1 Introduction

Relaxor ferroelectrics (RFEs) have attracted much interest recently due to their peculiar structure and unusual physical properties, which have the features of ferroelectrics, i.e. high polarizability. However, at the same time, they are characterized by slim hysteresis loops with a small coercive field Ec and remnant polarization Pr [1]. RFEs possessing high values of the quasistatic dielectric constants have broadened peaks and frequency-dependent maxima of the dielectric permittivity. These materials are suitable candidate materials for high-temperature capacitors [2, 3]. Furthermore, RFEs may also have excellent electrostriction, piezoelectric and electro-optical properties which make them attractive for energy storage applications [3]. Lacking long-range polar ordering and the appearance of polar nanoregions (PNRs) and nano-domains are characteristic features of relaxors [4, 5].

But what about relaxor multiferroics (RMFs)? RMFs have been found attractive because they possess multiferroic orders at the nanoscale, a broad range of frequency and temperature-dependent dielectric and magnetic susceptibility, mutual dependence of ferroic orders under the existence or absence of applied magnetic field, and destruction of ferroelectric long-range order in the existence of small magnetic fields [3]. Therefore, RMF has not only PNRs as in relaxor ferroelectric but it also has magnetic clusters which is featured in relaxor ferromagnet [4].

It is well-known that Pb (Zr, Ti) O3 (PZT) and Pb (Fe, W) O3 (PFW), have excellent relaxor ferroelectric properties with a high ferroelectric phase transition [4]. To exploit the basic ferroic properties of these materials, several types of dopants and adjusting compositions have been used to achieve relaxor multiferroic behavior [6]. Pb [(Zr0.53Ti0.47)0.6 (Fe 0.5Ta0.5)0.40] O3 (PZTFT) and Pb [(Zr0.53Ti0.47)0.4 (Fe0.67W0.33)0.60] O3 (PZTFW) systems are two examples [3, 4]. These systems exhibited a high dielectric constant (\(\sim \) 1380 at 1 kHz and 1385 at 100 Hz, respectively), diffused ferroelectric phase transition, low dielectric loss, low leakage current, slim hysteresis loop, and high temperature-dependent dielectric adjustability [3, 4]. Although, the usage of Pb oxides has made severe environmental concerns due to their high toxicity [7]. Therefore, recently, tremendous efforts have been focused on the research of exploring and evolving the new lead-free relaxor multiferroics as alternatives for environmental protection and human health as the usage of electronic products and high-temperature applications [8].

Among the eco-friendly Pb-free materials, BiFeO3 (BF) was considered to be the best-known room-temperature magnetoelectric (ME) multiferroics perovskite structure to substitute PZT materials due to its giant ferroelectric polarization ~ 100 µm/cm2 and high phase transition temperatures (Néel temperature “TNéel” ~ 643 K; Curie temperature “TCurie” ~ 1103 K) [9].

However, it is usually difficult to synthesize pure BiFeO3 without the presence of impurity phases because the perovskite BF has low structural stability [10]. Previous reports have shown that secondary phases in BiFeO3 (Bi2Fe4O9, Bi25FeO40) could be leading to strong hysteresis and high leakage currents, leading to low electrical properties [10]. Furthermore, BF is well-known to display weak ferromagnetism with relatively low magnetization [10]. Therefore, to improve its structural stability and multiferroic properties, different Pb-free groups of perovskite materials “ABO3”, such as K0.5Na0.5NbO3, (Bi0.5K0.5)TiO3, and BaTiO3 (BT), have been combined into BF [11]. The BFBT system has been designed for energy-storage capacitors and temperature-stable dielectric applications, whereas BT has a high resistivity and a stable phase structure. Therefore, the BFBT system is expected to show great promise because of the desired structural stabilization, its strong energy storage performance, and its electrical properties [1, 12].

Within top-down methods, one of the most common methods is ball milling. This method is known as an inexpensive and simple technique for the synthesis of nanostructured materials [13]. One of the most remarkable features of the mechanical milling technique is synthesizing nanocrystalline and amorphous materials [13]. During mechanical milling, the prepared structures are determined by a dynamic balance between the control of the diffusion recovery and mechanically induced disorder. If the recovery rate of the disorder is slow according to the disordering rate, so the defect structure and disorder could be produced by ball milling to lead to an amorphous state. Additionally, according to amorphization, as the milling time increases, the diffraction peaks broadening in the X-ray patterns increase, due to the decrease in crystallite size [14]. Amorphous materials produced by ball-milling have specific and useful properties which are not found in crystalline materials. Since, the amorphous material properties are characterized by a deficiency of long-range periodicity in the amorphous state and the prepared nanostructured have a homogeneous composition [11, 12]. Amorphous materials show high corrosion resistance, small magnetic losses, high strength and hardness, low deterioration of mechanical properties in radiation environments, and enhanced catalytic properties [13]. The amorphous materials that have this combination of mechanical and physical features have given rise to commercial applications, particularly in the field of magnetoelectric materials [13].

Based on the above, the influence of the ball milling technique on the structure, dielectric behavior, magnetization, relaxor characteristics and energy storage performance of 0.30BaTiO3–0.52Bi2O3–0.18Fe2O3 mol% (BFBT) has been examined. We have considered five different milling times in the prepared sample, 10, 20, 30, 50 and 75 h to determine the best milling time for the synthesis of the nanostructured (partially amorphous) BFBT sample. To the best of our knowledge, this is the first attempt to confirm the relaxor multiferroic properties and explore the electrical energy storage density and efficiency as a function of temperature for the BFBT sample by the application of mechanical milling.

2 Materials and methods

Nanostructured 0.30BaTiO3–0.52Bi2O3–0.18Fe2O3 mol% (BFBT) sample was prepared by the mechanical activation method using Bi2O3 (99%), Fe2O3 (99%) (Loba Chemie) and BaTiO3, (99%) (Aldrich Chemical) powder. The stoichiometric ratios from all the precursor powder were weighed out, then milled for 10, 20, 30, 50 and 75 h with adding 5 wt% Stearic acid as a control agent (owing to its low toxicity) in a high-energy ball milling attritor at room temperature using a ball-to-powder ratio of 20:1 and rotation speed of 500 rpm. The ball-milled BFBT powder was uniaxially pressed, after each milling, into pellets of about 4 mm in diameter at a pressure of 900 MP using a hydraulic press. X-ray diffraction patterns (XRD) of the as-milled powder were obtained from the Siemens D5000 X-ray diffractometer with Cu Kα radiation (= 0.154 nm) under an accelerating voltage of 40 kV and current of 30 mA, within the range of 10  < 2  < 70  using a step size of 0.05 deg/sec. The average particle size was determined by high-resolution transmission electron microscope (HRTEM, JEOL JEM-2100TEM). Fourier transform infrared spectroscopy analysis of samples was performed at room temperature on (FT-IR Bruker Alpha II ATR) with a frequency range of 400–2000 cm−1. The dielectric constant and loss of the as-milled pellet for 50 h were studied in the temperature range of 400–675 K at fixed frequencies using MICROTEST 6377 LCR meter. Polarization–electric field (P-E) hysteresis loops at a frequency of 50 Hz were measured using a Sawyer–Tower circuit. For these measurements, the silver paste was coated on both sides of the as-milled pellet for 50 h as top and bottom electrodes. The magnetic properties were measured by DC magnetic susceptibility measurements using Faraday’s method and a vibrating-sample magnetometer (VSM, Lake Shore model 7410, USA).

3 Results and analysis

3.1 Structure

3.1.1 X-ray diffraction analysis (XRD)

The Mechanical milling technique highly depends on several parameters such as milling time, milling temperature, size and size distribution of balls, the powder provided to operate the milling chamber, and the speed of milling [15]. In this work, the milling times are only changed, whereas other parameters were held constant. Figure 1a, b illustrate the XRD patterns of the as-received pure materials and their stoichiometric mixtures after various times of milling at room temperature. The XRD patterns of the as-received pure Bi2O3, Fe2O3, and BaTiO3 powder and the as-mix BFBT indicate the purity of raw materials. The characteristic peaks exhibited in the XRD pattern corresponded to the ICDD (JCPDS) Card No. 65-2366, Card No. 33-0664, and Card No. 79-2264 for Bi2O3, Fe2O3, and BaTiO3, respectively, as shown in Fig. 1a. The XRD pattern of the nanostructured BFBT sample obtained by the mechanical milling method at various milling times of 10, 20, 30, 50 and 75 h is observed in Fig. 1b. All diffraction peaks in the XRD pattern of powder samples were indexed with miller indices. The Bi4 (TiO4)3 phase (JCPDS12-0213) was found for the 10 h milled sample as shown in Fig. 1b. After 30 h of mechanical milling, the XRD patterns display the partially amorphous nature of the BFBT sample. We have increased the milling time to observe if there is a recrystallization phase or change in the amount of amorphous nature, which is possible in some systems. However, increasing the milling time to 75 h shows no change. Therefore, we have chosen the BFBT sample milled for 50 h to be an intermediate point between 30 and 75 h. According to the XRD analysis, the peak at 31.6° is attributable to BaTiO3 (JCPDS79-2264) in nanostructured BFBT samples milled for more than 10 h. The main peak of BiFeO3 at 2\(\uptheta \) = 31.8° with the (110) plane reported in JCPDS card No.72-2035 is very close to BaTiO3, so it could also be formed. According to Scherrer’s equation [16], the calculated crystalline size is decreased from 3.6 μm (0 h) to 22 nm (75 h). As can be seen by increasing the milling time, the intensity of the diffracted peak of nanostructured BFBT decreases and its width becomes broader gradually due to the refinement of crystallite size and improvement of lattice strain. This may be due to the strong collision of the mixed powders with the balls and the container’s walls during the milling process. While comparing the as-milled nanostructured BFBT with the as-mix powder, we see that all peaks are broadened, indicating the size reduction of particles into nano-scale and partially amorphous phase [17].

Fig. 1
figure 1

Room temperature X-ray diffraction patterns of (a) the as-received pure Bi2O3, Fe2O3, BaTiO3 and BFBT as mix powders and b the as-milled BFBT powders at various milling times

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3.1.2 High-resolution transmission electron microscope (HRTEM)

Figure 2 presents HRTEM pictures and consistent selected area electron diffraction (SAED) patterns of the nanostructured BFBT milled for 50 h. After milling for 50 h, the ball-milled nano-scale powder particles aggregate together to form large particles, as shown in Fig. 2a, owing to the strong active surface energy generated during ball milling. The average particle size of the as-milled nanostructured BFBT for 50 h is approximately 27.86 nm. Figure 2b, c shows the interlayer d-spacing for these particles. The d-spacing value is measured as 0.31 nm which is very close to d-spacing value of BT plane 110. The nanostructured (partially amorphous) nature of the sample is confirmed by the (SAED) pattern observed in Fig. 2c; the nano-structure of the agglomerated particles can hardly be observed. The SAED pattern indicates that the powder mixtures are almost nanostructured (partially amorphous) since there are sporadic diffraction spots with a halo pattern. As a result, the HRTEM result agreed well with the card’s XRD analysis (JCPDS79-2264).

Fig. 2
figure 2

HTEM micrographs of the nanostructured BFBT milled for 50 h (a) at 50 nm scale, b and c D-spacing and d SEAD pattern

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3.1.3 Fourier transform infrared (FTIR) spectra analysis

IR spectroscopy is measured to characterize the materials and determine the bond formation between various atoms found in the material [18]. FTIR spectrum constitutes the sample fingerprint containing a transmission peak that corresponds to the frequencies of vibrational bonds of the atoms existing in the material [18, 19]. Figure 3 represents the FTIR spectrum over a wide spectral range (400–1500 cm−1) of the as-mix and as-milled BFBT samples. From the FTIR spectra analysis, the characteristic peak displayed at the lowest wavenumber 418 cm−1 has been attributed to Ti–O–Ti vibrations. In addition, it is possible to generate a single Bi–O–Ti framework of bonds when BaTiO3 is introduced [18]. The peak around 442 cm−1 is attributed to Fe–O bending and stretching vibration. It is a characteristic vibration of the octahedral FeO6 complex group, which disappeared in BFBT milled after 50 and 75 h [19]. The transmission peak in area 505–555 cm−1 may be assigned to Bi–O–Bi and Bi–O stretching vibrations of [BiO6] octahedral structural units [20]. Besides, the absorption band at 539 cm − 1 is due to BaTiO3 [21]. However, there is a peak shift from 535 cm−1 (0 h BFBT sample) to 527 cm−1 (50 h BFBT sample) assigned to the octahedral distortion mode [O6] of BaTiO3 [18]. Moreover, the bands shift towards higher wavenumbers from 535 to 557 cm−1 in samples (0 h BFBT sample) to (75 h BFBT sample) may be because of the decrease in Bi2O3 content varying the local symmetry in [BiO6] polyhedral [18]. The bands appearing at 843–856 cm−1 can be assigned as symmetric stretching vibrations of links Bi–O related with polyhedral and pyramidal units [BiO3] [22]. Further, the bands at the vibrational frequency 1383–1445 cm−1 are assigned to O–H bending and stretching vibrations could be due to moisture absorption by nanostructured BaTiO3[23].

Fig. 3
figure 3

FTIR spectra of the as-mix BFBT and the as-milled BFBT samples at various milling times

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3.2 Dielectric properties

Temperature-dependent of dielectric permitivitty (ε') and loss tangent (tanδ) for the nanostructured BFBT pellet milled for 50 h is shown in Fig. 4. It exhibits strong frequency dispersion of dielectric constant with a broad peak below and above the dielectric maximum temperature (Tm) that shifts towards higher temperatures from 545 to 585 K with an increasing frequency from 0.5 kHz to 1 MHz. The dielectric loss exhibits the same anomaly in the temperature range of 575–585 K as well, suggesting the relaxor nature of the nanostructured BFBT sample. As understood from the figure, Tm is approximately 560 K, which is related to the ferroelectric–paraelectric phase transition at the Curie point. In our results, it is clear that there is a shift in the ferroelectric Curie temperature Tc of the BFBT sample from the usual transition temperature of BaTiO3 and BiFeO3. We think that the explanation of this shift is owing to the substitution that occurred between Fe3+ ions and Ti4+ ions at the B site in BaTiO3 and BiFeO3 crystal lattice throughout the mechanical milling process [17]. The dielectric properties of RFEs are widely believed to be governed by the dynamics of the polar nanoregions (PNRs). Higher concentration of grain boundaries and associated internal stresses in a nanostructured sample could influence the dynamics of these PNRs. In this respect, the enhancement in the relaxor behavior and dielectric properties is due to the residual crystalline phase and the reduction in grain size as a result of domain refinement and weakening of the long-range polar ordering in the fine-grained BFBT sample [24].

Fig. 4
figure 4

Temperature-dependence of dielectric permitivitty and loss (tanδ) for the nanostructured BFBT sample milled for 50 h at selected frequencies in the range 500 Hz–1 MHz

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The plot of 1/ε' against temperature is shown in Fig. 5a. It is seen that at 500 kHz, a linear region exceeds 554 K referring that the Curie–Weiss is obeyed at all temperatures exceeding the Curie temperature (TC). For normal ferroelectrics “displacive ones”, the dielectric constant data could be fitted via the Curie–Weiss relation when the temperature exceeds the Curie temperature TC [5]:

$$\frac{1}{{\varepsilon }^{^{\prime}}}= \frac{T-{T}_{\mathrm{C}}}{C} {T >T}_{\mathrm{C}},$$
(1)

where the Curie constant is defined as C which determines the ferroelectric transition nature, the Curie–Weiss temperature is TCW While the intersection that extrapolated of the high-temperature region of the graph with the temperature axis is To as plotted in Fig. 5a. But for relaxors, ε′ couldn’t obey the Curie–Weiss over TC till the Curie–Weiss temperature (TCW) is achieved through the dashed line [25]. C, TCW, and T0 are specified from fitting as 7.6 × 103, 558 K and 530 K, respectively. To determine the order of the ferroelectric transition, we obtain the ratio, n, of the two slopes, below and above TC, and that is (n = − 2.98), indicating a second-order phase transition. The diffuseness of the relaxor phase transition could be defined by using the modified Curie–Weiss law [26]:

$$\frac{1}{{\varepsilon }^{^{\prime}}}- \frac{1}{{\varepsilon }_{m}^{^{\prime}}}=\frac{{\left(T-{T}_{\mathrm{m}}\right)}^{\Upsilon}}{C},$$
(2)

where ε'm, Tm and \(\gamma \) refer to the maximum value of the dielectric constant at the transition temperature, the temperature corresponding to the dielectric maxima, and the diffusion coefficient, respectively. It is reported that the value of Υ between 1 “a normal ferroelectric” and 2 “relaxor ferroelectric” [5, 26]. The graph of ln (1/ε′ − 1/ε′m) against ln (T – Tm) curve at 1 kHz is shown in Fig. 5b, Υ = 1.78 is determined by linear fitting to the experimental data. This result implies that the nanostructured BFBT milled for 50 h sample exhibits the most obvious relaxor behavior.

Fig. 5
figure 5

a Fitting of the temperature dependence of 1/ε′ at 500 Hz, Curie–Weiss law is obeyed above TC, b shows modified Curie–Weiss law: the plot of ln(1/ε′ − 1/ε′m) vs. ln(T – Tm) curve at 1 kHz of the nanostructured BFBT sample milled for 50 h

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On the other hand, relaxor ferroelectrics are confirmed by linear fitting the Tm of the dielectric spectra with the Vogel–Fulcher (VF) relationship [3], as described in Fig. 6.

$$f={f}_{0}\mathrm{exp}\left[\frac{- {E}_{a}}{{k}_{B}({T}_{m}- {T}_{VF})}\right],$$
(3)

Which explains a process of freezing polar nanoregions (PNRs), where f denotes the experimental frequency, f0 denotes the pre-exponential factor, kB denotes the Boltzmann constant, Ea denotes the activation energy, and TVF denotes the characteristic Vogel–Fulcher freezing temperature. We found fo = 2.97 × 109 Hz, TVF = 447.5 K and Ea = 0.129 eV. The determined parameters confirm the presence of relaxor behavior in the as-milled nanostructured BFBT sample for 50 h. The temperature dependence of the real part of the σac conductivity for the nanostructured BFBT milled for 50 h at different frequencies is displayed in Fig. 7. The ac conductivity is equivalent to being nearly constant until 500 K, later increasing with increasing temperature. At the higher temperatures, σac exhibited a broad peak around 550 K, attributable to the relaxor ferroelectric phase transition. The peak in the spectra moved to the higher temperature region when the frequency increased and agrees well with the movement in the dielectric Tm, confirming the frequency-dependence phase transition temperature and relaxor nature of the sample. To find the nature of conductivity in the sample, we have plotted the real part of the ac conductivity and the angular frequency (ω) at different temperatures for the nanostructured BFBT milled for 50 h as shown in Fig. 8. The ac conductivity can be calculated using the following relation [27]:

$${\sigma }_{ac}={\varepsilon }^{^{\prime}}\,{ \varepsilon }_{0 }\,\omega \,\mathrm{tan}\delta ,$$
(4)

where ω (= 2\(\pi \)f) is the angular frequency and \({ \varepsilon }_{0}\) is the free space permittivity. σac observes a frequency-dependent conductivity up to 500 K over a wide range of frequencies. However, when the temperature increases, a flat plateau region below 1 MHz frequency has been observed, i.e., its behavior is nearly frequency independent. These regions depict the long-range order mobility of free mobile charge carriers i.e., space charge carriers and are usually observed in disordered systems. The investigated material followed the power law [4]:

$${\sigma }_{ac}={{\sigma }_{dc}+A\omega }^{S} \,0\hspace{0.17em}<\hspace{0.17em}S\hspace{0.17em}<\hspace{0.17em}1,$$
(5)

Where \({\sigma }_{dc}\) is the dc conductivity, which is referred to the long-range translational hopping, A is an empirical constant, which determines the strength of the polarisability, and S is the exponent that varies from 0 to 1, determining the interaction degree between mobile ions and the lattice around them. The experimental data were fitted with the law of power, with S representing the highly frequency-dependent conductivity ranging from 0.2 to 0.9. The temperature-dependent variation of the fitted parameter of exponents S is shown in Fig. 8 (inset). The S value exhibits a continuous decrease with the measured temperature for the nanostructured BFBT milled for 50 h. The characteristics of exponent S depend on frequency and temperature. The charge transport mechanism accurate nature could be proved by determining the variation of “S”. Depending on the correlated barrier hopping (CBH) model, the “S” value only reduces progressively with the increase in temperature. The current work’s experimental data are consistent with the CBH model’s prospects [27, 28].

Fig. 6
figure 6

Temperature of dielectric maximum (Tm) as a function of ln (f) for the nanostructured BFBT sample milled for 50 h (symbols represent the experimental data and the solid line is the fitting to the Vogel–Fulcher relation)

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Fig. 7
figure 7

The temperature dependence of σac conductivity for the nanostructured BFBT sample milled for 50 h at different frequencies

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Fig. 8
figure 8

AC conductivity as a function of frequency for the nanostructured BFBT sample milled for 50 h at different T regions fitted with a double power law. Inset: variation with temperature of exponent S

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3.3 Ferroelectric and ferromagnetic properties

Figure 9 displays a temperature-dependent polarization hysteresis loop of the nanostructured BFBT milled for 50 h. The hysteresis shape is slim, indicating relaxor ferroelectrics (RFEs) [3]. As illustrated by the P-E loops, they become slimmer and slimmer as the temperature increases, which is more favorable for energy storage applications [29]. The energy storage properties of nanostructured BFBT milled for 50 h are measured as seen in the schematic diagram in Fig. 10. The energy loss density (Wloss), recoverable energy density (Wrec), and energy storage efficiency (η) are calculated through the integration of the P–E hysteresis loop as follows [5]:

$${W}_{\mathrm{loss}}= {\int }_{0}^{{P}_{\mathrm{m}}}EdP,$$
(6)
$${W}_{\mathrm{rec}}= {\int }_{{P}_{\mathrm{r}}}^{{P}_{\mathrm{m}}}EdP,$$
(7)
$$\eta = \frac{{W}_{\mathrm{rec}}}{{W}_{\mathrm{rec}}+{W}_{\mathrm{loss}} } \times 100,$$
(8)

where Pm points to the maximum polarization, while Pr and E denote the remnant polarization and the applied electric field, respectively. It is understood from the figure that at room temperature, the PNRs have a random orientation, which becomes ordered when the electric field is applied. As the temperature increases, PNRs are also ordered by the induced electric field, resulting in a lower remnant polarization. It is widely believed that the small Pr refers that the random orientation is often recovered after E is decreased to zero, which is evidence that the material has relaxor behavior [1, 5]. The temperature dependence of the recoverable energy density (\({W}_{\mathrm{rec}}\)) and efficiency (\(\eta \)) of the nanostructured BFBT milled for 50 h is shown in Fig. 11. It is illustrated that with increasing temperature, the energy storage density and efficiency increase. The reason for this is that after applying the temperature to the ferroelectric material, it relaxes and disorients the domains. This decreases the reversible part of the ferroelectric switching that eventually decreases the area under the hysteresis loop. So, the sample energy loss is reduced while simultaneously increasing efficiency. This high-temperature dependence polarization change exhibits a large potential that is appropriate for temperature-stable dielectrics and energy storage capacitor applications [3, 5]. It is shown that \(\upeta \) increases to reach 89.3% at 360 K. Accordingly, the nanostructured BFBT milled for 50 h is considered to be a promising material for energy storage applications. The ferroelectric parameters for nanostructured BFBT milled for 50 h at different temperatures are given at 220 V and listed in Table 1.

Fig. 9
figure 9

Temperature dependence of P-E hysteresis loop of the nanostructured BFBT sample milled for 50 h

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Fig. 10
figure 10

Schematic description of the energy storage mechanisms in relaxor ferroelectrics

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Fig. 11
figure 11

Recoverable energy density (Wrec) and energy storage efficiency (η) as a function of the temperature of the nanostructured BFBT sample milled for 50 h

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Table 1 The ferroelectric parameters for the nanostructured BFBT sample milled for 50 h at different temperatures
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The temperature dependence of the inverse of DC molar magnetic susceptibility (1/χm) measured at 1070 Oe for the nanostructured BFBT milled for 50 h is plotted in Fig. 12. It is observed that as the temperature increases, 1/χm increases up to the Néel temperature (TN). This increase in 1/χm is because of the thermal energy that is responsible for the distribution of the oriented spins in the direction of the field. After reaching TN, 1/χm decreases sharply with temperature. Therefore, the Curie–Weiss law can be obeyed by extrapolating the straight line fit to the high-temperature region of the 1/χm data above TN, which is a typical characteristic of paramagnetic behavior. The Curie–Weiss law has been applied in the following form [30]:

$${\chi }_{\mathrm{m}}= \frac{C}{T-{\theta }_{C}}, {T >\theta }_{C}$$
(9)

Where C represents the Curie constant and is defined as the inverse of the slope of the straight line in the paramagnetic region, T is the absolute temperature, and θC is the Curie temperature. From the linear fitting of the paramagnetic region, C = 0.21 mol.Oe/emu.K, θC = 394 K. The positive θC value proves the existence of ferromagnetic interaction between spins [16]. The effective magnetic moment (μeff) value can be calculated from the following relation [30]:

Fig. 12
figure 12

The temperature dependence of the inverse of DC magnetic susceptibility (1/χm) measured at 1070 Oe (The inset shows the dependence of the derivative of magnetization (dM/dT) on the Temperature) of the nanostructured BFBT sample milled for 50 h

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$${\mu }_{\mathrm{eff}}= 2.83 \sqrt{C}.$$
(10)

The calculated μeff is 1.29 μB based on the determined C. The Néel temperature was determined from the plot of dM/dT versus temperature as illustrated in Fig. 12 (inset) and obtained to be 394 K which is smaller than that reported for pure BiFeO3. Finally, the magnetization versus an applied magnetic field M-H of the nanostructured BFBT milled for 50 h at room temperature is measured as shown in Fig. 13. The M-H data represent features of a weak ferromagnetic nature with remnant magnetization (Mr) of 24.33 × 10−3 emu/g, saturation magnetization (Ms) of 0.315 emu/g, and coercive fields (Hc) of 204.68 Oe [31]. Plastic deformation and nanostructure formation after mechanical treatment play a significant role in the magnetic properties [32]. This result agreeing with previously published literature, where the distributions of Ti4+ and Fe3+ ions in the octahedral positions influence the magnetization [33].

Fig. 13
figure 13

M-H hysteresis loop at room temperature for the nanostructured BFBT sample milled for 50 h (The inset represents the enlarged M-H loop)

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4 Conclusion

The present paper demonstrates that the synthesis of nanostructured 0.30BaTiO3–0.52Bi2O3–0.18Fe2O3 mol% (BFBT) ball milled for 50 h shows a frequency-dependent dielectric maximum. The result indicated by the XRD patterns and HRTEM confirms the fine structure formation in the sample. The average particle size from HRTEM is measured to be 27.86 nm. The dielectric permittivity of Tm follows linear VF relations, referring to the relaxor nature of the sample. The ac conductivity exhibits frequency-dependence near Tm. The present work’s experimental data is consistent with the prospects of the CBH model. The slim P-E hysteresis loop and a room-temperature weak ferromagnetism confirm the multiferroic relaxor nature. It is shown that energy storage efficiency (η) increases to reach 89.3% at 360 K. In this work, the mechanical milling technique is confirmed to be effective in enhancing the relaxor multiferroic properties of nanostructured BFBT for environmentally friendly potential magnetoelectric applications.