From conventional to inverse magnetocaloric effect in GdMn1-x Cr x O3

In this work, a phenomenological model (PM) is used to simulate the magnetocaloric effect (MCE) of GdMn1-x Cr x O3 (0 ≤x ≤0.4) (GMCO) samples through the modelling of experimental isofield thermo-magnetization curves. The results showed that the MCE of GMCO samples depends strongly on Cr content, achieving a conventional MCE for low Cr content (level of doping x ≤ 0.3). However, the MCE of a higher Cr content sample has an inverse MCE. The behaviour of MCE in GMCO samples indicated that GMCO compounds are interesting magnetocaloric materials and can fruitfully be functioned as cryogenic magnetic refrigerants below 40 K especially in radiation detectors for outer space research.


Introduction
With accelerated steps, the magnetic refrigerator (MR) has become a strong alternative to the gas refrigerator because of its more efficient cooling performance, less weight, more mechanical stability, less damage to the environment, and more energy savings [1][2][3][4][5][6]. MR is an essential requirement in aerospace applications, medical devices, space applications, and food cooling [7][8][9][10][11][12][13][14]. MR relies on the idea of applying the magnetocaloric effect (MCE) to magnetocaloric (MC) materials in the range of temperatures close to the temperature of the magnetic phase transition (T MPT ) [15][16][17][18][19][20]. The MCE is a phenomenon in which magnetic entropy change ( S M ) of MC materials occurs when they are subjected to a change in an external magnetic field (H exe ) [21][22][23][24][25][26][27][28]. In order for the development and improvement of MR to happen, the researchers studied different types of magnetic materials such as magnetic alloys, manganites and others [29][30][31][32][33][34][35][36]. In a traditional MCE, the cooling action in MC material happens as a reaction of adiabatic demagnetization process that is performed by a sudden eliminating of H exe [37]. Quite the reverse, MC materials can be cooled via adiabatic magnetization, which is done by increasing H exe on diamagnetic materials. This effect is termed an " inverse MCE" [38]. This inverse MCE appears clearly in antiferromagnetic (AFM) materials over the temperature range of the AFM transition.
There is a strong interest in studying rare earth manganites owing to unusual magnetic ordering such as AFM and spin reorientation as results of unusual spin arrangements [39][40][41]. Manganites of the RMnO 3 type usually contain undersized trivalent R ions, such as GdMnO 3 , and exhibit ferroelectricity caused by magnetic interaction competition, evoking an AFM spin ordering that results in lattice modulations [42]. Tiwari et al. investigated GdMn 1−x Cr x O 3 (GMCO) with 0 ≤ x ≤ 0.4 prepared via sol-gel technique showing well to do sequence of magnetic transitions and recommending that GMCO compounds can be functioned in the fields of magnetic switching, MR and spintronic applications [42]. From this practical point, the MCE of GMCO compounds is studied in this work. In this research, a phenomenological model (PM) is used to study the thermomagnetic properties through the work of simulated magnetization temperature curves for GMCO, concluding magnetic entropy change ( S M ), heat capacity change ( C P,H ), and relative cooling power (RCP).

Theoretical considerations
According to PM, described in [43,44], the dependence of magnetization (M) on temperature (T) is given by: where M i is an initial value of magnetization at a ferromagnetic (FM) or AFM-paramagnetic transition and M f is a final value of this transition as shown in Figure  1, where α = 2(β−γ )

Results and discussion
To simulate the MCE of GMCO, PM parameters for GMCO were determined directly from experimental data (magnetization vs. temperature) as in Ref. [42]. Figure 2 shows the magnetization vs. temperature for GMCO where the experimental data from Ref. [42] are symbolized by symbols, while the simulated data are symbolized by dashed lines. The PM parameters are tabulated in Table  1. We can realize from Figure A maximum of S M ( S Max ) can be determined when T = T MPT as follows: Figure 3 shows the simulated temperature dependence of S M for GMCO samples under H of 0.05 T, calculated by using Equation (2). Interestingly, ΔS M of GMCO depends strongly on Cr content, concluding that the thermomagnetic property of GMCO is characterized as a conventional MCE for low Cr content (level of doping x ≤ 0.3). However, the thermomagnetic property of GMCO is characterized as an inverse MCE when the level of doping x is 0.4 as a result of magnetization reversal below the compensation temperature (T comp ), at which the opposing magnetic moments are equal. But, above T comp , the thermomagnetic of GMCO characterizes as a conventional MCE with a very small value of S M as a result of a FM transition. This magnetization reversal, which causes an inverse MCE below T comp when the level of doping x is 0.4, is due to negative exchange interaction between Gd and Cr ions. At below T comp , when H exe is applied, the spins of Cr 3+ are directed along the H exe direction while the spins of Gd 3+ are directed opposite to that of the H exe direction. At the same time, the magnetic moment of Gd 3+ is more than magnetic moment of Cr 3+ ion. Therefore, the net magnetization of the highly Cr content sample is negative value. However, at above T comp , when H exe is applied, magnetic moment of Cr 3+ is greater than the magnetic moment of Gd 3+ resulting in a net positive magnetization, causing a conventional MCE.
The effectiveness of GMCO samples as MC material can be evaluated by RCP. This parameter is accounted for by the absolute value of S Max and full-width at halfmaximum (δT FWHM ) of the S M curve by following the formula: where δT FWHM can be obtained as follows:   Table 2 indicates that δT FWHM has moderate values and is ranged between 11.3 and 25.5 K for GMCO samples under H of 0.05 T. Furthermore, MC properties decrease dramatically with Cr content.
According to the PM model [43], the characterization of C P,H curves of the GMCO samples can be predicted as follows.     The oscillating temperature dependence of C P,H at temperatures is a reflection of S M behaviour due to different exchange interactions between Gd and Cr ions below and above 40 K as explained before [42].
To investigate further details about MCE in GMCO samples, the refrigerant capacity (RC) is well-thoughtout to judge the efficiency of GMCO samples as effective MC materials in MR. RC was calculated as follows [44]: Form Table 2, RC decreases with Cr content for a level of doping x ≤ 0.3. However, for further Cr content, RC improves with Cr content. Table 3 gives a comparative image of MCE parameters of GMCO samples with the corresponding ones of several MC materials in same value of H and higher ones in published works. Importantly, the MCE parameters of GMCO samples are higher or as good as the corresponding MCE parameters of these published works with the same value of H or larger. Interestingly, although the of H in the published Gd single crystal is twenty times of the best investigated GMCO sample, | S Max | of Gd single crystal is larger about 5.5 times. Moreover, GMCO samples are effective MC materials below 40 K due to the sequence of magnetic transitions below this temperature.
Finally, GMCO samples are interesting MC materials in cryogenic MR below 40 K especially for devices that function in very low temperatures. For instance, in very low temperature radiation detectors for outer space investigation, where the gravitational effect is non-effective, it is very hard to use 4 He− 3 He dilution refrigerators [45]. Therefore, these GMCO compounds are practical MC materials in this situation to overcome the difficulty of working 4 He− 3 He dilution refrigerators.

Conclusion
The simulation of MCE GMCO compounds containing different Cr content has been done via PM. The results of simulation show that this PM is a constructive model for the calculation of thermomagnetic properties for GMCO compounds. The MCE of GMCO samples depends strongly on Cr content, achieving a conventional MCE for low Cr content (level of doping x ≤ 0.3). However, the MCE of GMCO has a type of inverse MCE when the level of doping x is 0.4 below T comp . The opportunity to control from conventional MCE to inverse MCE via the level of Cr doping in GMCO increases the prospects for the design of MR. The behaviour of MCE in GMCO samples indicates that they are promising candidates in the cryogenic MR below 40 K.

Disclosure statement
No potential conflict of interest was reported by the author(s).