1 Introduction

Mn–Zn ferrite is the most important type of ferrimagnetic material, which has outstanding advantages such as high initial magnetic permeability, high magnetic saturation strength, and low magnetic loss at high frequency. Therefore, Mn–Zn ferrite is widely used in many fields such as inductor cores, transformers, recording instruments, antenna rods, electromagnetic interference devices (EMI), microwave devices, gas sensors, and magnetically constrained cancer treatment [1,2,3,4,5]. The method for preparing a Mn–Zn ferrite material generally employs a solid-state sintering process, which is easy to mass-produce. However, solid state sintering technique also has some disadvantages, such as longer heat treatment duration and higher sintering temperatures (above 1000 ℃) resulting in higher energy consumption and increased production costs. Especially in the high-temperature sintering process, the volatilization of Zinc element causes the deviation of the final product from the stoichiometric ratio and the creation of ferrous ions (Fe2+). It greatly reduces the electrical resistance of Mn–Zn ferrite and increases its high-frequency magnetic loss, which hinders its practical application [6,7,8]. In recent years, there have been more and more reports on the preparation of nano-sized Mn–Zn ferrite particles by chemical solution method, which mainly include: sol–gel method [9], solid state reaction method [10], chemical co-deposition method [11], hydrothermal method [12] and combustion synthesis method [13]. The chemical solution method generally obtains uniformly dispersed nano-sized Mn–Zn ferrite particles less than 100 nm at lower annealing temperature (far below 1000 ℃). Meanwhile, these nano-sized Mn–Zn ferrite particles have higher permeability, higher saturation magnetization, and lower power losses, which have great potential applications for the electronic devices, radar absorption, medical industry (magnetic hyperthermia or drug targeting), and wastewater decontamination, etc. [14,15,16,17,18]. Sun et al. [14] found that the addition of TiO2 could modulate the initial permeability, saturation magnetization and power loss of Mn–Zn ferrite (Mn0.71Zn0.23Fe2.06O4), making it suitable for switch mode power supply applications. Praveena et al. [16] synthesized Ni0.4Zn0.2Mn0.4Fe2O4 nanoparticles with high Curie temperature, which can be used for radar absorbing from MHz to 2 GHz. Kim et al. [17] found that Mn0.2Zn0.8Fe2O4 nanoparticles showed high temperature rise under induction magnetic field within a short time, revealing a potential application in the field of magnetic hyperthermia.

In order to reduce the cost of obtaining pure phase ferrite particles, a facile sol–gel method was introduced in this paper to prepare Mn–Zn ferrite (Mn0.6Zn0.4Fe2O4) particles at lower annealing temperature (minimum to 300 ℃). The crystalline structure, morphology and magnetic properties of these particles were investigated and analyzed in detail. The experimental results and analysis show that these particles exhibit pure phase with a spinel structure, and characteristic nanoparticle morphology with an average particle size between 10 and 30 nm. It is particularly interesting that Mn0.6Zn0.4Fe2O4 nanoparticles exhibit a size-dependent magnetic property, which is characterized by a decrease in the saturation magnetization (Ms) with the decreasing crystalline size. Finally, a possible mechanism or theoretical model was proposed in the paper to clearly demonstrate the size-dependent magnetic property of Mn–Zn ferrite nanoparticles. It is believed that the present works contribute to the advancement of comprehending the correlation between the structure and property in magnetic nanoparticles.

2 Experimental

2.1 Materials and reagents

All chemical reagents used in the experiment were of analytical grade with no further purification. Manganese (II) acetate tetrahydrate [Mn(CH3COO)2·4H2O] (Purity: 99.0%), Zinc acetate dihydrate [Zn(CH3COO)2·2H2O] (Purity: 99.0%), Iron (III) nitrate nonahydrate [Fe(NO3)3·9H2O] (Purity: 99.9% metals basis) and Ethylenediaminetetraacetic acid [(HO2CCH2)2NCH2CH2N(CH2CO2H)2] (Purity: ≥ 99.5%) were purchased from ALADDIN (China). Other chemical reagents were procured from RHAWN (China).

2.2 Synthesis of mn–zn ferrite nanoparticles

Mn0.6Zn0.4Fe2O4 nanoparticles were prepared by an ethylenediaminetetraacetic acid (EDTA) complexing sol–gel process, which was reported by our previous paper [19]. In a typical run, a certain amount of Manganese (II) acetate tetrahydrate [Mn(CH3COO)2·4H2O], Zinc acetate dihydrate [Zn(CH3COO)2·2H2O], and Iron (III) nitrate nonahydrate [Fe(NO3)3·9H2O] were weighed according to the stoichiometric ratio of the final product (Mn0.6Zn0.4Fe2O4), and initially dissolved in the deionized water to form a transparent solution. Ethylenediaminetetraacetic acid (EDTA) in a molar ratio of 1:1 with the metal nitrates was added to the above solution, stirring vigorously until a clarified solution was formed. Then, the above solution was heated at 130 ℃ with constant stirring until the solution was evaporated and turned into the fluffy brown powders. Finally, these brown powders were collected and then sintered at different temperatures (300–600 ℃) for 350 s in air by employing a rapid thermal processor (heating rate up to 80 ℃/s), so as to obtain pure Mn0.6Zn0.4Fe2O4 nanoparticles.

2.3 Characterizations

Powder X-ray diffraction (XRD) was conducted on the X-ray diffractometer (XRD, X’Pert PRO, Cu Kα source, wavelength λ = 1.5405 Å) to analyze the phase purity and crystalline structure of the samples. The Debye–Scherrer formula was used to estimate the average crystalline size of as-prepared nanoparticles. Microstructure and morphologies of the nanoparticles was observed by Field Emission Scanning Electron Microscope (FE-SEM, Quanta F250). A vibrating sample magnetometer (VSM, lakeshore 7410) was used to analyze the magnetic properties of the samples.

3 Results and discussion

3.1 Crystal structure analysis

Figure 1 shows the XRD patterns of all the samples. It is clear that pure phase Mn–Zn ferrite particles can be prepared by the EDTA complexing sol–gel process at a very low annealing temperature (300 °C). Compared with the standard diffraction XRD pattern of Mn–Zn ferrite (JCPDS No.: 85-1202), it is found that the diffraction peaks of all the samples prepared in this paper (the annealing temperature no more than 500 ℃) correspond exactly to the standard peaks, indicating that the particles prepared herein exhibit pure phase with a spinel cubic structure. As the annealing temperature increases, the intensity of the diffraction peak gradually increases, indicating that the increase of the annealing temperature promotes the further growth of the crystalline. Therefore, the crystallinity of the samples becomes better, and the crystalline size increases.

Fig. 1
figure 1

XRD diffraction patterns of the samples annealed at different temperature

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However, too high annealing temperature leads to decomposition of Mn–Zn ferrite nanoparticles and formation of impurity phase of Fe2O3. As shown in Fig. 1, the sample begins to decompose and form impurity phase of Fe2O3 at the annealing temperature of 550 °C. The dotted line in Fig. 1 shows the diffraction peak of Fe2O3, which are well consistent with the diffraction peak of standard Fe2O3 (JCPDS No.: 33-0664). In general, nanoparticles have a very small particle size and a very large specific surface area, leading to a relatively high chemical reactivity [20]. Excessively high annealing temperature contributes to the susceptibility of nanoparticles to react with each other, leading to the creation of other impure phases. Therefore, in order to obtain pure phase Mn–Zn ferrite nanoparticles, the annealing temperature is not likely to exceed 500 °C.

It is known that the crystalline size of magnetic nanoparticle has an important influence on the magnetism [21,22,23,24]. As the size of magnetic nanoparticle decreases, the specific surface area of the material gradually increases, leading to significant changes in its physical and chemical properties, with special attention to the changes in its magnetic properties. In order to probe deeply into the relationship between the crystalline size and magnetic properties, it is necessary to determine the crystalline size of magnetic nanoparticles. In this paper, Debye–Scherrer formula is used to calculate the average crystalline size of Mn0.6Zn0.4Fe2O4 nanoparticles on the basis of XRD results [25, 26]:

$$D_{G} = \, \left( {K\cdot \, \lambda } \right)/\beta \cdot\cos \theta$$
(1)

where DG is the average crystalline size of nanoparticles, K is the geometric size factor (K ~ 0.89), λ is the wavelength of X-ray (λ = 1.5405 Å), β is the half-width of the diffraction peak deducting the instrument broadening (the instrument broadening was corrected by measuring bulk LaB6), and θ is the Bragg diffraction angle corresponding to the diffraction peak.

The main diffraction peaks at 20° < 2θ < 70° from XRD patterns were selected for the calculation. The peak shape parameters (diffraction angle, diffraction intensity and half width) of each selected diffraction peak were obtained from the XRD patterns (Fig. 1) and substituted into Debye–Scherrer formula to calculate the corresponding crystalline size. Consequently, the average value is obtained to correspond the average crystalline size of Mn0.6Zn0.4Fe2O4 nanoparticles.

Table 1 summaries the average crystalline size of Mn0.6Zn0.4Fe2O4 nanoparticles prepared at different annealing temperatures. It can be found that as the annealing temperature increases, the crystal growth process accelerates, and thus the crystalline size gradually becomes larger. Meanwhile, the particles crystallinity is gradually increasing.

Table 1 Average crystalline sizes of Mn0.6Zn0.4Fe2O4 nanoparticles annealed at different temperatures
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It should be mentioned that the size broadening and strain broadening are the main reasons for the peak broadening in X-ray diffraction, in addition to the instrument broadening. Crystal defects, distortion and lattice mismatch are responsible for lattice strain in the powder samples, which can be estimated by the strain-distribution model [27, 28]. However, this internal micro-strain is often closely related to the preparation process for powder samples. For instance, the powder particles prepared by the ball milling process often exhibit significant internal micro-stain as a result of the mechanical strain introduced during the preparation procedure. However, the micro-strain is often very small in the free-standing nanoparticle system prepared by chemical solution method. Therefore, the size broadening is the most important reason for the broadening of X-ray diffraction peaks in the free-standing nanoparticle system prepared by chemical solution method. The effect of internal micro-strain is essentially negligible [27]. This paper focuses the size-dependent magnetic property of free-standing Mn–Zn ferrite nanoparticles synthesized by chemical solution method. Therefore, only the size broadening is considered in the calculation of Debye–Scherrer formula, while the strain broadening is ignored.

3.2 Morphology and microstructure analysis

Figure 2 shows SEM images and average particle size of Mn0.6Zn0.4Fe2O4 nanoparticles prepared at different annealing temperatures. It is clear that these nanoparticles agglomerated together to form large aggregates. However, a clear outline of the nanoparticles can still be observed in the high magnification SEM photographs. The particle sizes of these nanoparticles were roughly distributed in the range from 10 to 30 nm. The average particle sizes of the samples prepared at different annealing temperatures were obtained by statistical analysis of these nanoparticles with Gauss function fitting, as shown in the inset of Fig. 2. As seen in Table 2, the average sizes of Mn0.6Zn0.4Fe2O4 nanoparticles sintered at 300 °C, 400 °C, 450 °C, and 500 °C are 20 nm, 23 nm, 26 nm and 28 nm, respectively. Apparently, the nanoparticle diameters observed by SEM are not the same as the crystalline sizes calculated by Debye–Scherrer formula. This indicates that the nanoparticles observed by SEM are not single-crystal particles; it is a kind of secondary particles, which are composed of multiple primary particles (single-crystal particles). The crystalline size calculated by Debye–Scherrer formula in the previous section is the primary particles, which cannot be directly observed by SEM. Thus, its size is smaller than the secondary particle size observed by SEM [29, 30]. The magnetic properties of nanoparticles are strongly correlated with the crystalline size calculated by Debye–Scherrer formula.

Fig. 2
figure 2

SEM images and particle size distribution of Mn0.6Zn0.4Fe2O4 nanoparticles prepared at different annealing temperatures, a 300 °C, b 400 °C, c 450 °C, and d 500 °C

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Table 2 Average particle sizes of Mn0.6Zn0.4Fe2O4 nanoparticles annealed at different temperatures
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3.3 Magnetic properties analysis

It can be seen from XRD and SEM analysis that Mn0.6Zn0.4Fe2O4 nanoparticles prepared at the temperatures of 300~500 °C exhibit pure phase spinel structure with the crystalline size distribution between 10 and 30 nm. In order to explore the relationship between the crystalline size and magnetism, the magnetic properties of Mn–Zn ferrite nanoparticles were measured by a vibrating sample magnetometer. As shown in Fig. 3, all Mn0.6Zn0.4Fe2O4 nanoparticles exhibit remarkable soft magnetic properties, which is characterized by observation of the saturated magnetization at lower applied magnetic fields (H < 100 Oe). The highest saturation magnetization of 45.1 emu/g was observed at the Mn0.6Zn0.4Fe2O4 nanoparticles annealed at 500 ℃. However, the saturation magnetization of Mn0.6Zn0.4Fe2O4 particles annealed at 550 ℃ rapidly decrease to only 33.2 emu/g. It should be noted that XRD results above have confirmed that Mn0.6Zn0.4Fe2O4 nanoparticles are decomposed to form antiferromagnetic Fe2O3 as the annealing temperature beyond 500 °C. Fe2O3 as a classic antiferromagnet has no contribution to the magnetism of Mn–Zn ferrite nanoparticles, thus reducing the saturation magnetization.

Fig. 3
figure 3

Hysteresis loops (M-H curves) of Mn0.6Zn0.4Fe2O4 nanoparticles

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Moreover, the saturation magnetization Ms of the nanoparticles reduces with the decreasing crystalline size (as seen in Table 3), suggesting a classical size-dependent magnetic property. In general, the weakening saturation magnetization of magnetic nanoparticles with the decreasing particle size was attributed to the magnetic disorder structure of surface layer, since the remarkable surface or interfacial states have a great influence on the properties of nanoparticles [31, 32].

Table 3 Magnetic parameters of Mn0.6Zn0.4Fe2O4 nanoparticles annealed at different temperatures
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A careful observation on the hysteresis loops of Mn0.6Zn0.4Fe2O4 nanoparticles shows that both remanent magnetization and coercivity are not zero (Fig. 4a), revealing the ferrimagnetism nature in these nanoparticles rather than superparamagnetism. As shown in Fig. 4a, the magnified magnetic hysteresis loops near the zero-point show that the left and right coercivity (H−c and H+c) are obviously asymmetric about the zero point, presenting a typical bias behavior. The difference between the left and right coercivity (δHc =|H+c + H−c|) was calculated in the paper to reflect the bias phenomenon of the magnetic hysteresis loop. As shown in Fig. 4b and Table 3, the size of the nanoparticles gradually decreases with the decrease of the annealing temperature, while the bias of the hysteresis loop becomes more and more obvious. Generally, the hysteresis loop bias phenomenon originates from the competition between two different magnetic orders in the same material structure. A typical example is the exchange bias in the “ferromagnetic/antiferromagnetic” bilayer structure, which originates from the exchange coupling at the ferromagnetic/antiferromagnetic interface [33,34,35]. The hysteresis loop bias phenomenon indicates the existence of two different magnetic orders or different arrangement of magnetic moments in the Mn0.6Zn0.4Fe2O4 nanoparticles.

Fig. 4
figure 4

a Zoom-in M-H curves of Mn0.6Zn0.4Fe2O4 nanoparticles in the region of zero coordinates; b the bias of the magnetic hysteresis loop, δHc

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Moreover, the saturation magnetization (Ms) of as-prepared Mn–Zn ferrite nanoparticles increase gradually with increasing annealing temperature, which seems to be a specific relationship. As mentioned above, Mn–Zn ferrite nanoparticles prepared at different sintering temperatures have different average crystalline sizes. Therefore, the relationship between the saturation magnetization (Ms) and the average crystalline size (DG) of Mn0.6Zn0.4Fe2O4 nanoparticles was explored in the paper. As shown in Fig. 5, as the crystalline size DG increases (1/DG decreases in the figure), the saturation magnetization Ms gradually increases; meanwhile, it is found through curve fitting that Ms is linearly proportional to 1/DG.

Fig. 5
figure 5

Relationship between saturation magnetization (Ms) and crystalline size (D) of Mn0.6Zn0.4Fe2O4 nanoparticles

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In fact, Tang et al. [36] deduced an empirical formula for size-dependent saturation magnetization,

$$M_{S} \left( D \right) = M_{BS} \cdot\left( {1 - \frac{6t}{D}} \right)$$
(2)

where MS(D) is the size-dependent saturation magnetization of nanoparticles, MBS is the saturation magnetization of corresponding bulk material, D is the nanoparticle size, and t is the thickness of non-magnetic surface layer. Although this equation could explain the linear relationship between the saturation magnetization Ms and the reciprocal of the crystalline size D, it is not quite reasonable to assume that the thickness of non-magnetic surface layer (t) is a fixed value in the literature.

In order to well explain the relationship between the saturation magnetization (Ms) and the average crystalline size (DG) or size-dependent magnetic property of Mn0.6Zn0.4Fe2O4 nanoparticles, a possible theoretical model is proposed herein to demonstrate the experimental results, which is based on the Neel theory of ferrimagnetism [37]. In bulk material (ceramic or single crystal), the surface or interface states of the material are small proportion of the overall system, which has less influence on the properties of the material. Concerning nanoparticles system, the specific surface area of the nanoparticles increases rapidly as the particle size decreases, at which point the surface or interfacial states have a great influence on the material properties. As shown in Fig. 6, a nanoscale Mn0.6Zn0.4Fe2O4 crystalline could be assumed as a single domain magnetic particle. In this nanoscale system, the surface to volume ratio of the particle becomes very large. At this point, it can be analogized to a “core–shell” structure, which consists of a lattice-continuous inner “core” and a lattice-discontinuous outer “shell” [38,39,40,41,42,43]. Mn–Zn ferrite is a typical ferrimagnet, and its net magnetization arises from the arrangement of atomic magnetic moments between two sublattices with opposite orientations but unequal sizes. The inner “core” of this nanocrystalline has a continuously lattice as in bulk materials. Thus, the atomic magnetic moments in the inner “core” are arranged in a uniform orientation (so-called as “magnetic order core”), which contribute to the net magnetization of Mn–Zn ferrite nanoparticle. However, the presence of a large number of defect states leads to a discontinuous distribution of the lattice inside the surface layer of the nanoparticle. Therefore, the surface layer of Mn–Zn ferrite nanoparticle has a chaotic distribution of atomic magnetic moments, similar to paramagnetic disorder (so-called as “magnetic disorder shell”). It is obvious that the magnetic disorder shell has no any contribution to the net magnetization of Mn–Zn ferrite nanoparticle. Moreover, as the nanoparticle size decreases, the increasing share of magnetic disorder shell attenuates the contribution of magnetic order core to the net magnetization of Mn–Zn ferrite nanoparticles. Based on this theoretical model, it is easy to understand the experimental result that the magnetic properties of Mn–Zn ferrite gradually weaken as the annealing temperature decreases.

Fig. 6
figure 6

Schematic of the theoretical model for revealing the size-dependent magnetic property of Mn–Zn ferrite nanoparticles

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In addition, assuming that as-prepared nanoparticles are the spherical magnetic particles with single-domain, average “magnetic particle size” of these Mn0.6Zn0.4Fe2O4 nanoparticles can roughly be estimated according to an empirical formula as follows [44],

$$D_{M} = \sqrt[3]{{\frac{{6\mu_{i} }}{{\pi M_{S} }}}}$$
(3)

where the initial permeability (μi) and saturation magnetization (MS) can be obtained from the magnetization curves. The calculated average “magnetic particle size” of Mn0.6Zn0.4Fe2O4 nanoparticles are summarized in Table 4. Apparently, the average “magnetic particle size (DM)” of Mn0.6Zn0.4Fe2O4 nanoparticle is significant less than its crystalline size (DG) calculated by Debye–Scherrer formula. In this scenario, the “magnetic particle size (DM)” represents the diameter of the magnetic core (magnetic order core) without the surface layer. Therefore, the crystalline size (DG) containing the surface layer is naturally larger than the “magnetic particle size (DM)”. Meanwhile, it also implies that the magnetic disorder structure of the surface layer indeed exists in the Mn0.6Zn0.4Fe2O4 nanoparticles.

Table 4 Comparison of magnetic particle size and crystalline size of Mn0.6Zn0.4Fe2O4 nanoparticles annealed at different temperatures
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More interestingly, the difference between the crystalline size and magnetic particle size (DG–DM) increases with the decreasing crystalline size, as shown in Table 4. It suggests that the thickness of the magnetic disordered layer of Mn0.6Zn0.4Fe2O4 nanoparticles gradually increases with the decreasing crystalline size, which is consistent with the previous conclusion; That is, as the magnetic nanoparticle size continues to decrease, the surface to volume ratio of the particle gradually increases, resulting in the weakening magnetism of the nanoparticles.

At the same time, this theoretical model can also demonstrate the bias of the magnetic hysteresis loop, which is similar to the exchange bias phenomenon in the ferromagnetic/antiferromagnetic bilayer structure. In this “core–shell” structure, the magnetic moments of the atoms are easily reversed and orientated uniformly under the applied magnetic field in the “magnetic order core”. However, the atomic magnetic moments are difficult to be orientated uniformly in the “magnetic disorder shell” (paramagnetic-like). This results in a pinning effect at the “core–shell” interface, which prevents the reversal of the atomic magnetic moments at the interface [33, 45, 46]. From a macro perspective, it will lead to the bias of the magnetic hysteresis loop to some extent. Moreover, the lower the annealing temperature, the smaller the nanoparticle size, and thus the bias is approximately obvious.

4 Conclusion

In the paper, Mn–Zn ferrite (Mn0.6Zn0.4Fe2O4) nanoparticles were synthesized at relatively low annealing temperatures via an EDTA complexing sol–gel process. X-ray diffraction analyses show that pure phase Mn–Zn ferrite nanoparticles can be obtained at a wide spanning temperature range from 300 to 500 °C. These Mn–Zn ferrite nanoparticles exhibit excellent soft magnetic properties, possessing ultra-high saturation magnetization (Ms) up to 45.1 emu/g. Furthermore, the hysteresis loops of Mn0.6Zn0.4Fe2O4 nanoparticles show a significant bias, which imply that there are two types of magnetic moment arrangement in the nanoparticles. More interestingly, Mn0.6Zn0.4Fe2O4 nanoparticles exhibit unique size-dependent magnetic property. The smaller the crystalline size of the nanoparticles, the smaller the saturation magnetization (Ms). Moreover, Ms and the reciprocal of the crystalline size (1/D) present a linear proportional relationship. A novel theoretical model is proposed in the paper to address above issues. Magnetic nanoparticles with small crystalline size can be analogized to a “core–shell” structure. In the shell layer that consists of abundant surface or interfacial states, the magnetic moments are irregularly arranged, so-called ‘magnetic disorder shell’. In contrast, the ferrite “core” is a ferrimagnetically ordered region with regularly arranged magnetic moments, which is called as “magnetic order core”. As the crystalline size decreases, the proportion of the “magnetic disorder shell” increases. As a result, the magnetic properties of the whole ferrite nanoparticle are weakened, as well as the decreased saturation magnetization. At the same time, this theoretical model can also demonstrate the bias of the magnetic hysteresis loop, which should be attributed to the pinning effect at the interfaces.