Magnetic properties, magnetocaloric effect, and critical behaviors in Co1−xCrxFe2O4

This research work focuses on the magnetic properties, nature of the magnetic phase transition, magnetocaloric effect, and critical scaling of magnetization of various Co1−xCrxFe2O4 (x = 0, 0.125, 0.25, 0.375, and 0.5). The tunability of the magnetic moment, exchange interactions, magnetocrystalline anisotropy constant, and microwave frequency using Cr3+ content has been found. The nature of the magnetic phase transitions for all the Cr3+ concentrations exhibits as second order which has been confirmed from the analysis of critical scaling, universal curve scaling, and scaling analysis of the magnetocaloric effect. The critical exponent analysis for all samples was performed from the modified Arrott-, and Kouvel–Fisher-plots. These critical analyses suggest that x = 0.125, 0.250, and 0.375 samples show reliable results in the magnetocaloric effect with relative cooling power (RCP) values in the range of 128–145 J kg−1. On the other hand, x = 0.00, and 0.500 samples exhibit inconsistent RCP values. The universal curve scaling also confirms the reliability of the magnetocaloric effect of the investigated samples.


Introduction
Over the past few decades, research on iron oxide compounds like spinel ferrite, hexaferrite, and garnet has become a topic of discussion among scientists, due to their attractive practical applications in magneto-sensing devices, and biotechnology. [1][2][3][4][5][6][7] These key topics attracted scientists because these types of compounds exhibit unique magnetic and magnetocaloric effects (MCE). [1][2][3][8][9][10] Among all the ferrites, CoFe 2 O 4 (COF) has been focused on in recent years by academia, the medical sector, and industry due to its remarkable magnetic and MCE properties. [11][12][13][14][15][16][17][18][19] The structural, magnetic, and MCE properties of COF can be tuned by doping/substituting divalent or trivalent cations. 11,[18][19][20][21][22][23][24] The tunability of structural parameters due to Cr 3+ substitution in stoichiometric and non-stoichiometric COF is reported in our earlier literature. 25 It was observed that there are structural defects due to Cr 3+ substitution. In recent years, a large number of articles have been found on the study of magnetic eld (H) and temperature (T) dependent magnetization (M) of various ferrites. 1,4,26,27 Massoudi et al. have observed the non-collinear model on Ni-Zn-Al ferrite by comparing the theoretical and experimental magnetic moment calculated from the cation distribution and M-H hysteresis curve, respectively. 1 The paramagnetic moment followed by the Curie-Weiss law and magnetic phase transition temperature has been studied from the eld cooled (FC) and zero elds cooled (ZFC) magnetization. 23,28 Spinel ferrites also attracted scientists across the world due to their interesting MCE properties. 4,[29][30][31][32] The MCE is an intrinsic thermodynamic property of magnetic materials that causes a change in the temperature of the substance under the action of a magnetic eld. 33 In various literature, the MCE of such materials has been studied from the isothermal M-H over a wide temperature range near the magnetic phase transition. 4,34 The MCE values have been calculated from the change of magnetic entropy from the isothermal M-H curve based on Maxwell's thermodynamic relation. 34,35 The nature of magnetic phase transition has also been reported in various literature extracted from the isothermal M-H curves using the Arrott plot, 36 and the Arrott-Noakes model. 37 Law et al. have studied the nature of magnetic phase transition by calculating a critical exponent n from the change of entropy as a function of temperature. 38 Various reports have been conducted on the analysis of critical exponents to conrm the universal class of materials. 32, 34 Franco et al. have rst reported the phenomenological universal scaling curve taking the normalized entropy change as a function of rescaled temperature. 39 In this research, the effect of Cr 3+ substitution on magnetic and MCE properties of various Co 1Àx Cr x Fe 2 O 4 (x ¼ 0, 0.125, 0.25, 0.375, and 0.5) have been studied. A detailed investigation of magnetic properties has been carried out by analyzing the M-H hysteresis and FC-ZFC magnetization behaviors. The MCE properties of Cr 3+ substituted cobalt ferrite have been investigated by analyzing the M-H isotherms. The nature of magnetic phase transition has also been examined by analyzing the Arrott plot. The nature of the universal class of these materials has been analyzed by calculating the critical exponent followed by

Experimental
The nominal chemical compositions Co 1Àx Cr x Fe 2 O 4 (x ¼ 0, 0.125, 0.25, 0.375, and 0.5) have been synthesized by the standard solid-state reaction technique. The stoichiometric amount of Co 2 O 3 (98.0%), Cr 2 O 3 (99.9%), and Fe 2 O 3 (96.0%) have been mixed in a mortar with a pestle. Aer completing the mixing process for 2 hours for each composition, the mixtures were crushed using a planetary ball mill (MSK-SFM-1) for 12 h. To complete the solid-state reaction the milled powder has been calcined at 800 C for 6 h. Then the calcined powder of each composition has been pressed in the form of a pellet using a uniaxial pressure of 16 000 psi and then sintered at 1200 C for 6 h. Then a part of sintered pellets was re-crushed into ne powder for performing X-ray diffraction (XRD) to conrm the formation of spinel-type ferrite. The results of phase formation have been reported elsewhere. 25 Aer conrming the formation of spinel-type ferrites, these compositions are subjected to further investigation of their magnetic properties. The FC and ZFC magnetization were performed for the measurement of the phase transition temperature. The M-H hysteresis loop measurements were performed at room temperature for saturation magnetization and other relevant parameters, The M-H isotherms at a various temperatures above and below the magnetic phase transition for each composition have been conducted by using Quantum Design MPMS3 SQUID magnetometer. Then the MCE properties and critical scaling have been analyzed for each composition using standard method described in Section 3.4.

Magnetic properties
The saturation magnetization (M s ), remanent magnetization (M r ), and coercivity (H c ) are the most important parameters for a material to know its magnetic behavior. In general magnetization vs. applied magnetic eld (M-) hysteresis loop provide a reliable information about M s , M r , and H c . The M-H hysteresis loops for all samples have been illustrated in Fig. 1(a). From M-H hysteresis loops the values of M s , M r , and H c are extracted and listed in Table 1. From the Table 1, it evident that there is a decreasing trend of M s with increasing Cr 3+ content. However, H c and M r show the increasing trend up to x ¼ 0.375 then it decreases for further increase of x. The decreasing trend of M s may be due to the abnormal grain growth and pore blockage. The increasing trend of H c is perhaps due to the decrease in crystallite size (D) as calculated from the XRD data. 25 For x ¼ 0.500, the H c value does not shows the corresponding behavior as crystallite size which may be due to the excess ion as explained in the literature. To know the inter-grain exchange mechanism, the calculation of remanence ratio R (¼ M r /M s ) is most important. The calculated R values (Table 1) show less than 0.5 which indicates the existence of magnetic dipole interaction with random orientation. 40 According to Stoner-Wohlfarth theory, the anisotropy constant (K) value is related to the coercivity has been calculated using the following expression: 41 The calculated K values for all the Cr concentrations are tabulated in Table 1. It is observed from the Table 1 that K values increase with increasing Cr content up to x ¼ 0.375, indicating the increase of domain wall energy. Then it shows the decreasing value which may be due to the excess ions showing negative values of the vacancy parameter as explained in the literature. 25 The domain wall energy (s w ) can be calculated using the following expression: 42 where k B is Boltzmann constant, T C is Curie temperature and a is the lattice constant. The calculated domain wall energy for all samples has been tabulated in Table 1. From Table 1 the values for s w are found to be increasing with the increase of Cr content up to x ¼ 0.375 then it shows a decreasing trend.
To know the domain type of materials, the illustration of the dM/dH versus H plot is most important. 40 The dM/dH versus H for all samples have been depicted in Fig. 1(b). Multiple broad peaks near the zero magnetic eld observed for all samples indicate multi magnetic domain. To know the agreeable domain nature, determination of critical size by using the following expression is most important: 43 where, M SP is spontaneous magnetization. For all samples D m (Table 1) shows a lower value than the calculated D values from XRD, 25 which follows the particle spherical model. The D m < D for all samples reveals that the nanocrystallites have an incipient structure of magnetic domains. 43 The cation distribution results as presented in our previous article 25 clearly indicate that both Co 2+ and Co 3+ occupy the tetrahedral (A) and octahedral (B) sites, respectively whereas Fe 3+ occupied both the A-and B-sites for x ¼ 0. However, for x ¼ 0.125 to 0.500, the Cr 3+ is found in both the A-and B-sites in place of Co 2+ and Co 3+ , respectively. The calculated magnetic moment M A and M B for A-and B-sites are tabulated in Table 1. From Table 1 the values of M A are found to be decreasing with an increase of Cr 3+ which is due to the less magnetic moment of  Table 1. From Table 1, it is observed that the net magnetic moment show an increasing trend which shows inconsistency with the experimental M s . To know the reason behind the inconsistency the experimental number of Bohr magneton (n B ) is calculated from the value of M s using the following expression: 1 where, M is the molecular weight. The calculated values of n B are also tabulated in Table 1 The a YK values for all the samples are found to be in the range of 30 to 50 (Table 1) which conrms the non-collinear spin structure that indicates triangular spin arrangement in the B-sites. The lower values of Cr 3+ substitution indicate the decreasing trend of a YK but at higher values of Cr 3+ enhance the a YK . Although decreasing and increasing trends are evident but they do not show zero a YK . Therefore, the nonzero YK angle suggest that synthesized samples show YK magnetic ordering. The variation of a YK with Cr concentration also support the change in Curie temperature (T C ) as evident from Fig. 2.
The FC and ZFC magnetization plots for all samples were recorded in the presence of 10 mT eld in the temperature range of 300-900 K as shown in Fig. 2 (a-e le Y axis). It is evident that the magnetization (M) value in case of ZFC increases up to maximum at a certain temperature for all samples called blocking temperature (T B ) then it shows a decreasing trend with an increase of temperature while the FC magnetization decreases very slowly up to T C , then a sharp fall is observed for both cases. The values of T B of all samples are tabulated in Table 1 where maximum T B value is observed for x ¼ 0.125. To know the exact T C values dM/dT vs. T graphs are illustrated in Fig. 2(f), where a single peak for all samples conrms the single transition at T C without showing any spin frustration. The T C values are presented in Table 1. It is observed that T C show a maximum for x ¼ 0.125. With an increase in Cr content, there is a decrease in T C values. The variation of T C values and a YK angles of these comositions may be explained by the increasing and decreasing trend of calculated exchange interaction (J) using the following equation: where z is the coordination number (¼ 12) and s ¼ 1 2 . The values of J are presented in Table 1.
The inverse magnetic susceptibility (c À1 ) as a function of temperature (T) is depicted in Fig. 2 (a-e right Y axis) for all samples. From Fig. 2(a-e) it is found that the c À1 rises sharply when the magnetic state changes from the ferromagnetic to paramagnetic. In the paramagnetic region, susceptibility data follow the Curie-Weiss (CW) expression 34 where C is the Curie constant which can be obtained from the slope of the linear t of c À1 vs. T graph (Fig. 2) and Q CW is the CW temperature that also can be obtained from Fig. 2. The calculated values of C, and the estimated values of Q CW from the graphs are listed in Table 1. The estimated values Q CW are found to be lower than that of T C for the compositions up to x ¼ 0.375 which corresponds to the presence of long-range order. However, for x ¼ 0.500 the value of Q CW is found to be a larger than that of T C which corresponds to the short-range order which may originate from the excess ion. The experimental effective magnetic moment has been calculated by using C values according to the following expression: 1 The calculated values of m exp eff are tabulated in Table 1 where the decreasing trend of m exp eff with an increase of Cr content has been observed. The decrease in m exp eff with the increase of Cr 3+ content may refer to the decrease in ferromagnetic clusters present in the paramagnetic phase. 2 The microwave frequency (u m ) is an important parameter for any materials for high-frequency microwave applications. The u m can be evaluated by using the following expression: 1,2 Table 2 The obtained values of critical exponents (b, g, and d) and T C s from the modified Arrott plot (MAP), Kouvel- where g 1 is the gyromagnetic ratio (g 1 ¼ 2.

GðM; TÞ
where, A 1 , A 2 , and A 3 are Landau co-efficient. Neglecting the higher-order terms the above equation can be written as At the equilibrium condition, vG vM ¼ 0; then the magnetic equation of state can be written as The nature of FM-PM phase transition may be determine from the M 2 vs. m 0 H/M, known as Arrott plot. 36 The Arrott plots for all samples are depicted in Fig. 4. No negative slope has been found in Fig. 4, which conrms the second-order phase transition. It is worth noting that M 2 versus m 0 H/M plot should follow the equation of straight line passes through the origin. However, the above-mentioned behavior is not observed for all samples. Therefore, further analysis is performed for assumed second-order FM-PM phase transition using modied Arrott plots (MAP) according to Arrott-Noakes 37 as mentioned by the following expression: where b, and g are the critical exponents. The set of critical exponents (b, g, and d) are calculated by analyzing spontaneous magnetization (M SP ), zero-eld susceptibility (c 0 ), and magnetization isotherm at the T C using the following power-laws: 4,34 where, 3 ¼ T À T C T C is the reduced temperature, M 0 , G, and D 1 are the critical coefficients, and f + and f À are the scaling functions above and below T C , respectively. To calculate the values of b and g (using eqn (14) and (15)) the M SP vs. T, and 1/c 0 vs. T are presented in Fig. 5. From Fig. 5 the b and g values are estimated from the tting curve for all the samples that have been tabulated in Table 2. From Table 2   also has been tabulated in Table 2 Fig. 7. The d values are also determined from the previously calculated b, and g values according to statistical theory using Widom relation: 46 The estimated d values according to the above two cases for all samples are tabulated in Table 2    vs. T graph (Fig. 8) according to the following expressions: 34 T C values are extracted from the X-intercepts, and critical b and g values are obtained from the inverse of slopes of the tted straight line of Fig. 8. The estimated b, g, and T C values for all the samples according to KFPs are tabulated in Table 2, where b, g, and T C values are well-matched with the values as the mean-eld theory for x ¼ 0.125, 0.250, and 0.375. However, for x ¼ 0.00, and 0.500 the b, and g values calculated from KFPs show a remarkable difference compared to that of mean-eld theory.
To ensure the reliability of b, g, and T C values another robust method have been elucidated by plotting Mj3j Àb vs. m 0 Hj3j À(b+g) ) just above and below T C according to eqn (17). The Mj3j Àb vs. m 0 Hj3j À(b+g) ) have been plotted for all samples in Fig. 9. The inset in Fig. 9, each case displays the same data plotted on a loglog scale. From Fig. 9, it is evident that two separate groups of isotherms superimpose (one group greater than T C , and the other group less than T C ) for the samples x ¼ 0.125, 0.250, and 0.375. These results suggest the accuracy of b, g, and T C values from which it can be decided that these three compositions (x ¼ 0.125, 0.250, and 0.375) are a universal class of material. From the inset of Fig. 9, two branches (one below T C and other above T C ) show the linear behavior in the high eld region while in the low eld region show some deviation from linearity. These behaviors conrm that the scale theory gives more important data in higher elds. The isotherms for x ¼ 0.00, and 0.500 show different behavior that imply the non-universal class of the materials.

Magnetocaloric effect
The MCE properties is an intrinsic properties of magnetic materials that can be calculated by calculating the magnetic entropy change (DS m ) around T C . The DS m values are calculated from the isothermal M-H data based on Maxwell's thermodynamic relation: 34 The calculated DS m for all samples show negative values for all temperature and applied magnetic eld. The calculated ÀDS m values as a function of temperature are illustrated in Fig.  10 for all samples at different magnetic elds up to 5 T. From Fig. 10, the peak values of ÀDS m are dened as maximum entropy change jDS max m j are evident at T C or close to T C . From Fig. 10 it is observed that jDS max m j increases with an increase of magnetic eld are due to the spin ordering for all the samples. The jDS max m j values are tabulated in Table 3 for 5 T for all samples. Table 3 shows that for x ¼ 0.125, maximum entropy change is observed, however, a decreasing trend is found for further increasing of Cr content. The similar behavior is observed for M s .
Relative Cooling Power (RCP) is another important criterion that helps to characterize the MCE of such magnetic materials. The RCP for all samples has been calculated using the following relation: 4 where dT FWHM is the full width of the 0.5jDS max m j. The calculated RCP values are tabulated in Table 3, where very lower values of RCP with lower jDS max m j for x ¼ 0.00 are evident which may be due to the non-universal nature as explained in Section 3.3.  Table 3, RCP values are found to increase with the increasing Cr content and found a maximum of 145 J kg À1 for x ¼ 0.375 which is higher than the previously reported RCP values. 4,29-32 For x ¼ 0.500 the RCP values are found to be very low which may be due where, n is the exponent that depends on the magnetic state of the samples. The jDS max m j vs. m 0 H are plotted in the log-log scale and illustrated in Fig. 11(a) for all samples, and the values of n are obtained from the slope of the linear tting. The obtained n values have been tabulated in Table 2. To explain the reliability of this exponent, calculated the value of n at/near T C by using the following relation: 34 By applying the Widom relation eqn (18) and (25) can be rewritten as The calculated n exponents (using eqn (26)) have been tabulated in Table 2 for all compositions. In this case b and g values are considered from KFPs, and values of d are considered from the critical isotherms. The exponent calculated from eqn (26) are in good agreement with those obtained from the tted curve of Fig. 11(a) for x ¼ 0.125, 0.250, and 0.375. For x ¼ 0.00, and 0.500, there is a large difference in the value of n.
The d values have been obtained from the slope of the linear t of the RCP vs. m 0 H plot in the log-log scale ( Fig. 11(b)) according to eqn (24). The obtained d values by this method are tabulated in Table 2 39 According to this method normalized magnetic entropy as a function of re-scaled temperature q (eqn (27)) has been plotted at several magnetic elds which are depicted in Fig. 12.
where, T r 1 and T r 2 are two-temperatures corresponding to 0.5jDS max m j. In Fig. 12(b-d) it is observed that the results of various magnetic eld collapsed into a single master curve, which implies that synthesized samples of x ¼ 0.125, 0.250, and 0.375 are in universal class and show exact second-order phase transition. 34 However, from Fig. 12(a and e) it is evident that for x ¼ 0.00, and 0.500, samples are non-universal class.
To analyze the accuracy of MCE properties and order of phase transition, the value of n is calculated using the following expression: 38 nðT; m 0 HÞ ¼ d lnjDS m j d ln m 0 H The calculated n values as a function of T are illustrated in Fig. 13 for all samples where the inset depicted the jDS m j vs. m 0 H. From the Fig. 13(b-d), it is found that the n values for x ¼ 0.125, 0.250, 0.375 are close to 1 below T C which suggests that the dM/dT term in eqn (21) is weakly eld-dependent. 34 With an increase in temperature it is observed the decreasing trend and arrive the minimum n values of 0.684, 0.695, and 0.693 at T C for x ¼ 0.125, 0.25, and 0.375, respectively. These n values are consistent with the n values obtained from Fig. 11(a) and also from eqn (26) as tabulated in Table 2. Above T C , the n values are found to be the increasing trend but do not cross the critical value of 2 for x ¼ 0.125, 0.250, and 0.375. The minimum n values at T C and n < 2 above T C conrm the second-order phase transition which is explained by Law et al. 38 For x ¼ 0.00, and 0.500 as evident from Fig. 13(a and e) it is found the anomalous behavior of n-T curves shows the minimum n values at T C which is very different compared with that of the n values described in Sec. 3.3. This behavior for x ¼ 0.00, and 0.500 is non-universal class of materials showing the non-realistic MCE values. Although n-T shows anomalous behavior, however, n values less than 2 for all the temperature suggest that the samples (x ¼ 0.00, and 0.500) exhibit the second-order phase transition.

Conclusions
The effect of Cr 3+ substitution on the magnetic and MCE properties of various Co 1Àx Cr x Fe 2 O 4 prepared by the solid-state reaction technique have been evident in this report. The Arrott plot from the analysis of M-H isotherms exhibits the secondorder phase transition that has been perfectly conrmed from the critical analysis and scaling analysis of the MCE effect. The x ¼ 0.125, 0.250, and 0.375 samples demonstrate high RCP values in the range of 127-145 J kg À1 compared to that of other ferrites. The universal curve scaling and scaling analysis of the MCE effect conrms the universal class and the MCE values for x ¼ 0.125, 0.250, and 0.375 are reliable. The higher MCE values up to 145 J kg À1 are observed for x ¼ 0.375, which might be considered as potential candidates for the cooling technology. On the other hand, the higher microwave frequency for all compositions makes them a strong candidate for highfrequency microwave applications, especially in satellite communications and biomedical applications.

Conflicts of interest
There are no conicts to declare.