Structural, optical, magnetic properties and energy-band structure of MFe₂O₄ (M = Co, Fe, Mn) nanoferrites prepared by co-precipitation technique

Abstract

MFe₂O₄ (M = Co, Fe, Mn) nanoparticles were successfully formed through the chemical co-precipitation technique. X-ray diffraction, scanning electron microscopy, and energy-dispersive X-ray analysis were used to investigate samples' structural properties. The investigated structural properties included phases formed, crystallite size, cation distribution, hopping length, bond length, bond angle, edge length, and shared and unshared octahedral edge length. Scanning electron micrographs of the prepared samples demonstrated well-defined crystalline nanoparticles. The grain diameter was 15, 9, and 34 nm for CoFe₂O₄, Fe₃O₄, and MnFe₂O₄, respectively. The energy-dispersive X-ray analysis confirmed the existence of every element (Co, Fe, and O) and no discernible impurities in the samples. The optical properties were studied in detail through photoluminescence (PL) spectroscopy and Raman spectroscopy. The presence of active modes in Raman spectra confirmed the spinel structure of the MFe₂O₄ nanoparticles. The direct bandgap energy estimated through UV-visible spectroscopy was about 2.59–2.64 eV, corresponding with the energy-band structures of the octahedral site (1.70 eV) and the tetrahedral site (0.9 eV). This result was further confirmed by PL emission spectra. Based on Mie theory and UV-visible and PL spectral data, the mechanism of photothermal characterization for MFe₂O₄ nanoparticles was determined. Investigating the changes in temperature of magnetic parameters including coercivity, squareness ratio, and saturation magnetization for MFe₂O₄ samples showed the dominant influence of ion distribution and A–A, A–B, and B–B exchange interactions. This study also showed that strong anisotropy and weak dipolar interaction tended to increase the coercivity and squareness ratio of CoFe₂O₄. Conversely, weaker anisotropy and stronger dipolar interaction corresponded with the small coercivity and squareness ratio of Fe₃O₄ and MnFe₂O₄ samples.

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1. Introduction

The needs of emerging industries require various multifunctional materials that can be applied in diverse applications. Nanomaterials, especially nanosized spinel ferrites with the general formula MFe₂O₄ (M = divalent cation), exhibit many outstanding properties in the fields of magnetism, conductivity, and optics. Thus, they are a potential candidate in various fields from the electronics industry to biomedicine.^1–3 In general, the cation distribution between the A-site and B-site of an AB₂O₄ spinel structure decides the structural type of ferrites, i.e., whether it is normal or inverse or mixed.⁴ If all divalent M²⁺ ions occupy the tetrahedral site (A site) and the trivalent Fe³⁺ ions occupy the octahedral site (B site), one would have normal spinel ferrites, and the inverse spinel structure becomes possible only when all divalent ions occupy the B site. The trivalent Fe³⁺ ions can be equally divided between the A and remaining B sites.^5,6 Among nanoferrites, CoFe₂O₄, Fe₃O_4, and MnFe₂O₄ nanoferrites are excellent because of their unique physical properties and important applications.⁷ Bulk CoFe₂O₄, MnFe₂O_4, and Fe₂O₄ ferrites are predominantly inverse spinel ferrites in which Co²⁺/Fe²⁺/Mn²⁺ ions primarily occupy the B-site and Fe³⁺ ions are distributed equally among the A- and B-sites.⁸ However, in most cases, the preparation methods and the reduced size of spinels to the nanometer scale, the degree of inversion of the crystal structure becomes different from that of their bulk counterparts. Thus, the cation distribution directly affects the mechanical, chemical, and physical characteristics of spinel ferrite nanoparticles (NPs). This issue is well documented in the literature.⁹ However, owing to the cation exchange between tetrahedral and octahedral sites, the properties of some spinels become complicated; they are not the same even if they have the same chemical formula.¹⁰ This is a result of the quantum-confinement effect and the strength between the two A and B sublattice interactions.¹⁰ Therefore, controlling the fabrication parameters and choosing the appropriate sample-preparing method can confer beneficial properties for nanoferrite applications in engineering and biomedicine. The fabrication of CoFe₂O₄ and Fe₃O₄ NPs and most of their optical and magnetic properties have been thoroughly studied and are well documented.

Hunpratub et al.¹¹ synthesized CoFe₂O₄ NPs through a co-precipitation method and investigated the effect of cation distribution on their magnetic properties. They found that the predicted net magnetic moment values of all samples corresponded with the decrease in migration of Co²⁺ and Fe³⁺ between A and B sites. This result agreed with those of Yadav et al.¹² and Kumar et al.¹³ Besides cation distribution and particle size, they suggested that the noncollinear structure of spins on the surface also needs to be considered as an important factor affecting the magnetization of CoFe₂O₄ NPs.

Ansari et al.¹⁴ studied the structural, physical, chemical, electronic, and magnetic properties of solvothermally synthesized CoFe₂O₄ NPs. In particular, they found a doublet peak of T_2g(1) mode for CoFe₂O₄ with different sizes, consistent with the observation of Chandramohan et al.¹⁵ Á. de Jesús Ruíz-Baltazar et al.¹⁶ found that the T_2g(1), T_2g(2), and E_g modes are located at 312, 545, and 667 cm⁻¹, respectively, consistent with the strange modes at 453, 755, 870, and 950 cm⁻¹ for 40 nm-sized Fe₃O₄. They attributed the appearance of these unknown modes to iron oxides left over from the magnetite formation. However, many authors believe that the Raman characteristics of the inverse and mixed spinel structure are double peaks.^17–19 Therefore, 10 Raman peaks at most can be observed in Fe₃O₄ NPs.

It is clear that the previous reports of MFe₂O₄ only confined to analyze the structural, optical, magnetic, and the correlation between the cation distribution with the modification of its physical properties. In addition, in recent years, the MFe₂O₄ superparamagnetic nanomaterials were investigated for magnetic hyperthermia therapy. As a whole, this confirmed a fact that the role of dipolar interaction in the magnetic property were not investigated at length for MFe₂O₄ nanoferrites via the co-precipitation method. Therefore, an intention was carried out to study the above said property for MFe₂O₄ nanoferrites.

The photothermal effect in Fe₃O₄ NPs and its process based on photoluminescence emission in the near-infrared (NIR) region are attracting scientific research attention. Sadat et al.²⁰ used PL emission spectra in the NIR region and UV-visible (UV-Vis) spectra to establish the mechanism for interpreting the photothermal effect in (PAA)/Fe₃O₄, PS/Fe₃O₄, and Si/PS/Fe₃O₄ NPs. Zhao et al.²¹ studied the photothermal effect of Fe₃O₄ NPs in solution and thin-film forms under the influence of white light and NIR laser irradiation. Interestingly, they obtained a stronger photothermal effect for Fe₃O₄ under the stimulation of white-light irradiation. Their results are also explained based on PL and UV-Vis spectra and energy bands structure. However, the energy bands structure of ferrites is still a subject of debate. In general, the relatively complex electronic structure of transition metals has led to various confusing interpretations of the optical and magneto-optical Kerr spectra for ferrites owing to different assignments of the observed transitions.²² Specifically, the photothermal effect mechanism of CoFe₂O₄ and MnFe₂O₄ NPs has not yet been studied.

Recently, we have synthesized and determined the elastic parameters and nanocrystalline size of spinel ferrites MFe₂O₄ (M = Co, Fe, Mn, and Zn) through X-ray diffraction and infrared spectroscopy analyses.²³ In the present work, we continued to fabricate and study the structural, optical, and magnetic characteristics of the hard magnetic CoFe₂O₄ and the soft magnetic Fe₃O₄ and MnFe₂O₄ NPs. We focused on discussing the relationship between UV-Vis and PL spectra from building the energy-band structure diagram of the three samples, as well as the role of dipolar interaction in the magnetic property of MFe₃O₄ NPs.

2. Experimental detail

2.1. Synthesis of MFe₂O₄ (M = Co, Fe, Mn)

The co-precipitation technique was used to synthesize MFe₂O₄ NPs following a previous procedure.²³ Cobalt(II) chloride hexahydrate (CoCl₂·6H₂O, ≥98%), iron(III) chloride hexahydrate (FeCl₃·6H₂O, ≥98%), manganese(II) chloride tetrahydrate (MnCl₂·4H₂O, ≥98%), ferrous chloride tetrahydrate (FeCl₂·4H₂O, ≥98%), sodium hydroxide (NaOH, ≥98%) were purchased from Sigma-Aldrich, US. Each MFe₂O₄ sample was prepared from 50 mL of mixed aqueous solution by dissolving required amounts of CoCl₂·6H₂O and FeCl₃·6H₂O (for CoFe₂O₄), FeCl₂·4H₂O and FeCl₃·6H₂O (for Fe₃O₄), and MnCl₂·4H₂O and FeCl₃·6H₂O (for MnFe₂O₄) with 2 : 1 molar ratio of Fe³⁺ to M²⁺, respectively. Chemical precipitation was achieved at room temperature under vigorous stirring by adding 200 mL of 2 M NaOH solution and mixing for 60 min at pH 11 to 13. Afterward, the black precipitate of each sample was cooled to room temperature, the precipitates were separated with a permanent magnet, and they were washed with deoxygenated distilled water until neutral pH. Finally, MFe₂O₄ was washed with acetone and dried in an oven at 70 °C for 120 min. All steps of preparing MFe₂O₄ nanoferrites are clearly illustrated in Fig. 1.

Fig. 1 Schematic for synthesis MFe₂O₄ (M = Co, Fe, Mn) nanoferrites using co-precipitation method.

2.2. Characterization

The crystalline phase of the synthesized MFe₂O₄ NPs was examined using a Rigaku X-ray diffractometer equipped with Cu Kα (λ = 1.5406 Å). The 2θ range of 10° to 80° and a step size of 0.02° with a dwell time of 0.05 s were used during the measurement. Morphology and particle size were analyzed by scanning electron microscopy (SEM; Hitachi S-4800). The chemical compositions of MFe₂O₄ NPs were obtained by the energy-dispersive X-ray (EDX) analysis using a Hitachi S-4800 FESEM equipped with an EDX energy-dispersive spectroscopy attachment.

Optical properties were assessed by PL emission. Measurements were made at an excitation wavelength of 310 nm using a fluorescence spectrophotometer (Fluorolog 3 FL3-22; Horiba). Raman spectra were obtained using an XploRA PLUS Micro Raman spectrometer (Horiba) with a laser wavelength of 785 nm as an excitation source. UV-Vis spectroscopy (JASCO V-770) was used to determine the bandgap of MFe₂O₄ NPs. A 3 T Physical Property Measurement System VersaLab (model VL173; USA) was used to analyze magnetic properties.

3. Results and discussion

3.1. Structural characteristics of MFe₂O₄ NPs

Fig. 2 illustrates the X-ray diffraction (XRD) pattern and Rietveld refinement of CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ compatible with the JCPDS card number of CoFe₂O₄ (No. 156-5326), Fe₃O₄ (No. 151-3304), MnFe₂O₄ (No. 230-0618). Their patterns showed typical diffraction peaks assigned to the (111), (220), (311), (222), (400), (422), (511), (440), (620), and (533) planes of the cubic spinel structure with the Fd3̄m space group of ferrites. No additional peaks arising from impurities were detected within the limits of XRD measurement. The goodness factor (χ²) was less than 2, which lay within the confidence threshold (1 ≤ χ² < 2), demonstrating high refinement reliability.

Fig. 2 The Rietveld refinement of XRD patterns for MFe₂O₄ nanoparticles (observed – black line, calculated – red line and difference – blue line).

The experimental lattice constant (a_exp), unit-cell volume (V) and oxygen positional parameter (u) were determined from Rietveld refinement and are listed in Table 1. The lattice constant and oxygen positional parameter of Fe₃O₄ were less than those of CoFe₂O₄ and MnFe₂O₄, which was attributed to the redistribution of cations (Fe³⁺, Mn²⁺, and Co²⁺) between the two interstitial sites.^11,24 Conversely, the disparity of the oxygen positional parameter with ideal value (u = 0.375 Å) may be due to anion displacement from the ideal position to expand the tetrahedral interstices.²⁵ However, these experimental values were close to the values obtained for the ferrite system in the literature.^26,27

Table 1 Cation distribution, experimental and theoretical lattice parameter (a_exp, a_th), unit cell volume (V) and oxygen positional parameter (u) mean ionic radii of A site and B site , crystalline grain diameter (d), grain size (D) of MFe₂O₄ nanoparticles

Sample	Cation distribution		aexp (Å)	ath (Å)	V (Å^3)	u			d (nm)	D (nm)
	A site	B site
CoFe2O4	Co0.08^2+Fe0.92^3+	Co0.92^2+Fe1.08^3+	8.4093	8.4093	594.67	0.3806	0.4972	0.6910	14.6	14.7
Fe3O4	Fe^3+	Fe^2+Fe^3+	8.3628	8.3628	584.86	0.3762	0.4900	0.7125	7.0	9.0
MnFe2O4	Fe^3+	Mn^2+Fe^3+	8.4981	8.4981	613.71	0.3808	0.4900	0.7375	27.4	34.3

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The cation distribution at the A and B sites was determined based on the assumption that the occupancies of the cations with a sum of cations at A site equal to 1 and a sum of cations at B site equal to 2 of the three samples are represented in Table 1. Thus, according to the cation distribution, the theoretical lattice constant (a_th) was calculated using the following expression:^11,26,28

where r_A and r_B are the ionic radius of tetrahedral and octahedral sites, respectively, and were calculated using the equations:where R_O is the radius of oxygen ion (1.38 Å). The (a_th) values of three samples are listed in Table 1. The lattice constants a_th and a_exp well agreed with each other, indicating the migration of Co²⁺, Mn²⁺, and Fe³⁺ ions into the tetrahedral and octahedral sites, respectively.²⁸

Moreover, the mean cation radii at the tetrahedral and octahedral sites were determined based on the following cation distribution:

where C is the ionic concentration at the tetrahedral and octahedral sites; r(M_A²⁺) are the ionic radii of Co²⁺ (0.58 Å) and Fe³⁺ (0.49 Å) ions in the tetrahedral sites, respectively; r(M_B²⁺), r(Fe_B³⁺) are the ionic radii of Co²⁺ (0.745 Å), Mn²⁺ (0.83 Å), Fe²⁺ (0.78 Å), and Fe³⁺ (0.645 Å) ions in the octahedral sites, respectively.²⁹ The values of three samples are shown in Table 1. The average ionic radius of the tetrahedral site was reduced whereas the average ionic radius of the octahedral site increased in the order Co > Fe > Mn ions in MFe₂O₄. This finding was because Fe³⁺ ions (0.49 Å) with a smaller ionic radius were replaced by Co²⁺ ions (0.58 Å) with a larger ionic radius. Conversely, in (B) sites, all Mn²⁺ ions (0.83 Å) and Fe²⁺ ions (0.78 Å) with larger ion radii migrated to the B site, so the average ion radius of the B site increased.

The hopping length (L_A and L_B), tetrahedral and octahedral bond lengths (d_AO and d_BO), tetrahedral edge length (d_AOE), and shared and unshared octahedral edge length (d_BOE and d_BOEU) of the three samples were computed according to the equations below,^28,30,31 and the results are listed in Table 2.

Table 2 Hopping length (L_A and L_B), tetrahedral bond length (d_AO), octahedral bond length (d_BO), tetrahedral edge length (d_AOE), shared octahedral edge length (d_BOE), unshared octahedral edge length (d_BOEU) of MFe₂O₄ nanoparticles

Sample	Hopping length		Edges
	LA (Å)	LB (Å)	dAO (Å)	dBO (Å)	dAOE (Å)	dBOE (Å)	dBOEU (Å)
CoFe2O4	3.641	2.973	1.902	2.056	3.106	2.840	2.975
Fe3O4	3.621	2.957	1.828	2.081	2.985	2.928	2.957
MnFe2O4	3.680	3.005	1.925	2.076	3.144	2.865	3.006

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In ferrites, the distances between intrinsic ions (cation–anion and cation–cation distances) and bond angles between metal ions directly affect the strength of the magnetic exchange interactions, thus, to gain deeper insight into the magnetic properties of the samples, the inter-ionic length and bond angles between ions were calculated, and results are provided in Tables 3 and 4. The interionic distances were determined using relations,^32–35

for cation–anion length,for cation–cation distances.

Table 3 Values of the interionic distances of MFe₂O₄ nanoparticles

Sample	Me–O (Å)				Me–O (Å)
	p	q	r	s	b	c	d	e	f
CoFe2O4	2.0554	1.9019	3.6419	3.6684	2.9731	3.4863	3.6413	5.4620	5.1496
Fe3O4	2.0807	1.8280	3.5003	3.6270	2.9567	3.4670	3.6212	5.4318	5.1211
MnFe2O4	2.0752	1.9253	3.6866	3.7082	3.0045	3.5231	3.6798	5.5197	5.2040

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Table 4 Values of the interionic angles (in degrees) of MFe₂O₄ nanoparticles

Sample	Bond angle
	θ1	θ2	θ3	θ4	θ5
CoFe2O4	123.47	145.55	92.65	125.87	74.84
Fe3O4	124.87	152.52	90.55	125.39	78.82
MnFe2O4	123.40	145.23	92.75	125.90	74.65

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All these parameters of the three samples are given in Table 3. The change in values of these intrinsic distances well agreed with the unit-cell volume of the samples (Table 1).

The bond angles (θ₁; θ₂; θ₃; θ₄; θ₅) between metal ions were computed from values of cation–cation and cation–anion lengths using expressions.^32–35 Results are listed in Table 4.

The values of θ₁, θ_2, and θ₅ for Fe₃O₄ exceeded those of CoFe₂O₄ and MnFe₂O₄, whereas those of θ₃ and θ₄ for Fe₃O₄ were smaller than those of CoFe₂O₄ and MnFe₂O₄. Generally, the greater decrease in θ₃ and θ₄ corresponded with stronger interactions in B–B; conversely, the greater increment in θ₁, θ₂, and θ₅ corresponded with weaker interactions in A–B.³³ Fig. 3 illustrates the visualization of the arrangement of ion pairs with the intrinsic lengths between them and the angles formed by these lengths in the spinel structure. The influence of changing the value of bond angles on the magnetic properties of the samples is discussed in detail in the following section.

Fig. 3 Schematics of the configuration of the ion pairs in ideal spinel ferrites with advantageous distances and angles for effective magnetic interactions.²⁶

To minimize the total absolute value of errors, the average crystallite size (D) of the prepared MFe₂O₄ samples was calculated using the full width at half-maximum (β) values and angle values corresponding with all peaks, as proposed by Monshi et al.³⁴

The ln β vs. graphs of the samples are drawn in Fig. 4. The obtained average crystallite size (D) was 14.5, 7.0, and 27.4 nm for CoFe₂O₄, Fe₃O₄, and MnFe₂O₄, respectively, as listed in Table 1. The large difference in crystal size between samples showed the role of pH and reaction time on phase formation and crystal size. This finding was consistent with that of Saf et al.²⁸ They found that CoFe₂O₄ samples synthesized at low pH have smaller crystallite sizes with poor crystallinity, whereas CoFe₂O₄ samples synthesized at high pH have a larger crystallite size with a greater degree of crystallinity.

Fig. 4 Modified Scherrer for MFe₂O₄ nanoparticles.

3.2. FE-SEM and EDX of MFe₂O₄ NPs

The morphology of MFe₂O₄ NPs was examined by the FE-SEM technique, and results are shown in Fig. 5(a–c). Fig. 5 reveals particle agglomeration with spherical NPs and a narrow particle-size distribution. Analysis results of SEM images using ImageJ software demonstrated the average particle size to be 14.7, 9.0, and 34.3 nm for CoFe₂O₄, Fe₃O₄, and MnFe₂O₄, respectively. Experimental studies have determined the critical sizes for superparamagnetic (SPM) behavior of CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ NPs to be 10–17 nm,³⁶ 20 nm,^37,38 and 43 nm,³⁹ respectively. Therefore, our samples satisfy the conditions for the existence of the SPM state at temperature T > T_B and blocked state at temperature T < T_B (here T_B is a blocking temperature). The values of the average particle sizes from SEM images well agreed with those of the average crystallite sizes determined from the XRD patterns. The structure of MFe₂O₄ NPs was further investigated by EDX spectroscopy. The results are presented in Fig. 5(d–f). Co, Fe, Mn, and O elements were present without any impurity elements from precursors. As shown in the inset of Fig. 5(d–f), the observed atomic percentage of metal cations and anions determined from the average value of three different positions of each sample was close to the standard metal cation-to-anion ratio (3 : 4) of ferrites. This finding further confirmed that all precursors reacted very well with one another to finally form the desired nanoferrite samples.

Fig. 5 (a–c) HR-SEM image, particle size distribution histogram and (d–f) corresponding EDX spectra of MFe₂O₄ nanoparticles.

3.3. Raman spectra of MFe₂O₄ NPs

Raman spectroscopy measurements were performed at room temperature to further study the surface and structural properties of materials, and results are presented in Fig. 6(a). Calculations from group theory predict A_1g + E_g + 3T_2g Raman modes in the normal cubic spinel structure.^40,41 Thus, the phonon-mode assignment for our samples was conducted in accordance using the vailable literature.^{17,19,40–47} The highest frequency A_1g mode above 600 cm⁻¹ was assigned to the symmetric breathing mode of oxygen atoms concerning Co, Fe, and Mn ions in the tetrahedral void (AO₄).⁴² The second-highest frequency T_2g(3) mode from 450 cm⁻¹ to below 600 cm⁻¹ was due to as either the antisymmetric breathing mode of the AO₄ unit^40,43 or to the asymmetric bending motion of oxygen bonded to the tetrahedral cation.⁴⁴ The T_2g(2) mode from 330 cm⁻¹ to below 480 cm⁻¹ was due to a translation along one direction of the lattice, with the cation and oxygen atoms moving in opposite directions.^40,45 The E_g mode from 140 cm⁻¹ to below 330 cm⁻¹ was ascribed to the symmetric bending motion of oxygen anions within the AO₄ unit.⁴⁰ The lowest frequency Raman-active mode T_2g(1) was a complete translation of the tetrahedral void within the spinel lattice.^40,42,45 However, all Raman modes were asymmetric (or dissociated). The deconvoluted Raman spectrum for all major peaks of the samples along with their fits are presented in Fig. 6(b–d), whereas parameters derived are summarized in Table 5. These results showed that each peak can be presented like a doublet, which was characteristic of the inverse or mixed spinel structure,^19,46,47 similar to the peak positions reported for NiFe₂O₄ and ZnFe₂O₄ NPs.¹⁷ This variation in the relative Raman intensities of CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ was due to ionic-radius differences in Co²⁺, Fe²⁺, Fe³⁺, and Mn²⁺. Moreover, the downshifts and broadening of the Raman peaks in CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ NPs primarily originated from the strain growth and quantum-confinement effect of optical phonons in nanosized crystalline particles.⁴⁸

Fig. 6 (a) Experimental Raman spectrum under excitation wavelength of 785 nm and (b–d) Raman active modes fitted with cumulative fit are displayed for MFe₂O₄ nanoparticles.

Table 5 Assigned Raman modes for MFe₂O₄ nanoparticles

Sample	T2g(1) (cm^−1)	T2g(1) (cm^−1)	Eg (cm^−1)	Eg (cm^−1)	T2g(2) (cm^−1)	T2g(2) (cm^−1)	T2g(3) (cm^−1)	T2g(3) (cm^−1)	A1g (cm^−1)	A1g (cm^−1)
CoFe2O4	78.49	105.51	146.89	307.24	360.10	472.37	490.37	560.81	630.03	685.45
Fe3O4	77.92	106.90	147.48	329.72	373.56	397.73	478.60	509.08	665.32	717.67
MnFe2O4	80.86	109.78	150.31	307.14	337.33	413.11	453.70	595.30	619.36	666.88

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3.4. Optical properties and electronic band structure of MFe₂O₄ NPs

To investigate the optical-absorption properties of the materials, UV-Vis-NIR absorption spectra of the MFe₂O₄ NP samples were measured using a spectrophotometer (JASCO V-770) within the wavelength range of 300 nm to 900 nm. Fig. 7 shows the absorbance spectra of MFe₂O₄ NPs. As shown in Fig. 7, the Fe₃O₄ NPs exhibited an absorbance peak at 390 nm, following the finding of Zhao et al.²¹ Meanwhile, those of MnFe₂O₄ NPs were about 410 nm, and a wide absorbance band was observed for CoFe₂O₄ NPs. This result was due to the change in optical-energy bandgap and the effect of nanoferrites' surface roughness.^48,49 The absorbance spectra were related to the electronic structure and were thus used to determine the relation of crystallite size to the optical bandgap. The optical energy bandgap of CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ NPs were evaluated using the Tauc model⁵⁰ with the help of UV-Vis data.

αhν = A(hν − E_g)ⁿ (11)

where n represents the nature of transitions with n = 1/2 for allowed direct transitions and n = 2 for indirect transitions, respectively; A is a constant that depends on the transition probability, and E_g is the optical bandgap. Fig. 8(a–c) displays (αhν)² vs. hν for the samples. Direct bandgap values were obtained by extrapolating the linear regions of these plots along the X-axis. The obtained bandgap energy was found to be 2.63, 2.64, and 2.59 eV for the CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ NPs, respectively, showing a decrease in the bandgap from Fe₃O₄ to MnFe₂O₄. The decrease in bandgap energy was due to the size effect and the redistribution of anion ions from octahedral to tetrahedral sites. The obtained direct bandgap value of CoFe₂O₄ and Fe₃O₄ NPs were comparable with previously reported values.^51–53 Fig. 8(d–f) shows the plot of the absorption coefficient (α) against photon energy for the three samples based on the absorbance data from Fig. 7. The absorption coefficient α is depicted by the relation⁵⁴where α₀, E, and E₀ are the absorption coefficient, incident energy, and onset of absorption, respectively. Notably, α₀ and E₀ depends on the nature of the material. E_u = k_BT/σ is specified as the Urbach energy (also known as Urbach “tail”), where σ is the steepness parameter and k_B is the Boltzmann constant. Generally, the Urbach energy is governed by the crystal defect originating from thermal fluctuations, impurities, and crystalline-lattice disordering.^20,55 The Urbach energy is determined from the reciprocal of the slope of the linear portion's fitting line in the ln(α) versus photon energy (hν) plot. The Urbach energy values obtained for CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ were 0.94, 1.32, and 1.20 eV, respectively. E_u increased with decreased grain size, indicating that electrons experienced a larger extent of disorder while passing across the bandgap. Thus, the behavior of the optical bandgap was consistent with the characterization of Urbach energy. The E_u obtained for our samples also well agreed with those in the literature.²¹

Fig. 7 Room temperature UV-Vis absorbance spectrum of MFe₂O₄ nanoparticles.

Fig. 8 (a–c) Plot of (Eα)² vs. (hν), in which red line represents the experimental absorption spectra, and dashed black line is the fit to the linear portion of the data, where the intercept of that curve in the x-axis gives estimating the optical band gap and (d–f) the red line represents the experimental absorption curve, and dashed black line is the fit to the linear portion of the curve; and the Urbach energy and onset of absorption are calculated from the fitting for MFe₂O₄ nanoparticles.

To study the energy-transfer mechanism of MFe₂O₄ NPs in detail, we proposed an energy-band structure model based on theoretical calculations from the literature^{20,21,56–60} and the results of our PL and UV-Vis measurements. According to Camphausen et al.,⁵⁶ the energy gap between the O(2p) to the empty M(4s) in the 3d transition-metal oxides is about 6 eV, whereas Zaanen et al.⁵⁷ found that this bandgap ranges within 4–6 eV. Between these bands is the octahedral and tetrahedral crystal-field energies generated from the orbitals of 3d metals. The splitting energy between lower t_2g and higher e_g levels in an octahedral crystal field is Δ_cf,O ∼1.75 eV, and those of lower e and higher t₂ levels in a tetrahedral crystal field is Δ_cf,t ∼1.55 eV.⁵⁸ Moreover, the energy gap between the t_2g level (octahedral crystal field) and e level (tetrahedral crystal field) and the valence band of O(2p), respectively, approaches 0.9 eV.^22,59,60 From the above analysis, the energy-band structure of MFe₂O₄ NPs is illustrated in Fig. 9. Using this approach, the bandgap values for the CoFe₂O₄ (2.63 eV), Fe₃O₄ (2.64 eV), and MnFe₂O₄ (2.59 eV) determined from the absorbance data (Fig. 8(a–c)) were consistent with the bandgap (∼2.65 eV) between the O(2p) and e_g level of the octahedral site. Fig. 8(d–f) also shows that the onset of absorption was very close to the crystal-field splitting energy for the octahedral site (1.75 eV).

Fig. 9 Schematic of the energy bands for MFe₂O₄ system.

Sadat et al.²⁰ obtained the PL spectra for 10 nm to 5 μm-sized Fe₃O₄ NPs. They found three main peaks near 565 nm (2.2 eV), 690 nm (1.79 eV), and 840 nm (1.47 eV) when the samples were illuminated with laser light at a wavelength of 407 nm (3.05 eV). They suggested that the peak at 565 nm is due to the radiative recombination of mobile electrons from t_2g → e_g on the octahedral site, the peak at 690 nm to the recombination of trapped electrons from the octahedral site to O(2p), and the peak at 840 nm to the electron traps on the tetrahedral site. These observations were consistent with the energy-band structure that they proposed. However, Zhao et al.²¹ observed only the peak at 674 nm (1.84 eV) for 10 nm-sized Fe₃O₄ NPs illuminated with a laser of 449 nm (2.76 eV). This large difference may be due to the influence of the quantum-confinement effect originating from the nanoscale of particle size and the use of dissimilar excitation light when measuring the PL spectra of Fe₃O₄.

PL measurements of MFe₂O₄ NPs were performed using Xe light under an excitation wavelength of 310 nm (4.0 eV). Fig. 10(a) depicts the PL spectra of CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ NPs. As shown in Fig. 10(a), the emission spectrum at each sample primarily comprised three dominated clear emission peaks at around 469 nm (2.64 eV), 728 nm (1.70 eV), and 823 nm (1.51 eV). The peak of UV emission at around 469 nm was consistent with the energy gaps caused by the e_g orbital of the octahedral site and the O(2p) level, whereas the visible peak at around 728 nm and NIR peak at around 823 nm were attributed to the crystal-field splitting energy for the octahedral and tetrahedral sites, respectively.

Fig. 10 Photoluminescence spectrum (a), the wavelength dependence of extinction (Q_ext), absorption (Q_abs), scattering (Q_sca) efficiencies (b–d) for MFe₂O₄ system. The values are calculated based to the Mie theory using the complex refractive indices of the samples.

According to Mie's theory, to evaluate the photothermal performance of a material, the ratio of absorption efficiency (Q_abs) to extinction efficiency (Q_ext) needs to be analyzed and considered in detail, where Q_ext is the imaginary part of the complex refractive index, related to light absorption. Therefore, the absorption (Q_abs), extinction (Q_ext), and scattering (Q_sca) efficiencies as a function of wavelength for MFe₂O₄ NPs were calculated using equations:^20,21

Q_abs = Q_ext − Q_sca (13c)

in which a_n and b_n are the coefficients, is the size parameter with D being the mean radius of the particles, n_med is the refractive index of the media, and λ₀ is the vacuum wavelength. Using the code “Mie-Plot” written by Philip Laven,⁶¹ the numerical values of Q_ext, Q_sca, and Q_abs were calculated as a function of wavelength for MFe₂O₄ NPs and are shown in Fig. 10(b–d). Fig. 10(b) reveals that the extinction efficiency of Fe₃O₄ NPs was mostly absorption-dominated, whereas those CoFe₂O₄ and MnFe₂O₄ NPs were not [Fig. 10(b) and (d)]. Thus, CoFe₂O₄ and MnFe₂O₄ had lower Q_abs/Q_ext than Fe₃O₄ (1.00) and less efficient photothermal performance. This finding was consistent with the fact that smaller particles have higher surface area, so photothermal heating is more efficient.²⁰ However, the scattering efficiency of MnFe₂O₄ exceeded those of CoFe₂O₄ and Fe₃O₄ owing to the difference in their grain size. This finding indicated good agreement with Mie's scattering theory because with decreased particle size, the pattern of the diffracted and scattered light spread away from the forward direction. Consequently, the larger grain size of MnFe₂O₄ contributed to a greater scattering efficiency.

3.5. Magnetic properties of MFe₂O₄ NPs

Magnetic measurements of the MFe₂O₄ NPs were performed at different temperatures. Fig. 11 shows the magnetization (M–H) loops (3 T ≤ H ≤ 3 T) measured at 55, 100, 150, 200, 250, and 300 K. The magnetic parameters of all three samples including saturation magnetization (M_s), remnant magnetization (M_r), coercivity (H_c), exchange bias field (H_ex), effective anisotropy constant (K_eff), and squareness ratio (R = M_r/M_s) at various temperatures were extracted from these hysteresis loops and are listed in Table 6. For CoFe₂O₄, the value of M_s at 55 K (83.58 emu g⁻¹ or 3.51 μ_B) was slightly higher than the theoretical value of ideal inverted structural ferrite (3.0 μ_B per F.U.),^62,63 bulk counterpart (80 emu g⁻¹),^62,63 and some nanosized CoFe₂O₄ samples in literature.⁶² The increment in M_s may be due to the distribution of Co and Fe ions at tetrahedral and octahedral sites compared with the ideal spinel structure. This was also confirmed by Ansari et al.¹⁴ in oleic acid-coated CoFe₂O₄, that is, even the organic coating does not significantly affect the high value of M_s in an inverted-structure CoFe₂O₄. For MnFe₂O₄, M_s was 88.87 emu g⁻¹ (3.67 μ_B per F.U.) at 55 K, which was significantly higher than that of its bulk counterpart (3.3–4.5 μ_B per F.U.).^64–68 Notably, Mn²⁺ has an isoelectronic configuration (3d⁵), which was zero crystal-field stabilization energy; thus, the influence of the inverse parameter on the magnetic properties of MnFe₂O₄ was less pronounced than that of other spinel ferrites such as MgFe₂O₄ or Fe₃O₄.⁶⁹ Consequently, similar to CoFe₂O₄, the M_s of MnFe₂O₄ was larger than those of an inverted structural spinel (3.0 μ_B per F.U.).^64,65 In contrast to the above two samples, M_s at 55 K (59.22 emu g⁻¹) of Fe₃O₄ was smaller than that of its bulk counterpart (90 emu g⁻¹).⁷⁰ This was due to the spin-canting phenomena originating from the enhancement in the surface-to-volume ratio of the smaller-sized (9 nm) Fe₃O₄. This characteristic has been mentioned in most studies on the magnetic properties of nanosized Fe₃O₄.⁷⁰ The value of M_s in our samples was consistent with the change in the value of bond angles determined in Section 3.1. As shown in Table 5, the decrease in bond angle θ₃ and increase in length between the metal ions at the octahedral site and anion (p) for Fe₃O₄ indicated the weakening of the B–B interaction. Likewise, the bond angle θ₅ for Fe₃O₄ increased, again confirming a weak interaction between A–B and A–A. Therefore, according to ref. 71, M_s of Fe₃O₄ was smaller than that of the remaining samples. Table 6 reveals that the H_c of all samples monotonically decreased with increased temperature.

Fig. 11 M(H) loops measured at different temperatures for the MFe₂O₄ system.

Table 6 Saturation magnetization (M_s), remnant magnetization (M_r), coercivity (H_c) exchang bias field (H_ex), effective anisotropy constant (K_eff), squareness ratio (M_r/M_s) and a magnetic grain size (D_m) at various temperature of MFe₂O₄ nanoparticles

Sample	Temp. (K)	Ms (emu g^−1)	Ms (μB)	Mr (emu g^−1)	Hc (Oe)	Hex (Oe)	Keff (erg cm^−3)	Mr/Ms	Hdip (kOe)	Dm (nm)
CoFe2O4	55	83.58	3.51	59.74	9129.04	191.52	6.25 × 10^6	0.71	0.29	5.6
	100	83.48	3.51	54.59	6902.64	55.86	4.72 × 10^6	0.66	0.38	6.2
	150	82.47	3.46	47.13	4782.10	69.70	3.23 × 10^6	0.58	0.55	7.0
	200	80.07	3.36	38.09	3024.39	7.98	1.98 × 10^6	0.48	0.87	8.2
	250	76.59	3.22	29.02	1843.36	7.98	1.16 × 10^6	0.39	1.43	9.9
	300	72.06	3.03	22.26	1015.72	21.32	5.99 × 10^5	0.31	2.60	12.3
Fe3O4	55	59.22	2.45	9.24	233.02	−9.00	1.13 × 10^5	0.16	138.47	23.0
	100	58.18	2.41	6.88	167.12	11.00	7.98 × 10^4	0.12	193.07	25.8
	150	56.54	2.34	5.45	136.50	0.50	6.34 × 10^4	0.10	236.38	27.9
	200	54.39	2.25	4.32	94.01	2.00	4.20 × 10^4	0.08	343.22	32.0
	250	51.88	2.15	4.36	94.02	3.00	4.01 × 10^4	0.08	343.18	32.5
	300	48.88	2.03	0.63	41.32	3.00	1.65 × 10^4	0.01	780.89	43.6
MnFe2O4	55	88.87	3.67	13.40	266.00	4.00	1.84 × 10^5	0.15	1.67	22.8
	100	85.29	3.52	11.80	243.50	3.50	1.62 × 10^5	0.14	1.82	23.7
	150	80.05	3.31	9.17	198.50	1.50	1.24 × 10^5	0.11	2.24	26.0
	200	74.10	3.06	9.17	177.50	0.50	1.08 × 10^5	0.12	2.37	27.2
	250	67.86	2.80	8.73	138.50	2.50	9.44 × 10^4	0.13	2.49	28.4
	300	61.52	2.54	7.51	118.00	1.00	8.06 × 10^4	0.12	2.64	30.0

---

Furthermore, the coercivity values of CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ at 55 K reached 9.13 kOe, 233 Oe, and 266 Oe, respectively. For CoFe₂O₄, H_c was remarkably higher than that for bulk CoFe₂O₄ (∼5 kOe at 5 K),⁷² whereas the low coercivity of Fe₃O₄ and MnFe₂O₄ indicated the soft magnetic nature of two these samples. The temperature variation of H_c for three samples can be fitted according to Keller's law, i.e.,

H_c = H_c0[1 − (T/T_B)^1/2] (14)

where H_c0 is coercivity at 0 K and T_B is the blocking temperature. The experimental data and fitting curves for CoFe₂O₄, Fe₃O₄ and MnFe₂O₄ are revealed in Fig. 12. H_c0 was 15 230, 356, and 389 Oe for CoFe₂O₄, Fe₃O₄ and MnFe₂O₄, respectively, and their T_B values were 328, 399 and 630 K, respectively. The good fitting according to Keller's law proved that three samples were non-interacting, randomly oriented spherical particles with cubic anisotropy. Thus, the relationship among coercivity, saturation magnetization, and effective anisotropy constant (K_eff) can be given by the formula H_c = 0.64K_eff/M_s. The obtained K_eff values are listed in Table 6. The effective anisotropy constant values at 55 K were compatible with those of CoFe₂O₄, Fe₃O₄, and MnFe₂O₄ published in the literature.^73–75 Indeed, according to the Stoner–Wohlfarth (SW) model,⁷⁶ if R < 0.5, magnetostatic interactions between particles would dominate, whereas if R = 0.5, the particles would have no exchange–coupling and become randomly oriented uniaxial anisotropic ferromagnets. Finally, 0.5 < R < 1.0 confirmed the existence of exchange–coupling between particles. Therefore, from the low value of R (<0.2), we assumed that the interaction was much stronger between MnFe₂O₄ and Fe₃O₄ particles⁷⁷ and that the particle assembly was multidomain.⁷¹ Accordingly, the SW model was less relevant for MnFe₂O₄ and Fe₃O₄ NPs. CoFe₂O₄ is known to exhibit cubic magnetic anisotropy with the first crystalline magnetic anisotropy constant K₁ > 0, so the theoretical value of R should be 0.832. However, R in our CoFe₂O₄ sample was smaller than the theoretical value, so we inferred that the contribution of surface spins to the value of K_eff was negligible. Thus, the SW model was less consistent for explaining the high values of H_c and K_eff in this case. These findings indicated that for our MFe₂O₄ samples, surface anisotropy and dipole interactions were the main factors affecting H_c and K_eff.

Fig. 12 Temperature dependence of coercivity (solid circle) and the fitting line according to Kneller's law (solid black line) for MFe₂O₄ samples.

To evaluate in more detail the role of these factors on the magnetic properties of the system, their magnetic grain volume (V_m) is calculated by the formula V_m = 25k_BT_B/K_eff,⁷⁸ where k_B is the Boltzmann constant. Subsequently, the diameters of magnetic grains (D_m) for the three samples, respectively, were obtained; results are listed in Table 6. Table 6 shows that D_m increased monotonically with temperature. Moreover, the D_m of both CoFe₂O₄ and MnFe₂O₄ samples were smaller than the average physical size, indicating the significant influence of canted surface spins around CoFe₂O₄ and MnFe₂O₄ NPs; whereas D_m of Fe₃O₄ NPs were larger than the average size because of the collective behavior of several particle moments stimulated by the dipole–dipole interparticle interactions.⁷³ Thus, for CoFe₂O₄ and MnFe₂O₄ NPs, the K_eff mainly originates from the surface spins. Conversely, for Fe₃O₄ NPs, surface spins do not provide K_eff but those induce magnetocrystalline anisotropy and dipole interaction. The strength of dipolar interparticle interactions for MFe₂O₄ samples was calculated using the maximum dipolar field H_dip between nearest-neighbor particles through the relation H_dip = 2μ/d³, in which μ is the particle moment (μ = M_s × V_m) and d is the distance between the centers of two particles.⁷⁹ The H_dip parameters obtained for the samples are presented in Table 6. Notably, H_dip for Fe₃O₄ was larger than that for the remaining samples, indicating that H_dip increased rapidly for larger-sized magnetic particles. Given that the magnetic particle size of Fe₃O₄ was larger than that of MnFe₂O₄ and CoFe₂O₄, more canted surface spins occurred around the CoFe₂O₄ and MnFe₂O₄ particles. Thus, the surface anisotropy in Fe₃O₄ was weaker than that of MnFe₂O₄ and CoFe₂O₄. Meanwhile, the smaller values of H_dip in CoFe₂O₄ and MnFe₂O₄ compared with that of Fe₃O₄ confirmed that the magnetocrystalline anisotropy was weaker and the dipole–dipole interaction was stronger in CoFe₂O₄ and MnFe₂O₄. However, a comparison of H_c and R for all three samples and the H_dip calculated for three samples suggested that the dipolar interactions between particles for MFe₂O₄ NPs were predominant. Fe₃O₄ and CoFe₂O₄ particles synthesized with organic precursors also show similar behaviors.^{14,73,79–81}

4. Conclusions

MFe₂O₄ (M = Co, Fe, Mn) NPs were synthesized successfully using co-precipitation technique. The distribution of cations within interstitial sites was determined by X-ray diffraction, which showed that Co²⁺ and Fe³⁺ions were distributed over both A and B sites for CoFe₂O₄. Conversely, Fe²⁺ and Mn²⁺ ions occupied the tetrahedral site only for Fe₂O₄, and MnFe₂O₄, respectively. The cation distribution within A and B sites induced changes in structural parameters, such as mean radii of tetrahedral and octahedral sites, edge lengths, bond lengths, and bond angles in the unit cell of MFe₂O₄. SEM images showed that MFe₂O₄ NPs had a narrow size distribution with an average size of 14.7, 9.0, and 34.3 nm for CoFe₂O₄, Fe₃O₄, and MnFe₂O₄, respectively. The observation of doublet phenomena of characteristic Raman peaks confirmed the feature of the inverse or mixed spinel structure for MFe₂O₄ NPs. The optical energy bandgap calculated through UV-Vis spectroscopy was 2.63, 2.64, and 2.59 eV for CoFe₂O₄, Fe₃O₄, and MnFe₂O₄, respectively. Based on the bandgap energy determined from UV-Vis and PL measurements, the energy-band structure diagram of MFe₂O₄ was established, in which the bandgap energy determined from the PL spectrum corresponded with the crystal-field splitting energy for the octahedral (1.70 eV) and tetrahedral (1.51 eV) sites. The mechanism of photothermal behavior of MFe₃O₄ NPs was further determined based on the PL emission and UV-Vis spectra. Magnetic parameters such as coercivity, remanence, squareness ratio, and effective anisotropy constant were deduced from the magnetization loops under an applied magnetic field of 3 T ≤ H ≤ 3 T. The change in magnetic saturation of MFe₂O₄ samples was explained based on the cation distribution and A–A, A–B, and B–B interactions. The large coercivity and squareness ratio of CoFe₂O₄ corresponded with strong effective anisotropy primarily due to the surface spins. Meanwhile, the smaller coercivity and squareness ratio of MnFe₂O₄ and Fe₃O₄ resulted from the weaker anisotropy and stronger dipolar interaction.

Data availability

The data that support the findings of this study are available within the article.

Author contributions

Tran Dang Thanh: writing – original draft, review & editing, investigation, project administration. Tran Thi Ngoc Nha: writing – original draft, investigation. Tran Thi Ha Giang: investigation. Dang Ngoc Toan: resources, investigation. Pham Hong Nam: resources, investigation. Dinh Thanh Khan: resources, investigation. Do Hung Manh: writing – review & editing, investigation. Pham Thanh Phong: writing – review & editing, investigation.

Conflicts of interest

The authors confirm that there are no known conflicts of interest associated with this publication.

Acknowledgements

This work was supported by an excellent research team development program grant funded by the Vietnam Academy of Science and Technology (VAST) under Project Code NCXS01.04/22–24.

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