1 Introduction

In recent years, scientists have attached great importance to the study of magnetic materials [1]. Nano-magnetic materials are widely used in computer chips, information storage, sensors, optical devices, and devices in the electronic industry [2]. In particular, some semiconductors are also doped with magnetic elements, which make the materials have dual characteristics, such as Fe, Ni magnetic metal elements are doped in SiO2 [3, 4]. Therefore, the research of magnetic materials is of great significance. The ferrite nano-magnetic materials compared with other materials have higher saturation magnetization and higher coercive force. So the material has a good research value. Usually in the research of ferrite nano-magnetic materials, we change the magnetism by doping or substituting. In the preparation process of nano-magnetic materials, solid phase reaction, microemulsion method, chemical reaction method, sol–gel method, ball milling method, and coprecipitation method were adopted [5]. In this field, many scholars have also studied. Rahman et al. [6] prepared nano-magnetic materials of Ni–Cu–Zn ferrite substituted by Al3+ ions with solid-state method. It found that the average crystallite size, linear density, saturation magnetization, and initial permeability of Ni–Cu–Zn ferrite nano-magnetic materials decrease with the increase of Al3+ ion substitution. Qing Lin et al. [7] studied the effect of Al substitution for CoFe2O4 ferrite on its structure and magnetic properties. The results show that the lattice constant decreases with the increase of Al3+ ions. The coercivity and magnetization increase first and then decrease. Ramkrishna H. Kadam et al. [8] studied the magnetic and electrical properties of Cr3+ ions-doped Ni0.4Cu0.4Zn0.2Fe2O4 ferrite. It found that the magnetic properties of Ni–Cu–Zn ferrite were affected by the increase of Cr3+ ions content. Daniel Fritsch et al. [9, 10] have paid more attention to the study of cation distribution and the change of structure and magnetic properties of (A)[B]2O4 spinel ferrite. Rajesh Babu et al. [11] studied the nanostructure and magnetic properties of Ni0.5Zn0.5Fe2-xAlxO4 ferrite system, the lattice constants and porosity decreased with the increase of Al substitution. Matli Penchal Reddy et al. [12] studied Co-doped Mn–ferrite and found that the lattice constant increased and the saturation magnetization decreased with the increase of Co2+ doping amount.

In the present study, Al3+ ion substituted Ni–Mg–Co ferrite nano-magnetic material prepared by sol–gel auto-combustion method. We try to adjust the magnetic properties of Ni–Mg–Co ferrite by adjusting wt%. If the suitable Al3+ ion is selected to replace Fe3+ ion, the lattice structure and magnetic properties of Ni–Mg–Co ferrite may be affected. The purpose of this work is to understand the relationship between the increase of Al3+ ion doping content and the magnetic properties of Ni–Mg–Co ferrite nano-magnetic materials. For example, the changes of residual magnetization (MR), saturation magnetization (MS), coercive force (HC), anisotropy constant (k), magnetic moment (μB) and Yafet–Kittel angle (αYK) are of great significance.

2 Experimental process

2.1 Preparation materials and methods

In this study, Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0.0, 0.1, 0.2 and 0.3) ferrite nano-magnetic materials were prepared by sol–gel auto-combustion method. Figure 1 is the flowchart of the sample preparation experiment. The samples were prepared with high-purity citric acid (C6H8O7·H2O, relative molecular weight 210.14), nickel nitrate [Ni(NO3)2·6H2O, relative molecular weight 290.79], magnesium nitrate [Mg(NO3)2·3H2O, relative molecular weight 256.41], cobalt nitrate [Co(NO3)2·6H2O, relative molecular weight 291.03], aluminium nitrate [Al(NO3)3·9H2O, relative molecular weight 3375.13], and iron nitrate [Fe(NO3)3.9H2O, relative molecular weight 404.00]. By calculating the amount of each sample with a certain molar mass, and then stirring, drying, sintering, annealing etc., Al3+ ion was synthesized to substitute Ni–Mg–Co ferrite. The experimental process is divided into two steps. The first, sample preparation; second, sintering to annealing.

  1. 1.

    Preparing process: The quality of each sample is calculated by a certain molar mass. The measured samples are poured into a clean beaker, and 150 mL distilled water and stirred evenly. The chemical reaction process is as follows:

    $$ \begin{gathered} {\text{2Ni(NO}}_{{3}} {)}_{{2}} {\text{ + 2Mg(NO}}_{{3}} {)}_{{2}} {\text{ + 6Co(NO}}_{{3}} {)}_{{2}} { + }10x{\text{Al(NO}}_{{3}} {)}_{{3}} + 10(2 - x){\text{[Fe(NO}}_{{3}} {)}_{{3}} {\text{] + C}}_{{6}} {\text{H}}_{{6}} {\text{O}}_{{8}} \hfill \\ {\text{ } + \text{ 2H}}_{{2}} {\text{O } + \text{ 2NH}}_{{3}} \cdot {\text{H}}_{{2}} {\text{O}}\mathop{\longrightarrow}\limits{{\text{Heat } + \text{ Stir}}}{\text{10Ni}}_{{{0}{\text{.2}}}} {\text{Mg}}_{{{0}{\text{.2}}}} {\text{Co}}_{{{0}{\text{.6}}}} {\text{Fe}}_{{{2} - x}} {\text{Al}}_{x} {\text{O}}_{4} + {\text{40NO}}_{{2}} \uparrow + {\text{21N}}_{2} \uparrow \hfill \\ + {\text{6CO}}_{2} \uparrow + {\text{5O}}_{2} \uparrow + 1{\text{0H}}_{{2}} {\text{O}} \uparrow . \hfill \\ \end{gathered} $$

    When ammonium hydroxide was added into the solution after uniform stirring, the solution showed neutrality (pH = 7). The prepared solution is continuously stirred and heated under the operation of the magnetic stirrer (the temperature is controlled at 80℃) until the solution becomes a viscous gel state. Then it is dried in the drying box for 2 h (the temperature is controlled at 120℃).

  2. 2.

    Sintering to annealing process: First, the dried sample is preheated by alcohol lamp. After preheating, continue heating, evaporate all water molecules, and grind the sample into powder state. The powder samples were then annealed in a resistance furnace on 900℃ for 2 h. Finally, the samples were ground into powder again and cooled slowly at room temperature.

Fig. 1
figure 1

Preparation of Al substitution for Ni–Mg–Co ferrites nano-magnetic material by sol–gel method

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2.2 Sampling technology

Measuring samples: X-ray diffraction (XRD, Bruch, Germany) was used to measure the crystal plane spacing and half-width; Fourier-transform infrared spectroscopy (FT-IR, WQF-510, China) was used to observe and measure the movement of ν1 and ν2 peaks. The morphology of the samples was observed by transmission electron microscopy (TEM, JEM-2010, Japan). Analysis of chemical composition of samples by energy dispersive spectrometer (EDS < SEM > , Jsm-6700F, Japan). Vibration sample magnetometer (VSM, Lakeshore 7304, USA) was used to measure the magnetic properties of samples.

3 Results and discussion

3.1 X-ray diffraction analysis

Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0, 0.1, 0.2 and 0.3) ferrite nano-magnetic materials were prepared by sol–gel auto-combustion method. Figure 1 is the flowchart of the sample preparation experiment. Figure 2a is an XRD diagram of Al3+ ions substituting Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0, 0.1, 0.2 and 0.3) ferrite sintered to 900℃. Figure 2a shows that the prepared Al3+ ion-substituted Ni–Mg–Co ferrite samples have characteristic spinel peaks (111), (220), (311), (222), (400), (422), (511), (440), and (533). It is confirmed that the samples have spinel peak structure. In addition, the spectra show that the diffraction peaks of Al3+ ion-substituted Ni–Mg–Co ferrite match well with the characteristic peaks of spinel structure, and it indicated that the prepared samples had a single-phase structure, no impurity peaks were observed in the measured diffraction peaks. Meanwhile, it showed that all Al3+ ions enter the octahedral B-sites of Ni–Mg–Co ferrite, which is consistent with the results in some literatures [13, 14]. Figure 2b shows that the characteristic peaks (311) of samples move toward higher angles. This phenomenon is mainly due to the uniform strain and elastic deformation of the lattice caused by the excitation structure of Al3+ ions when Al3+ ions (0.53 Å) with smaller ionic radius replace Fe3+ ions (0.65 Å) with larger ionic radius. The crystal plane spacing and lattice constants decrease. The lattice of Ni–Mg–Co ferrite will shrink. Average crystallite size is calculated by (FWHM) of XRD diffraction (311) peak and Debye–Scherrer formula. The standard formula is as follows [15]:

$$ D = {{1.0747\lambda } \mathord{\left/ {\vphantom {{1.0747\lambda } {(\beta \cos \theta )}}} \right. \kern-\nulldelimiterspace} {(\beta \cos \theta )}}. $$
(1)
Fig. 2
figure 2

The XRD pattern of Ni0.2Mg0.2Co0.6AlxFe2-xO4 (x = 0.0, 0.1, 0.2, and 0.3) nanoferrite along with the figure showing the shifting of the most intense (311) peak

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In the formula, ‘λ’ is the X-ray diffraction wavelength (λ = 1.5406), ‘β’ is the half-width of the diffraction reflex by XRD. ‘θ’ is the diffraction angle of (311) peak. The scanning area of XRD data ranges from 10° to 80°. The lattice constants of samples are calculated by the following formula [15]:

$$ a = d\sqrt {h^{2} + k^{2} + l^{2} } . $$
(2)

Among them ‘d’ is the interplanar space and (h, k, l) are the Miller indices. The data in Table 1 are obtained by calculating formulas (1 and 2) above. The data in Table 1 and Fig. 3a show that the lattice constants of the prepared samples decrease with the increase of Al3+ ions substitution. The lattice constant decreases from 8.3708 to 8.3251 (Å) (Table 1). This is attributed to the substitution of smaller radius Al3+ ions (0.53 Å) [16, 17]. In addition, the average crystallite size of Ni–Mg–Co ferrite decreased from 69.7129 to 61.358 nm with the increase of Al3+ ion content. The X-ray density [18,19,20]. is also calculated by the following formula:

$$ \rho_{x} = {{8M} \mathord{\left/ {\vphantom {{8M} {(N_{A} a^{3} )}}} \right. \kern-\nulldelimiterspace} {(N_{A} a^{3} )}}, $$
(3)
Table 1 Parameters obtained from XRD data for Ni0.2Mg0.2Co0.6AlxFe2-xO4(x = 0.0, 0.1, 0.2, and 0.3) nanoferrite particles
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Fig. 3
figure 3

Variation of latitice constant, average crystallite size, and porosity versus of Al content x

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where, ‘a’, ‘M’ and ‘NA’ are, respectively, expressed as the lattice constants of the sample, the molecular mass of the sample and the Avogadro constants (NA = 6.022 × 1023 mol−1). Considering the shape of the tablet, the following formula is used to calculate the volume density [19]:

$$ \rho_{b} = {m \mathord{\left/ {\vphantom {m {(\pi R^{2} l)}}} \right. \kern-\nulldelimiterspace} {(\pi R^{2} l)}}, $$
(4)

In the formula, ‘m’, ‘R’ and ‘l’ are the quality, radius and thickness of the tablet, respectively. Considering the effect of the porosity of Ni–Mg–Co ferrite substituted by Al3+ ions on the average crystallite size, the porosity of the sample was analyzed. It is well known that the porosity of samples is caused by intergranular pore (Pinter) and intragranular pore (Pintra) [20]. The porosity is equal to the sum of intergranular pore and intragranular pore (P% = Pinter + Pintra). The formula for calculating porosity is as follows [21]:

$$ P(\% ) = {{(\rho_{x} - \rho_{b} )} \mathord{\left/ {\vphantom {{(\rho_{x} - \rho_{b} )} {\rho_{x} }}} \right. \kern-\nulldelimiterspace} {\rho_{x} }}, $$
(5)

The dislocation can be calculated by the following formula:

$$ \delta = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {D^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${D^{2} }$}} $$
(6)

Formulae (3), (4) and (5) are calculated to obtain the data in Table 2. The data show that the linear density and bulk density decrease with the increase of Al3+ ions substitution. This may be due to the fact that the relative atomic mass of Al3+ ion (26.98gm·mol−1) is less than that of Fe3+ ion (55.85gm·mol−1), which leads to the decrease of the lattice constant of the Ni–Mg–Co ferrite. The mass of the sample decreased faster than the cell volume, and the X-ray density decreased. Figure 3b shows that the porosity (P%) increases with the increase of Al3+ ions substitution. This conclusion is consistent with the results of Al3+ substitution of Ni–Zn ferrite by Rajesh Babu et al. [11]. The porosity of Al3+ ion substitution Ni–Mg–Co ferrite increases, resulting in the decrease of the average crystallite size. Because the increase of porosity will hinder the movement of crystallite boundaries, thus preventing grain growth [22]. Through the above conclusions, the results of the change of the average crystallite size measured by XRD are more forcefully explained. Similarly, Table 1 data show that the dislocation lines increase. This indicates that Al3+ ions may occupy the lattice gap position in Ni–Mg–Co ferrite crystals. Al3+ ion replaces the original basic ion and occupies the grid site in the crystal. The above conclusion can be drawn. In addition, the aberration of crystal structure can activate the accelerated mass transfer of Al3+ ions in the crystal when Al3+ ions replace Fe3+ ions.

Table 2 Various structural parameters of Ni0.2Mg0.2Co0.6AlxFe2-xO4(x = 0.0, 0.1, 0.2 and 0.3) nanofrrite particles
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3.2 FT-IR spectra analysis

Figure 4a is the Fourier-transform infrared spectra of Ni0.2Mg0.2Co0.6Fe2-xAlxO4 ferrite (x = 0, 0.1, 0.2 and 0.3). The measuring range of the instrument and equipment is 300–4000 cm−1. Two characteristic peaks ν1 and ν2 were observed in the spectra. The peaks were 588 cm−1 and 389 cm−1, respectively. These peaks are mainly caused by the tensile vibration of the high-frequency and low-frequency absorption \(\overleftrightarrow {{\text{A}} - {\text{site}} \leftarrow {\text{Zn}}^{{2 + }} {,}\,{\text{Cd}}^{{2 + }} {,}\,{\text{Ga}}^{{3 + }} {,}\,{\text{In}}^{{3 + }} {,}\,{\text{Ge}}^{{3 + }} {,}\,{\text{Mn}}^{{2 + }} {,}\,{\text{Fe}}^{{3 + }} {,}\,{\text{V}}^{{3 + }} {,}\,{\text{Cu}}^{{1 + }} {,}\,{\text{Fe}}^{{2 + }} {,}\,{\text{Mg}}^{{2 + }} {,}\,{\text{Li}}^{{2 + }} {,}\,{\text{Al}}^{{3 + }} {,}\,{\text{Cu}}^{{2 + }} {,}\,{\text{Co}}^{{2 + }} {,}\,\,{\text{Mn}}^{{3 + }} {,}\,{\text{Ti}}^{{4 + }} {,}\,{\text{Sn}}^{{3 + }} {,}\,{\text{Ni}}^{{2 + }} {,}\,{\text{Cr}}^{{3 + }} \, \to {\text{B}} - {\text{site}}}\) bands passing through tetrahedral A-site and octahedral B-site [23]. In the lattice structure of spinel ferrite, the site of different metal cations is different. Because the chemical bond vibration between A-site metal ion and oxygen atom, chemical bond vibration between B-site metal ion and oxygen atom produces the same absorption peak at high and low frequency [24]. In addition, other absorption peaks of 1050 cm−1, 1380 cm−1, 1630 cm−1, and 2350 cm−1 were also found. Characteristic peak ν1 (588 cm−1) and another characteristic peak ν2 (389 cm−1) are shown in the Fig. 4. The peak ν2 is not obvious because the wave number exceeds the instrumental range. The absorption peak of 1050 cm−1 is caused by nitrate group, which is similar to that caused by sulfonate group in some references [25]. The absorption peak at 1380 cm−1 may be the tensile vibration of carboxylate group and the tetrahedral ferrite complex [26]. The absorption peak at 1630 cm−1 may be caused by the stretching vibration of –OH bond in H2O molecule [27].

Fig. 4
figure 4

a FT-IR spectra of Ni0.2Mg0.2Co0.6AlxFe2-xO4 (x = 0.0, 0.1, 0.2, and 0.3) nanoferrite system. b Variation of force constant k0 and kt with Al content x

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Table 3 shows the position of absorption peaks of Ni0.2Mg0.2Co0.6Fe2-xAlxO4 ferrite (x = 0, 0.1, 0.2 and 0.3) substituted by Al3+ ion annealed to 900℃. It found that the absorption peak ν1 shifts to high frequency with the increase of Al3+ ion substitution. This is because Fe3+ ions with larger radius of ferrite generally occupy the octahedral B-site, and the same is true for Al3+ ions in the octahedral B-site. Fe3+ ions with larger ion radius are replaced by Al3+ ions with smaller ion radius. As a result, the bond lengths of metal cations and oxygen ions (-O2−) at octahedral B-site change, resulting in the difference of frequency bands between ν1 and ν2. Meanwhile, the lattice constant changes due to the distribution in the lower frequency band [28]. It is also mentioned in some literatures that the absorption peak at ν2 is caused by the imbalance of tetrahedral A-site charge, which may be caused by the movement of oxygen ions. Usually longer bond lengths are formed, but oxygen ions can also form stronger bonds with metal ions [29]. Therefore, the existence of metal cations (M2+ or M3+) and oxygen ions (-O2−) in the material was confirmed by FT-IR, which further explained the structure of X-ray diffraction spinel. The force constants at tetrahedral A-site and octahedral B-site are also analyzed. Analysis of metal cations occupying A-site and B-site obtained:

$$ {\text{(Fe}}^{{3 + }} )_{A} {\text{[(Ni}}_{{{0}{\text{.2}}}}^{{{2} + }} {\text{M}}g_{0.2}^{2 + } {\text{Co}}_{0.6}^{2 + } {\text{ )Al}}_{x}^{{{3} + }} {\text{ Fe}}_{{{1 - }x}}^{{{3} + }} {]}_{{\text{B}}} . $$
Table 3 Frequency bands ν1 and ν2 of the samples of the prepared Ni0.2Mg0.2Co0.6AlxFe2-xO4 (x = 0.0, 0.1, 0.2, and 0.3)
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XRD analysis shows that the lattice constant decreases with the increase of Al3+ ion substitution (Fig. 2a). Meanwhile, we all know that with the decrease of the bond length, the influence frequency increases. Finally, the force constant is calculated by the formula. According to the force constants of tetrahedral A-site and octahedral B-site are the second derivatives of potential energy, the standard formulas are as follows [30]:

$$ k_{0} = \tfrac{{0.92128M1V_{2}^{2} }}{M1 + 32}, $$
(7)
$$ k_{t} = \frac{{0.04416M2v_{1}^{2} }}{(u/V) + 3}. $$
(8)

Of which:

$$ u = \frac{{2k_{0} }}{{(v_{1}^{2} M_{1} - 2k_{0} )}};\,V = \frac{{64.2M_{1} u}}{{M_{2} }}. $$

In the formula, ‘k0′ is the force constant of B-site of octahedron and ‘kt’ is the force constant of A-site of tetrahedron. ν1 and ν2 are the frequency bands caused by the bond length vibration between metal cations and oxygen ions at tetrahedral A-site and octahedral B-site, respectively. ‘M1. and ‘M2’ are the molecular mass of tetrahedral site and octahedral site, respectively. Table 3 shows the force constants ‘kt’ and ‘k0’ with the increase of Al3+ ions substitution. Figure 4b and Table 3 show that the force constants ‘kt’ and ‘k0′ decrease with the increase of Al3+ ions substitution. The main reason is that less energy is needed to disconnect the bond length between metal cations and oxygen ions at tetrahedral A-sites and octahedral B-sites. Therefore, the force constant decreases with the increase of Al3+ ions. This conclusion is similar to that mentioned in the literature [31]. In the literature, when Co2+ ion substituted Li ferrite, the force constant increases with the increase of Co2+ ions substitution, and the result of Al3+ ion is contrary. Because the radius of Co2+ ion (0.74 Å) is larger than that of Fe3+ ion (0.65 Å) and the radius of Al3+ ion (0.53 Å) is smaller than that of Fe3+ ion (0.65 Å), resulting in the decrease of force constant with the increase of Al3+ ions content.

3.3 TEM analysis

The morphology and particle size distribution of Ni–Mg–Co ferrite nanomaterials annealed to 900 ℃ were observed by TEM. Figure 5 shows the TEM image of Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0.0 and x = 0.025) ferrite. It is observed in TEM that the Ni–Mg–Co ferrite particles are cubic and spherical like. Meanwhile, it is observed that some magnetic nanoparticles adhere to each other and form clusters at high annealing temperature. It is mainly due to the strong interaction between the van der Waals binding energy and the magnetic couple moment between the magnetic nanoparticles [32, 33]. It is also found that the average size of nanoparticles annealed to 900 ℃ in Ni–Mg–Co ferrite (x = 0 and x = 0.1) is in the range of 117.2 ~ 122.6 nm. The shape and size of the nanoparticles are uniform. In addition, Ni–Mg–Co ferrite was characterized by selected area electron diffraction. Figure 6a and b show selected area electron diffraction patterns (SAED) of x = 0 and x = 0.1. It is found that the structure of pure samples and samples containing Al3 + ions (x = 0 and x = 0.1) is similar to that of spinel. It is preliminarily inferred that the Ni–Mg–Co ferrite substituted by Al3+ ions has spinel structure. These results correspond well with the XRD data.

Fig. 5
figure 5

TEM images and particle size distribution diagrams of Al3+ ions-substituted Ni0.2Mg0.2Co0.6AlxFe2-xO4 nanoferrite particles (annealed to 900 ℃): a, b represents x = 0.0 and c, d represents x = 0.1

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Fig. 6
figure 6

SAED images of Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0.0 and 0.1) nanoferrite particles

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3.4 Analysis of the chemical composition

The chemical composition of Al3+ ion-substituted Ni–Mg–Co ferrite was analyzed by EDS. The chemical formula for the synthesis of nano-magnetic materials is Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0.0, 0.1, 0.2, and 0.3). Figure 7 is a typical EDS spectrum. The presence of Ni, Mg, Co, Fe, Al and O ions plasma in Ni0.2Mg0.2Co0.6Fe2-xAliO4 (x = 0.0, 0.1, 0.2 and 0.3) nano-magnetic materials was confirmed by EDS spectra. It also found that the X-ray counting (K cnt) increases with the increase of Al3+ ions substitution. When ‘x = 0’, the weight percentage Wt% of Fe3+ ions is 52.80, and the atomic substitution At% is 33.63. With the increase of Al3+ ion substitution, the weight percentage of Fe3+ ion Wt% and atomic hundred substitution At% decrease. When ‘x = 0.3’, Wt% and At% of Fe3+ ions decrease to 45.55 and 27.73, respectively, while Wt% and At% of Al3+ ions increase from 0 to 3.96 and 4.99, respectively. It is further explained that a large number of Al3+ ions replace Fe3+ ions, thus the lattice constant decreases, and the (311) peak of XDR moves to the high angle. Al3+ ion substitution has also been successfully realized (Table 4).

Fig. 7
figure 7

The EDS diagram of Ni0.2Mg0.2Co0.6AlxFe2-xO4 (x = 0.0, 0.1, 0.2, and 0.3) nanoferrite sintered on 900 ℃. The doping amount of each element can be observed in the diagram

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Table 4 SThe A-site and B-site magnetic moments and αYK angles of Ni0.2Mg0.2Co0.6AlxFe2-xO4 (x = 0.0, 0.1, 0.2 and 0.3) nanoferrite
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3.5 Magnetic properties

The physical properties of the samples were measured by VSM. Figure 8 shows the hysteresis loops of Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0.0, 0.1, 0.2 and 0.3) ferrite annealed to 900℃. As shown in Fig. 8, the results show that Al3+ ion substitution Ni–Mg–Co ferrite has the characteristics of ferromagnetism, and for ferrite with spinel structure, the main reason affecting their magnetic properties is the type and distribution of cations. Saturation magnetization (Ms), residual magnetization (Mr) and coercivity (Hc) can be observed directly from Fig. 9 (Table 5). In addition, the rectangular ratio (Mr/Ms), dM/dH and 2Hm were measured (shown in Table 5 and Table 6). From Fig. 9a and b and Table 5, it showed that Mr, Ms and Hc, respectively, decrease with the increase of Al3+ ions substitution. This may be due to ion superexchange in tetrahedral A-sites and octahedral B-sites in spinel ferrite [34, 35]. However, it is also found from Fig. 9b and Table 5 that when x = 0.1, the change of Hc has a reverse effect. Hc increased from 523.25 to 604.30 (Oe). It shows that the Hc of x = 0.1 is larger than that of pure sample. This result is attributed to the uneven magnetization caused by the coupling of strong exchange action in different directions of the surrounding grains when the average crystallite size is the same as the thickness of the domain wall, which ultimately leads to an increase in Hc [36]. Meanwhile, From Table 5, it found that Mr decreases from 28.66 to 21.49 emu g−1; Ms decreases from 55.22 to 43.32 emu g−1; Hc decreases from 523.25 to 230.95 Oe. For Co ferrite, Co2+ ions mainly fill the octahedral B-sites, and lead to the anisotropy of magnetic lattice through the spin–orbit coupling characteristics of Co2+ ions. Well known, Al3+ ions mainly occupy octahedron B-sites. With the increase of Al3+ ion content, more magnetic cations (Fe3+ ions and Co2+ ions) in the octahedron B-sites are replaced by Al3+ ions. Octahedral B-site Co2+ ions are forced to migrate to tetrahedral A-site, while the decrease of Co2+ concentration at octahedral B-site leads to the decrease of the anisotropy of magnetic lattice at B-site. Therefore, Mr, Ms and Hc all decreased. It is also stated in some references that the probability of Al3+ ion occupying the B-site of octahedron is greater than that of Co2+ ion and Fe3+ ion. Figure 10 shows the distribution of metal cations and oxygen ions in tetrahedral A-sites and octahedral B-sites. It is a known fact that the relationship between Ms and Mr is well explained by Néel’s theory. The spin of the electrons in A-site and B-site eliminates the reverse parallel magnetic moment when the cations in A-site and B-site vibrate [28]. The distribution of metal cations at A-sites and B-sites in Ni–Mg–Co ferrite is as follows:

$$ \frac{{(\overrightarrow {{{\text{Fe}}^{{3 + }} )}} }}{{\text{Tetrahedral A } - \text{ site}}} \, \frac{{\overleftarrow {{{\text{[(Ni}}_{{{0}{\text{.2}}}}^{{{2} + }} {\text{M}}g_{0.2}^{2 + } {\text{Co}}_{0.6}^{2 + } {\text{ )Al}}_{x}^{{{3} + }} {\text{ Fe}}_{{{2} - x}}^{{{3} + }} {\text{]O}}_{{4}}^{{2 - }} }} }}{{\text{ Octahedral B } - \text{ site}}}. $$
Fig. 8
figure 8

M–H Hysteresis loops of Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0.0, 0.1, 2, and 0.3) nanoferrite system

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Fig. 9
figure 9

Variation of saturation magnetization, remanent magnetization, anisotropy constant, saturation magnetization, and coercivity with Al content x

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Table 5 Magnetic parameters of the prepared ferrite samples
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Table 6 The values of magnetic moment, magnetic anisotropy, dM/Dh, and Hm calculated from M(H) data of Ni0.2Mg0.2Co0.6AlxFe2-xO4 (x = 0.0, 0.1, 0.2, and 0.3) nanoferrite
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Fig. 10
figure 10

The lattice structure of tetrahedral A-site and octahedral B-site and the distribution of metal cations and oxygen ions

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Similarly, in Ni–Mg–Co ferrite, the Al3+ ion occupies the octahedral B-site, and the magnetic moment μB of the Al3+ ions is ‘0’. Then a large number of non-magnetic moment Al3+ ions replace the Co2+ ions on the original octahedral B-sites, which reduces the anisotropic constant of the Co2+ ions on the B-sites, resulting in a reduction of the magnetic moment on the octahedral B-sites. The anisotropic constant K is determined by Hc and Ms. The calculation formula is as follows [37]:

$$ K = \frac{1}{2}\mu_{0} {\text{HcMs}}{.} $$
(9)

In the formula, ‘μ0′ is the permeability in vacuum (μ0 = 1) and dimensionless in Gauss units (CGS). ‘Hc’ is the coercivity of the sample and ‘Ms’ is the saturation magnetization. Figure 9(c) shows that the anisotropy constant decrease with the increase of Al3+ ions content. The anisotropy constant of Ni–Mg–Co ferrite is mainly affected by Co2+ ion. Because the electron configuration of Co2+ ion is 3d7 [38], the anisotropy is attributed to the octahedral B-site, resulting in the coupling of frozen orbital angular momentum and spin [39]. Fe3+ ion is substituted by Al3+ ion, and the spin–orbit coupling at B-site is weakened. Therefore, the anisotropy decreases of Al3+ ion-substituted Ni–Mg–Co ferrite. From formula (9), we know that Hc is inversely proportional to Ms, which is consistent with our results. Another observation in Table 5 shows that the coercive force increases sharply with the increase of Al3+ ions content. This can be attributed to the large orbital angular momentum anisotropy of Al3+ ions. This is the same as the conclusion put forward by Patange et al. [40]. In addition to the inhomogeneous magnetization caused by the superexchange coupling, the relationship between the anisotropy constant and coercive force is explained more forcefully. It is well known that the magnitude of the magnetic moment of the sample is affected by the magnetic moment of the A-site and B-site. The magnetic moment μB(Cal.) is equal to the difference between the magnetic moments of A-sites (MA) and B-sites (MB). Its physical expression is [41].

$$ \mu_{B} (Cal) = M_{B} - M_{A} . $$
(10)

In addition, the Bohr magnetic moment of Al3+ ions substituting for Ni–Mg–Co ferrite can be calculated by the following formula [31]:

$$ \mu_{B} = \frac{{M_{W} \times M_{s} }}{5585}. $$
(11)

In the formula, ‘Mw’ is the relative atomic mass of the sample, ‘Ms’ is the saturation magnetization. The data in Table 5 and Table 6 are obtained from the above formulas (9), (10), and (11). The reason why the anisotropic constant and magnetic moment decrease with the increase of Al3+ ions substitution is explained more fully. Table 6 and Fig. 9c shows that the magnetic moment decreases from 2.25μB to 1.69μB.The Hc in the formula (9) is affected by the anisotropy constant K. The magnetic moments of Ni2+, Mg2+, Co2+, Fe3+ ,and Al3+ ions mentioned in some literatures are 2μB, 0μB, 3μB, 5μB and 0μB, respectively. Table 4 shows the changes of magnetic moments at A-sites and B-sites of Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0, 0.1, 0.2, and 0.3) ferrite doped with Al3+ ions. Meanwhile, the magnetic moments of A-sites and B-sites and Yafet–Kittle model are used to calculate the magnetic moments of the samples, which are in unit μB(Cal) [42, 43]. The calculation formula is as follows:

$$ \mu_{B(\exp )} = M_{B} \cos \alpha_{YK} - M_{A} . $$
(12)

So Y–K angle \(\alpha_{YK}\) can be written as:

$$ \cos \alpha_{YK} = \frac{{\mu_{B(\exp )} + M_{A} }}{{M_{B} }}. $$
(13)

The magnetic moments of A-sites and B-sites [44], namely:

$$ \begin{gathered} (Fe_{{}}^{3 + } )_{A} [Ni_{0.2}^{2 + } Mg_{0.2}^{2 + } Co_{0.6}^{2 + } Al_{x}^{3 + } Fe_{2 - x}^{3 + } O_{4}^{2 - } ]_{B}^{{}} , \hfill \\ M_{A} = 1 \times 5\,\mu_{B} , \hfill \\ M_{B} = [0.2 \times 2 + 0.2 \times 0 + {0}.{6} \times {3} + (2 - x) \times 5 + x \times 0]\mu_{B} , \hfill \\ = (12.2 - 5x)\mu_{B} . \hfill \\ \end{gathered} $$

The Y–K angle \(\alpha_{YK}\) obtained from the above formulas (12), (13) and the above theoretical calculations. It is shown in Table 4, the magnetic stability and magnetic recording of Ni0.2Mg0.2Co0.6Fe2-xAlxO4 (x = 0, 0.1, 0.2 and 0.3) ferrite nano-magnetic materials can also be described by the rectangular ratio (S) and the switching field distribution (SFD).

$$ \left. {\frac{dM}{{dH}}} \right| = \frac{{M_{r} }}{Hc\,(1 - S*)}, $$
(14)
$$ SDF\, = \,\frac{\Delta H}{{Hc}}. $$
(15)

The data in Table 6 and Fig. 11. show that the width of dM/dH curve decreases and the 2Hm peak increases with the increase of Al3+ ions substitution. The results show that there are very few stable particles in the sample and the rewriting performance is good. The stable particles retain the original direction of magnetization when the magnetic record is not easily written to the field reversal. It is also found that the peak value of 2Hm is symmetrical at H = 0. The values of H → Hm at x = 0, 0.1, 0.2 and 0.3 are 62.32 × 10–3 emu g−1 Oe, 63.19 × 10–3 emu g−1 Oe, 68.20 × 10–3 emu g−1 Oe and 74.59 × 10–3 emu g−1 Oe, respectively. Compared with pure samples, a large number of unstable regions appear in the superparamagnetic region with the increase of Al3+ ion substitution. This may be largely influenced by the structure of nano-magnetic lattices. However, from the magnetic record rewriting energy, the performance of the sample is enhanced.

Fig. 11
figure 11

Field dependence of dM/dH of different samples. 2Hm measures the magnetic field that separates two peaks

Full size image

4 Conclusions

Al3+ ion-substituted Ni–Mg–Co ferrite nanomaterials were prepared by sol–gel auto-combustion method:

The XRD diffractogram show that all Al3+ ion-substituted Ni–Mg–Co ferrite samples have single-phase spinel structure. It was also found that the peak (311) shifted to the higher angle, indicating that the average crystallite size decreased with the increase of Al3+ ions substitution. The average crystallite size was calculated by Debye–Scheller formula. The average crystallite size decreased from 69.7129 to 61.5358 nm. The spinel structure of Al3+ ion-substituted Ni–Mg–Co ferrite was confirmed by Fourier-transform infrared spectroscopy (FT-IR). It is also found that the absorption peaks ν1 at 588 cm−1 and ν2 at 389 cm−1 move in high-frequency direction. The force constants k0 and kt caused by two bond lengths are calculated by two absorption peaks. It is found that the force constants decrease with the increase of Al3+ ion substitution. The morphology of the samples was observed by scanning electron microscopy, and the average particle size was calculated. The average particle size ranged from 117.2 to 122.6 nm. The chemical composition of Al3+ ion-substituted Ni–Mg–Co ferrite was analyzed by EDS. It was explained that Al3+ ions were successfully doped into Ni–Mg–Co ferrite. In addition, Ni, Mg, Co, Fe, A,l and O were found in the samples. The ferromagnetic properties of Ni–Mg–Co ferrite substituted by Al3+ ions were confirmed by VSM measurements. By VSM analyzing, with the increase of Al3+ ion content, the Mr decreases from 28.66 to 21.49 emu g−1, the Ms decreases from 55.22 to 43.32 emu g−1, the Hc decreases from 604.30 to 230.95 Oe, the anisotropic constant (K) decreases from 1.45 × 104 to 0.5 × 104 erg g−1, magnetic moment (μB) ecreases from 2.25 to 1.69, the \(\alpha_{YK}\) decrease ftom 53.540 (degree) to 51.301 (degree).