(1 − x)BaTi_0.5Mn_0.5O₃ + (x)Ni_0.6Zn_0.4Fe_1.85Sm_0.15O₄ composite multiferroics: Analyzing the customizing effect on conductive and magnetic properties of BaTi_0.5Mn_0.5O₃ by substituting Ni_0.6Zn_0.4Fe_1.85Sm_0.15O₄ at different concentrations Open Access

A large portion of recent technologies relies on both ferroelectricity and magnetism. Their technological and fundamental demands motivate researchers to search for such a new material in which these two phenomena are coupled intimately. After many trials, an exhilarating advanced material has been developed by researchers.^1–4 This material is multiferroic in which a feeble electromagnetic interaction can exhibit striking cross-coupling effects when it induces magnetic order in an electrically ordered state and vice versa.¹ These materials have great influences on tunable multifunctional devices such as wave modulators, novel memory devices, switches, optical diodes, spin-wave generators, and amplifiers.² 

There are two categories of multiferroics: single-phase multiferroics and composites, which consist of two or more phases simultaneously. The single-phase materials are rare in nature and hardly used in practical applications because the magnetoelectric (ME) response of these materials is either comparatively weak or occurs at low temperatures.³ Therefore, these materials may not be suitable for producing room temperature multifunctional devices. To overcome these technological problems, an alternative approach has been developed involving composites. These composite materials exhibit enhanced ME coupling above room temperature. Because of their high efficiency, cost effectiveness, outstanding mechanical strength, remarkable springiness, easier fabrication, and potential to control the molar ratio of the phase, the composite multiferroics are more convenient than the single-phase materials. Consequently, these materials have recently attracted the attention of researchers.⁴ 

Because of the high potential applications, various researchers have synthesized many composite systems, such as BaTiO₃ + Ni(Co,Mn)Fe₂O₃,⁵ BaTiO₃ + CoFe₂O₄,⁶ BaTiO₃ + NiFe₂O₄,⁷ BaTiO₃ + LiFe₅O₈,⁸ and Bi₄Ti₃O₁₂ + CoFe₂O₄,⁹ and investigated their characteristics. To the best of our knowledge, the NiZnFe₂O₄ + BaTiO₃ (NZFO + BTO) ceramic is the most studied composite material for noticeable dielectric and magnetic properties. BaTiO₃ is one of the best and most studied perovskite ferroelectric compounds due to its high dielectric constant, high remanence polarization, excellent mechanical and chemical properties, and low loss.^10,11 Another compound NiZnF₂O₄ is attractive for its excellent magnetic properties, high mechanical strength, great chemical stability, and low cost of raw materials.^12,13 The Jahn–Teller ion, Zn, increases the magnetostriction coefficient of NiFe₂O₄ because of its high coupling coefficient.¹⁴ Hence, in this research, BaTiO₃ has been selected as the ferroelectric phase and NiZnFe₂O₄ as the ferromagnetic phase to develop a composite of a high magnetoelectric (ME) coupling coefficient. To obtain good mechanical stability, high density materials should be produced by proper synthesis. Accordingly, the standard double sintering technique is followed as it is cost effective and easy to follow, and the samples become homogeneous with low porosity.

Substitution of highly resistive elements is observed to stop the increase in leakage current. Therefore, studies have been performed to enhance resistivity by substituting a dopant in both phases. Researchers have tried to adjust the electromagnetic characteristics by adding a few transition metal elements such as Mn, Fe, Co, and Ni with BaTiO₃.^15–18 They reported that Cr, Mn, and Fe doped BaTiO₃ are good candidates for ferromagnetic fabrication.^19,20 Few of them also proposed to substitute Co and Mn in BTO to produce magnetic order at low temperature.^20,21 Moreover, doping of Mn in BTO ceramics reduces the grain size and contributes to increase the electrical resistivity.²² Mn has three valance states: Mn²⁺, Mn³⁺, and Mn⁴⁺. Among them, Mn⁴⁺ ions are more likely to take the place of Ti⁴⁺ ion sites and participate in the collective motion in the BTO lattice. Ti⁴⁺ ions are less reducible than Mn³⁺ and Mn⁴⁺ ions. As a result, when some Ti⁴⁺ ion sites are occupied by Mn⁴⁺ ions, the electrons get trapped at these sites. The sites, taken by the Mn ions, are far apart from each other when the concentration of Mn ions is low. Thus, the hopping of trapped electrons from one site to another becomes almost impossible. Consequently, doping of Mn into BTO may minimize its conductance. Therefore, Mn is preferred as a substitute in BTO to decrease the leakage current. In contrast, the interesting features of NZFO spinel ferrites strongly depend on their ion distribution.^23,24 Rare earth (RE) elements (Yb, Er, Sm, TB, Gd, Dy, and Ce) have larger ionic radii than that of Fe. Hence, substitution of REs by a very small quantity can deform the crystalline lattice and drastically change the magnetic as well as electric properties of ferrites.^25,26 Among various RE elements, doping of Y, Eu, and Gd in Ni–Zn ferrites decreases the Curie temperature and increases the coercivity.²⁷ 

Sattar et al. performed a comparative study of substituting RE elements (RE = La, Nd, Sm, Gd, and Dy) in ferrites.²⁸ They reported that doping of Sm³⁺ reduced the relative density of ferrites through the formation of SmO₂.²⁸ This SmO₂ usually promotes grain growth at the time of sintering. Increased grain growth favors the permeability enlargement. Moreover, the formation of SmO₂ accelerates the growth of inner porosity. The reported porosity was about 7.5% for Sm, which was much larger than those of other REs.²⁸ As a consequence, resistivity enlargement is presumed. The leakage current is also expected to be reduced by the doping of Sm. The authors also claimed promising results for saturation magnetization compared to other REs with the substitution of Sm (1.5 µ_B) in place of Fe (5 µ_B).²⁸ This may happen for the reduced canting angle. On these grounds, Sm is considered to substitute in NZFO ferrites.

Considering all of the above studies, the impact of magnetostrictive NZFSO substitution in various ratios to piezoelectric BTMO is studied to understand the ME effects in the composites. The properties obtained from the composites (1 − x)BTMO + (x)NZFSO have been reported here. It is observed that the coalition of these two phases produces interfacial lattice strain in the composite, which modifies the dielectric, magnetic, and ME properties. The present study may lead to the enhancement of magnetoelectric effects, which are important in making phase shifters, switching devices, and sensors.

A series of multiferroic composites (1 − x)BTMO + (x)NZFSO with x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 were prepared by the conventional double sintering ceramic method. First, raw materials such as BaCO₃, TiO₂, and MnO₂ with 99.99% purity were used for preparing the BTMO ceramic and Ni₂O₃, ZnO, Fe₂O₃, and Sm₂O₃ with 99.99% purity were used to prepare the NZFSO ferrite. These basic oxide and carbonate powders were measured according to the stoichiometric proportion. Then, the powders were hand milled in an agate mortar for at least 6 h. These well mixed powders were then shaped into disks using a steel die and a hydraulic press with a uniaxial pressure of 7000 psi. At this time, a small drop of polyvinyl alcohol (PVA) was used as a binder for the powders. After that, the disks were calcined in a furnace at 700 °C for 4 h to ensure reaction between precursors. These calcined disks were again crushed for 2.5 h to obtain homogeneity. Powders of spinel and perovskite phases were then weighed and combined together according to their weight percentages in the composites. These powders were then again hand milled for 1.5 h to produce a homogeneous mixture. These were finally formed into toroids and pellets under 15 kN and 20 kN, respectively, using a single drop of PVA and were sintered in air at 1250 °C for 4 h.

The phase formation and structural analysis of the calcined powders were carried out using an x-ray diffractometer with Cu-Kα radiation (λ = 1.5406 Å) at room temperature. The scanning speed was 1°/min, and the data were collected with an angle of 2θ ranging from 15° to 70°. The lattice parameters were calculated from the x-ray diffraction (XRD) data using the following relations:

Here, h, k, and l are the Miller indices of the crystal plane.

Disk shaped pellets were used to measure the bulk density. The equation utilized is $ρB=mπr2h$⁠, where m is the mass of the sintered pellets, r is the radius of the pellets, and h indicates the thickness of the pellets. The following equation

was utilized to calculate the x-ray density of the samples. In this equation, a and c indicate the lattice parameters, M is the molecular weight, N_A represents Avogadro’s number (6.023 × 10²³), and Z stands for the number of molecules per unit cell (Z = 1 for the ferroelectric phase and Z = 8 for the ferrite phase). The microstructure of the sintered samples was examined by Field Emission Scanning Electron Microscopy (FESEM). The grain size was measured from the average of ∼40 grains of the corresponding composition. The dielectric measurements were carried out by varying the frequency from 1 kHz to 10² MHz at room temperature with the help of a Wayne Kerr Impedance Analyzer (6500B). At this time, both sides of pellets were coated with conducting silver paste. This assures good electrical contacts. The dielectric constant was calculated from the relation $ε′=Cdε0A$⁠. Here, C is the capacitance of the pellet, A represents the cross sectional area of the electrode, and ɛ₀ (= 8.85 × 10⁻¹² F/m) represents the permittivity of the free space. P–E loop measurements were done by a Radiant Precision Multiferroic Test system (model No. P-PMF with Trek 609B voltage amplifier). The M–H loops of the samples were obtained with the help of a vibrating sample magnetometer. The real and imaginary parts of complex initial permeability with frequency variation ranging from 10 kHz to 1 MHz were measured using a Wayne Kerr impedance analyzer (6500B). The used equations were $μi′=LsL0$ and μ″_i = $μi′$ tan θ, where L_s is the self-inductance of the sample core and $L0=μ0N2Sπd̄$⁠, with N indicating the number of turns of the coil, S standing for the cross sectional area, and $d̄=d1+d22$ being the mean diameter. In the expression for $d̄$⁠, d₁ and d₂ indicate the inner and outer diameters of the toroid shaped samples, respectively.

The structures of the (1 − x)BaTi_0.5Mn_0.5O₃ + (x)Ni_0.6Zn_0.4Fe_1.85Sm_0.15O₄ composites (where x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0) are characterized from the XRD patterns shown in Fig. 1. The samples were sintered at 1250 °C for 4 h to investigate the presence of any interfacial phases. The XRD peaks are quite clear and matched with Miller indices according to JCPDS card no. 86-1570 for BTMO and JCPDS card no. 52-0277 for NZFSO.²⁹ The presence of these indices reveals the co-existence of ferroelectric and ferromagnetic phases. The peak intensities and their positions confirm that a chemical reaction has occurred between these two phases during the sintering. SmFeO₃ peaks are observed as secondary phases. The number and the intensity of peaks depend on the individual phase percentages. The peak intensity of NZFSO becomes prominent when ferrites replace BTMO with a higher concentration.

The lattice parameters were calculated using the Nelson–Riley (N–R) function. The N–R function $Fθ$ was measured by the equation $Fθ=12cos2θsin2θ+cos2θθ$⁠, with θ being Bragg’s angle. Finally, the lattice parameter was taken from the y-intercept of the lattice parameter vs $Fθ$ graph. The pure BTMO is tetragonal in shape, and the lattice parameters are a = 4.008 Å and c = 4.029 Å with tetragonality c/a = 1.005. On the other hand, the lattice parameter for simple cubic NZFSO is 8.393 Å. The variation in lattice parameters in composites may be attributed to the amount of dopant ions and the large difference of the ionic radius between the dopants and the parent atoms in the lattice.^30–32 The chemical chain of both phases breaks during sintering because of the thermal excitation of ions. Hence, redistribution of ions takes place. Ni²⁺ and Zn²⁺ ions of NZFSO substitute Ba²⁺ of BTMO in the A-site. The total ionic radii of Ni²⁺(0.69 Å) and Zn²⁺(0.68 Å) are smaller than that of Ba²⁺(1.47 Å).³³ Therefore, the A-site is expected to shrink itself. On the other hand, Ti⁴⁺ and Mn³⁺ of BTMO are substituted by Fe³⁺ and Sm³⁺ of NZFSO in the B-site. The total ionic radii of Fe³⁺(0.64 Å) and Sm³⁺(1.02 Å) are slightly larger than the total radii of Ti⁴⁺(0.68 Å) and Mn³⁺(0.645 Å), which may expand the B-site.³³ Such variation in ionic radii of co-substitutes at the A- and B-sites may have increased the tension between the two sites. Accordingly, the lattice parameter reduces in the composites due to the increased stress, and the peak is observed to shift toward larger angles. Adhlakha and Yadav also reported a similar trend in the variation of lattice parameters.³⁴ The average crystal sizes of the constituent phases are calculated from the most intense (110) and (311) diffraction peaks using the well-known Debye–Scherrer’s equation D = 0.9λ/β cos θ, where β is the full width half maximum (FWHM), λ indicates the x-ray wavelength, and θ denotes the Bragg angle.

The variation in the x-ray density and the bulk density with the increasing ferrite content is shown in Fig. 2. It is clear from Fig. 2 that the highest and the lowest x-ray densities are found for the parent BTMO and NZFSO, respectively. As a result, the substitution of comparatively low density NZFSO to the high density BTMO reduces the x-ray density. The densities of Ni²⁺(8.90 g cm⁻³), Zn²⁺(7.14 g cm⁻³), Sm³⁺(7.52 g cm⁻³), and Fe³⁺(7.87 g cm⁻³) constituents of NZFSO are much larger than those of Ba²⁺(3.59 g cm⁻³), Ti⁴⁺(4.51 g cm⁻³), and Mn³⁺(7.4 g cm⁻³) constituents of the BTMO phase.³⁵ Therefore, the substitution of NZFSO into BTMO causes an increase in the bulk density of the composites following the sum rule of mixtures. However, the bulk density is relatively lower than the x-ray density. This happens from the formation of pores in the composites during the sample preparation process. The increasing bulk density ensures better densification, which is in agreement with Fig. 3. Figure 3 shows that the porosity decreases with the increased substitute concentration. The values of lattice parameter, bulk density, x-ray density, and porosity of all the samples are listed in Table I.

The phase fractions of BTMO and NZFSO in the composites were calculated utilizing the intensity of the most intense peaks of the two constituent phases with the following formulas:

and

These values are plotted in Fig. 4 with the variation in the NZFSO concentration. The obtained $PNZFSO%$ and $PBTMO%$ are in justifiable agreement with the required weight percentage of NZFSO and BTMO for preparing the composites, respectively. Unfortunately, the obtained results vary slightly from the actual values, which indicates the formation of a secondary phase. This secondary phase formation can be confirmed from the extra peak of SmFeO₃, as shown in Fig. 1.

Microstructural properties play an influential role in the various characteristics of ferromagnetic and ferroelectric ceramics. Scanning electron microscopy (SEM) images of the samples are shown in Fig. 5, and the average grain sizes are listed in Table I. The SEM images indicate that the microstructures are non-homogeneous and the grain size increases with the increasing ferrite content except at x = 0.4. Although this grain size is smaller than other composites, it is still larger than that of pure BTMO. The grain size of parent BTMO is smaller than that of parent NZFSO. Hence, the grain size increases with the addition of large sized NZFSO according to the sum rule of mixtures. Moreover, the addition of large grain-sized ferrites creates a variation in the diffusion process of the grain boundary growth mechanism.³⁶ As a result, the grain size shows an increasing trend with the increasing ferrite concentration, as shown in Fig. 6. With the increasing grain size, the porosity of the samples decreases, which supports the XRD data. The reduction in porosity relates to the enhancement of uniformity and densification.

Dielectric materials store energy in an electric field by polarizing the medium. This energy storing ability is denoted as dielectric constant. The real part of the dielectric constant is measured to obtain the storage capability. The loss factor is also observed to know about the energy loss of the prepared samples. These characteristics of all the composites are studied with the variation in frequency from 1 kHz to 10² MHz. Figure 7(a) shows that all the samples have a high dielectric constant in the low frequency range. Then, the values gradually decrease with the increasing frequency, exhibiting a large dielectric dispersion. However, they become almost steady at high frequencies. The dielectric constant corresponding to this region is called the static dielectric constant. The space charge polarization or the dielectric polarization mechanism can explain the dispersive behavior of the dielectric constant.³⁷ In polycrystalline ceramics, perfectly conducting grains are generally separated by insulating grain boundaries. Hence, the displacement of charge carriers occurs, and they tend to align themselves with the application of an external field. Consequently, space charge polarization is produced, which is governed by the available free charge carriers. Furthermore, the total dielectric constant depends on dipolar, ionic, atomic, and electronic polarizations. All these polarizations are frequency dependent, and they significantly happen at low frequencies. This results in higher values of the dielectric constant at lower frequencies. The net polarization drops at high frequencies as each polarization mechanism ceases to contribute, except the electronic one. This happens because only the non-massive electrons can keep pace with the fast changing applied field. As a result, the dielectric constant drops to a static value.

Figure 7(b) shows the variation in the real part of the dielectric constant with the ferrite content. The dielectric constant gradually decreases with the increasing ferrite content. The summation rule presumes that the values of ɛ′ in composites should be the weighted sum of the values of ɛ′ in its parent components. This rule is followed fairly closely by all the samples of the present composites. Moreover, the decrease in dielectric constant indicates an increase in conductivity with the increasing ferrite content. An increase in conductivity may happen with an increase in grain boundary or grain size. As a result, the obtained variation is consistent with the observed grain size variation in Fig. 6.

Figure 8 manifests the variation in dielectric loss (tan δ) with the frequency. The value of tan δ decreases as the frequency increases, and all composites possess low tan δ in the high frequency domain. When a field is applied, the dipoles tend to align with the field direction. However, their alignments are constrained by the other dipole effects. Hence, some energy is lost. Dipolar, ionic, and electronic losses are active in the samples at low frequencies. Consequently, the losses are high at that region. However, only the electronic losses remain in the high frequency region. Hence, the loss decreases with the increasing frequency. There are also peaks at high frequencies for each of the ferrite contents. Maximum resonance occurs at the resonance frequency, producing resonance peaks. Similar results are observed in other investigations as well.^38,39 Because of the low loss in the high frequency region, the composition with x = 0.8 shows the potential for employing in the high frequency microwave devices.

The electric modulus reveals the real dielectric relaxation process. The complex electric modulus M^* = M′ + jM″ provides an approach to inspect the electrical response of the materials. Therefore, researchers adopt it to study the relaxation phenomena in ceramics.^40–42 The electric modulus also confirms the obscurity appearing from the grain or grain boundary effect, which probably cannot be perceived from complex impedance plots. In Fig. 9(a), the real part of the electric modulus increases with the increasing frequency owing to the conduction of charge carriers. The values of M′ decrease with the increasing concentration of the substitute. The reduction in M′ largely depends on the increased grain size. The enhancement of grain sizes is observed in Fig. 6. These two results support each other’s conclusions.

From Fig. 9(b), M″ peaks are observed for all the samples at low frequencies except at x = 0.8. The low frequency region or the left side of the M″ peak indicates the frequency range of long distance movement within which ions can hop from one neighboring site to another. At the same time, the right side of the M″ peak that covers the high frequency portion represents the frequency range of ion confinement to the potential well. The frequency range for successful hopping between adjacent sites is larger than that of all the other composites for x = 0.8. This exceptional result may be the reason of significantly higher polarization at x = 0.8 than the other composites, which is shown in Fig. 11(a). The shifting of the M″ peak frequency acts as relaxation of conductivity.⁴² 

The grain, grain boundary, and electrode effects on the total resistance of a sample are determined using the real and the imaginary parts of impedance. Figure 10 shows the variation in reactance Z″ as a function of resistance Z′. Such plots are named as Cole–Cole plots or Nyquist plots. In general, a Cole–Cole plot consists of three semi-circular arcs. At low frequencies, the electrode effects become prominent and produce the first semicircular arc. The grain effects produce a semicircle at the high frequency region. In between these two semicircular arcs, another one can be observed due to the grain boundaries.⁴³ 

In the present Cole–Cole plot, there are two clearly visible arcs with an incomplete third semi-circular arc at x = 0.6. These three arcs indicate the contribution of grains, grain boundaries, and electrodes to the total resistance at x = 0.6. All the other compositions except x = 0.0 exhibit one semicircular arc followed by another incomplete one confirming the effect of grains and grain boundaries. The non-conductive properties at x = 0.0 arise from the grain effects only. The substantial decrease in the diameter of the semicircular shape happens by the change in ferrite concentration, which signifies the reduction in resistive properties. Previously, grain size values obtained from SEM images were found to increase. The increased grain size enlarges the conduction path and reduces the resistance consequently. This, in turn, decreases the arc radii of Cole–Cole plots.

The P–E loop is mainly a plot of the charge or polarization developed against the applied field at a given frequency. Figure 11(a) exhibits the variation in electric polarization as a function of the electric field at a constant line frequency of 50 Hz. All the composites show unsaturated oval shaped hysteresis loops with high remanent polarization. The remanent polarization P_r enhances with the increasing ferrite concentration except at x = 0.6. The maximum remanent polarization is P_r = 258.123 µC/cm² at x = 0.8. Since high P_r composites are suitable for switching applications, (0.2)BTMO + (0.8)NZFSO can be a potential candidate for the same. The enhancement of P_r depends on the electromechanical coupling, reduction in dielectric losses, and conduction between ferroelectric and ferromagnetic interfaces.⁴⁴ Figure 8 reveals that the dielectric loss is significantly low at x = 0.8. This may be the reason behind the large P_r at x = 0.8.

The composite loops indicate that the non-dielectric NZFSO has affected the polarization of BTMO. The hysteresis loop area is quite large for the NZFSO phase. Consequently, the enlarged loop area of the composites may be the result of the non-conductive nature of the ferrite phase.⁴⁵ The observed approximately roundish hysteresis loops result from the large leakage current, overshadowing the electrical dipolar contributions. Consequently, the coercive field E_c enhances as the composites become more difficult to be polarized. The large E_c indicates that the ferrite hinders domain wall movement because of the pinned and hindered domain wall motion. Similar results were observed for various compositions by many researchers.^3,46 At higher fields, the polarization characterizations become much more difficult. This may be the result of high resistive ferrite phases. The P–E hysteresis loops of (1 − x)BTMO + (x)NZFSO composites at line frequency with respect to different breakdown voltages are presented in Fig. 11(b). It shows that both P_r and E_c are enhanced with the increasing concentration of the substitute.

The shape of loops becomes more and more roundish with the increasing concentration of resistive NZFSO along with lower P_max values. E_c is found very small for BTMO, which indicates the paraelectric behavior, and it increases significantly with NZFSO substitution. Therefore, it can be claimed that the composites transit from the para- to the ferro-electric phase. The substitution of NZFSO may have delocalized the carriers. Consequently, interfacial polarization improves both P_r and E_c. From Figs. 11(a) and 11(b), it is clear that the prepared composites cannot be considered as ferroelectric.⁴⁶ The roundish shape of loops indicates the absence of saturation polarization. This recommends the certainty of more studies such as the M–E effect.

Figure 12 presents the magnetic hysteresis curves or M–H curves of (1 − x) BaTi_0.5Mn_0.5O₃ + (x) Ni_0.6Zn_0.4Fe_1.85Sm_0.15O₄ samples with x = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. The shape of the loops clearly indicates a soft ferromagnetic nature with a low coercive field. The magnetic signal basically comes from the NZFSO phase present in the samples.

The findings from M–H loop measurements of the samples are shown in Table I. Importantly, the coercive field measures the magnetic field strength. It can be affected by various factors such as magnetocrystallinity, microstrain, size distribution, and the magnetic domain size. Additionally, coercivity is inversely proportional to the grain size in the multidomain region. The large sized grains make the movement of the domain wall easier. From Fig. 5, it is seen that the grain size increases with the increasing ferrite contents. This indicates the reduction in coercivity. The hysteresis loop relocates toward the magnetization axis with the increasing ferrite content. This kind of variation can be explained by using the sum rule of mixtures,⁴⁷ 

The saturation magnetization increases with the increasing ferrite content, indicating that the ferrite phase exists independently in the composites. The ferrite grains act as a source of magnetic moments in the composites and contribute to the magnetization of the samples.^48,49 As the ferrite content increases, the magnetization increases in composites due to the increase in magnetic contacts between the ferrite grains following the sum rule of mixtures.

The remanent magnetization also increases with the increase in ferrite content because of the increasing concentration of ferromagnetic NZFSO following the sum rule of mixtures. The anisotropy constant (K) was calculated by using the formula K $=HcMs2$⁠. The increase in anisotropy constant may be due to the increase in saturation magnetization. Enhancement of K indicates that a large magnetic field will have to be applied for aligning the magnetic dipoles toward a hard direction from an easy crystalline axis direction. Hence, a large external magnetic field is required to produce magnetization. The studied composite materials can be used as soft ferromagnets for having low anisotropy with a small coercive field. Therefore, they can be used to make magnetic cores for transformers and inductors.

The complex permeability can be expressed as $μi*=μi′+iμi″$⁠, where, μ_i′ and μ″_i represent the real and the imaginary parts of complex permeability, respectively. μ_i′ indicates the storage of energy and μ″_i represents the dissipation of energy. Materials with high saturation magnetization, large grain size, and low porosity generally exhibit high permeability.⁵⁰ Moreover, the value of μ_i′ depends on stoichiometry, composition, impurity concentration, magnetostriction, and crystal anisotropy.⁵¹ In Figs. 6 and 12, it is observed that the samples show larger grain size and higher saturation magnetization, respectively. Hence, the samples are expected to have high permeability as well.

The real part of the initial permeability as a function of frequency is shown in Fig. 13(a). The initial permeability remains almost constant at low frequencies. After a certain high frequency, known as the cut-off frequency, the value of μ_i′ decreases to 71% of its initial value. At this frequency, the initial permeability starts to fall sharply, which is in agreement with the Globus model.^14,52 Spin and domain wall motions are the two striking mechanisms of obtaining magnetization. These two mechanisms get reduced significantly at high frequencies. Their reduction happens because both intra-granular pores and impurities behave as catalysts.⁵³ Hence, permeability decreases drastically and loss increases. From x = 0.0 to x = 0.4, the real part of permeability remains approximately constant as long as there is no phase lag between the applied field and the domain wall displacement. However, for x = 0.6 to x = 1.0, the permeability is observed to be nearly constant over a large range of frequencies before it begins to decline. This constant permeability reflects the fact that no structural relaxations and resonances are taking place in the observed frequency spectra. This also indicates how long the samples can remain magnetically stable with the applied field variation. The synthesized samples are expected to be applicable in a wide frequency range (from 10⁴ Hz to 2.4 × 10⁷ Hz), as they demonstrate impressive stable permeability.

The variation in μ_i′ as a function of content is plotted in Fig. 13(b). The spin may start to align with the lower concentration of NZFSO, resulting in a small Yafet–Kittel (Y–K) angle. As a result, μ_i′ increases with the addition of the ferrite content until x = 0.2. Further substitution of ferrites may have disturbed the spin alignment, and μ_i′ decreases slightly until x = 0.6 for the enlargement of the canting angle. After a certain concentration of NZFSO (x > 0.6), disordered spins may begin to be ordered again. Consequently, the initial permeability μ_i′ increases for further addition of NZFSO following the alignment of spins.

Figure 14 shows the variation in tan δ_M as a function of frequency. It represents the inefficiency of the magnetic system. The loss is observed to decrease gradually with the increase in frequency. The low frequency region corresponds to high resistivity due to the grain boundary, which reduces the eddy current losses.⁵⁴ As a result, the loss is high in that region. On the other hand, the high frequency region corresponds to low resistivity due to grains. Therefore, the eddy current losses become high⁵⁵ and magnetic losses decrease. The loss is the lowest for the ferrite content, x = 1.0. It decreases gradually in the composites with the increasing ferrite content, which is expected according to the sum rule of composites.

A comparative study on the synthesis route and various properties of our studied composite with the previously reported composite samples is shown in Table II. From this table, we can observe that our samples exhibit improved dielectric and enhanced magnetic properties, which indicate better application values of the prepared composites. Hence, they can be applicable in computer hard drive magnets, headphones, security systems, microphones, loudspeakers, sensors, magnetic suspensions, motors such as drills, washing machines, and generators, for example, wind turbines, turbo generators, and wave power.

In this study, composite multiferroics (1 − x)BaTi_0.5Mn_0.5O₃ + (x)Ni_0.6Zn_0.4Fe_1.85Sm_0.15O₄ are successfully synthesized at 1250 °C for 4 h. The XRD studies confirm the coexistence of the cubic spinel and tetragonal perovskite phase. Although the x-ray density decreases, the bulk density is enhanced with the reduction of piezoelectric phases. The decreasing porosity assures better densification. Surface morphology reveals that the grain size increases. All the samples show normal dielectric dispersive behavior with frequencies, explainable by the electron hopping mechanism. Moreover, this mechanism is responsible for both polarization and conduction. Both the dielectric constant and the dielectric loss are reduced at high frequencies, which makes the relative quality factor quite impressive. The Cole–Cole plot illustrates the effect of electrode, grain, and grain boundary in resistive properties. The increase in permeability supports the enlargement of grain sizes. The increasing saturation magnetization and the decreasing coercive field indicate that the composites are soft ferrites. Remanent polarization rises up to x = 0.8 except at x = 0.6 with the increasing ferrite concentration. Among all the composites, (0.2)BTMO + (0.8)NZFSO exhibits significantly large remanent polarization, maximum polarization, saturation magnetization, coercive field, and initial permeability. It also shows the possibility of being highly conductive at high frequencies due to a low dielectric constant. In addition, this composite shows high permeability with low loss within a quite impressive frequency range. Hence, (0.2)BTMO + (0.8)NZFSO has higher application values in switching and microwave appliances than the other studied composites.

I. N. Esha and K. N. Munny contributed equally to this work.

The authors express their heart-felt gratitude to the Materials Science Division of the Bangladesh Atomic Energy Centre and Centre for Advanced Research in Sciences, University of Dhaka. We are greatly indebted to the Nano Research Laboratory, Department of Physics, and the Faculty of Glass and Ceramics of Bangladesh University of Engineering and Technology for supporting us in carrying ferroelectric measurements. We are sincerely thankful to the Nano and Advanced Materials Laboratory, Department of Physics, University of Dhaka. The authors thank the Bose Centre for Advanced Study and Research in Natural Science, University of Dhaka for supporting this work. We also acknowledge Jared D. Friedl of the Department of Physics and Astronomy, University of Toledo, Toledo, USA, for giving us linguistic support.

The data that support the findings of this study are available from the corresponding author upon reasonable request.