 and 5 T. (b) represents the magnetoresistance percentage MR%.
rðTÞ ¼ r0 þ r2 T 2 þ r9=2 T 9=2 mechanism [24] according to the following formula:
rPM ðTÞ ¼ ra T exp
Ea kB T
(11)
wherera is a pre-exponential coefficient independent from T. Ea is the activation energy and kB is Boltzmann constant. In the low temperature ferromagnetic phase, a dependence of rðTÞ is approximated by an expression that includes some scattering mechanisms:
(12)
Where r0 arises from grain or domain boundaries, the term r2 T2 [25,26] indicates the resistivity due to the electron-electron scattering process [27] and it’s generally the most dominant up to 100 K [28]. Finally, the term r9=2 T9/2 is a combination of electronelectron, electron-magnon and electron-phonon scattering processes [29,30].
6. Percolation model None of the mentioned models can explain the prominent change of the P-T curves nearTMSc, which is based upon an
Table 2 Values of different parameters used to fit the experimental data to equations (4) and (5). Samples
Region
LGSMIO
T T T T T T
PCMCoO PCMCrO
< > < > < >
TM-Sc TM-Sc TM-Sc TM-Sc TM-Sc TM-Sc
A
B
a
Ref.
0.00225 ± 6.7E-5 0.0099 ± 0.003 0.00238 ± 0.001 - 0.00572 ± 0.000597 0.00219 ± 1.52451E-4 0.01109 ± 3.69503E-4
0.0057 ± 1.43E-5 0.002 ± 8.3E-4 1.02438 ± 0.012 0.941 ± 0.00471 3.20123 ± 0.15264 0.00493 ± 4.72716E-4
0.453 0.461 0.115 0.115 0.001 0.001
This work This work [21] [21] [21] [21]
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Fig. 10. The temperature dependence of electrical resistivity of the LGSMIO sample under different magnetic fields applied. Symbols are the experimental results and solid lines are the graphs fitted by Eq. (18).
[32]:
rðTÞ ¼ rFM f þ rPM ð1 f Þ
(13)
wheref and ð1 f Þ are the volume concentration of the (PM) domains and (FM) regions, respectively and are expressed as [33]:
f ¼
Fig. 9. (a):ln(dr/dT) versus ln [(T-TM-Sc)/TM-Sc] below TM-Sc for LGSMIO under 5 T. (b) ln(dr/dT) versus ln [(T-TM-Sc)/TM-Sc] above TM-Sc for LGSMIO under 5 T.
approach that the system consists in phase separation between ferromagnetic metallic and paramagnetic insulating regions. Li et al. [31] proposed that in CMR materials, a metallic conductivity exists in the ferromagnetic regions and that a semiconductor-like conductivity above TC exists in the paramagnetic regions. In such a phaseefoliated system, the metal-insulator transition is a percolation phase transition and the behavior of rðTÞ can be explained on the basis of the percolation theory. Thus, considering the percolation character of the metal-insulator transition and assuming that the competition between the ferromagnetic and paramagnetic regions plays an important role in the formation of CMR effects in manganites, resistivity can be written as follows
1
1 þ exp KDBUT
(14)
exp kDUT B f’ ¼ 1 f ¼ 1 þ exp kDUT
(15)
B
Where
DU ¼ U0 1
T
! (16)
TCmod
DU is the difference in energy between the FM and PM states and TCmod means the temperature used in the model near/equal to TC. The total ðrÞ can be represented as:
rðTÞ ¼ r0 þ r2 T 2 þ r9=2 T 9=2 f þ ra T expðEa =kB TÞð1 f Þ (17)
Table 3 Estimated critical exponents for La0.57Gd0.1Sr0.33Mn0.9In0.1O3 and theoretical models. System
a and a’
b
g
d
Ref.
La0.57Gd0.1Sr0.33Mn0.9In0.1O3 La0.57Gd0.1Sr0.33Mn0.9In0.1O3 Pr0.6Ca0.4Mn0.96Co0.04O3 Pr0.6Ca0.4Mn0.96Cr0.04O3 3D Heisenberg model Mean field model Ising model Tricritical mean-field theory
0.453(T < TM-Sc) 0.461(T > TM-Sc) 0.115 0.001 0.115 0 0 0
0.272 ± 0.004 e 0.336 ± 0.04 0.502 ± 0.03 0.365 0.5 0.325 0.25
e 1.067 ± 0.002 1.304 ± 0.02 1.192 ± 0.01 1.336 1.0 1.24 1.0
e e e e 4.8 3.0 4.82 5.0
This work This work [21] [21] [7,8] [9,10] [14] [14]
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Table 4 Obtained parameters from the best fit to the Eq. (18) of the experimental data of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 sample at 0, 2 and 5 T. H (T)
r0 (U.cm)
r2 (U.cm/K2)
r9=2 (U.cm/K2)
ra (U.cm)
Ea =kB ðKÞ
DU=kB ðKÞ
TCmod ðKÞ
R2
0T 2T 5T
0.0015 0.0012 0.0055
9.20E-6 8.24E-6 7.23E-6
5.61E-13 4.65E-13 3.63E-13
4.94 E7 2.36 E7 2.14 E7
2882 1528.25 1196.035
5639 5516 5328
324 320 318
0.9996 0.9998 0.9999
The estimated critical exponents are (b ¼ 0.272 ± 0.004; g ¼ 1.067 ± 0.002 and a ¼ 0.453). The values deduced from the critical exponents are close to the theoretical prediction of tricritical mean-field model. These results are in good agreement with the analysis of the critical exponents from magnetization measurements. The conduction mechanism was explained by a small polaron hopping in the insulating region, and by electron scattering mechanisms in the metallic region. Then, to understand the transport mechanism in the entire temperature range, we used the phenomenological percolation model, which is based on the phase segregation of ferromagnetic metallic clusters and paramagnetic semiconductor regions.
References
Fig. 11. Variation of the volume fraction of the FM phase f vs. T.
rðTÞ ¼ ðr0 þ r2 T 2 þ r9=2 T 9=2 Þ$
2
1
3
6U0 ð1
1 þ exp6 4 0
2
T T mod C
Þ7
kB T
T T mod C
1
7 5
31
B exp4 U0 ð kB T Þ5 C C Ea B C B þ ra T expð Þ$B 3C 2 C kB T B A @ T 6U0 ð1T mod Þ7 C 7 6 1 þ exp4 5 k T
(18)
B
Fig. 10 displays the simulated (red line) and experimental results for the rðTÞ curve obtained for zero field for LGSMIO. The best fit parameters are also given in Table 4. It’s worth mentioning that this percolation model is suitable for all the samples. Their excellent agreement confirms that FM domains and PM regions coexist at near TC. The temperature dependence on the volume concentration of the ferromagnetic phase f is shown in Fig. 11. It is clear that f (T) remains equal to 1 below the metale semiconductor transition temperature, which confirms the strong dominance of the FM fraction in this range. Then, the FM volume fraction begins to decrease to 0 from the ferromagnetic metallic state to a paramagnetic-semiconductor state, confirming the validity of the percolation approach which assumes a convertion from ferromagnetic to paramagnetic regions. 7. Conclusion To conclude, we have examined the critical behavior of resistivity in LGSMIO sample under different applied magnetic fields.
[1] N. Abelmoula, E. Dhahri, N. Fourati, L. Reversat, J. Alloys. Comp. 365 (2004) 25e30. [2] M. Khlifi, M. Bejar, O. EL Sadek, J. Alloys. Comp. 509 (2011) 7410e7415. [3] Y. Tokura, A. Urushibara, Y. Moritomo, et al., J. Phys. Soc. Jpn 63 (1994) 3931. [4] A. Asamitsu, Y. Moritomo, Y. Tomioka, et al., Nature 373 (1995) 407. [5] N. Zaidi, S. Mnefgui, A. Dhahri, J. Dhahri, E.K. Hlil, J. Alloys. Comp. 616 (2014) 378e384. [6] J.M.D. Coey, M. Viret, S. vonMolnar, J. Adv. Phys. 48 (1999) 167. [7] S. Taran, B.K. Chaudhuri, S. Chatterjee, H.D. Yang, S. Neeleshwar, Y.Y. Chen, J. Appl. Phys. 98 (2005) 103903. €ßler, U.K. Ro €ßler, K. Do € rr, [8] B. Padmanabhan, H.L. Bhat, S. Elizabeth, S. Ro K.H. Müller, Phys. Rev. B 75 (2007) 024419. [9] Ch.V. Mohan, M. Seeger, H. Kronmüller, P. Murugaraj, J. Maier, J. Magn. Magn.Mater 183 (1998) 348e355. [10] M. Ziese, J. Phys. Condens. Matter 13 (2001) 2919e2934. [11] D. Kim, B.L. Zink, F. Hellman, J.M.D. Coey, Phys. Rev. B 65 (2002) 214424. [12] P.N. Lisboa, A.W. Mombro, H. Pardo, E.R. Leite, W.A. Ortiz, J. Solid State Commun. 130 (2004) 31. [13] S.K. Banerjee, Phys. Lett. 12 (1964) 16. [14] M. Abassi, N. Dhahri, J. Dhahri, K. Taibi, E.K. Hlil, J. Rare Earths 33 (2015) 263e270. [15] H.E. Stanley, Rev. Mod. Phys. 71 (1999) 358. [16] R. Choithrani, M.A. Bhat, N.K. Gaur, J. Magn. Magn.Mater 361 (2014) 19. [17] E. Dhahri, K. Guidara, A. Cheikhrouhou, J.C. Joubert, J. Pierre, Phase Transitions A Multinatl. J. 66 (1998) 99e107. [18] M.E. Fisher, J.S. Langer, Phys. Rev. Lett. 20 (1968) 665. [19] D.J.W. Geldart, T.G. Richard, Phys. Rev. B 12 (1975) 5175. [20] D. Kim, B. Revaz, B.L. Zink, F. Hellman, J.J. Rhyne, J.F. Mitchell, Phys. Rev. Lett. 89 (2002) 227202. [21] R. Thiyagarajan, S. Esakki Muthu, G. Kalaiselvan, R. MahendiranArumugam, J. Alloys. Comp. 618 (2015) 159e166. [22] Y. Suezaki, Y. Mori, J. Prog. Theor. Phys. 41 (1969) 1177. [23] Igor Herbut, Cambridge University Press Science”(2007). [24] M. Ziese, C. Srinitiwarawong, Phys. Rev. B 58 (1998) 11519. [25] A. Urushibara, Y. Moritomo, T. Arima, A. Asamitsu, G. Kido, Y. Tokura, Phys. Rev. B 51 (1995) 14103e14109. [26] D.C. Worledge, G.J. Snyder, M.R. Beasley, T.H. Geballe, J. Appl. Phys. 80 (1996) 5158e5161. [27] M. Viret, L. Ranno, J.M.D. Coey, Phys. Rev. B 55 (1997) 8067e8070. [28] P.T. Phong, N.V. Khiem, N.V. Dai, D.H. Manh, L.V. Hong, N.X. Phuc, J. Magn. Magn. Mater 321 (2009) 3330e3334. [29] G.J. Synder, R. Hiskes, S. DiCarolis, M.R. Beasley, T.H. Geballe, Phys. Rev. B 53 (1996) 14434e14444. [30] G. Venkataiah, P.V. Reddy, J. Solid. State. Commun. 136 (2005) 114e119. [31] G. Li, H.-D. Zhou, S.L. Feng, X.-J. Fan, X.G. Li, J. Appl. Phys. 92 (2002) 1406. [32] Za. Mohamed, Mounira. Abassi, J. Dhahri, E.K. Hlil, J. Alloys.Comp 646 (2015) 23e31. [33] P.T. Phong, N.V. Dai, D.H. Manh, T.D. Thanh, N.V. Khiem, L.V. Hong, N.X. Phuc, J. Magn. Magn. Mater 322 (2010) 2737e2741.
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Critical scaling and percolation model in La0.57Gd0.1Sr0.33Mn0.9In0.1O3 manganite Mounira Abassi a, *, Asma Zaidi a, J. Dhahri a, E.K. Hlil b a b
Laboratoire de la Mati ere Condens ee et des Nanosciences, Universit e de Monastir, 5019, Tunisia Institut N eel, CNRS et Universit e Joseph Fourier, B.P. 166, 38042, Grenoble, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 April 2016 Received in revised form 29 June 2016 Accepted 30 June 2016 Available online 2 July 2016
We investigated the influence of indium (In) doping on the structural, magnetic and electrical properties of polycrystalline sample La0.57Gd0.1Sr0.33Mn0.9In0.1O3, prepared by solegel method. The crystallographic study showed that the sample crystallizes in the rhombohedral system with R3c space group. The relationship between the electrical and magnetic properties of manganites was traced through analyzing the dependence of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 resistivity on temperature. The measurement of resistivity versus the temperature r(T) of the sample showed that a ferromagnetic (FM) to a paramagnetic (PM) transition at Curie temperature (TC) was usually accompanied by a simultaneous metal-insulator (M-I) transition. Resistivity decreased with increasing the applied magnetic field. The values of TMSc were found to move towards a high temperature side with increasing the magnetic field. The estimated critical exponents are (b ¼ 0.272 ± 0.004; g ¼ 1.067 ± 0.002 and a ¼ 0.453). The values of these exponents obtained from resistivity were very close to those predicted by the tricritical mean-field model. These results were in good agreement with the analysis of the critical exponents from magnetization measurements. The electrical resistivity was fitted with the phenomenological percolation model, which is based on the phase segregation of the ferromagnetic metallic regions and the paramagnetic semiconductor regions. So, we found that the estimated results were in accordance with the experimental data. © 2016 Elsevier B.V. All rights reserved.
Keywords: Manganites Metal-insulator transition Resistivity Critical exponents Percolation model
1. Introduction Rare-earth perovskite manganites have been extensively studied thanks to their interesting electrical and magnetic properties [1,2]. Mixed manganites R1xAxMnO3 (R being trivalent rare earth ions, A ¼ Ca, Ba, Sr) have attracted much interest because these materials exhibit a rich phase diagram as a function of temperature [3], magnetic field [4], magnetocaloric effect (MCE) and illustrate the metal-insulator as well as the phenomenon of colossal magnetoresistance (CMR) [5]. The physical properties of perovskite manganites for technological applications such as magnetic refrigeration, electrical resistivity and very high-density magnetic recording (read heads for hard disks or magnetic memories (MRAM) non-volatile) have been studied in the framework of double exchange (DE) interaction for the whole doped manganite [6]. An important part of the work on manganites aimed to present
* Corresponding author. Tel.: þ216 25 012 603; fax: þ216 73 500280. E-mail address: [email protected] (M. Abassi). http://dx.doi.org/10.1016/j.jallcom.2016.06.293 0925-8388/© 2016 Elsevier B.V. All rights reserved.
the effect of CMR. However, the study of the anomalies of the various thermo physical properties, like the specific heat in the vicinity of the magnetic phase transition, with investigating the values of the universal critical parameters, has not been given the sufficient importance in research projects done on manganites. As far as manganites are concerned, there have been a lot of experiments to estimate the critical exponents of ferromagnetic manganites. Systems showing a second order metal insulator phase transition obey one of the common universality classes. By contrast, the experimental estimates of the critical exponents are still controversial including those for short-range Heisenberg interaction [7,8], the mean-field values [9,10] and those which cannot be classified into any universality class ever known [11]. Although a lot of work has been conducted to study the Mn-site doping effects, the critical exponents as very important parameters for magnetic transition are less available for the Mn-site doping. In this study, we carried out a systematic and detailed analysis of the magnetization measurement of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 polycrystalline sample near the ferromagneticeparamagnetic phase transition temperature. We have also determined the critical
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temperature TC and the critical parameters b and g. Then, we tried to investigate the critical behavior in LGSMIO at its semiconductoremetallic transition via the measurements of resistivity. The values of these exponents obtained from resistivity are very close to those predicted by the tricritical mean-field model. Finally, we intend to study the dependence of the electrical resistivity properties as a function of the LGSMIO, volume fraction, f ; which is analyzed in the framework of the percolation theory. A percolation model based on the mixed phase consisting of itinerant electrons and localized magnetic polarons has been proposed to explain the observed results. 2. Experimental details
Scherer formula:
DS ¼
kl bhkl cos q
(1)
where b is the breadth of the observed diffraction line at its half intensity maximum, K is the so-called shape factor (¼0.9) and l is the X-ray wavelength used. The value of (DS) is 35 nm. The average size of the particle of this sample is determined using the Scanning Electron Microscope (SEM) images by finding the minimum and maximum dimensions of the large number of particles (seen in the inset of Fig. 2). The value is 102 nm. The grain sizes observed by SEM were larger than those calculated by Scherrer’s formula. This can be explained by the fact that each particle observed by SEM is formed by several crystallized grains. In order to check the existence of all elements in this sample, energy dispersive X-ray analysis (EDAX) was carried out at room temperature. EDAX spectrum represented in Fig. 2 reveals the presence of La, Gd, 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 this sample is represented in Table 1. This measurement confirmed the cationic composition of this sample and the absence of secondary phases in the X-rays.
The microstructure of ceramic materials in general and CMR materials in particular was highly affected by the preparation routes and heat treatments. Sol-gel routes are known to produce very high quality, homogeneous and fine particle materials [12]. Bulk polycrystalline sample with a compositional formula La0.57Gd0.1Sr0.33Mn0.9In0.1O3 was prepared by the sol-gel process. In this method, the metal nitrates taken in stoichiometric ratios were dissolved in an aqueous solution. Citric acid was added in 1:1 ratio and the pH was adjusted to a value between 6.5 and 7.0 by adding ammonia. After getting a sol-gel on slow evaporation, a gelating reagent-ethylene glycol- was added and heated between 160 and 180 C to get a gel. On further heating, this solution yielded a dry fluffy porous mass (precursor), which was calcined at 700 C for 6 h. Then, the powder was pressed into circular pellets. These pellets were sintered at 900 C in air for 12 h. Phase purity, homogeneity and cell dimensions were determined by powder XRD. The phase analysis was carried out using FULLPROF program based on Rietveld method. 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 (EDX). Magnetizations (M) vs. temperature (T) were measured using a BS1 magnetometer. The isotherm magnetization vs. magnetic field (m0H) was recorded by the SQUID-VSM system. This latter was measured in the range of 0e5 T and with a temperature interval of 2 K in the vicinity of Curie temperature (TC). To obtain the metal-insulator transition temperature (TM-Sc) and to study the influence of the magnetic field on resistivity, the electrical resistivity and magnetoresistance measurements were done by standard dc four-probe technique using a closed cycle helium refrigerator cryostat in applied fields of 0 T, 2 T and 5 T.
Fig. 3(a) shows the temperature dependence of magnetizations using an applied field of 0.05 T. This curve reveals that the La0.57Gd0.1Sr0.33Mn0.9In0.1O3 oxide exhibits a magnetic transition from a ferromagnetic (FM) to a paramagnetic (PM) state when increasing temperature. This transition occurs at the Curie temperature (TC) which is obtained from the peak of the dM/dT curves. The Curie temperature TC is found to be 336 K, which is close to room temperature. In order to clarify the nature of the FMePM phase transition, we measured the magnetic-field dependence of magnetization at different temperatures (MeHeT), around Curie temperature TC. Fig. 3(b) shows the evolution of magnetization versus the applied magnetic field obtained at different temperatures (isothermal magnetization). Obviously, the M(H) curves were found to increase with decreasing temperature in the selected temperature range, where thermal fluctuation of spins decreased with decreasing temperature. Moreover, from the Arrot-plot (inset of Fig. 3(b)) in which all curves M2 vs. H/M have positive slopes, the magnetic phase transition was found to be of a second order according to Banerjee’s criterion [13].
3. Results and discussion
4. Critical parameters determined from magnetization data
3.1. Structural properties
We determined the values of the critical exponents of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 sample from the magnetization data versus temperature and magnetic field M (H, T), to understand their magnetic and magnetocaloric properties. Experimental results revealed that this sample exhibits a continuous (second-order) paramagnetic (PM) to ferromagnetic (FM) phase transition. In mean field theory at near TC, M2 vs H=M at various T should form a progression of parallel straight lines. The line for T ¼ TC passes through the origin on this plot. Moreover, the intersections of these curves for T_TC with the H=M axis gave the values of 1=c0 ðTÞ at H ¼ 0. In the present case the curves in the Arrott plot are non linear, indicating that the mean-field theory (b ¼ 0.5, g ¼ 1 and d ¼ 3) Fig. 4(a) characteristics of systems with long-range interactions is invalid for the present phase transition. Therefore, the values of MS ðTÞ and c1 0 ðTÞ were determined using a modified Arrott plot (Also called modified Arrott plots
The X-ray diffraction (XRD) pattern of the La0.57Gd0.1Sr0.33Mn0.9In0.1O3 oxide recorded at room temperature is shown in Fig. 1. Such a diffraction indicates that the sample crystallized in a rhombohedral structure with an R3c space group, in which the La/ Gd/Sr atoms are at 6a (0, 0, 1/4) position, Mn/In at 6b (0, 0, 0) position and O at 18 e (x, 0, 1/4) position. The cell parameters are found to be a ¼ b ¼ 5.41 Å and c ¼ 13.13 Å. The quality of the agreement is evaluated through the adequacy of the fit indicator c2 ¼ 1.5. 3.2. Morphological properties From the reflection of 2q values of XRD profile, we can also calculate the average grain size (DS) from the XRD peaks using
3.3. Magnetic properties
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Fig. 1. X-ray diffraction pattern (solid curve) and Rietveld refinement result (open symbols) for La0.57Gd0.1Sr0.33Mn0.9In0.1O3 at room temperature. The difference between the data and the calculation is shown at the bottom.
Fig. 2. EDAX spectrum for La0.57Gd0.1Sr0.33Mn0.9In0.1O3. The insets shows the typical SEM.
Table 1 Results of EDAX analysis of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 sample. Element
Typical cationic composition from EDX
Wt%
At.%
Nominal composition
La Gd Sr Mn In
0.568 0.101 0.332 0.901 0.102
33.876 6.886 12.452 21.205 4.979
11.361 2.021 6.640 18.021 2.042
La0.57Gd0.1Sr0.33Mn0.9In0.1O3
(MAP)). In this technique, the M ¼ f ðHÞ data are converted into series of isotherms ðM1=b ¼ f ðH=MÞ1=g Þ depending on the following relationship [14]:
ðH=MÞ1=g ¼
ðT TC Þ þ ðM=M1 Þ1=b T1
(2)
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the estimated critical exponents are (b¼ 0.282 ± 0.004; g ¼ 0.975 ± 0.002 at TC ¼ 332.57 ± 0.11). The deduced values of the critical exponents were close to the theoretical prediction of tricritical mean-field model. Here, the obtained critical exponents values can be verified with the prediction of the scaling theory using the MðH; εÞ ¼ εb f ±ðH=εbþg Þ equation, where ε is the reduced C temperature ε ¼ TT TC and fþ and f are regular analytical functions above and below TC [15]. According to this scaling law, the plots of M=jεjb as a function of H=jεjbþg fall into two universal curves; one for temperatures above TC and the other for temperatures below TC. Fig. 6(d) shows the scaling plot on a linear scale and the inset shows the same data in a logarithmic scale. Here in, we can clearly see that all points follow two curves, one for T < TC and the other for T > TC. As a consequence, the obtained values of the critical exponents are reliable and in agreement with the scaling hypothesis. 5. Electrical properties 5.1. Magnetoresistance property The temperature dependence of the electrical resistivity under zero and 5 T magnetic field is shown in Fig. 7(a). We noticed an electric transition from a metallic (dr/dT > 0) to an insulator (dr/ dT < 0) state. This figure reveals a ferromagnetic-metallic to paramagnetic semiconductor transition near TMSc (the temperatures’ dr values obtained from the inflexion point of dT plots). Moreover, we are interested in the magnetoresistance property in the LGSMIO compound which is defined by the reduction of the electrical resistivity of the material via an applied magnetic field. The normalized resistivity decrease (MR %) is calculated using the following formula:
MRð%Þ ¼ Fig. 3. (a) Magnetic isotherms measured at different temperatures for La0.57Gd0.1Sr0.33Mn0.9In0.1O3 sample close to the Curie temperature (TC ¼ 336 K). Inset shows the plot of dM/dT vs T. (b). Isothermal magnetization in the temperature range under a magnetic field up to 5 T. Inset shows the Arrot plot (M2 vs. H/M) of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 compound.
The MAP is constructed using the critical exponents of 3DHeisenberg ðb ¼ 0:365; g ¼ 1:336 and d ¼ 4:80Þ Fig. 4(b), tricritical mean field theories ðb ¼ 0:250; g ¼ 1 and d ¼ 5Þ Fig. 4(c) and 3D-Ising ðb ¼ 0:325; g ¼ 1:241 and d ¼ 4:82Þ Fig. 4(d). Based on these curves, all models present quasi straight and nearly parallel lines in the high field region. Thus, it is somehow difficult to distinguish which one of them is the best for the determination of the critical exponents. In order to compare these results and select the best model which describes this system, we calculated their relative slopes (RS) which are defined as RS ¼ SðTÞ=SðTC Þ. If the modified Arrott plots show a series of absolute parallel lines, the relative slope should be kept to 1 irrespective of temperature [14]. As shown in Fig. 5, the RS of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 using mean-field, 3D-Heisneberg and 3D-Ising model clearly deviates from RS ¼ 1, but the RS of tricritical mean-field is close to it. The critical properties near the ferromagnetice paramagnetic phase transition temperature were analyzed from the data of static magnetization measurements for the sample, using various techniques such as modified Arrott plot (MAP) Fig. 6(a), KouveleFisher (KF) Fig. 6(b) method and critical isotherm (CI) analysis Fig. 6(c). The details of calculation are reported in Ref. [14]. The exponents b, g and d, are, respectively, associated with the spontaneous magnetization (MS), the inverse of the initial susceptibility (c1 0 ) and magnetization isotherm (M H at TC) [14]. In the present work,
rðH ¼ 0Þ rðHÞ 100 rðH ¼ 0Þ
(3)
where r (H ¼ 0) is the zero field resistivity and r (H) is the resistivity under an external magnetic field of 5 T. The MR dependence of the temperature is shown in Fig. 7(b). This curve displays the presence of peaks around the semiconductor-metal transition temperature (324 K) with a value of MR% around 67%, which is higher than that found in other manganite oxides [16,17]. Fig. 8(a) depicts the temperature derivative of resistivity normalized with respect to its value at Curie temperature. It is plotted as versus temperature from 290 to 360 K for the LGSMIO compound. The results illustrate qualitative features (the divergence near TC) relevant to transport theory. 5.2. Critical parameters determined from electrical resistivity For a better understanding of the nature of phase transition, we tried to determine the values of the critical exponents close to second order metal-semiconductor transition and to assign one of these models to second order systems. Depending on Fisher-Langer theory [18], specific heat at a constant pressure (Cp) and at the phase transition temperature is proportional to the temperature derivative of the resistivity at T ¼ TMSc. In ferromagnetic systems and other materials, the derivative of the resistivity presents a divergence of the type:
CP f
dr dT
¼
dr dh
¼ ha
(4)
where a is the specific heat critical exponent, Cp is the specific heat and h ¼ ½T TMSc =TMSc is the reduced temperature. The two power law forms of equation (4) below and above TM-SC given by
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Fig. 4. Modified Arrott plots: isotherms of M1/bvs.(m0H/M)1/g with (b) 3D-Heisenberg model, (c) tricritical mean-field model and (d) 3D-Ising model for La0.57Gd0.1Sr0.33Mn0.9In0.1O3 compound.
CðTÞ ¼
i 1 dr Aþ h ¼ ð hÞa 1 þ Bþ for T < TMSc rðTC Þ dT a (6)
Fig. 5. Normalized slope RS [defined as RS ¼ S(T)/S(TC)] as a function of temperature using several methods.
Geldert et al. [19] are:
CðTÞ ¼
i 1 dr Aþ h ð hÞa 1 þ Bþ for T < TMSc ¼ rðTC Þ dT a (5)
where A and B are constants, a and a’ are specific heat critical exponents below and above TMSc .The temperature derivative of resistivity normalized with respect to its value at TMSc ½ð1=rðTC Þðdr=dTÞ againsth is shown in Fig. 8(b) for LGSMIO sample and the equations (5) and (6) are fitted below and above TMSc at the same figure. The solid lines passing through the data are the best fits in the two regions. The values of constants A, B and the specific heat critical exponents below and above TMSc (a anda’ ) are obtained from the fitting analysis for the sample (present work) and the values of other manganites are listed in Table 2 for comparison. It is clear from Table 2 that the specific heat critical exponents for LGSMIO sample (a and a’ ) are found to be 0.453 which agrees well with that obtained using tri-critical mean-field model [20]. The specific heat critical exponents for Co doped sample (a anda’ ) are found to be 0.115 and this value is consistent with that obtained using 3D Heisenberg model. For the Cr doped samples, the plots below and above TMSc are agreement with the values of 0.001 close to the mean field theory, suggesting long range ferromagnetic order [21]. In addition, other critical exponents b and g are calculated from the Suezaki-Mori model [22] which relates the temperature derivative of the electrical resistivity to the reduced temperature (h) magnetic ordering as follows:
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Fig. 6. (a) The spontaneous magnetization MS(T, 0) and the inverse initial susceptibility 1=c0 as a function of a temperature, respectively for the LGSMIO sample. (b) KouveleFisher plots for the spontaneous magnetization MS(T, 0) and the inverse initial susceptibility 1=c0 ðTÞ. The solid lines are the linear fits. (c) Isothermal M vs m0H plot of LGSMIO at TC ¼ 336 K; the inset shows the same plot in log-log scale and the solid line is the linear fit. (d) Scaling plots M|ε|bvs. m0H|ε|(bþg), indicating two universal curves below and above TC for LGSMIO sample. Inset shows the same plots on a logelog scale.
dr ¼ Bþ hðaþg1Þ dT
for
T > TMSc
(7)
dr ¼ Bg hðaþlÞ=2 þ B hðaþg1Þ dT
for
T < TMSc
(8)
where the constants Bþ and B- incorporate the term involving the zone boundary energy gap Bg and ½ðT TMSc Þ=TMSc is the reduced temperature, as already denoted. Taking natural logarithm on both sides, equation (7) can be rewritten as,
Ln
dr ¼ ða þ g 1ÞLnðhÞ dT
forT > TMSc
(9)
The slope of ln(dr/dT) versus ln(hÞ plot gives the value of (a þ g 1). As a is obtained from the fit with Fisher-Langer method, we can obtain the value of g. Fig. 9(a) shows ln(dr/dT) vs ln(hÞ above TMSc and gives the value of a þ g 1 for LGSMIO sample. The slope is estimated to be 0.520± 0.02. Substituting the value of a (derived from the figure in this relation), the critical exponent g was obtained with a value equal to 1.067 ± 0.02. The first term in equation (8) involving Bg will be dominant at temperatures less than TM-SC because the scaling law gives ða þ gÞ=2 ða þ g 1Þ ¼ b. As done earlier, the equation (8) can also be rewritten by taking natural logarithm on both sides as:
Ln
dr ¼ dT
aþg 2
ða þ g 1Þ LnðhÞ
forT < TMSc
(10)
Using the scaling relation, (ð½a þ g=2Þ ða þ g 1Þ) is equal to the value of the critical exponent b. Hence, the direct slope of the plot ln(dr/dT) versus ln(h) below TMSc will give b value. Fig. 9(b) presents ln(dr/dT) versus ln(h) below TMSc for LGSMIO sample and the value of slope is obtained as b ¼ 0.272 ± 0.004. The critical exponents are determined for LGSMIO (present work) and theoretical models are listed in Table 3. Finally, one can note that Rushbrooke scaling relation a þ 2b þ g ¼ 2 (theory) which gives 2.064 for LGSMIO sample is also satisfied [23]. Generally, for LGSMIO, the exponents’ values obtained from the resistivity are very close to the ones predicted from the tri-critical mean-field model. These results are in agreement with the analysis of critical exponents from the magnetization measurements. This study shows a descriptive report of the correlation between magnetic and electrical transport properties. In light of this qualitative agreement, a strong correlation between electrical and magnetic properties in manganite of LGSMIO type near the phase transition temperature is proved. In order to understand the transport mechanism in our sample, we will use the theoretical model describing it. Taking LGSMIO sample into consideration, the resistivity rises when the temperature decreases from 324 K and reaches a maximum at TM-Sc. This region (T>TMSc ) is characterized by an insulator behavior which is described by the adiabatic small polaron hopping
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Fig. 8. (a): The thermal derivative of resistivity normalized with respect to resistivity value at TC for LGSMIO. (b) The thermal derivative of resistivity normalized with respect to resistivity value at TM-Sc, as a function of reduced temperature [h¼ (T-TM-Sc)/ TM-Sc] LGSMIO.
Fig. 7. (a): Variation of resistivity as function of temperature of the LGSMIO under 0 and 5 T. (b) represents the magnetoresistance percentage MR%.
rðTÞ ¼ r0 þ r2 T 2 þ r9=2 T 9=2 mechanism [24] according to the following formula:
rPM ðTÞ ¼ ra T exp
Ea kB T
(11)
wherera is a pre-exponential coefficient independent from T. Ea is the activation energy and kB is Boltzmann constant. In the low temperature ferromagnetic phase, a dependence of rðTÞ is approximated by an expression that includes some scattering mechanisms:
(12)
Where r0 arises from grain or domain boundaries, the term r2 T2 [25,26] indicates the resistivity due to the electron-electron scattering process [27] and it’s generally the most dominant up to 100 K [28]. Finally, the term r9=2 T9/2 is a combination of electronelectron, electron-magnon and electron-phonon scattering processes [29,30].
6. Percolation model None of the mentioned models can explain the prominent change of the P-T curves nearTMSc, which is based upon an
Table 2 Values of different parameters used to fit the experimental data to equations (4) and (5). Samples
Region
LGSMIO
T T T T T T
PCMCoO PCMCrO
< > < > < >
TM-Sc TM-Sc TM-Sc TM-Sc TM-Sc TM-Sc
A
B
a
Ref.
0.00225 ± 6.7E-5 0.0099 ± 0.003 0.00238 ± 0.001 - 0.00572 ± 0.000597 0.00219 ± 1.52451E-4 0.01109 ± 3.69503E-4
0.0057 ± 1.43E-5 0.002 ± 8.3E-4 1.02438 ± 0.012 0.941 ± 0.00471 3.20123 ± 0.15264 0.00493 ± 4.72716E-4
0.453 0.461 0.115 0.115 0.001 0.001
This work This work [21] [21] [21] [21]
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M. Abassi et al. / Journal of Alloys and Compounds 688 (2016) 1251e1259
Fig. 10. The temperature dependence of electrical resistivity of the LGSMIO sample under different magnetic fields applied. Symbols are the experimental results and solid lines are the graphs fitted by Eq. (18).
[32]:
rðTÞ ¼ rFM f þ rPM ð1 f Þ
(13)
wheref and ð1 f Þ are the volume concentration of the (PM) domains and (FM) regions, respectively and are expressed as [33]:
f ¼
Fig. 9. (a):ln(dr/dT) versus ln [(T-TM-Sc)/TM-Sc] below TM-Sc for LGSMIO under 5 T. (b) ln(dr/dT) versus ln [(T-TM-Sc)/TM-Sc] above TM-Sc for LGSMIO under 5 T.
approach that the system consists in phase separation between ferromagnetic metallic and paramagnetic insulating regions. Li et al. [31] proposed that in CMR materials, a metallic conductivity exists in the ferromagnetic regions and that a semiconductor-like conductivity above TC exists in the paramagnetic regions. In such a phaseefoliated system, the metal-insulator transition is a percolation phase transition and the behavior of rðTÞ can be explained on the basis of the percolation theory. Thus, considering the percolation character of the metal-insulator transition and assuming that the competition between the ferromagnetic and paramagnetic regions plays an important role in the formation of CMR effects in manganites, resistivity can be written as follows
1
1 þ exp KDBUT
(14)
exp kDUT B f’ ¼ 1 f ¼ 1 þ exp kDUT
(15)
B
Where
DU ¼ U0 1
T
! (16)
TCmod
DU is the difference in energy between the FM and PM states and TCmod means the temperature used in the model near/equal to TC. The total ðrÞ can be represented as:
rðTÞ ¼ r0 þ r2 T 2 þ r9=2 T 9=2 f þ ra T expðEa =kB TÞð1 f Þ (17)
Table 3 Estimated critical exponents for La0.57Gd0.1Sr0.33Mn0.9In0.1O3 and theoretical models. System
a and a’
b
g
d
Ref.
La0.57Gd0.1Sr0.33Mn0.9In0.1O3 La0.57Gd0.1Sr0.33Mn0.9In0.1O3 Pr0.6Ca0.4Mn0.96Co0.04O3 Pr0.6Ca0.4Mn0.96Cr0.04O3 3D Heisenberg model Mean field model Ising model Tricritical mean-field theory
0.453(T < TM-Sc) 0.461(T > TM-Sc) 0.115 0.001 0.115 0 0 0
0.272 ± 0.004 e 0.336 ± 0.04 0.502 ± 0.03 0.365 0.5 0.325 0.25
e 1.067 ± 0.002 1.304 ± 0.02 1.192 ± 0.01 1.336 1.0 1.24 1.0
e e e e 4.8 3.0 4.82 5.0
This work This work [21] [21] [7,8] [9,10] [14] [14]
M. Abassi et al. / Journal of Alloys and Compounds 688 (2016) 1251e1259
1259
Table 4 Obtained parameters from the best fit to the Eq. (18) of the experimental data of La0.57Gd0.1Sr0.33Mn0.9In0.1O3 sample at 0, 2 and 5 T. H (T)
r0 (U.cm)
r2 (U.cm/K2)
r9=2 (U.cm/K2)
ra (U.cm)
Ea =kB ðKÞ
DU=kB ðKÞ
TCmod ðKÞ
R2
0T 2T 5T
0.0015 0.0012 0.0055
9.20E-6 8.24E-6 7.23E-6
5.61E-13 4.65E-13 3.63E-13
4.94 E7 2.36 E7 2.14 E7
2882 1528.25 1196.035
5639 5516 5328
324 320 318
0.9996 0.9998 0.9999
The estimated critical exponents are (b ¼ 0.272 ± 0.004; g ¼ 1.067 ± 0.002 and a ¼ 0.453). The values deduced from the critical exponents are close to the theoretical prediction of tricritical mean-field model. These results are in good agreement with the analysis of the critical exponents from magnetization measurements. The conduction mechanism was explained by a small polaron hopping in the insulating region, and by electron scattering mechanisms in the metallic region. Then, to understand the transport mechanism in the entire temperature range, we used the phenomenological percolation model, which is based on the phase segregation of ferromagnetic metallic clusters and paramagnetic semiconductor regions.
References
Fig. 11. Variation of the volume fraction of the FM phase f vs. T.
rðTÞ ¼ ðr0 þ r2 T 2 þ r9=2 T 9=2 Þ$
2
1
3
6U0 ð1
1 þ exp6 4 0
2
T T mod C
Þ7
kB T
T T mod C
1
7 5
31
B exp4 U0 ð kB T Þ5 C C Ea B C B þ ra T expð Þ$B 3C 2 C kB T B A @ T 6U0 ð1T mod Þ7 C 7 6 1 þ exp4 5 k T
(18)
B
Fig. 10 displays the simulated (red line) and experimental results for the rðTÞ curve obtained for zero field for LGSMIO. The best fit parameters are also given in Table 4. It’s worth mentioning that this percolation model is suitable for all the samples. Their excellent agreement confirms that FM domains and PM regions coexist at near TC. The temperature dependence on the volume concentration of the ferromagnetic phase f is shown in Fig. 11. It is clear that f (T) remains equal to 1 below the metale semiconductor transition temperature, which confirms the strong dominance of the FM fraction in this range. Then, the FM volume fraction begins to decrease to 0 from the ferromagnetic metallic state to a paramagnetic-semiconductor state, confirming the validity of the percolation approach which assumes a convertion from ferromagnetic to paramagnetic regions. 7. Conclusion To conclude, we have examined the critical behavior of resistivity in LGSMIO sample under different applied magnetic fields.
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