Rare-Earth Doped Gd_3−xRE_xFe₅O₁₂ (RE = Y, Nd, Sm, and Dy) Garnet: Structural, Magnetic, Magnetocaloric, and DFT Study

Abstract

The study reports the influence of rare-earth ion doping on the structural, magnetic, and magnetocaloric properties of ferrimagnetic Gd_3−xRE_xFe₅O₁₂ (RE = Y, Nd, Sm, and Dy, x = 0.0, 0.25, 0.50, and 0.75) garnet compound prepared via facile autocombustion method followed by annealing in air. X-Ray diffraction (XRD) data analysis confirmed the presence of a single-phase garnet. The compound’s lattice parameters and cell volume varied according to differences in ionic radii of the doped rare-earth ions. The RE³⁺ substitution changed the site-to-site bond lengths and bond angles, affecting the magnetic interaction between site ions. Magnetization measurements for all RE³⁺-doped samples demonstrated paramagnetic behavior at room temperature and soft-ferrimagnetic behavior at 5 K. The isothermal magnetic entropy changes (−ΔS_M) were derived from the magnetic isotherm curves, M vs. T, in a field up to 3 T in the Gd_3−xRE_xFe₅O₁₂ sample. The maximum magnetic entropy change ($-∆S_M^{max}$) increased with Dy³⁺ and Sm³⁺substitution and decreased for Nd³⁺ and Y³⁺ substitution with x content. The Dy³⁺-doped Gd_2.25Dy_0.75Fe₅O₁₂ sample showed $-∆S_M^{max}$~2.03 Jkg⁻¹K⁻¹, which is ~7% higher than that of Gd₃Fe₅O₁₂ (1.91 Jkg⁻¹K⁻¹). A first-principal density function theory (DFT) technique was used to shed light on observed properties. The study shows that the magnetic moments of the doped rare-earths ions play a vital role in tuning the magnetocaloric properties of the garnet compound.

1. Introduction

Magnetic refrigeration (MR) technology based on the magnetocaloric effect (MCE) principle has been considered a promising alternative to replace conventional vapor compression cooling technology [1,2,3,4]. It is an intrinsic magneto-thermal response of magnetic materials [5]. Materials exhibit MCE by inducting adiabatic heating or cooling in the applied magnetic field. One of the quantitative parameters to characterize magnetocaloric materials (MCM) is the isothermal magnetic entropy change (ΔS_M), which is induced by a change in an applied magnetic field (ΔH) [1]. Recently, a Gd-based garnet, Gd₃Fe₅O₁₂, has received attention due to its high MCE at low temperatures (below 50 K), making it suitable for liquefaction processes, cryogenic technology, and space applications [2,3,4]. Gd₃Fe₅O₁₂ belongs to an essential class of iron garnet materials due to their significant magnetocaloric [6], magneto-optic [7,8], recording device [9], microwave device [10], sensing [11], and magnetic properties [10,11].

Gd₃Fe₅O₁₂ is a complex ceramic oxide, having the chemical formula A₃B₂C₃O₁₂ (where A = RE³⁺ ion, B and C = Fe³⁺ ions). The unique crystal symmetry of a garnet plays a crucial role in its physical properties. The garnet structure holds a wide variety of cations. The structure consists of three different crystallographic sites, namely, dodecahedral (c), octahedral (a), and tetrahedral (d), where 24A ions reside in the (c) site, 16B ions in the (a) site, and 24C ions in the (d) site. The unit cell of the garnet structure contains eight formula units of {Gd₃}[Fe1₂](Fe2₃)O₁₂ arranged as a framework of metal-oxygen polyhedra formed from (a) and (d) site cations. Here, { }, [ ], and ( ) represent the three different cationic sublattices. These cations are located at the centers of the corresponding polyhedrons, as shown in Figure 1. Gd³⁺ ions occupy the dodecahedral (Figure 1a) site with position 24c, while Fe1 and Fe2 ions occupy octahedral and tetrahedral sites with positions 16a and 24d (Figure 1b,c). The arrangements of different polyhedral and oxygen ions are given in Figure 2a for the garnet structure. The crystal structure of Gd₃Fe₅O₁₂ with eight formula units per cell is shown in Figure 2b.

In Gd₃Fe₅O₁₂, two sub-lattices of ferric ions couple anti-ferromagnetically in the superexchange interaction via oxygen anions. The formula unit consists of three Fe³⁺ cations on tetrahedral sites and two Fe³⁺ cations on octahedral sites. The Gd³⁺ ions are also anti-ferromagnetically coupled to the net moment of the Fe³⁺ ions, but this coupling is weaker than that between Fe³⁺ ions. Since the Gd³⁺ ions are disordered at room temperature, the ferri-magnetic properties of the material at high temperatures are governed by the moments of the Fe³⁺ ions [12,13,14]. It is known that Fe³⁺ ions at octahedral and tetrahedral sites provide a positive and negative contribution to the compound’s net magnetic moment. At low temperatures, however, the Gd³⁺ lattice becomes ordered and dominates the material’s magnetic properties due to the more significant magnetic moment of Gd³⁺ (below 90 K [3]) compared with Fe³⁺ ions. The magnetic and magnetocaloric properties largely depend on the total angular momentum quantum number (J). The increase in the J value is expected to increase the magnetic moment of each magnetic cluster in the garnet and lead to a rise in the ΔS_M value. The bulk garnet magnetization as a function of temperature can be written as

$$M\left(T\right)=M_c\left(T\right)-[M_d\left(T\right)-M_a\left(T\right)]$$

where $M_c\left(T\right)$ is the magnetization of the Gd³⁺ sublattice, and $M_d\left(T\right)$ and $M_a\left(T\right)$ are the magnetizations of Fe³⁺ at tetrahedral and octahedral sublattices, respectively. At low temperatures, the magnetization of RE³⁺ sublattices is more than that of the Fe³⁺ sublattices. As $T$ increases, the magnetization of the RE³⁺ sublattices decreases faster than Fe³⁺ sublattices and reaches a point where the net moment is zero. The temperature at this point is called the compensation temperature (T_comp). Above the compensation temperature, the net magnetization of the iron sublattices $[M_d\left(T\right)-M_a\left(T\right)]$ exceeds that of the RE³⁺ sublattice, resulting in a rise in the magnetization [15]. This is because the rare-earth and iron sublattice moments randomize at different temperatures. For example, Ho₃Fe₅O₁₂ shows T_comp~127 K [16], and Er₃Fe₅O₁₂ shows T_comp at 186 K [17].

The Gd₃Fe₅O₁₂ compound displays high magnetocaloric properties at low temperatures associated with intrinsic magnetic frustration and magnetic ordering of the Gd³⁺ sublattice [3]. The intrinsic magnetic properties of the garnet are affected by the partial substitution for Gd³⁺ or Fe³⁺ sites or both. Nguyet et al. studied the crystallization and magnetic characterization of (Dy, Ho)₃Fe₅O₁₂ nanopowders prepared using a sol-gel technique [18]. They reported a sizeable magnetic susceptibility and coercivity compared to the corresponding values for bulk samples, a trend attributed to the disordered nature of the surface spin of single-domain particles. Jie et al. studied the structural and magnetic properties of Ca- and Sr-doped Nd₃Fe₅O₁₂ nanopowders prepared using a hydrothermal method [19]. The particle size of Nd_3−x(Ca, Sr)_xFe₅O₁₂ decreased with the concentration of Ca and Sr, while the saturation magnetization value decreased due to the weak exchange interaction. Li et al. studied the MCE in heavy rare-earth iron garnets (Ho₃Fe₅O₁₂ and Er₃Fe₅O₁₂) [20]. Ho₃Fe₅O₁₂ and Er₃Fe₅O₁₂ displayed a compensation effect characterized by a zero magnetization at 134 K and 80 K, respectively. The reported maximum magnetic entropy change value at the 5 T field is 4.72 Jkg⁻¹K⁻¹ for Ho₃Fe₅O₁₂ at 34 K and 4.94 Jkg⁻¹K⁻¹ for Er₃Fe₅O₁₂ at 24 K, respectively. Aparnadevi et al. studied the structural and magnetic behavior of Bi-doped Gd₃Fe₅O₁₂ prototype garnet synthesized via the ball milling method [21]. A shift in the Curie point towards the high-temperature region was observed and ascribed to the stabilizing effect of Bi ion on magnetic ordering. Canglong Li et al. studied the magnetocaloric effect in RE₃Fe₅O₁₂ (RE = Gd, Dy) synthesized using a sol-gel method [22]. The maximum value of −ΔS_M achieved 3.40 Jkg⁻¹K⁻¹ at 40 K and 3.51 Jkg⁻¹K⁻¹ at 58 K, for RE = Gd and Dy, respectively, reflecting the influence of the difference in magnetic moments of Gd³⁺ and Dy³⁺.

The ionic radii of these rare-earth ions are Dy³⁺~0.912 Å, Nd³⁺~0.983 Å, Sm³⁺~0.958 Å, and Y³⁺~0.90 Å [23], and their corresponding magnetic moments are 10 µ_B for Dy³⁺ [24], 1.14 µ_B for Nd³⁺ [25], 0.74 µ_B for Sm³⁺ [26], and 0 for Y³⁺ [27]. Considering these ionic radii and magnetic moment trends, the rare-earth substitution in Gd garnet is expected to bring a new magnetic order in the compound. A detailed study of the RE³⁺ doping effect on the structural, magnetic, and magnetocaloric properties of Gd₃Fe₅O₁₂ garnet is lacking. Suitable RE³⁺ substitution in Gd₃Fe₅O₁₂ is expected to bring changes in the lattice structure, magnetic moment, and exchange-coupling, affecting the compound’s magnetic and magnetocaloric properties. The present work reports a detailed study on the effect of RE³⁺ substitution in Gd_3−xRE_xFe₅O₁₂, (RE³⁺ = Y, Nd, Sm, and Dy, x = 0.0, 0.25, 0.50, and 0.75) garnet compound. For example, in Gd³⁺-rich Gd₃Fe₅O₁₂, the Gd³⁺ ion is an ⁸S_7/2 -state (J = 7/2, L = 0), and the magnetic moment per ion is 7 μ_B. Thus, the Gd³⁺ ion is not affected by the crystalline field. The system is isotropic, with the Gd³⁺ moment following the applied magnetic field. Therefore, it is easy to align other substituted anisotropic ions towards the hard direction of magnetization when Gd³⁺ is replaced with other rare-earth ions in a small amount. However, in RE₃Fe₅O₁₂ compounds other than Gd, with RE = Ho³⁺ (a non-S state ion), the crystalline electric field causes quenching of the orbital angular momentum L. The exchange field plus the crystal electric field will cause the RE³⁺ moments to assume a conical arrangement relative to the easy magnetization direction, which is also possible [28,29,30].

In the present work, we report the results of detailed structural, magnetic, magnetocaloric, and Mossbauer spectral studies of rare-earth ion substituted Gd_3−xRE_xFe₅O₁₂ (RE³⁺ = Y, Nd, Sm, and Dy) garnet, with the compounds being synthesized using an autocombustion technique. The chosen rare-earth ions Y³⁺ (zero magnetic moments), Sm³⁺ and Dy³⁺ (positive magnetic moment), and Nd³⁺ (negative moment) are expected to have a marked influence on the exchange-interaction and dipole–dipole interaction in the ferrimagnetic Gd_3−xRE_xFe₅O₁₂ compound.

2. Experimental Details

2.1. Synthesis

Gd_3−xRE_xFe₅O₁₂, RE³⁺ = Y, Nd, Sm, and Dy, x = 0.0, 0.25, 0.50, and 0.75 samples were synthesized via an autocombustion method using glycerin as a chemical reagent. Nitrate salts of rare-earth, Gd(NO₃)₃·6H₂O and Fe(NO3)₃·9H₂O, were mixed in the stoichiometric amount into deionized water. Glycine-to-metal nitrate molar ratios of 1:8 were mixed as a combustion reagent fuel. The solution was ultrasonicated for 60 min. Glycine complexes the metal cations, thereby preventing selective precipitation and oxidizing by nitrate anions, thereby serving as a fuel for combustion [31]. The mixture was heated to 80 °C until a brown viscous gel formed. Instantaneously, the gel ignited, forming copious amounts of gas, resulting in a lightweight, voluminous powder. The resulting “precursor” powder was calcined at 1100 °C for 12 h to obtain pure RE³⁺ doped Gd_3−xRE_xFe₅O₁₂ iron garnet.

Table 1. Stoichiometry of chemicals used in the synthesis of the Gd_3−xRE_xFe₅O₁₂ compound.

Gd3−xRExFe5O12		Gd(NO3)3. 6H2O	Fe(NO3)3. 9H2O	Dy(NO3)3. H2O	Nd(NO3)3. 6H2O	Sm(NO3)3. 6H2O	Y(NO3)3. 5H2O	Glycine
RE3+	x
		Weight in gm.
Dy	0.00	0.718	1.071	0.000	-	-	-	0.318
	0.25	0.657	1.069	0.046	-	-	-	0.318
	0.50	0.596	1.068	0.092	-	-	-	0.317
	0.75	0.536	1.066	0.138	-	-	-	0.317
Nd	0.00	0.718	1.071	-	0.000	-	-	0.318
	0.25	0.661	1.075	-	0.058	-	-	0.319
	0.50	0.602	1.078	-	0.117	-	-	0.321
	0.75	0.544	1.082	-	0.176	-	-	0.321
Sm	0.00	0.718	1.071	-	-	0.000	-	0.318
	0.25	0.659	1.073	-	-	0.059	-	0.319
	0.50	0.601	1.075	-	-	0.118	-	0.319
	0.75	0.541	1.077	-	-	0.177	-	0.321
Y	0.00	0.718	1.071	-	-	-	0.000	0.318
	0.25	0.671	1.091	-	-	-	0.049	0.324
	0.50	0.621	1.111	-	-	-	0.100	0.33
	0.75	0.569	1.132	-	-	-	0.153	0.336

provides the stoichiometry of the chemicals used in the synthesis.

2.2. Characterization

X-ray diffraction (XRD) experiment was conducted with CuKα₁ (λ~1.5406 Å) radiation using a D8 Advance diffractometer (Bruker, Germany) to examine the phase purity and structural characteristics of the prepared sample. The powder X-ray data were collected in the 2θ range from 20° to 70° with a step size of 0.042° and collection time of 0.2 s/step using a solid-state Vantec detector (Bruker). The morphology of the samples was analyzed using scanning electron microscopy, SEM (Phenom model number PW-100-015 at 10 kV. The magnetic properties of samples viz. hysteresis and field-cooled (FC) and zero-filed-cooled (ZFC) measurements were performed using a physical property measurement system (PPMS, Quantum Design, San Diego, CA, USA) as a function of temperature in the range 5–300 K and field up to 3 T. The sample was cooled down to 5 K for the ZFC measurement without an external field. Then, the magnetic field of 100 Oe was applied to the system, followed by magnetization measurement as a function of temperature from 5 K to 300 K. The FC measurement was performed by lowering the system temperature to 5 K in a 100 Oe field. The FC magnetization as a function of temperature was recorded in warming-up conditions from 5 to 300 K. To calculate magnetic entropy change, isothermal magnetization curves were collected in a field up to 3 T in a temperature step of 7 K.

2.3. Density Functional Theory

First-principles density functional theory (DFT) calculations were performed for the self-consistent calculations and geometry optimizations. The DFT+U method [32] was used with the Perdew–Burke–Ernzerhof (PBE) [33] version of the exchange-correlation functional. The calculations were performed using the Vienna ab initio simulation package (VASP) [34] under the projected-augmented wave (PAW) [35] pseudo-potential. The structure Gd_3−xRE_xFe₅O₁₂ considered in the calculation has a size of 80 atoms in total. Respective structures with the variable x ranging from 0.0 to 1.0 along with a step of 0.25, were considered in the calculations. All the data is taken from the relaxed structures after optimization. Note that in the Dudarev approach of DFT+U [32], the parameters U and J do not enter calculations separately; instead, an effective Coulomb-exchange interaction U_eff = U − J is used.

3. Results and Discussion

3.1. Structural Properties

3.1.1. Phase Analysis

The XRD was performed on a powder sample and finally spread on a zero background sample holder. Figure 3 shows the room temperature XRD pattern of Gd_3−xRE_xFe₅O₁₂ (RE = Y, Nd, Sm, and Dy, x = 0.0, 0.25, 0.50, and 0.75). Single-phase garnet structure (ICDD card no. 01-083-1027) with a cubic crystalline phase group Ia$\stackrel{-}{3}d$ was evident for x < 0.75. An impurity, GdFeO₃, appears at higher doping content, x = 0.75, for Nd³⁺, Sm^3+, and Y³⁺. The RE³⁺ substitution shows a gradual shift in the XRD peaks compared to pure Gd₃Fe₅O₁₂ (inset Figure 3). The observed shifts are in accordance with the difference in ionic radii of substituted RE³⁺ compared to Gd³⁺ (r~0.938 Å) in octahedral symmetry where Dy³⁺ (r = 0.912 Å) and Y³⁺ (r = 0.90 Å) ions are smaller, and Nd³⁺ (r = 0.983 Å) and Sm³⁺ (r = 0.958 Å) are bigger than Gd³⁺ ion [23].

The structural analysis was carried out via the Rietveld [36] refinement technique using GSAS [37] software. The fitted powder profile of Gd_3-xRE_xFe₅O₁₂ is presented in Figure 4, Figure 5, Figure 6 and Figure 7, and the structural parameters derived from Rietveld refinement are listed in

Table 2. Structural parameters derived from Rietveld refinement of powder XRD data of the Gd_3−xRE_xFe₅O₁₂ compound.

Gd3−xRExFe5O12		a (Å)	V (Å3)	O(x)	O(y)	O(z)	Density	Rwp (%)	χ2
	x						(g/cm3)
Dy	0.00	12.4693(11)	1938.629(5)	−0.0296	0.0538	0.1467	6.487	1.194	1.64
	0.25	12.4676(14)	1937.697(6)	−0.0283	0.0562	0.1492	6.489	1.622	2.82
	0.50	12.4647(12)	1936.299(5)	−0.0297	0.0563	0.1470	6.485	1.351	2.55
	0.75	12.4616(13)	1935.367(18)	−0.0279	0.0561	0.1496	6.497	1.515	4.21
Nd	0.00	12.4693(11)	1938.629(5)	−0.0296	0.0538	0.1467	6.487	1.194	1.64
	0.25	12.482213)	1944.699(6)	−0.0292	0.0524	0.1483	6.419	1.269	1.88
	0.50	12.4899(12)	1948.422(5)	−0.0306	0.0545	0.1472	6.430	1.207	1.80
	0.75	12.5018(13)	1953.594(6)	−0.0273	0.0567	0.1449	6.346	1.575	2.67
Sm	0.00	12.4693(11)	1938.629(5)	−0.0296	0.0538	0.1467	6.487	1.194	1.64
	0.25	12.4746(13)	1940.963(6)	−0.0282	0.0558	0.1481	6.474	1.249	1.97
	0.50	12.4848(14)	1946.077(7)	−0.0298	0.0554	0.1461	6.423	1.586	3.50
	0.75	12.4925(19)	1949.377(9)	−0.0307	0.0573	0.1492	6.480	1.982	4.38
Y	0.00	12.4693(11)	1938.629(5)	−0.0296	0.0538	0.1467	6.487	1.194	1.64
	0.25	12.4648(13)	1936.298(6)	−0.0284	0.0558	0.1460	6.392	1.406	1.72
	0.50	12.4550(11)	1932.136(5)	−0.0295	0.0537	0.1481	6.235	1.458	2.24
	0.75	12.4466(32)	1928.386(2)	−0.026	0.0580	0.1460	6.100	1.256	1.64

.

3.1.2. Bond Angle and Bond Length

The Rietveld refinement reveals a good match (R-weighted parameter (R_{wp %}) < 2.0%) of the observed and calculated profiles for all the samples. The lattice parameters, density, oxygen coordinate, and Rwp (%) obtained from the Rietveld refinement are listed in Table 2. It can be seen that the value of the lattice parameter a and unit cell volume V changes with the RE³⁺ content x. The changes in the lattice parameter with x obey Vegard’s law [38].

Each of the three positive ion positions in the garnet structure is associated with a different coordination polyhedron of oxygen ion. In Gd₃Fe₅O₁₂, Fe³⁺ Octa (Figure 8a), Fe³⁺ tetra (Figure 8b), and Gd³⁺ dodeca (Figure 8c) have a regular polyhedral with respect to edge length. The atom-to-atom angles (Gd–O–Fe1(Oct.), Gd–O–Fe2(Tetra.), and Fe1–O–Fe2 and site-to-site bond distances (Gd–Fe1, Gd–Fe2, Fe1–O, Fe2–O, and Fe1–Fe2) are listed in

Table 3. (A) Rietveld refinement and DFT derived bond-angle for Gd_3−xRE_xFe₅O₁₂ compound. (B) Rietveld refinement and DFT derived bond-distance for the Gd_3−xRE_xFe₅O₁₂ compound.

(A)
Gd3−xRExFe5O12		Bond Angle (°)
		Gd–O–Fe2(tetra)		Gd–O–Fe2(tetra)		Gd–O–Fe1(Octa)		Gd–O–Fe1(Octa)		Fe1–O–Fe2
	x	Theory	Expt.	Theory	Expt.	Theory	Expt.	Theory	Expt.	Theory	Expt.
Dy3+	0.00	93.37	92.36	122.95	121.36	101.54	101.78	104.17	104.43	126.19	129.04
	0.25	93.19	93.10	123.3	122.05	102.02	101.94	103.99	104.12	125.86	127.82
	0.50	93.18	91.87	123.11	122.51	101.52	102.46	103.99	103.66	125.92	127.66
	0.75	93.08	93.23	122.04	122.85	101.43	102.55	103.65	104.15	126.6	126.67
Nd3+	0.00	93.37	92.36	122.95	121.36	101.54	101.78	104.17	104.43	126.19	129.04
	0.25	93.38	93.36	122.72	121.19	101.71	100.82	103.95	104.73	127.26	128.36
	0.50	93.55	92.28	121.62	121.24	100.35	101.66	102.56	103.73	128.73	127.92
	0.75	93.41	91.21	122.03	121.29	100.88	101.66	102.67	104.88	128.2	127.03
Sm3+	0.00	93.37	92.36	122.95	121.36	101.54	101.78	104.17	104.43	126.19	129.04
	0.25	93.46	92.60	123.42	122.10	101.02	102.21	103.89	104.32	127.44	128.75
	0.50	93.21	91.71	122.11	122.32	100.84	102.48	103.36	104.04	128.13	128.34
	0.75	93.37	92.43	122.04	122.71	100.79	101.71	103.47	102.72	128.16	127.92
Y3+	0.00	93.37	92.36	122.95	121.36	101.54	101.78	104.17	104.43	126.19	129.04
	0.25	93.22	91.72	123.52	121.61	101.02	101.83	104.25	104.51	127.44	128.93
	0.50	93.53	92.95	123.56	121.71	100.86	101.25	103.83	104.33	127.64	128.82
	0.75	93.63	93.06	123.58	122.21	100.85	101.34	103.88	103.92	127.53	127.69
(B)
Gd3−xRExFe5O12					Bond Lengths (Å)
		Gd–Fe2(Tetra)		Gd–Fe1(Octa)		Fe2(Tetra)–O		Fe1–Fe2		Fe1(Octa)–O
	x	Theory	Expt.	Theory	Expt.	Theory	Expt.	Theory	Expt.	Theory	Expt.
Dy3+	0.00	3.117	3.117	3.485	3.485	1.878	1.876	3.484	3.485	2.029	1.984
	0.25	3.111	3.117	3.482	3.485	1.875	1.877	3.467	3.485	2.043	1.993
	0.50	3.104	3.116	3.456	3.483	1.867	1.884	3.462	3.4838	2.028	1.997
	0.75	3.102	3.115	3.456	3.483	1.864	1.875	3.454	3.483	2.015	2.022
Nd3+	0.00	3.117	3.117	3.485	3.485	1.878	1.876	3.484	3.485	2.029	1.984
	0.25	3.112	3.121	3.473	3.4888	1.876	1.872	3.469	3.4888	2.002	1.997
	0.50	3.121	3.123	3.466	3.4911	1.863	1.871	3.474	3.4911	1.996	1.998
	0.75	3.11	3.126	3.472	3.4944	1.863	1.889	3.464	3.4944	2.043	1.928
Sm3+	0.00	3.117	3.117	3.485	3.485	1.878	1.876	3.484	3.485	2.029	1.984
	0.25	3.113	3.119	3.462	3.4868	1.865	1.886	3.465	3.4868	1.997	2.007
	0.50	3.114	3.121	3.469	3.4896	1.865	1.891	3.471	3.4896	2.003	1.986
	0.75	3.115	3.123	3.472	3.4918	1.865	1.867	3.472	3.4918	1.995	2.034
Y3+	0.00	3.117	3.117	3.485	3.485	1.878	1.876	3.484	3.485	2.029	1.984
	0.25	3.111	3.116	3.467	3.4846	1.866	1.87	3.466	3.4846	1.999	1.981
	0.50	3.113	3.114	3.463	3.4813	1.865	1.864	3.475	3.4813	1.995	1.996
	0.75	3.108	3.111	3.463	3.4789	1.864	1.863	3.477	3.4789	1.998	2.012

and plotted in Figure 9a,b, Figure 10a,b, respectively. The bond lengths Gd–Fe1 decreased with the Dy³⁺ and Y³⁺ substitution, whereas Nd3+ and Sm3+ substitution increased. The bond length of RE–Fe1 is similar to the bond length calculated by S. Geller et al. for the Y₃Fe₅O₁₂ garnet [39], as listed in Table 3. Due to the high centrosymmetric nature of the cubic garnet structure, the bond angles, such as Fe1–Gd–Fe2 (56.8°) and Gd–O–Gd, etc., remain largely unaltered. In contrast, the bond angle Fe1–O–Fe2 decreases, and the bond angle Gd–O–Fe1 increases with RE³⁺ substitution following the ionic size difference between the Gd³⁺ and RE³⁺ ions. The magnetic interactions between ions largely depend on bond-angle and bond-length values. Magnetic interactions cannot occur via the conduction of electrons in garnet due to their insulating nature. The magnetic ions of garnet are separated from each other by the large oxygen ions, and this separation is too large to give rise to an appreciable direct exchange [40]. However, this situation can give rise to important superexchange interactions. These indirect exchange interactions depend on the bond length and bond angle. Moreover, indirect exchange interactions decrease with the magnetic ions separation and increase as the angle formed by triplet Fe³⁺–O²⁻–Fe³⁺ tends to 180° [41]. Table 3 shows the bond angle between Fe1³⁺–O²⁻–Fe2³⁺ decreases from 129.04° to 126.67° for Dy³⁺, 127.67° for Nd³⁺, 127° for Sm³⁺, and 127.69° for Y³⁺ doped Gd garnet samples at x = 0.75. This arrangement favors weak superexchange interaction between the tetrahedral and octahedral sublattices. The bond length Fe1-Fe2 decreased for Dy³⁺ and Y³⁺ and increased for Nd³⁺ and Sm³⁺ doped samples from 3.485 Å (x = 0.0) to 3.483 Å for Dy³⁺ (x = 0.75), 3.478 Å for Y³⁺ (x = 0.75), 3.494 Å for Nd³⁺ (x = 0.75) and 3.492 Å for Sm³⁺ (x = 0.75), respectively. Thus, the strength of superexchange interaction between Fe³⁺–O²⁻–Fe³⁺ increased for Dy³⁺ and Y³⁺ doped and decreased for Nd³⁺ and Sm³⁺ doped garnet samples.

The angle between Fe1³⁺–O²⁻–Fe2³⁺ is greater than between Gd³⁺–O²⁻–Fe1³⁺ and Gd³⁺–O²⁻–Fe2³⁺. Therefore, the superexchange interaction Fe1³⁺–O²⁻–Fe2³⁺ could be weaker than Gd³⁺–O²⁻–Fe1³⁺ and Gd³⁺–O²⁻–Fe2³⁺. The exchange interaction of two magnetic ions and Gd³⁺ ions is through an oxygen ion bridging between them [42]. In this interaction, the overlap of the 2p electrons (with dumbbell-shaped distribution) of the oxygen ion with the electronic distribution of the magnetic ions is an important feature. The interaction increases with the overlap and, accordingly, will be greatest for short Fe³⁺/Gd³⁺-O²⁻ distances and Gd³⁺/Fel³⁺-O²⁻-Fe2³⁺ angles near 180°. Therefore, in Gd₃Fe₅O_12, the strongest interaction (Table 3A) will probably occur between Fe³⁺(octa) and Fe³⁺(tetra), for which the Fe1³⁺-O²⁻-Fe2³⁺ angle is 129.04° and bond length is ~3.485 Å. In Gd_3−xRE_xFe₅O_12, the RE³⁺-O²⁻-Fe³⁺(tetra) angle is 121.36° (Table 3A), and the bond length is 3.117 Å (Table 3B).

Table 4. Atomic site occupancy derived from Rietveld refinement for the Gd_3−xRE_xFe₅O₁₂ compound.

Gd3−xRExFe5O12		Gd	RE	Fe2(tetra.)	Fe1(Octa.)	Chemical Formula
RE	x
Dy3+	0.00	1.0052	-	0.9982	1.0093	Gd3.02Fe2.99Fe2.02O12
	0.25	0.9264	0.0899	0.9923	1.0587	Gd2.78Dy0.27Fe2.98Fe2.12O12
	0.50	0.8288	0.1492	1.0006	1.0067	Gd2.49Dy0.45Fe3.00Fe2.01O12
	0.75	0.7784	0.2181	1.0033	1.0108	Gd2.34Dy0.65Fe3.01Fe2.02O12
Nd3+	0.00	1.0052	-	0.9982	1.0093	Gd3.02Fe2.99Fe2.02O12
	0.25	0.9245	0.0806	0.9900	0.9917	Gd2.77Nd0.24Fe2.97Fe1.98O12
	0.50	0.849	0.1649	0.9985	1.0022	Gd2.55Nd0.49Fe3.00Fe2.00O12
	0.75	0.7438	0.2526	1.0022	1.0111	Gd2.23Nd0.76Fe3.01Fe2.02O12
Sm3+	0.00	1.0052	-	0.9982	1.0093	Gd3.02Fe2.99Fe2.02O12
	0.25	0.9111	0.0829	1.011	1.0186	Gd2.73Sm0.25Fe3.03Fe2.04O12
	0.50	0.8215	0.1604	0.9946	1.0073	Gd2.47Sm0.48Fe2.98Fe2.01O12
	0.75	0.7688	0.2269	1.0576	1.049	Gd2.31Sm0.68Fe3.17Fe2.10O12
Y3+	0.00	1.0052	-	0.9982	1.0093	Gd3.02Fe2.99Fe2.02O12
	0.25	0.9020	0.0873	1.0512	1.0705	Gd2.71Y0.26Fe3.15Fe2.01O12
	0.50	0.8171	0.1682	0.9917	0.9977	Gd2.45Y0.51Fe2.98Fe2.00O12
	0.75	0.7890	0.2156	1.0220	0.9900	Gd2.37Y0.65Fe3.07Fe1.98O12

lists the atomic site occupancy for Gd³⁺, RE³⁺, Fe³⁺, and O²⁻ determined from the Rietveld refinement. As reported in Reference [43], the crystallographic structure is defined in the space group Ia-3d (#230) with four independent atoms: gadolinium, oxygen, and the two iron atoms successively at 24c, 96h, 16a, and 24d sites. Gd (dodecahedral site) has a maximum occupancy at x = 0.0, which decreases with the x content of RE³⁺. The compound’s chemical formula derived from the atomic site occupancy is listed in Table 4. The derived composition of the compound matches the assumed starting stoichiometry of the compound.

3.2. Structural Parameters

The site radii (r_A and r_B), bond length (R_A and R_B), shared edges (d_AE and d_BE) length, and unshared edges (d_BEU) length for tetrahedral and octahedral of Gd_3−xRE_xFe₅O₁₂ compound are calculated using Bertaut method [44]. The subscripts A and B refer to octahedral and tetrahedral sites.

$$r_A=\left[u-\frac{1}{4}\right]a\sqrt{3}-R_o$$

(1)

$$r_B=\left[\frac{5}{8}-u\right]a-R_o$$

(2)

$$R_A=a\sqrt{3}\left(\mathrm{Ƃ}+\frac{1}{8}\right)$$

(3)

$$R_B=a\sqrt{\left(\frac{1}{16}-\frac{\mathrm{Ƃ}}{2}+3{\mathrm{Ƃ}}^2\right)}$$

(4)

$$d_{AE}=a\sqrt{2}\left(2u-\frac{1}{2}\right)$$

(5)

$$d_{BE}=a\sqrt{2}\left(1-2u\right)$$

(6)

$$d_{BEU}=a\sqrt{\left(4u^2-3u+11/16\right),}$$

(7)

where R_o is the radius of the oxygen ion (1.32 Å), u is a positional parameter, u_ideal is 0.375 Å, and ƃ = u − u_ideal, ƃ is the deviation of oxygen parameters [45]. The positional parameter (u) is calculated from the relation [46];

$$u=\frac{\frac{1}{2}{R}^2-\frac{11}{12}+{\left(\frac{11}{48}{R}^2-\frac{1}{18}\right)}^2}{2R^2-2},$$

(8)

where R = (Fe2-O)/(Fe1-O). The calculated r_A, r_B, R_A, R_B, d_A, d_BE, and d_BEU values for RE³⁺ doped garnets are listed in

Table 5. Site-radii (r), bond-length (R), and share-edges (d) of the Gd_3−xRE_xFe₅O₁₂ compound.

			Site Radii		Bond Length		Shared Edges		Unshared Edges	Tolerance Factor
RE	x	u	rA (Å)	rB (Å)	RA (Å)	RB (Å)	dAE (Å)	dBE (Å)	dBEU (Å)	t
Dy3+	0.00	0.382	1.542	1.704	2.862	3.026	4.673	4.143	4.412	0.669
	0.25	0.381	1.509	1.721	2.829	3.043	4.620	4.195	4.410	0.658
	0.50	0.382	1.536	1.705	2.856	3.028	4.664	4.149	4.410	0.668
	0.75	0.381	1.503	1.723	2.823	3.045	4.611	4.201	4.408	0.656
Nd3+	0.00	0.382	1.542	1.704	2.862	3.026	4.673	4.143	4.412	0.669
	0.25	0.382	1.528	1.716	2.848	3.038	4.651	4.175	4.416	0.663
	0.50	0.382	1.527	1.720	2.847	3.042	4.649	4.182	4.418	0.662
	0.75	0.385	1.617	1.672	2.937	2.998	4.796	4.043	4.427	0.694
Sm3+	0.00	0.382	1.542	1.704	2.862	3.026	4.673	4.143	4.412	0.669
	0.25	0.382	1.531	1.712	2.851	3.034	4.656	4.164	4.413	0.665
	0.50	0.383	1.559	1.700	2.878	3.023	4.701	4.127	4.418	0.674
	0.75	0.380	1.491	1.741	2.811	3.063	4.591	4.242	4.418	0.649
Y3+	0.00	0.382	1.542	1.704	2.862	3.026	4.673	4.143	4.412	0.669
	0.25	0.383	1.569	1.686	2.889	3.010	4.717	4.096	4.412	0.679
	0.50	0.382	1.515	1.714	2.835	3.036	4.629	4.177	4.406	0.661
	0.75	0.381	1.497	1.721	2.817	3.043	4.601	4.201	4.403	0.655

. The unshared edges, d_AE, and d_BE for Dy³⁺ and Y³⁺ decreased, while for Nd³⁺ and Sm³⁺ doping, the value increased [47]. Similarly, site radii r_A and bond-length R_A decreased, and r_B and R_B increased for Dy³⁺ and Y³⁺ doping, while for Nd³⁺ and Sm^3+, the opposite trend is observed. These variations in structural parameters are per ionic radii differences between doped RE³⁺ and Gd³⁺ ions. The u value is observed to remain unaffected by doping due to the centrosymmetric structure of the compounds.

3.3. Crystallite Size and Density

The crystallite size and strain were also calculated using Halder-Wagner-Langford’s (HWL) method [48]. The HWL equation relates the FWHM of peaks, β, with the mean crystallite size,“D,” and the micro-deformation of a grain, ε (strain parameter), as follows,

$${\left(\frac{{\beta }^{*}}{d^{*}}\right)}^2=\frac{1}{D}\left(\frac{{\beta }^{*}}{d^{*2}}\right)+{\left(\frac{\epsilon }{2}\right)}^2,$$

(9)

where β* is given by β* = (β/λ) cos (θ), where λ is the X-ray wavelength and d* is given as d* = (2/λ) sin (θ).

The plot of (β*/d*)² vs. β*/d*² is a straight line, for which the intercept and the slope allow the values of the microstrain (ε) and the crystallite size (D) to be determined. The HWL plot has the advantage that data for reflections at low and intermediate angles are given more weight than those at higher diffraction angles, which are often less reliable. Figure 11 shows the HWL plots to compute the crystallite size and strain of the sample using Equation (2). The average crystallite size of the doped Gd_3−xRE_xFe₅O₁₂ samples obtained from HWL plots is listed in

Table 6. Average crystallite size, strain, and X-ray density for the Gd_3−xRE_xFe₅O₁₂ compound.

Gd3−xRExFe5O12		Average Crystallite Size (nm)	Grain Size from SEM (nm)	Strain from HWL Method	X-ray Density, ρ (g/cm3)
RE	x	HWL Method		×10−4
Dy3+	0.00	70.82	1000	3.7	6.46
	0.25	68.55		12.5	6.47
	0.50	66.34		5.6	6.48
	0.75	65.48	1000	2.8	6.49
Nd3+	0.00	70.82	1000	3.7	6.46
	0.25	59.32		22.3	6.42
	0.50	58.68		23.1	6.39
	0.75	60.78	500	16.8	6.34
Sm3+	0.00	70.82	1000	3.7	6.46
	0.25	62.46		6.0	6.44
	0.50	67.24		7.1	6.41
	0.75	61.03	1500	7.1	6.39
Y3+	0.00	70.82	1000	3.7	6.46
	0.25	70.81		17.6	6.35
	0.50	70.88		12.7	6.25
	0.75	74.70	1000	13.4	6.14

. The crystallite size of the pure sample obtained from Scherrer’s method was 67 nm and decreased with x content from 67 nm to 64 nm for Dy³⁺ (x = 0.75), 61 nm for Nd³⁺ (x = 0.75), 63 nm for Sm³⁺ (x = 0.75) and 53 nm for Y³⁺ (x = 0.75) doped compound. The observed grain refinement upon RE³⁺ substitution can result (1) from the increased microstrain due to the size difference between Gd³⁺ and RE³⁺ [49,50], (2) RE³⁺ diffusion to the boundaries, which could restrain the grain growth [51], and (3) the reduction in the unit cell volume accompanied by shortening the diffusion path between nearby grains could result in smaller grains during calcination. Similar grain refinement is observed upon RE³⁺ substitution in other ferrites [52,53]. The HWL strain increased with RE³⁺ content and reached a value of 2.77 × 10⁻⁴, 1.68 × 10⁻⁴, 7.11 × 10⁻⁴, and 2.77 × 10⁻⁴ for x = 0.75. The observed positive slopes in the HWL plots in all samples indicate the presence of strain. Due to the complex and inhomogeneous nature of the substituted oxide sample, the single origin of strain is difficult to pin. The observed strain may have its origin in ionic size differences, vacancies, and random distribution of ions on the atomic sites.

The X-ray density was calculated using the relation,

ρ_x = 8 M/N_Aa³

(10)

where M is the relative molecular mass, N_A is Avogadro’s number, and a is the lattice parameter. Table 6 listed the X-ray density for Gd_3−xRE_xFe₅O₁₂. The calculated density increased for the Dy³⁺ (from 6.487 g/cm³ to 6.496 g/cm³), whereas it decreased for the Nd³⁺, Sm³⁺, and Y³⁺ doped compounds. The change in density observed in the RE³⁺ doped garnet is due to the doped element’s different atomic radii and mass. The observed variation in X-ray density is due to the lower atomic mass of Nd (144.24 u), Sm (150.40 u), and Y (88.91 u), replacing Gd (157.20 u), and the higher atomic mass of Dy (162.50 u) replacing Gd in Gd_3−xRE_xFe₅O₁₂. However, the contribution of defects to the X-ray density cannot be ignored.

3.4. Microstructural Analysis

The surface morphologies of Gd_3−xRE_xFe₅O₁₂, x = 0.0, and 0.75 obtained via SEM are shown in Figure 12. The parent compound consists of irregularly shaped grains with well-defined boundaries and voids. The grain size measurement histogram is also shown in Figure 12. The average grain size, listed in Table 6, is obtained by fitting the size distribution histogram to the log-normal distribution function as reported by Odo [54].

$$f\left(d,\mu ,\sigma \right)=\frac{1}{d\sigma \sqrt{2\pi }}exp\left[-\frac{{\left(\ln \left(d\right)-\mu \right)}^2}{2{\sigma }^2}\right],$$

(11)

where d is the cross-sectional length of the particle, µ and σ are the logarithmic mean and standard deviation, respectively. It is noted that the Dy³⁺ and Y³⁺ substitution garnet has a grain size similar to that of the parent compound Gd₃Fe₅O₁₂, whereas the grain size decreased upon Nd³⁺ and increased with Sm³⁺ substitution. The average length of particles for the Gd₃Fe₅O₁₂, Dy³⁺ (x = 0.75), and Y³⁺ (x = 0.75) substituted samples is ~1.0 μm, and the average length reduced to ~500 nm for the Nd³⁺ (x = 0.75) doped samples. The observed decrease in particle size with Nd³⁺ substitution can result from the increased microstrain due to the higher ionic radii of Nd³⁺. Moreover, the diffusion of Nd³⁺ to the boundaries restrains grain growth. The average length of particles increased to 1.5 μm for the Sm³⁺ (x = 0.75) samples. The increased grain size with Sm³⁺ substitution is due to nearly the same ionic radii as Gd^3+, allowing the easy long-range diffusion of Sm³⁺. However, the grain size also depends on the porosity, sintering temperature, and grain boundaries found in the substituted garnet compounds.

3.5. Theoretical Study

Ab initio density functional theory (DFT) calculations are performed using the VASP [33] simulation package for geometry optimization and post-processing calculations. The pseudo-potential constructed under the projector augmented wave (PAW) [55] method describes the valence electrons. The exchange-correlation functional of the Perdew–Burke–Ernzerhof (PBE)+U [32,56] type is considered in the total energy calculations, where the wave function expansion is carried out by considering the plane-wave basis set having the energy cutoff of 400 eV. The spin and the orbital part of the total magnetic moment are calculated by considering the spin–orbit interactions. A gamma-centered k-point mesh sampled at 2 × 2 × 2 is used to integrate the Brillouin zone. We used the total energy criteria for both electronic self-consistency and geometry optimization. The electronic self-consistency is achieved when the total energies of two consecutive electronic steps are smaller than 10⁻⁴ eV. The structures are allowed to relax along with the lattice parameters until the total energies of two consecutive ionic steps are smaller than 10⁻³ eV.

The total density of states (TDOS) of Gd₃Fe₅O₁₂ is shown in Figure 13a, along with the spin-up and spin-down components, and the orbital contributions from Gd, Fe, and O to the TDOS are shown in Figure 13b–d, respectively. From Figure 13b, it is clear that the f-orbital of the Gd atom has a major contribution to the spin-up component of the conduction band and the spin-down component of the valence band in TDOS, along with the small contribution from its d-orbital. Similarly, in Figure 13c,d, the d-orbitals of Fe³⁺ atoms and the p-orbitals of the oxygen (O) seem to have a small contribution to the TDOS as well. The magnetism in the material arises because of the asymmetric nature of spin-up and spin-down components in the density of states, which is seen in Figure 14.

The Gd_3−xRE_xFe₅O₁₂ structure considered in the calculations consists of 80 atoms (Figure 1a), which is four times larger than its functional unit (f. u). The calculations are carried out for the four rare-earth (RE) elements, namely, Dy, Nd, Sm, and Y, which are used as a dopant on the Gd sites with values ranging from x = 0 to 1, with a step size of 0.25. The corresponding values for the total magnetic moments (spin and orbital) per formula unit of the optimized structures are obtained from the calculations. The effective Coulomb-exchange interaction (U_eff) value is set to be 6 eV for the 4f orbitals in Dy, Nd, Sm, and Gd, whereas it is 4 eV for the d orbitals in Fe and Y [57,58]. The initial values of the magnetic moment of each element were taken from the literature [5,59,60]. The calculated values of the individual elements’ orbital and spin magnetic moments after the optimization are listed in

Table 7. DFT calculated values of orbital and spin magnetic moments of RE³⁺, Fe³⁺, and O²⁻ ions.

Ions	Orbital Moment (μL)	Spin Moment (μS)	Total Moment (μT)
Dy3+	4.07	4.98	9.05
Nd3+	−4.43	2.89	−1.54
Sm3+	−2.43	4.98	2.55
Y3+	0.00	0.00	0.00
Gd3+	0.00	6.99	6.99
Fe3+ (Tetra)	0.00	−4.06	−4.06
Fe3+ (Octa)	0.00	4.16	4.16
O2−	0.00	0.00	0.00

below.

From Table 7, the orbital magnetic moment of Dy³⁺ has a positive value, whereas Nd³⁺ and Sm³⁺ have negative values. The higher negative μ_L value in Nd³⁺ makes the total moment (μ_T) negative. Moreover, Fe³⁺, devoid of the orbital moment, has only the spin magnetic moment (μ_S), with octahedral Fe³⁺ having a positive value and tetrahedral Fe³⁺ having a negative value. The elements Y³⁺ and O²⁻ have zero magnetic moments. At low temperatures with a 3d⁵, S = 5/2 configuration, we expect the magnetic moment to be 5 μB per iron ion. The Fe sublattices are anti-ferromagnetically coupled, giving a maximum net magnetization of 3 × (−5 μB) = −15 μB for the tetrahedral sublattice and 2 × (5 μB) = 10 μB for the octahedral sublattice per formula unit. The spontaneous magnetization direction of the Gd sublattice is taken to be positive. From the Gd sublattice, with Gd³⁺, S = 7/2 ions, we can expect a maximum magnetization of 3 × 7μB = 21 μB. This gives the expected saturation magnetization of 16 μB for Gd garnet, a value confirmed experimentally earlier [28].

The variation of the magnetic moment formula unit as a function of doping concentration, x, is shown in Figure 14 for four different RE³⁺ ions mentioned above. When x = 0.0, i.e., in the Gd₃Fe₅O₁₂ sample, the ${\mathsf{\mu }}_B$~16 agrees with the previous calculations [41,61,62]. With the increase in the x value, the total magnetic moment increases in the case of Dy³⁺ doping. This is because the magnetic moment of Dy³⁺ is larger than that of Gd³⁺. Meanwhile, the magnetic moment decreases in the other three doping cases (Nd, Sm, and Y). The decreasing rate is consistent with the order of their magnetic moments (Nd < Y < Sm), which is also consistent with the experimental results. Furthermore, structural parameters for Gd_3−xRE_xFe₅O₁₂, Table 3A, B, derived from DFT calculations, validate the values obtained from the XRD Rietveld refinement

3.6. Magnetic Properties

The Curie temperature Tc for Gd_3−xRE_xFe₅O₁₂ compounds was measured using a thermogravimetric analyzer (TGA) with a permanent magnet (Figure 15a–d). Figure 15 shows that the weight of the sample increased with temperature due to increased magnetic force on the sample. This implies that the net magnetic moment of the sample increased with the temperature up to Tc. The tetrahedral site iron ions have a positive moment, while the octahedral site has a negative moment. At elevated temperatures due to thermal energy, these moments are canted with respect to the z-axis. The sum of the projection of these moments on the z-axis determines the net moment of the compound. The observed increase in magnetization with temperature indicates that the octahedral site moment dominates the net moment. At temperature Tc, thermal energy dominates the magnetic spins, and the material exhibits paramagnetic behavior. The observed T_C value is 570 K for x = 0.0 and remains unaltered upon RE³⁺ doping. The Tc value is dictated by the strength and the number of superexchange Fe³⁺–O²⁻–Fe³⁺ interactions, which largely remain unaffected by the RE³⁺ doping. The observed Tc with RE³⁺ substitution is listed in

Table 8. T_comp and T_c values of garnet compounds.

Compounds	Tcomp (K)	TC (K)	Reference
Er3Fe5O12	87		[63]
Er3Fe4.2Al0.8O12	139		[63]
Tb3Fe5O12	244		[64]
Ho3Fe5O12	137		[65]
Gd3Fe5O12	288		[66]
Gd3Fe5O12	280	570	Present work
Gd2.25Dy0.75Fe5O12	287	572	Present work
Gd2.25Nd0.75Fe5O12	263	574	Present work
Gd2.25Sm0.75Fe5O12	260	572	Present work
Gd2.25Y0.75Fe5O12	238	567	Present work

.

The magnetization as a function of temperature for RE³⁺ doped Gd_3−xRE_xFe₅O₁₂ was investigated in the temperature range below room temperature. Figure 16 displays the temperature dependence of magnetization M(T) curves for Gd_3−xRE_xFe₅O₁₂ under zero-field cooled (ZFC) and field-cooled (FC) conditions at a 100 Oe field. A distinct characteristic of M(T) curves is the presence of a compensation temperature, T_comp. The temperature at which the magnetization crosses zero is called the compensation temperature, T_comp. This occurs because the magnetization of RE³⁺ ion at the c sites is equal and opposite to the net magnetization of Fe³⁺ ion sublattice at the a and d sites. i.e., 3Fe(d)-[2M_Fe(a) + 3M_RE(c)]. Table 8 lists the comparison of T_comp values for different doped garnet compounds. In the case of Gd_3−xRE_xFe₅O₁₂, T_comp values vary between 280 K and 238 K, depending upon the type of RE³⁺ doping.

This behavior may be explained in terms of the temperature dependence of the magnetization of the three magnetic sublattices (dodecahedral, octahedral, and tetrahedral) balance each other [61,62]. Neel et al. [41] reported that the magnetic properties of RE₃Fe₅O₁₂ can be explained by assuming that the three sublattice ferrimagnetism is due to positive RE³⁺ spin on dodecahedral sites, positive Fe³⁺ spin on octahedral site, and negative Fe³⁺ ion tetrahedral sites. The exchange interaction between RE³⁺ ions is almost negligible, so the moment of RE³⁺ ions should be aligned with the exchange interaction with Fe³⁺ ions. Structural evidence favors the strong magnetic interactions of the rare-earth ions with the tetrahedral Fe³⁺ ions (down magnetic moment) [41]. The strong interaction of the Gd³⁺ ion moments with those of the tetrahedral Fe³⁺ ion moments leads to random canting of the Gd³⁺ ion moments, thereby reducing the contribution of the dodecahedral sublattice to the net spontaneous magnetization of the garnet [67]. This could be the reason for the slow increase in the net magnetization value with temperature lowering.

A cusp is observed in the ZFC curves at a temperature where the RE³⁺ moment aligns parallel to the net Fe³⁺ moment. For Gd₃Fe₅O_12, the cusp is observed at temperature (T_F) = 50 K, while for other RE³⁺ substituted compounds, T_F shifts to higher temperatures. Further, a decrease in magnetization value is observed at a temperature below T_F. This decrease in magnetization value occurs because of a strong RE³⁺-O²⁻-Fe³⁺(Tetra.) interaction, which flips RE³⁺ moments parallel to the Fe³⁺(Tetra.) in a negative direction. The exchange interaction between the 4f rare-earth electrons and the 3d iron electrons is not direct but occurs indirectly via the oxygen ions [68]. This interaction is strengthened in the presence of RE³⁺ ions with non-zero orbital angular momenta, such as Nd³⁺, Sm³⁺, and Dy³⁺. This conclusion is further corroborated by noticing the absence of a cusp in M vs. T for Y³⁺ doped garnet, where Y³⁺ does not possess any orbital angular momentum. The increase in T_F value with RE³⁺ substitution results from the increased number and strength of superexchange interactions RE³⁺–O²⁻–Fe³⁺(Tetra.) ensuing from increased bond-angle in favor of strengthening the interaction, Figure 10b. The M vs. T curve cusp is more prominent for the Dy³⁺ doped sample. Because of negative moments of Nd³⁺ (−1.54 μB), the magnetization attains a negative value below T_comp. Meanwhile, Sm3+ (2.55 μB) shows a positive magnetization value with a cusp below T_comp. This discussion concludes that RE³⁺ with a finite orbital angular momentum couple strongly with Fe³⁺ sublattice moment via superexchange interaction.

The temperature-dependent magnetization process is depicted in Figure 17. At T_comp, the three sublattice moment adds to zero moments. Below T_comp, magnetization slowly peaks with rare-earth contributing positively to the net moment, while at low temperatures below T_F, increased RE³⁺ moments canting due to strong RE³⁺–O²⁻–Fe³⁺(Tetra.) superexchange interaction leads to net negative moments. The T_F value shifts to a higher temperature depending on the strength and number of superexchange interaction pairs. At x = 0.75, at low temperatures, below T_comp, the magnetic anisotropy of Nd³⁺, Sm³⁺, and Gd³⁺ exceeds that of iron, with their moment being aligned along the easy axis, thus increasing the net moment of the compound.

Figure 18a–d shows the magnetization vs. field curves, M vs. H, for Gd_3−xRE_xFe₅O₁₂ measured at 5 K. All samples show the ferromagnetic behavior at 5 K. The magnetic curve tends to saturate below the applied field of 0.5 T for all samples. The saturation magnetization, Ms, value reached ~92.3 emu/g for x = 0.0 and decreased with x content for Nd³⁺, Sm³⁺, and Y³⁺ except for Dy³⁺. The saturation magnetization value for all samples matched the trend of theoretically derived values in

Table 9. Magnetic properties of Gd_3−xRE_xFe₅O₁₂. K1 and a2 are calculated using Equations (14) and (15).

Gd3−xRExFe5O12		Ms (emu/g)	Bohr Magneton	(α)Y-K (Degrees)
	x		(µB)
	0.00	92.3	2.40	-
Dy3+	0.25	90.1	2.35	12.70
	0.50	94.8	2.47	40.08
	0.75	99.3	2.59	55.34
Nd3+	0.25	79.4	2.08	21.11
	0.50	70.6	1.84	48.13
	0.75	61.7	1.62	64.83
Sm3+	0.25	88.0	2.29	14.89
	0.50	76.8	2.01	46.07
	0.75	73.6	1.91	62.15
Y3+	0.25	85.6	2.22	17.23
	0.50	81.0	2.11	44.81
	0.75	70.5	1.84	62.74

. The total magnetic moment in the Gd_3−xRE_xFe₅O₁₂ garnet is due to the contribution of three different magnetic sublattices. The total magnetic moment is represented as:

3M_Fe(tetra) − [2M_Fe(octa) + (3 − x)M_Gd(dodec) + xM_RE(dodec)]

(12)

As obtained from the DFT study, the magnetic moment of the Dy³⁺ ion has a maximum value (10.48 µ_B), and low values for Nd³⁺ (3.62 µ_B), Sm³⁺ (0.65 µ_B), and Y³⁺ (0 µ_B) compared to Gd³⁺ (6.9 µ_B). Therefore, the net moment for the Gd_3−xRE_xFe₅O₁₂ compound decreases for Nd³⁺, Sm³⁺, and Y³⁺ doped garnet but improves with Dy³⁺ doping. The experimental magnetic moment of the RE³⁺ substitution Gd_3−xRE_xFe₅O₁₂ sample is calculated in terms of Bohr magneton using the equation below and listed in Table 9.

$$\mathrm{Bohr}\mathrm{magneton}\left({\mu }_B\right)=\frac{M\times M_s}{5585\times {\rho }_{x-ray}}$$

(13)

where ρ_x−ray is the X-ray density Equation (11), where M is the molecular weight of the samples, and M_s is the saturation magnetization in emu/g of Gd_3−xRE_xFe₅O₁₂. Bohr magneton value for Gd₃Fe₅O₁₂ is observed at 2.40 µB and changes with RE³⁺ substitution. The magnetic moment of Dy³⁺ doped garnet increases from 2.4 µB to 2.59 µB, whereas Nd³⁺, Sm^3+, and Y³⁺ doped samples show a decreasing trend. Change in Bohr magneton value with RE³⁺ substitution is due to the different magnetic moment of RE³⁺ ions and matches the theoretical study’s trend.

Yafet and Kittle (Y-K) angles describe the direction of spin of iron ions in ferrites. The Yafet and Kittle (${\alpha }_{Y-K}$) angles of RE³⁺ doped Gd_3−xRE_xFe₅O₁₂ are calculated by using the following equation:

$${\mu }_B=\left(6+x\right)Cos{\alpha }_{Y-K}-5\left(1-x\right),$$

(14)

where ${\mu }_B$ is the Bohr magnetons calculated experimentally. The ${\alpha }_{Y-K}$ arises due to the non-collinear direction of the moment between tetrahedral and octahedral sublattices. Table 9 shows the linear increase in the ${\alpha }_{Y-K}$ angle with RE³⁺ substitution. In addition, RE³⁺ doped samples show Y-K type canting of local moments. The linear trend in the ${\alpha }_{Y-K}$ angle with Rex is due to the split of sublattices having magnetic moments equal in magnitude and each making an angle ${\alpha }_{Y-K}$ with the direction of net magnetization.

Figure 19a–d shows the M vs. H curves for Gd_3−xRE_xFe₅O₁₂ powder at 300 K. With the increase in temperature to 300 K, due to thermal fluctuation, the ferrimagnetic order is lost, and the Gd_3−xRE_xFe₅O₁₂ system attains paramagnetic order (it is not PM, it is unusual that there is remanence magnetization). To further investigate the effect of RE³⁺ on the magnetic and magnetocaloric behaviors of Gd_3−xRE_xFe₅O₁₂, isothermal magnetization as a function of the applied field, M(H), was measured from 11 K to 210 K with a temperature step of 7 K up to 3T field. The isothermal plots for Dy³⁺, Nd³⁺, Sm^3+, and Y³⁺ substitution Gd_3−xRE_xFe₅O₁₂ are shown in Figure 20, Figure 21, Figure 22 and Figure 23. The isothermal magnetization curve shows the ferromagnetic ordering at low temperatures and paramagnetic at elevated temperatures. The magnetization increases sharply and saturates immediately at the low field, which is a sign of ferromagnetic behavior. Magnetization increases gradually with an increasing field and does not show any sign of saturation, thus displaying the paramagnetic behavior.

3.7. Magnetocaloric Study

Our primary focus is to study the magnetocaloric effect of RE³⁺ doped Gd_3−xRE_xFe₅O₁₂. The change in magnetic entropy (ΔS_M) is the most recommended parameter to evaluate the efficiency of magnetocaloric materials. It is calculated using the magnetic isothermal data (Figure 20, Figure 21, Figure 22 and Figure 23) near the vicinity of the transition temperature. The isothermal magnetic entropy change has been computed using the thermodynamic Maxwell relation [69],

$$∆S_M={\mu }_o{\int }_{H_i}^{H_f}{\left(\frac{\partial M}{\partial T}\right)}_{H}dH$$

(15)

$$∆S_M=\frac{{\mu }_o}{∆T}\left[{\int }_0^{H_f}M\left(T+∆T,H\right)dH-{\int }_0^{H_f}M\left(T,H\right)dH\right]$$

(16)

It is numerically calculated as;

$$-∆S_M\left(H,T\right)=\sum \frac{M_i–M_{i+1}}{T_{i+1}–T_i}∆H_i,$$

(17)

where H_i and H_f are the initial and final external applied fields, and µ_o is the permeability of free space. −ΔS_M is calculated from the isothermal magnetization curve of Figure 20, Figure 21, Figure 22 and Figure 23.

The magnetic entropy change has a maximum value near transition temperature, Tc, and its value decreases with a further increase or decrease in temperature. The sign of the magnetic entropy change is negative, which means heat is released when the magnetic field is changed adiabatically [70].

Figure 24, Figure 25, Figure 26 and Figure 27 show the magnetic entropy change curve −ΔS_M(T) as a function of temperature for Gd_3−xRE_xFe₅O₁₂. The maxima ($-∆S_M^{max}$) for the −ΔS_M(T) curve is observed to be independent of temperature and field, as shown in Figure 28. As shown in Figure 24a–d, the $-∆S_M^{max}$ value for Dy³⁺ doped garnet increases with x content. The optimum value of magnetic entropy change reached 2.04 J.Kg⁻¹K⁻¹ for x = 0.75, which is ~7% higher than that of the x = 0.0 sample. An increase in magnetic entropy change with Dy³⁺ substitution is due to the replacement of Gd³⁺ (6.99 ${\mathsf{\mu }}_B$) having a smaller magnetic moment by Dy³⁺ (9.05 ${\mathsf{\mu }}_B$) having a significant magnetic moment (from the DFT calculation above). The variation in $∆S_M$ is mainly due to the superexchange interaction between Fe–O–Fe ions.

By doping Dy³⁺, the Dy-Fe superexchange interactions become strong, enhancing the −ΔS_M value [71]. The ($-∆S_M^{max}$) value decreases with x content for the Nd, Sm, and Y doped garnet except for Sm (x = 0.75), as shown in Figure 25, Figure 26 and Figure 27. The decreasing behavior of the magnetocaloric effect with RE³⁺ doped garnet can be explained based on the magnetic moment of an individual element. From the theoretical observations, the magnetic moment of Nd (−1.54 ${\mathsf{\mu }}_B$), Sm (2.55 ${\mathsf{\mu }}_B$), and Y (0 ${\mathsf{\mu }}_B$) are smaller than the magnetic moment of Gd (6.99 ${\mathsf{\mu }}_B$). The $-∆S_M^{max}$ increases for Sm³⁺ doped garnet for x = 0.75. The $∆S_M\left(T\right)$ plots show the broad curve covering a large temperature range with RE³⁺ doped samples. Figure 28a–d shows the summary of the $-∆S_M^{max}$ value of Gd_3−xRE_xFe₅O₁₂ as a function of field. The maxima value ($-∆S_M^{max}$) shows the proportional relation with the applied field. The $-∆S_M^{max}$ values of some garnets are summarized in

Table 10. Comparison of the magnetocaloric parameters, ∆S_M^max and RCP, of a selection of garnets.

Compound	Tpeak	ΔSMmax	H (T)	RCP	R	Reference
Ho3Fe5O12	34	4.72	5	136		[16]
Er3Fe5O12	24	4.94	5	103		[16]
Gd3Fe5O12	40	1.99	3	193		[22]
Dy3Fe5O12	58	2.03	3	165		[18]
Gd3Ga2.5Fe2.5O12	12	~1.70	1			[72]
Gd2.25Dy0.75Ga2.5Fe2.5O12	10	~1.55	1			[18]
Gd3Fe5O12 (bulk)	40	0.45	1			[73]
Gd3Fe5O12 (50 nm)	25	1.49	3			[28]
Gd3Fe5O12	40	1.91	3	219	1.10	Present work
Gd2.25Dy0.75Fe5O12	54	2.04	3	234	1.13	Present work
Gd2.25Nd0.75Fe5O12	40	1.25	3	140	1.14	Present work
Gd2.25Sm0.75Fe5O12	54	2.03	3	234	1.13	Present work
Gd2.25Y0.75Fe5O12	44	1.62	3	180	1.12	Present work

to compare our results.

The relative cooling power (RCP) is a metric that quantifies the performance of magnetocaloric materials. The RCP value depends on the −ΔS_M and magnetocaloric materials’ operating temperature range. The RCP is calculated as follows,

$$RCP=\left|∆S_M^{max}\right|\times \delta T_{FWHM}$$

(18)

where δT_FWHM is the full-width-half-maxima obtained from the −∆S_M vs. T plots of Figure 24, Figure 25, Figure 26 and Figure 27.

Figure 29 shows the evolution of the RCP of Gd_3−xRE_xFe₅O₁₂ as a function of the applied magnetic field. The RCP value for x = 0.0 is ~31 J/kg at H = 0.5 T, which increases with the applied field and becomes ~219 J/kg at H = 3.0 T. The calculated RCP value for Gd_3−xRE_xFe₅O₁₂ is higher even at low fields than the other garnets reported in the literature [20,74,75]. The low field high RCP value of the Gd_3−xRE_xFe₅O₁₂ is very promising for the magnetic refrigeration application for low-temperature applications. The influence of the magnetic field on RCP may be estimated according to the formula,

RCP = A H^R

(19)

The R exponents obtained from the numerical fit of RCP are listed in Table 10. The R-value for Gd₃Fe₅O₁₂ is 1.10 and increases with RE³⁺ substitution. The maximum R-value is obtained for the Nd³⁺ (0.75) doped garnet. The R-value describes the field dependency of RCP. An R-value close to 1 implies the linear increase of RCP with the applied field.

4. Conclusions

The synthesis of RE³⁺ doped Gd_3−xRE_xFe₅O₁₂ (x = 0.0, 0.25, 0.50, and 0.75, RE³⁺ = Y, Nd, Sm, and Dy) was conducted successfully via the sol-gel autocombustion method. The substitution of RE³⁺ ions on the Gd³⁺ site of garnet brings in a structural and magnetic change in the compound. The XRD analysis shows the formation of a garnet structure with the Ia-3d space group. The Rietveld refinement shows that the lattice parameter decreased with Dy³⁺ and Y³⁺ substitution and increased with Nd³⁺ and Sm³⁺ substitution in accordance with the ionic radii of corresponding RE³⁺ ionic radii. The bond angle between RE³⁺-O²⁻-Fe³⁺ increased, Fe(Oct.)³⁺–O²⁻–Fe(Tetra.)³⁺ decreased, and the bond length between RE³⁺-O⁻² decreased with the Dy³⁺ and Y³⁺ doped sample. These structural changes have an essential influence on the magnetic structure of the compound. Magnetic studies reveal that the Dy³⁺ substitution garnet shows higher saturation magnetization with a maximum value of 99 emu/g for x = 0.75, whereas all other RE³⁺ show a decrease in saturation magnetization value. The temperature-dependent magnetization study reveals that RE³⁺ ions with non-zero magnetic moments couple strongly with the Fe³⁺(Tetra.) site. The Dy³⁺ doped garnet shows the highest magnetic entropy change value compared to other RE³⁺ doped garnets. The maxima value for Dy³⁺ doped garnet achieved (ΔS_M^max~2.00 Jkg⁻¹K⁻¹) is due to the compound’s sizeable magnetic moment. In summary, a substantial change in magnetic entropy value shows that rare-earth doped garnets could be suitable magnetocaloric materials for low-temperature cooling technology.