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Critical behavior of Zn0.6−xNixCu0.4Fe2O4 ferrite nanoparticles
Journal of Alloys and Compounds 656 (2016) 676e684
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Critical behavior of Zn0.6xNixCu0.4Fe2O4 ferrite nanoparticles Elaa Oumezzine a, *, Sobhi Hcini a, b, Mohamed Baazaoui a, E.K. Hlil c, Mohamed Oumezzine a a
Laboratory of Physical Chemistry of Materials, Faculty of Science of Monastir, Department of Physics, 5019, University of Monastir, Monastir, Tunisia Buraydah of Technical College at Al-Asyah, Al-Asyah - King Abdulaziz Road, PB: 304, Al-Asyah 51971, Kingdom of Saudi Arabia c Neel Institute, CNRS and Joseph Fourier University, PB: 166, Grenoble 38042, France b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 July 2015 Received in revised form 6 September 2015 Accepted 30 September 2015 Available online 9 October 2015
We have investigated the critical behavior of Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles near the ferromagnetic-paramagnetic (FM-PM) phase-transition temperature (TC). Experimental results reveal that all samples undergo a second-order phase transition. Through various techniques such as modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, the estimated critical exponents are close to those expected for three-dimensional Heisenberg class for x ¼ 0 (b ¼ 0.386 ± 0.002, g ¼ 1.271 ± 0.012 and d ¼ 4.387 at TC ¼ 305 K). Whereas for a high amount of Ni, these exponents belong to a different universality class (b ¼ 0.716 ± 0.063, g ¼ 0.807 ± 0.008 and d ¼ 2.010 at TC ¼ 565 K for x ¼ 0.2 sample) and (b ¼ 0.785 ± 0.004, g ¼ 0.797 ± 0.002 and d ¼ 2.061 at TC ¼ 705 K for x ¼ 0.4 sample). This is due to the fact that the substitution of Zn2þ (non-magnetic ions) by Ni2þ (magnetic ions) increases the AeB interaction sites of AB2O4 spinel ferrite which in turn increases magnetic disorder when increasing Ni content. Using the magnetic entropy change equation: themax DS ¼ aðHÞn , we have studied the relationship between the exponent n and the critical exponents of M our samples. The obtained n value are 0.641, 0.843 and 0.873 for x ¼ 0, 0.2 and 0.4, respectively. These values are in good agreement with those deducted from the critical exponents using the KF method. © 2015 Elsevier B.V. All rights reserved.
Keywords: Ferrite nanoparticles Second order phase transition Critical behavior
1. Introduction Spinel ferrite compounds with the general molecular formula MFe2O4 (where M2þ ¼ Mn2þ, Fe2þ, Co2þ, Ni2þ, Cu2þ, Zn2þ etc…) belong to an important family of materials. These materials are successfully used in different fields like in the microwave industries, disk recording, electrical devices, ferrofluids [1e6]. Shortly, the ferrite materials have also been found to exhibit the large magnetocaloric effect (MCE) under a moderate applied magnetic field revealing that these compounds are possible candidates for magnetic refrigeration applications [7e10]. So, the study of the MCE of these ferrite materials is not only important from the point of view of potential applications but also because it provides a tool to understand the intrinsic properties of these materials. In particular, the details of the magnetic phase transition and critical behavior can be obtained through the MCE. In fact, the analysis of the critical behavior in the vicinity of the magnetic phase transition is a powerful tool to investigate in details the mechanisms of the
* Corresponding author. E-mail address: [email protected] (E. Oumezzine). http://dx.doi.org/10.1016/j.jallcom.2015.09.269 0925-8388/© 2015 Elsevier B.V. All rights reserved.
magnetic interaction responsible for the transition [11,12]. The study of critical exponents allows determining very accurately the Curie temperature (TC) and the correlation of these exponents with those of the appropriate model, existing in the literature, allows the determination of the exchange integral in the ferromagnetic state. Experimental studies of the critical behavior near the PM-FM phase transition, using a variety of techniques, have yielded a wide range of values for the critical exponent b of the magnetization [13e15]. The values range between 0.25 and 0.5, which includes the mean-field (b ¼ 0.5), the 3D isotropic nearest-neighbor Heisenberg (b ¼ 0.365), the 3D Ising (b ¼ 0.325) and tricritical mean-field (b ¼ 0.25) estimates. In addition to b, static dc-magnetization measurements also yield critical exponents g and d for the initial susceptibility c(T) and the critical isotherm M(TC, H), respectively [16,17]. Recently, we have reported in our previous work [18] the structural, magnetic and magnetocaloric properties of Zn0.6xNixCu0.4Fe2O4 (0 x 0.6) ferrite nanoparticles prepared using the Pechini solegel technique. The X-ray diffraction results indicate that the ferrite samples have a cubic spinel type structure without any impurity phase. Structural analysis shows that the lattice constant and the unit cell volume decrease with increasing Ni
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content. Magnetization measurements and Arrott plot analysis reveal a second-order paramagnetic (PM)-ferromagnetic (FM) phase transition with an increase in Curie temperature TC with increasing Ni content. The magnetocaloric effect has also been assessed by means of magnetic entropy change DSM which is basically determined from magnetic field dependences of magnetization at different temperatures M(H, T) near TC. We have shown that the doping of a high amount of Ni increased the maximum . Respectively, the relative cooling power magnetic entropy DSmax M (RCP) values are relatively high which makes our samples promising materials to be used in ecologically friendly magnetic refrigeration technology. Due to the lack of many investigations on critical behavior of ferrite materials, we detailed in the present work the critical behavior in the vicinity of the PM-FM phase transition for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles by analyzing the critical exponents through various techniques, such as the modified Arrott plot (MAP) and the Kouvel-Fisher (KF) methods. 2. Results Fig. 1 shows the representative isothermal magnetization M(H, T) curves recorded at temperatures around the Curie temperature TC for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) samples in the applied field range of H ¼ 0e50 kOe. Below TC, M(H, T) curves show a non linear behavior with a sharp increase for low field values and a tendency to saturation as field increases reflecting a ferromagnetic behavior. However for T > TC, a drastically decrease of M(H, T) is observed with an almost linear behavior reflecting a paramagnetic behavior, due to the thermal agitation which disrupts the arrangement of the magnetic moments. In order, to determine the nature of the magnetic phase transition (first or second order), we presented in Fig. 2 the Arrott plots (M1/b vs. (H/M)1/g with b ¼ 0.5 and g ¼ 1 [19]) for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) samples. The M2 vs. H/M curves should be straight lines, and the intercept of these straight lines on the H/M axis determines the order of the magnetic transition: a positive slope corresponds to a second order transition while a negative slope corresponds to a first-order transition. The (M2 vs. H/M) curves presented in Fig. 2 exhibit, in the vicinity of TC, a positive slope indicating that the PM-FM phase transition observed for our samples is of second-order according to Banerjee criteria [20]. According to the mean field model, such curves should give a series of straight lines in the high field region at different temperatures, and the line at TC should cross the origin [21]. It is clear in Fig. 2 that these conditions are not accurate for our samples which indicate that the mean field theory is invalid. Therefore, in order to obtain the correct b and g critical exponents for x ¼ 0, 0.2 and 0.4 samples, the M(H, T) data was analyzed using the MAP method [22], which is generalized by the scaling equation of state: 1
1
ðH=MÞg ¼ c1 ε þ c2 M b
Fig. 1. M (H, T) curves near TC Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles.
(1)
where c1 and c2 are temperature dependent parameters and ε ¼ (T TC)/TC is the reduced temperature. Fig. 3 shows the MAP at different temperatures by using three models of critical exponents for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) samples: the 3D-Heisenberg model (b ¼ 0.365, g ¼ 1.336) in Fig. 3(aec); the 3D-Ising model (b ¼ 0.325, g ¼ 1.24) in Fig. 3(def) and the tricritical mean-field model (b ¼ 0.25, g ¼ 1) in Fig. 3(gei). It is clear from Fig. 3 that the conditions that all isotherms are almost parallel in high fields, the intercept of these lines on the (H/ M)1/g axis is negative/positive below/above TC and the line of M1/b
vs (H/M)1/g at TC ¼ 305 K pass through the origin, are more accurate for the 3D-Heisenberg model for x ¼ 0 sample (see Fig. 3a). However, the curves of x ¼ 0.2 and x ¼ 0.4 samples are non linear and show a down curvature even in the high field region, which indicates that these three models are also unable to describe the critical behavior for these two samples. The analysis of MAP by using the appropriate model, show that x ¼ 0.2 and 0.4 samples exhibit unconventional exponents. For new construction of MAP, we need to find other sets of the critical exponent values reflecting more frankly the critical behavior of
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(b ¼ 0.716 and g ¼ 0.807) for x ¼ 0.2 and (b ¼ 0.751 and g ¼ 0.781) for x ¼ 0.4. Following a standard procedure, the spontaneous magnetization versus temperature, Ms(T), would be obtained from the linear extrapolation in the high-magnetic field region for the isotherms to the coordinate axis of M1/b. Similarly, the inversely initial magnetic susceptibility versus temperature, c1 0 (T), is also obtained from the intersections with the (H/M)1/g axis. According to the scaling hypothesis [21], for a second-order phase transition around TC, critical exponents b (associated with the spontaneous magnetization Ms(H ¼ 0) below TC) and g (associated with the initial susceptibility c ¼ vM=vHjH¼0 above TC) are given as:
Ms ðTÞ ¼ M0 ð εÞb ;
ε<0
g c1 0 ðTÞ ¼ ðh0 =M0 Þε ;
ε>0
(2) (3)
here M0 and h0/M0 are the critical amplitudes. The Ms(T) and c0(T) data obtained from the linear extrapolation are then fitted to Eqs. (2) and (3), respectively, to achieve better b, g and TC values, as can be seen from Fig. 5. The corresponding fits give:
For x ¼ 0 sample;
b ¼ 0:375±0:004 with TC ¼ 305:15±0:100 g ¼ 1:314±0:027 with TC ¼ 305:082±0:009
For x ¼ 0:2 sample;
b ¼ 0:719±0:091 with TC ¼ 564:708±1:728 g ¼ 0:782±0:050 with TC ¼ 564:120±0:071
and
For x ¼ 0:4 sample;
b ¼ 0:766±0:002 with TC ¼ 705:396±0:001 g ¼ 0:774±0:003 with TC ¼ 705:256±0:004
On the other hand, these critical exponents can be also determined more accurately according to the Kouvel-Fisher (KF) method [23]:
Ms ðTÞ
dMs ðTÞ 1 ðT TC Þ ¼ dT b
c1 0 ðTÞ
" #1 dc1 ðT TC Þ 0 ðTÞ ¼ dT g
(4)
(5)
According to these equations, using the obtained Ms(T) and
Fig. 2. Arrott plots around TC for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles. According to the mean field model (values of critical exponents b ¼ 0.5 and g ¼ 1 should generate the regular Arrott plots, M2 vs. H/M).
1 c1 and c1 0 (T) vs. T from Fig. 5, the plotting of Ms(T)[dMs(T)/dT] 0 (T) 1 [dc1 against temperature in Fig. 6, yields straight lines 0 (T)/dT] with slope 1/b and 1/g, respectively, and the intercepts on the T axis
are equal to TC. The linear fit to the plots following the KF method gives:
these samples. We tried to change the values of the critical exponents, and this process was repeated several times until the iterations converge. The convergence was reached with the values of (b ¼ 0.716 and g ¼ 0.807 at TC ¼ 565 K, for x ¼ 0.2) and (b ¼ 0.751 and g ¼ 0.781 at TC ¼ 705 K, for x ¼ 0.4). Using these critical exponents, it is clear from Fig. 4 that all isotherms are almost parallel in high fields region, the intercept of these lines on the (H/M)1/g axis is negative/positive below/above TC and the line of M1/b vs (H/M)1/g at TC pass through the origin, implying that the estimated values of the critical exponents b and g are reliable. Based on the description mentioned above, we have chosen initial values of the critical exponents of the 3D-Heisenberg model (b ¼ 0.365 and g ¼ 1.336) for x ¼ 0 and the critical exponents
For x ¼ 0 sample;
b ¼ 0:386±0:002 with TC ¼ 305:012±0:001 g ¼ 1:271±0:012 with TC ¼ 305:036±1:155
For x ¼ 0:2 sample;
b ¼ 0:716±0:063 with TC ¼ 565:080±0:063 g ¼ 0:807±0:008 with TC ¼ 563:42±0:022
and
For x ¼ 0:4 sample;
b ¼ 0:785±0:004 with TC ¼ 705:651±0:006 g ¼ 0:797±0:002 with TC ¼ 705:964±0:003
It is notable that the values of the critical exponents as well as of TC, calculated using the both methods are consistent with the 3D-
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Fig. 3. Modified Arrott plots (MAP): isotherms of M1/b vs. (H/M)1/g for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles; (aec) 3D-Heisenberg model (b ¼ 0.365, g ¼ 1.336), (def) 3D-Ising model (b ¼ 0.325, g ¼ 1.24) and (gei) Tricritical mean-field model (b ¼ 0.25, g ¼ 1).
Heisenberg model for x ¼ 0 sample and with the corrected MAP for x ¼ 0.2 and 0.4 samples. This confirms that the estimated values of the critical exponents are self-consistent and unambiguous. Concerning the third exponent d, it can be determined directly from the critical isotherm of M(TC, H). Fig. 7 shows the plot of the isothermal magnetization M(H) vs. H at T ¼ TC, and in the inset the same plot has been presented in logelog scale. According to the following formula:
M ¼ DH
1=d
;
ε¼0
(6)
where D is the critical amplitude, log(M) vs. log(H) plot would give a straight line with slope 1/d. From the linear fitting in the inset of Fig. 7, we have obtained d ¼ 4.387, 2.010 and 2.061 for x ¼ 0, 0.2 and 0.4 respectively). Furthermore, according to the statistical theory, the three critical exponents obey the Widom scaling relation [24]:
d ¼ 1 þ g=b
(7)
Using the above values of b and g, Eq. (7) yields (d ¼ 4.504, 2.087 and d ¼ 2.010 for x ¼ 0, 0.2 and 0.4 respectively) when b and g are evaluated according to Fig. 5 and (d ¼ 4.292, 2.127 and d ¼ 2.015 for x ¼ 0, 0.2 and 0.4 respectively) when b and g are obtained by the KF method (Fig. 6). Thus, the critical exponents obtained in this study
obey the Widom scaling relation, implying that the obtained b and g values are reliable. We have then compared our data with the prediction of the scaling theory. According to the critical region theory, the isothermal magnetization can be described by Ref. [21]:
. MðH; εÞ ¼ jεjb f± H jεjbþg
(8)
where fþ for T > TC and f for T < TC are regular analytical functions. Eq. (8) implies that Mjεjb as a function of HjεjðbþgÞ produces two universal curves, one for temperatures above TC and the other for temperatures below TC. With the obtained critical exponents, the scaling performance of Mjεjb vs. HjεjðbþgÞ curves (see Fig. 8 and their insets) reveals that the M-H data points at high-magnetic fields falling into two fþ and f universal branches for T > TC and T < TC, respectively. These results prove the reliability of the critical exponents obtained from our work. In order to demonstrate the influence of the critical exponents on the magnetocaloric effect MCE, the field dependence of the entropy change was also analyzed. According to Oesterreicher et al., the field dependence of the maximum magnetic entropy change of materials with a second-order phase transition can be expressed as [25]:
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Fig. 4. Modified Arrott plots (MAP): isotherms of M1/b vs. (H/M)1/g with the critical exponents obtained for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles.
max DS ¼ aðHÞn M
(9)
where a is a constant and the exponent n depends on the magnetic state of the samples. It can be locally calculated as follows:
n¼
dlnDSM dlnH
(10)
Particularly, at T ¼ TC, the exponent n becomes field independent [26]. In this case:
n¼1þ
b1 bþg
(11)
where b and g are the critical exponents. With bd ¼ (b þ g), the relation (11) can be written as:
n¼1þ
1 1 1 d b
(12)
Using the order parameters b, g and d at TC ¼ 305 K, 565 K and 705 K respectively for x ¼ 0, 0.2 and 0.4 samples, obtained from the KF method, the values of n calculated from the above relations are found to be:
Fig. 5. Spontaneous magnetization Ms(T) (left-axis) and inverse initial susceptibility c1 0 (T) (right-axis) as a function of temperature for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles, along with fits (red solid lines) based on the power laws defined by Eqs. (2) and (3), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
8 < n ¼ 0:629 for x ¼ 0 sample From Eq: ð11Þ : n ¼ 0:813 for x ¼ 0:2 sample : n ¼ 0:864 for x ¼ 0:4 sample 8 < n ¼ 0:637 for x ¼ 0 sample From Eq: ð12Þ : n ¼ 0:802 for x ¼ 0:2 sample : n ¼ 0:867 for x ¼ 0:4 sample In the mean field approach, the value of n at T ¼ TC is predicted to be 2/3 [27]. Consequently, the experimental values of the critical parameters studied previously for our compounds confirm the invalidity of the mean field model. vs. H at each On the other hand, by fitting the data of DSmax M
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Fig. 6. Kouvel-Fisher plots for the spontaneous magnetization Ms(T) (left-axis) and the inverse initial susceptibility c1 0 (T) (right-axis) for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles, red solid lines are the fits of the model according to Eqs. (4) and (5), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
temperature to Eq. (9) (inset of Fig. 9), we obtain the value of n as a function of temperature as depicted in Fig. 9 for x ¼ 0, 0.2 and 0.4 samples. It can be noted from Fig. 9 that the value of n exhibits a sudden change around TC. It decreases with increasing temperature in the FM region (below TC) with a minimum value near TC and increases with increasing temperature in the PM region (above TC). The obtained n value are 0.641, 0.843 and 0.873 for x ¼ 0, 0.2 and 0.4 samples, respectively. These values are in good agreement with those obtained from the critical exponents using Eqs. (11) and (12). This confirms the good correlation between the critical behavior and the magnetocaloric effect.
681
Fig. 7. Isothermal M(TC, H) plot for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles (the insets shows the same plots in logelog scale and the red solid lines are the linear fits following Eq. (6), for the determination of the critical exponent d. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Discussion The values of the critical exponents for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) nanoparticles samples (present work), those of some other ferrite compounds available in literature [28e31], besides the theoretical values based on various models [14,21], are summarized in Table 1 for comparison. We can see from Table 1 that the values of the critical exponents of our samples depend on the Ni concentration. Indeed, the values of critical exponent b
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Fig. 8. Scaling plots for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles below and above TC using b and g determined by the Kouvel-Fisher method. The insets show the same plots on a logelog scale.
increase however the g values decrease when increasing Ni content in Zn0.6xNixCu0.4Fe2O4 samples. For x ¼ 0 sample, b ¼ 0.386 and g ¼ 1.271 are close to the 3D-Heisenberg theory. However, x ¼ 0.2 and 0.4 samples exhibit unconventional exponents (b ¼ 0.716 and 0.785, respectively). From Table 1, one can notice that the critical exponents of undoped sample (x ¼ 0) are found to be comparable to those reported in Refs. [28,29] where mostly exponents are comparable to the 3D-Heisenberg values. However, the values of the critical exponents for x ¼ 0.2 and 0.4 samples are different to those found in Refs. [30,31] for ZnxNi1xFe2O4 system. In these references, it is
Fig. 9. Temperature dependence of exponent n of Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles. The insets present the field dependence of entropy change.
observed that the critical behavior of samples cannot be affected by slight increase of Ni magnetic ions (with 0.25 Ni at.%). In particular, the authors in Refs. [30,31] show that the critical exponent values, within the error limits, are consistent with the 3D-Heisenberg values, with no obvious concentration dependence. This difference observed between our results and those found in Refs. [30,31] implies that an addition of a high amount of Ni magnetic ions changes the critical behavior of our samples. This can best be explained, in the case of our samples, as follows: Taking in charge neutrality into consideration, the general molecular formula (AB2O4 spinel ferrite) of our samples can be written 2þ 3þ 2þ 3þ 2 as: ðCu2þ 0:4 Zn0:6x Fex ÞðNix Fe2x ÞO4 (0 x 0.4) [18]. According
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Table 1 Comparison of the values of the critical exponents of Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles (this work) with those the theoretical values based on the standard models and with some other ferrites. Material
TC (K)
b
g
d
Reference
Zn0.6Cu0.4Fe2O4 Zn0.4Ni0.2Cu0.4Fe2O4 Zn0.2Ni0.4Cu0.4Fe2O4 Mean-field model Tricritical mean-field model 3D-Heisenberg model 3D-Ising model Zn0.2Fe2.804 Ti0.2Fe2.8O4 Zn0.75Ni0.25Fe2O4 Zn0.5Ni0.5Fe2O4 Zn0.25Ni0.75Fe2O4 NiFe2O4
305 565 705
0.386 ± 0.002 0.716 ± 0.063 0.785 ± 0.004 0.5 0.25 0.365 ± 0.003 0.325 ± 0.002 0.40 ± 0.01 0.47 _ 0.442 0.405 0.376
1.271 ± 0.012 0.807 ± 0.008 0.797 ± 0.002 1.0 1.0 1.336 ± 0.004 1.24 ± 0.002 1.35 ± 0.01 1.28 1.420 1.386 1.392 1.330
4.387 2.010 2.061 3.0 5.0 4.80 ± 0.04 4.82 ± 0.02 _ _ _ _ _ _
This work This work This work [21] [14] [21] [21] [28] [29] [30,31] [30,31] [30,31] [30,31]
763.5 602 295.2 553.5 729.5 865.4
to this formula, the Cu content was kept constant at 40 atom % of A sites, the magnetic Ni2þ ions are known to occupy the octahedral (B) sites and the non-magnetic Zn2þ ions have a preference for the tetrahedral (A) sites, while Fe3þ ions are distributed across both the sites [7,9,32,33]. As the concentration of magnetic Ni2þ ions in the B sites increases, the magnetic Fe3þ ions are pushed from the B to the A sites. The increase of magnetic Fe3þ ions in the A sites increases the AeB interaction and in turn increases the magnetic disorder which lead to the increase of the critical exponent b and the decrease of g when increasing Ni content. Consequently, the critical exponents change from the 3D-Heisenberg values for x ¼ 0 sample to unconventional exponents for x ¼ 0.2 and 0.4 samples. Theoretically, these results can reflect the following useful information: the critical exponents in these materials are governed by lattice dimension (3D), the dimension of order parameter (n ¼ 3, magnetization), and the range of interaction (short range, long range or infinite) [34]. In homogeneous magnets the universality class of the magnetic phase transition depends on the range of the exchange interaction [34]:
. JðrÞ ¼ 1 r dþs
(13)
where r, d and s are the distance, the dimension of the system and the range of the exchange interaction, respectively. It has been argued that if s 2, the 3D-Heisenberg exponents (b ¼ 0.365, g ¼ 1.336, d ¼ 4.8) are valid for a three-dimensional isotropic ferromagnet and J(r) decreases with distance r faster than r5 [16,35]. Whereas when s is less than 3/2, the mean-field exponents (b ¼ 0.5, g ¼ 1 and d ¼ 3) are valid, which indicates that J(r) decreases with r slower than r4.5. For intermediate range, i.e., for J(r) z r3s with 3/2 s 2, the exponents belong to a different universality class which depends upon s. In the case of our samples, the values of the critical exponents are in agreement with the 3DHeisenberg model for x ¼ 0 sample however x ¼ 0.2 and 0.4 samples exhibit unconventional exponents. So the exchange interaction J(r) decreases with distance r faster than r5 for x ¼ 0 sample however it ranges from r5 and r4.5 for x ¼ 0.2 and 0.4 samples. 4. Conclusion In summary, we have studied the critical behavior of Zn0.6xNixCu0.4Fe2O4 around TC values. Arrott plots reveal a second order PM-FM phase transition. The values of critical exponents as well as TC calculated using both modified Arrott plot and KouvelFisher plot are reasonably well. For x ¼ 0 sample, the estimated critical exponents are consistent with the 3D-Heisenberg model,
however the exponents belong to a different universality class for x ¼ 0.2 and 0.4 samples. The change in the universality class from the 3D-Heisenberg model to a different universality class is due to the substitution of Zn2þ (non-magnetic ions) by Ni2þ (magnetic ions). The field and temperature-dependent magnetization behavior follows the scaling hypothesis and all the data point fall into two distinct branches, one for T < TC and another for T > TC, indicating that the critical exponents obtained in our work are can accurate. Interestingly, around dependences of DSmax M TC, field ¼ aðHÞn. The n values obtained be described by a power law DSmax M from the power-law fitting are in good agreement with those obtained from the critical exponents using the KF method. References [1] R.D. McMichael, R.D. Shull, I.J. Swartzendruber, L.H. Bennett, J. Magn. Magn. Mater. 111 (1992) 29. [2] A. Ghasemi, A. Hossienpour, A. Morisako, A. Saakhi, M. Saleki, J. Magn. Magn. Mater. 302 (2006) 429. [3] Y. Sahoo, M. Cheon, S. Wang, H. Luo, E.P. Furlani, P.N. Prasad, J. Phys. Chem. B 108 (2004) 3380. € [4] N. Kasapoglu, A. Baykal, Y. KOseoģlu, M.S. Toprak, Scr. Mater. 57 (2007) 441. € [5] T. Ozkaya, A. Baykal, M.S. Toprak, Y. KOseoģlu, Z. Durmus, J. Magn. Magn. Mater. 321 (2009) 2145. [6] Y. Cheng, Y. Zheng, Y. Wang, F. Bao, Y. Qin, Synth. J. Solid State Chem. 178 (2005) 2394. [7] M.S. Anwar, Faheem Ahmed, Bon Heun Koo, Acta. Mater. 71 (2014) 100. [8] Shahida Akhter, D.P. Paul, S.M. Hoque, M.A. Hakim, M. Hudl, R. Mathieu, P. Nordblad, J. Magn. Magn. Mater. 367 (2014) 75. [9] N. Chau, N.K. Thuan, D.L. Minh, N.H. Luong, VNU J. Sci. Math. Phys. 24 (2008) 155. [10] P. Poddar, J. Gass, D.J. Rebar, S. Srinath, H. Srikanth, S.A. Morrison, E.E. Carpenter, J. Magn. Magn. Mater. 307 (2006) 227. [11] R. Cabassi, F. Bolzoni, A. Gauzzi, F. Licci, Phys. Rev. B 74 (2006) 184425. €hnle, W.U. Kellner, H. Kronmüller, Phys. Rev. B 35 (1987) 3640. [12] M. Fa [13] C.S. Hong, W.S. Kim, N.H. Hur, Phys. Rev. B 63 (2001) 092504. [14] H.S. Shin, J.E. Lee, Y.S. Nam, H.L. Ju, C.W. Park, J. Solid State Commun. 118 (2001) 377. [15] A. Schwartz, M. Scheffler, S.M. Anlage, Phys. Rev. B 61 (2000) R870. [16] K. Ghosh, C.J. Lobb, R.L. Greene, S.G. Karabashev, D.A. Shulyatev, A.A. Arsenov, Y. Mukovskii, Phys. Rev. Lett. 81 (1998) 4740. [17] Ch.V. Mohan, M. Seeger, H. Kronmüller, P. Murugaraj, J. Maier, J. Magn. Magn. Mater. 183 (1998) 348. [18] Elaa Oumezzine, Sobhi Hcini, Mohamed Baazaoui, El-Kebir Hlil, Mohamed Oumezzine, Powder Technol. 278 (2015) 189. [19] A. Arrott, Phys. Rev. 108 (1957) 1394. [20] S.K. Banerjee, Phys. Lett. 12 (1964) 16. [21] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, London, 1971. [22] A. Arrott, J.E. Noakes, Phys. Rev. Lett. 19 (1967) 786. [23] J.S. Kouvel, M.E. Fisher, Phys. Rev. 136 (1964) A1626. [24] B. Widom, J. Chem. Phys. 43 (1965) 3898. [25] H. Oesterreicher, F.T. Parker, Magnetic cooling near Curie temperatures above 300 K, J. Appl. Phys. 55 (1984) 433. [26] V. Franco, A. Conde, M.D. Kuz'min, J.M. Romero-Enrique, J. Appl. Phys. 105 (2009) 07A917. [27] M. Pe˛ kała, J. Appl. Phys. 108 (2010) 113913. [28] C.J. Tinsley, J. Magn. Magn. Mater. 15 (1980) 459.
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[29] A.G. Flores, V. Raposo, J. Iniguez, J.M. Garcia-Beneytez, J. Arcas, M. Vazquez, Phys. Status Solidi 191 (2002) 555. [30] M. Haug, M. Fahnle, H. Kronmuller, F. Haberay, J. Magn. Magn. Mater. 69 (1987) 163. [31] M. Haug, M. Fahnle, H. Kronmuller, F. Haberay, Phys. Status Solidi 144 (1987) 411.
[32] P.K. Roy, J. Bera, J. Mater. Process. Tech. 197 (2008) 279. [33] A. Verma, T.C. Goel, R.G. Mendiratta, P. Kishan, J. Magn. Magn. Mater. 208 (2000) 13. [34] M.E. Fisher, S.K. Ma, B.G. Nickel, Phys. Rev. Lett. 29 (1972) 917. [35] M.H. Phana, V. Franco, N.S. Bingham, H. Srikanth, N.H. Hur, S.C. Yu, J. Alloy. Compd. 508 (2010) 238.
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Critical behavior of Zn0.6xNixCu0.4Fe2O4 ferrite nanoparticles Elaa Oumezzine a, *, Sobhi Hcini a, b, Mohamed Baazaoui a, E.K. Hlil c, Mohamed Oumezzine a a
Laboratory of Physical Chemistry of Materials, Faculty of Science of Monastir, Department of Physics, 5019, University of Monastir, Monastir, Tunisia Buraydah of Technical College at Al-Asyah, Al-Asyah - King Abdulaziz Road, PB: 304, Al-Asyah 51971, Kingdom of Saudi Arabia c Neel Institute, CNRS and Joseph Fourier University, PB: 166, Grenoble 38042, France b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 July 2015 Received in revised form 6 September 2015 Accepted 30 September 2015 Available online 9 October 2015
We have investigated the critical behavior of Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles near the ferromagnetic-paramagnetic (FM-PM) phase-transition temperature (TC). Experimental results reveal that all samples undergo a second-order phase transition. Through various techniques such as modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, the estimated critical exponents are close to those expected for three-dimensional Heisenberg class for x ¼ 0 (b ¼ 0.386 ± 0.002, g ¼ 1.271 ± 0.012 and d ¼ 4.387 at TC ¼ 305 K). Whereas for a high amount of Ni, these exponents belong to a different universality class (b ¼ 0.716 ± 0.063, g ¼ 0.807 ± 0.008 and d ¼ 2.010 at TC ¼ 565 K for x ¼ 0.2 sample) and (b ¼ 0.785 ± 0.004, g ¼ 0.797 ± 0.002 and d ¼ 2.061 at TC ¼ 705 K for x ¼ 0.4 sample). This is due to the fact that the substitution of Zn2þ (non-magnetic ions) by Ni2þ (magnetic ions) increases the AeB interaction sites of AB2O4 spinel ferrite which in turn increases magnetic disorder when increasing Ni content. Using the magnetic entropy change equation: themax DS ¼ aðHÞn , we have studied the relationship between the exponent n and the critical exponents of M our samples. The obtained n value are 0.641, 0.843 and 0.873 for x ¼ 0, 0.2 and 0.4, respectively. These values are in good agreement with those deducted from the critical exponents using the KF method. © 2015 Elsevier B.V. All rights reserved.
Keywords: Ferrite nanoparticles Second order phase transition Critical behavior
1. Introduction Spinel ferrite compounds with the general molecular formula MFe2O4 (where M2þ ¼ Mn2þ, Fe2þ, Co2þ, Ni2þ, Cu2þ, Zn2þ etc…) belong to an important family of materials. These materials are successfully used in different fields like in the microwave industries, disk recording, electrical devices, ferrofluids [1e6]. Shortly, the ferrite materials have also been found to exhibit the large magnetocaloric effect (MCE) under a moderate applied magnetic field revealing that these compounds are possible candidates for magnetic refrigeration applications [7e10]. So, the study of the MCE of these ferrite materials is not only important from the point of view of potential applications but also because it provides a tool to understand the intrinsic properties of these materials. In particular, the details of the magnetic phase transition and critical behavior can be obtained through the MCE. In fact, the analysis of the critical behavior in the vicinity of the magnetic phase transition is a powerful tool to investigate in details the mechanisms of the
* Corresponding author. E-mail address: [email protected] (E. Oumezzine). http://dx.doi.org/10.1016/j.jallcom.2015.09.269 0925-8388/© 2015 Elsevier B.V. All rights reserved.
magnetic interaction responsible for the transition [11,12]. The study of critical exponents allows determining very accurately the Curie temperature (TC) and the correlation of these exponents with those of the appropriate model, existing in the literature, allows the determination of the exchange integral in the ferromagnetic state. Experimental studies of the critical behavior near the PM-FM phase transition, using a variety of techniques, have yielded a wide range of values for the critical exponent b of the magnetization [13e15]. The values range between 0.25 and 0.5, which includes the mean-field (b ¼ 0.5), the 3D isotropic nearest-neighbor Heisenberg (b ¼ 0.365), the 3D Ising (b ¼ 0.325) and tricritical mean-field (b ¼ 0.25) estimates. In addition to b, static dc-magnetization measurements also yield critical exponents g and d for the initial susceptibility c(T) and the critical isotherm M(TC, H), respectively [16,17]. Recently, we have reported in our previous work [18] the structural, magnetic and magnetocaloric properties of Zn0.6xNixCu0.4Fe2O4 (0 x 0.6) ferrite nanoparticles prepared using the Pechini solegel technique. The X-ray diffraction results indicate that the ferrite samples have a cubic spinel type structure without any impurity phase. Structural analysis shows that the lattice constant and the unit cell volume decrease with increasing Ni
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content. Magnetization measurements and Arrott plot analysis reveal a second-order paramagnetic (PM)-ferromagnetic (FM) phase transition with an increase in Curie temperature TC with increasing Ni content. The magnetocaloric effect has also been assessed by means of magnetic entropy change DSM which is basically determined from magnetic field dependences of magnetization at different temperatures M(H, T) near TC. We have shown that the doping of a high amount of Ni increased the maximum . Respectively, the relative cooling power magnetic entropy DSmax M (RCP) values are relatively high which makes our samples promising materials to be used in ecologically friendly magnetic refrigeration technology. Due to the lack of many investigations on critical behavior of ferrite materials, we detailed in the present work the critical behavior in the vicinity of the PM-FM phase transition for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles by analyzing the critical exponents through various techniques, such as the modified Arrott plot (MAP) and the Kouvel-Fisher (KF) methods. 2. Results Fig. 1 shows the representative isothermal magnetization M(H, T) curves recorded at temperatures around the Curie temperature TC for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) samples in the applied field range of H ¼ 0e50 kOe. Below TC, M(H, T) curves show a non linear behavior with a sharp increase for low field values and a tendency to saturation as field increases reflecting a ferromagnetic behavior. However for T > TC, a drastically decrease of M(H, T) is observed with an almost linear behavior reflecting a paramagnetic behavior, due to the thermal agitation which disrupts the arrangement of the magnetic moments. In order, to determine the nature of the magnetic phase transition (first or second order), we presented in Fig. 2 the Arrott plots (M1/b vs. (H/M)1/g with b ¼ 0.5 and g ¼ 1 [19]) for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) samples. The M2 vs. H/M curves should be straight lines, and the intercept of these straight lines on the H/M axis determines the order of the magnetic transition: a positive slope corresponds to a second order transition while a negative slope corresponds to a first-order transition. The (M2 vs. H/M) curves presented in Fig. 2 exhibit, in the vicinity of TC, a positive slope indicating that the PM-FM phase transition observed for our samples is of second-order according to Banerjee criteria [20]. According to the mean field model, such curves should give a series of straight lines in the high field region at different temperatures, and the line at TC should cross the origin [21]. It is clear in Fig. 2 that these conditions are not accurate for our samples which indicate that the mean field theory is invalid. Therefore, in order to obtain the correct b and g critical exponents for x ¼ 0, 0.2 and 0.4 samples, the M(H, T) data was analyzed using the MAP method [22], which is generalized by the scaling equation of state: 1
1
ðH=MÞg ¼ c1 ε þ c2 M b
Fig. 1. M (H, T) curves near TC Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles.
(1)
where c1 and c2 are temperature dependent parameters and ε ¼ (T TC)/TC is the reduced temperature. Fig. 3 shows the MAP at different temperatures by using three models of critical exponents for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) samples: the 3D-Heisenberg model (b ¼ 0.365, g ¼ 1.336) in Fig. 3(aec); the 3D-Ising model (b ¼ 0.325, g ¼ 1.24) in Fig. 3(def) and the tricritical mean-field model (b ¼ 0.25, g ¼ 1) in Fig. 3(gei). It is clear from Fig. 3 that the conditions that all isotherms are almost parallel in high fields, the intercept of these lines on the (H/ M)1/g axis is negative/positive below/above TC and the line of M1/b
vs (H/M)1/g at TC ¼ 305 K pass through the origin, are more accurate for the 3D-Heisenberg model for x ¼ 0 sample (see Fig. 3a). However, the curves of x ¼ 0.2 and x ¼ 0.4 samples are non linear and show a down curvature even in the high field region, which indicates that these three models are also unable to describe the critical behavior for these two samples. The analysis of MAP by using the appropriate model, show that x ¼ 0.2 and 0.4 samples exhibit unconventional exponents. For new construction of MAP, we need to find other sets of the critical exponent values reflecting more frankly the critical behavior of
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(b ¼ 0.716 and g ¼ 0.807) for x ¼ 0.2 and (b ¼ 0.751 and g ¼ 0.781) for x ¼ 0.4. Following a standard procedure, the spontaneous magnetization versus temperature, Ms(T), would be obtained from the linear extrapolation in the high-magnetic field region for the isotherms to the coordinate axis of M1/b. Similarly, the inversely initial magnetic susceptibility versus temperature, c1 0 (T), is also obtained from the intersections with the (H/M)1/g axis. According to the scaling hypothesis [21], for a second-order phase transition around TC, critical exponents b (associated with the spontaneous magnetization Ms(H ¼ 0) below TC) and g (associated with the initial susceptibility c ¼ vM=vHjH¼0 above TC) are given as:
Ms ðTÞ ¼ M0 ð εÞb ;
ε<0
g c1 0 ðTÞ ¼ ðh0 =M0 Þε ;
ε>0
(2) (3)
here M0 and h0/M0 are the critical amplitudes. The Ms(T) and c0(T) data obtained from the linear extrapolation are then fitted to Eqs. (2) and (3), respectively, to achieve better b, g and TC values, as can be seen from Fig. 5. The corresponding fits give:
For x ¼ 0 sample;
b ¼ 0:375±0:004 with TC ¼ 305:15±0:100 g ¼ 1:314±0:027 with TC ¼ 305:082±0:009
For x ¼ 0:2 sample;
b ¼ 0:719±0:091 with TC ¼ 564:708±1:728 g ¼ 0:782±0:050 with TC ¼ 564:120±0:071
and
For x ¼ 0:4 sample;
b ¼ 0:766±0:002 with TC ¼ 705:396±0:001 g ¼ 0:774±0:003 with TC ¼ 705:256±0:004
On the other hand, these critical exponents can be also determined more accurately according to the Kouvel-Fisher (KF) method [23]:
Ms ðTÞ
dMs ðTÞ 1 ðT TC Þ ¼ dT b
c1 0 ðTÞ
" #1 dc1 ðT TC Þ 0 ðTÞ ¼ dT g
(4)
(5)
According to these equations, using the obtained Ms(T) and
Fig. 2. Arrott plots around TC for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles. According to the mean field model (values of critical exponents b ¼ 0.5 and g ¼ 1 should generate the regular Arrott plots, M2 vs. H/M).
1 c1 and c1 0 (T) vs. T from Fig. 5, the plotting of Ms(T)[dMs(T)/dT] 0 (T) 1 [dc1 against temperature in Fig. 6, yields straight lines 0 (T)/dT] with slope 1/b and 1/g, respectively, and the intercepts on the T axis
are equal to TC. The linear fit to the plots following the KF method gives:
these samples. We tried to change the values of the critical exponents, and this process was repeated several times until the iterations converge. The convergence was reached with the values of (b ¼ 0.716 and g ¼ 0.807 at TC ¼ 565 K, for x ¼ 0.2) and (b ¼ 0.751 and g ¼ 0.781 at TC ¼ 705 K, for x ¼ 0.4). Using these critical exponents, it is clear from Fig. 4 that all isotherms are almost parallel in high fields region, the intercept of these lines on the (H/M)1/g axis is negative/positive below/above TC and the line of M1/b vs (H/M)1/g at TC pass through the origin, implying that the estimated values of the critical exponents b and g are reliable. Based on the description mentioned above, we have chosen initial values of the critical exponents of the 3D-Heisenberg model (b ¼ 0.365 and g ¼ 1.336) for x ¼ 0 and the critical exponents
For x ¼ 0 sample;
b ¼ 0:386±0:002 with TC ¼ 305:012±0:001 g ¼ 1:271±0:012 with TC ¼ 305:036±1:155
For x ¼ 0:2 sample;
b ¼ 0:716±0:063 with TC ¼ 565:080±0:063 g ¼ 0:807±0:008 with TC ¼ 563:42±0:022
and
For x ¼ 0:4 sample;
b ¼ 0:785±0:004 with TC ¼ 705:651±0:006 g ¼ 0:797±0:002 with TC ¼ 705:964±0:003
It is notable that the values of the critical exponents as well as of TC, calculated using the both methods are consistent with the 3D-
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Fig. 3. Modified Arrott plots (MAP): isotherms of M1/b vs. (H/M)1/g for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles; (aec) 3D-Heisenberg model (b ¼ 0.365, g ¼ 1.336), (def) 3D-Ising model (b ¼ 0.325, g ¼ 1.24) and (gei) Tricritical mean-field model (b ¼ 0.25, g ¼ 1).
Heisenberg model for x ¼ 0 sample and with the corrected MAP for x ¼ 0.2 and 0.4 samples. This confirms that the estimated values of the critical exponents are self-consistent and unambiguous. Concerning the third exponent d, it can be determined directly from the critical isotherm of M(TC, H). Fig. 7 shows the plot of the isothermal magnetization M(H) vs. H at T ¼ TC, and in the inset the same plot has been presented in logelog scale. According to the following formula:
M ¼ DH
1=d
;
ε¼0
(6)
where D is the critical amplitude, log(M) vs. log(H) plot would give a straight line with slope 1/d. From the linear fitting in the inset of Fig. 7, we have obtained d ¼ 4.387, 2.010 and 2.061 for x ¼ 0, 0.2 and 0.4 respectively). Furthermore, according to the statistical theory, the three critical exponents obey the Widom scaling relation [24]:
d ¼ 1 þ g=b
(7)
Using the above values of b and g, Eq. (7) yields (d ¼ 4.504, 2.087 and d ¼ 2.010 for x ¼ 0, 0.2 and 0.4 respectively) when b and g are evaluated according to Fig. 5 and (d ¼ 4.292, 2.127 and d ¼ 2.015 for x ¼ 0, 0.2 and 0.4 respectively) when b and g are obtained by the KF method (Fig. 6). Thus, the critical exponents obtained in this study
obey the Widom scaling relation, implying that the obtained b and g values are reliable. We have then compared our data with the prediction of the scaling theory. According to the critical region theory, the isothermal magnetization can be described by Ref. [21]:
. MðH; εÞ ¼ jεjb f± H jεjbþg
(8)
where fþ for T > TC and f for T < TC are regular analytical functions. Eq. (8) implies that Mjεjb as a function of HjεjðbþgÞ produces two universal curves, one for temperatures above TC and the other for temperatures below TC. With the obtained critical exponents, the scaling performance of Mjεjb vs. HjεjðbþgÞ curves (see Fig. 8 and their insets) reveals that the M-H data points at high-magnetic fields falling into two fþ and f universal branches for T > TC and T < TC, respectively. These results prove the reliability of the critical exponents obtained from our work. In order to demonstrate the influence of the critical exponents on the magnetocaloric effect MCE, the field dependence of the entropy change was also analyzed. According to Oesterreicher et al., the field dependence of the maximum magnetic entropy change of materials with a second-order phase transition can be expressed as [25]:
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Fig. 4. Modified Arrott plots (MAP): isotherms of M1/b vs. (H/M)1/g with the critical exponents obtained for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles.
max DS ¼ aðHÞn M
(9)
where a is a constant and the exponent n depends on the magnetic state of the samples. It can be locally calculated as follows:
n¼
dlnDSM dlnH
(10)
Particularly, at T ¼ TC, the exponent n becomes field independent [26]. In this case:
n¼1þ
b1 bþg
(11)
where b and g are the critical exponents. With bd ¼ (b þ g), the relation (11) can be written as:
n¼1þ
1 1 1 d b
(12)
Using the order parameters b, g and d at TC ¼ 305 K, 565 K and 705 K respectively for x ¼ 0, 0.2 and 0.4 samples, obtained from the KF method, the values of n calculated from the above relations are found to be:
Fig. 5. Spontaneous magnetization Ms(T) (left-axis) and inverse initial susceptibility c1 0 (T) (right-axis) as a function of temperature for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles, along with fits (red solid lines) based on the power laws defined by Eqs. (2) and (3), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
8 < n ¼ 0:629 for x ¼ 0 sample From Eq: ð11Þ : n ¼ 0:813 for x ¼ 0:2 sample : n ¼ 0:864 for x ¼ 0:4 sample 8 < n ¼ 0:637 for x ¼ 0 sample From Eq: ð12Þ : n ¼ 0:802 for x ¼ 0:2 sample : n ¼ 0:867 for x ¼ 0:4 sample In the mean field approach, the value of n at T ¼ TC is predicted to be 2/3 [27]. Consequently, the experimental values of the critical parameters studied previously for our compounds confirm the invalidity of the mean field model. vs. H at each On the other hand, by fitting the data of DSmax M
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Fig. 6. Kouvel-Fisher plots for the spontaneous magnetization Ms(T) (left-axis) and the inverse initial susceptibility c1 0 (T) (right-axis) for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles, red solid lines are the fits of the model according to Eqs. (4) and (5), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
temperature to Eq. (9) (inset of Fig. 9), we obtain the value of n as a function of temperature as depicted in Fig. 9 for x ¼ 0, 0.2 and 0.4 samples. It can be noted from Fig. 9 that the value of n exhibits a sudden change around TC. It decreases with increasing temperature in the FM region (below TC) with a minimum value near TC and increases with increasing temperature in the PM region (above TC). The obtained n value are 0.641, 0.843 and 0.873 for x ¼ 0, 0.2 and 0.4 samples, respectively. These values are in good agreement with those obtained from the critical exponents using Eqs. (11) and (12). This confirms the good correlation between the critical behavior and the magnetocaloric effect.
681
Fig. 7. Isothermal M(TC, H) plot for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles (the insets shows the same plots in logelog scale and the red solid lines are the linear fits following Eq. (6), for the determination of the critical exponent d. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Discussion The values of the critical exponents for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) nanoparticles samples (present work), those of some other ferrite compounds available in literature [28e31], besides the theoretical values based on various models [14,21], are summarized in Table 1 for comparison. We can see from Table 1 that the values of the critical exponents of our samples depend on the Ni concentration. Indeed, the values of critical exponent b
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Fig. 8. Scaling plots for Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles below and above TC using b and g determined by the Kouvel-Fisher method. The insets show the same plots on a logelog scale.
increase however the g values decrease when increasing Ni content in Zn0.6xNixCu0.4Fe2O4 samples. For x ¼ 0 sample, b ¼ 0.386 and g ¼ 1.271 are close to the 3D-Heisenberg theory. However, x ¼ 0.2 and 0.4 samples exhibit unconventional exponents (b ¼ 0.716 and 0.785, respectively). From Table 1, one can notice that the critical exponents of undoped sample (x ¼ 0) are found to be comparable to those reported in Refs. [28,29] where mostly exponents are comparable to the 3D-Heisenberg values. However, the values of the critical exponents for x ¼ 0.2 and 0.4 samples are different to those found in Refs. [30,31] for ZnxNi1xFe2O4 system. In these references, it is
Fig. 9. Temperature dependence of exponent n of Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles. The insets present the field dependence of entropy change.
observed that the critical behavior of samples cannot be affected by slight increase of Ni magnetic ions (with 0.25 Ni at.%). In particular, the authors in Refs. [30,31] show that the critical exponent values, within the error limits, are consistent with the 3D-Heisenberg values, with no obvious concentration dependence. This difference observed between our results and those found in Refs. [30,31] implies that an addition of a high amount of Ni magnetic ions changes the critical behavior of our samples. This can best be explained, in the case of our samples, as follows: Taking in charge neutrality into consideration, the general molecular formula (AB2O4 spinel ferrite) of our samples can be written 2þ 3þ 2þ 3þ 2 as: ðCu2þ 0:4 Zn0:6x Fex ÞðNix Fe2x ÞO4 (0 x 0.4) [18]. According
E. Oumezzine et al. / Journal of Alloys and Compounds 656 (2016) 676e684
683
Table 1 Comparison of the values of the critical exponents of Zn0.6xNixCu0.4Fe2O4 (x ¼ 0, 0.2 and 0.4) ferrite nanoparticles (this work) with those the theoretical values based on the standard models and with some other ferrites. Material
TC (K)
b
g
d
Reference
Zn0.6Cu0.4Fe2O4 Zn0.4Ni0.2Cu0.4Fe2O4 Zn0.2Ni0.4Cu0.4Fe2O4 Mean-field model Tricritical mean-field model 3D-Heisenberg model 3D-Ising model Zn0.2Fe2.804 Ti0.2Fe2.8O4 Zn0.75Ni0.25Fe2O4 Zn0.5Ni0.5Fe2O4 Zn0.25Ni0.75Fe2O4 NiFe2O4
305 565 705
0.386 ± 0.002 0.716 ± 0.063 0.785 ± 0.004 0.5 0.25 0.365 ± 0.003 0.325 ± 0.002 0.40 ± 0.01 0.47 _ 0.442 0.405 0.376
1.271 ± 0.012 0.807 ± 0.008 0.797 ± 0.002 1.0 1.0 1.336 ± 0.004 1.24 ± 0.002 1.35 ± 0.01 1.28 1.420 1.386 1.392 1.330
4.387 2.010 2.061 3.0 5.0 4.80 ± 0.04 4.82 ± 0.02 _ _ _ _ _ _
This work This work This work [21] [14] [21] [21] [28] [29] [30,31] [30,31] [30,31] [30,31]
763.5 602 295.2 553.5 729.5 865.4
to this formula, the Cu content was kept constant at 40 atom % of A sites, the magnetic Ni2þ ions are known to occupy the octahedral (B) sites and the non-magnetic Zn2þ ions have a preference for the tetrahedral (A) sites, while Fe3þ ions are distributed across both the sites [7,9,32,33]. As the concentration of magnetic Ni2þ ions in the B sites increases, the magnetic Fe3þ ions are pushed from the B to the A sites. The increase of magnetic Fe3þ ions in the A sites increases the AeB interaction and in turn increases the magnetic disorder which lead to the increase of the critical exponent b and the decrease of g when increasing Ni content. Consequently, the critical exponents change from the 3D-Heisenberg values for x ¼ 0 sample to unconventional exponents for x ¼ 0.2 and 0.4 samples. Theoretically, these results can reflect the following useful information: the critical exponents in these materials are governed by lattice dimension (3D), the dimension of order parameter (n ¼ 3, magnetization), and the range of interaction (short range, long range or infinite) [34]. In homogeneous magnets the universality class of the magnetic phase transition depends on the range of the exchange interaction [34]:
. JðrÞ ¼ 1 r dþs
(13)
where r, d and s are the distance, the dimension of the system and the range of the exchange interaction, respectively. It has been argued that if s 2, the 3D-Heisenberg exponents (b ¼ 0.365, g ¼ 1.336, d ¼ 4.8) are valid for a three-dimensional isotropic ferromagnet and J(r) decreases with distance r faster than r5 [16,35]. Whereas when s is less than 3/2, the mean-field exponents (b ¼ 0.5, g ¼ 1 and d ¼ 3) are valid, which indicates that J(r) decreases with r slower than r4.5. For intermediate range, i.e., for J(r) z r3s with 3/2 s 2, the exponents belong to a different universality class which depends upon s. In the case of our samples, the values of the critical exponents are in agreement with the 3DHeisenberg model for x ¼ 0 sample however x ¼ 0.2 and 0.4 samples exhibit unconventional exponents. So the exchange interaction J(r) decreases with distance r faster than r5 for x ¼ 0 sample however it ranges from r5 and r4.5 for x ¼ 0.2 and 0.4 samples. 4. Conclusion In summary, we have studied the critical behavior of Zn0.6xNixCu0.4Fe2O4 around TC values. Arrott plots reveal a second order PM-FM phase transition. The values of critical exponents as well as TC calculated using both modified Arrott plot and KouvelFisher plot are reasonably well. For x ¼ 0 sample, the estimated critical exponents are consistent with the 3D-Heisenberg model,
however the exponents belong to a different universality class for x ¼ 0.2 and 0.4 samples. The change in the universality class from the 3D-Heisenberg model to a different universality class is due to the substitution of Zn2þ (non-magnetic ions) by Ni2þ (magnetic ions). The field and temperature-dependent magnetization behavior follows the scaling hypothesis and all the data point fall into two distinct branches, one for T < TC and another for T > TC, indicating that the critical exponents obtained in our work are can accurate. Interestingly, around dependences of DSmax M TC, field ¼ aðHÞn. The n values obtained be described by a power law DSmax M from the power-law fitting are in good agreement with those obtained from the critical exponents using the KF method. References [1] R.D. McMichael, R.D. Shull, I.J. Swartzendruber, L.H. Bennett, J. Magn. Magn. Mater. 111 (1992) 29. [2] A. Ghasemi, A. Hossienpour, A. Morisako, A. Saakhi, M. Saleki, J. Magn. Magn. Mater. 302 (2006) 429. [3] Y. Sahoo, M. Cheon, S. Wang, H. Luo, E.P. Furlani, P.N. Prasad, J. Phys. Chem. B 108 (2004) 3380. € [4] N. Kasapoglu, A. Baykal, Y. KOseoģlu, M.S. Toprak, Scr. Mater. 57 (2007) 441. € [5] T. Ozkaya, A. Baykal, M.S. Toprak, Y. KOseoģlu, Z. Durmus, J. Magn. Magn. Mater. 321 (2009) 2145. [6] Y. Cheng, Y. Zheng, Y. Wang, F. Bao, Y. Qin, Synth. J. Solid State Chem. 178 (2005) 2394. [7] M.S. Anwar, Faheem Ahmed, Bon Heun Koo, Acta. Mater. 71 (2014) 100. [8] Shahida Akhter, D.P. Paul, S.M. Hoque, M.A. Hakim, M. Hudl, R. Mathieu, P. Nordblad, J. Magn. Magn. Mater. 367 (2014) 75. [9] N. Chau, N.K. Thuan, D.L. Minh, N.H. Luong, VNU J. Sci. Math. Phys. 24 (2008) 155. [10] P. Poddar, J. Gass, D.J. Rebar, S. Srinath, H. Srikanth, S.A. Morrison, E.E. Carpenter, J. Magn. Magn. Mater. 307 (2006) 227. [11] R. Cabassi, F. Bolzoni, A. Gauzzi, F. Licci, Phys. Rev. B 74 (2006) 184425. €hnle, W.U. Kellner, H. Kronmüller, Phys. Rev. B 35 (1987) 3640. [12] M. Fa [13] C.S. Hong, W.S. Kim, N.H. Hur, Phys. Rev. B 63 (2001) 092504. [14] H.S. Shin, J.E. Lee, Y.S. Nam, H.L. Ju, C.W. Park, J. Solid State Commun. 118 (2001) 377. [15] A. Schwartz, M. Scheffler, S.M. Anlage, Phys. Rev. B 61 (2000) R870. [16] K. Ghosh, C.J. Lobb, R.L. Greene, S.G. Karabashev, D.A. Shulyatev, A.A. Arsenov, Y. Mukovskii, Phys. Rev. Lett. 81 (1998) 4740. [17] Ch.V. Mohan, M. Seeger, H. Kronmüller, P. Murugaraj, J. Maier, J. Magn. Magn. Mater. 183 (1998) 348. [18] Elaa Oumezzine, Sobhi Hcini, Mohamed Baazaoui, El-Kebir Hlil, Mohamed Oumezzine, Powder Technol. 278 (2015) 189. [19] A. Arrott, Phys. Rev. 108 (1957) 1394. [20] S.K. Banerjee, Phys. Lett. 12 (1964) 16. [21] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, London, 1971. [22] A. Arrott, J.E. Noakes, Phys. Rev. Lett. 19 (1967) 786. [23] J.S. Kouvel, M.E. Fisher, Phys. Rev. 136 (1964) A1626. [24] B. Widom, J. Chem. Phys. 43 (1965) 3898. [25] H. Oesterreicher, F.T. Parker, Magnetic cooling near Curie temperatures above 300 K, J. Appl. Phys. 55 (1984) 433. [26] V. Franco, A. Conde, M.D. Kuz'min, J.M. Romero-Enrique, J. Appl. Phys. 105 (2009) 07A917. [27] M. Pe˛ kała, J. Appl. Phys. 108 (2010) 113913. [28] C.J. Tinsley, J. Magn. Magn. Mater. 15 (1980) 459.
684
E. Oumezzine et al. / Journal of Alloys and Compounds 656 (2016) 676e684
[29] A.G. Flores, V. Raposo, J. Iniguez, J.M. Garcia-Beneytez, J. Arcas, M. Vazquez, Phys. Status Solidi 191 (2002) 555. [30] M. Haug, M. Fahnle, H. Kronmuller, F. Haberay, J. Magn. Magn. Mater. 69 (1987) 163. [31] M. Haug, M. Fahnle, H. Kronmuller, F. Haberay, Phys. Status Solidi 144 (1987) 411.
[32] P.K. Roy, J. Bera, J. Mater. Process. Tech. 197 (2008) 279. [33] A. Verma, T.C. Goel, R.G. Mendiratta, P. Kishan, J. Magn. Magn. Mater. 208 (2000) 13. [34] M.E. Fisher, S.K. Ma, B.G. Nickel, Phys. Rev. Lett. 29 (1972) 917. [35] M.H. Phana, V. Franco, N.S. Bingham, H. Srikanth, N.H. Hur, S.C. Yu, J. Alloy. Compd. 508 (2010) 238.