Figure 1 shows the XRD patterns of BFO and xBST-(1 - x)BFGO (x = 0.00, 0.10, 0.20 and 0.25) ceramics sintered at 900 and 950 ° C, respectively. Two impurity peaks indicated by the symbols ‘ * ’ and ‘ Δ ’ have been observed in the XRD pattern of BFO (Fig. 1a) because of the slow phase formation kinetics of BFO. The impurity peaks might be due to the formation of Bi₂Fe₄O₉ and Bi₂₄Fe₂O₃₉. The XRD patterns of BFO exhibit rhombohedrally distorted perovskite structure, and the lattice parameters are found to be a = b=5.58 Å and c = 13.65 Å . On the other hand, no impurity phase is seen in the XRD patterns of xBST-(1 - x)BFGO ceramics (Fig. 1b), and no peak of BST is found which indicates that BST has mixed with BFGO properly. As shown in Fig. 1b, XRD peaks shift toward lower angle side with increasing BST, but there is no indication of change in basic structure of perovskite. Figure 1c shows the enlarge view of (104) and (110) peaks for all compositions. The splitting of (110) peak is observed for all compositions showing a characteristic of rhombohedral structure. The ionic radius of Gd is larger than Fe, but the radius of Ti is similar to Fe. As a result, the mixed occupation of Ba and Sr in A-site and Gd and Ti in B-site makes a random displacement. The random displacement can cause different 2θ reflection and lead to the splitting effect. All peaks of the XRD pattern have been indexed following rhombohedrally distorted perovskite structure which is equivalent to hexagonal symmetry [21]. The lattice parameters a andc are calculated by following the formula of hexagonal equivalent [21] and using the intense peaks (012) and (110). The results are in good agreement with the phase diagram of xBaTiO₃-(1 - x)BiFeO₃ [8, 21]. Kumar et al. [4] stated that in xBaTiO₃-(1 - x) BiFeO₃ solid solutions, the phase is rhombohedral when x ≤ 0.3 and is cubic when x ≥ 0.4 and finally to tetragonal for x ≥ 0.93. The lattice parameters a andc are listed in Table 1. It is observed that the value of a is increasing with increasing BST content. It is also observed that unit cell volume V is increasing with the addition of BST. Our results are consistent with the previous reports of BT-BFO system [21, 22].

Fig. 1

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	Fig. 1 XRD patterns of BFO sintered at 900 ° C a, various xBST-(1 - x)BFGO ceramics sintered at 950 ° C b and enlarged view of (104) and (110) peaks c

Table 1

Table 1

Table 1 Lattice parameters, unit cell volume (V), bulk density (ρ B), theoretical density (ρ th), tolerance factor (t), porosity (P) and average grain size (D) x a (Å ) c (Å ) V (Å 3) ρ B (g/cm3) ρ th (g/cm3) t P (%) D (μ m) 0.00 5.5457 13.7038 367 7.60 8.83 0.847 13.9 2.17 0.10 5.5790 13.6028 368 7.38 8.54 0.855 13.5 2.60 0.20 5.5859 13.5698 368 7.10 8.28 0.864 13.0 2.47 0.25 5.6001 13.7960 374 7.06 8.03 0.868 12.0 2.42

Table 1 Lattice parameters, unit cell volume (V), bulk density (ρ B), theoretical density (ρ th), tolerance factor (t), porosity (P) and average grain size (D)

As mentioned above that most of the intense peaks of XRD shift toward lower angle side, which indicates the increase in unit cell volume of the prepared solid solution on the addition of BST. This may also produce a distortion in the perovskite structure. The degree of distortion in the structure is calculated in terms of tolerance factor t as defined by the following equation [23]:

(1)

whereR_A, R_B and R_O are the ionic radii of A-site atom, B-site atom and oxygen, respectively. For the case of xBa_0.95Sr_0.05TiO₃-(1 - x)BiFe_0.9Gd_0.1O₃ compound, the above equation can be written as:

(2)

where R_Ba²⁺ , R_Sr²⁺ , R_Bi³⁺ , R_Ti⁴⁺ , R_Fe³⁺ , R_Gd³⁺ and R_O^2- are the ionic radii of Ba²⁺, Sr²⁺, Bi³⁺, Ti⁴⁺, Fe³⁺, Gd³⁺ and O^2-, respectively. The calculated value of t for xBST-(1 - x)BFGO is listed in Table 1. The value of t varies from 0.847 to 0.868, and it is obvious that the value of t is increasing, which indicates that the perovskite structure of xBST-(1 - x)BFGO becomes more stable with increasing BST content. Structure of BST is more stable than BFGO, hence the overall stability increases with increasing BST concentration.

Figure 2a shows the variation of bulk density (ρ _B) and theoretical density (ρ _th) of xBST-(1 - x)BFGO ceramics sintered at 950 ° C for 1 h as a function of BST content. It is seen that both ρ _B and ρ _th decrease with the addition of BST content. It is also appeared that the value of ρ _B for all the compositions is lower than that of the corresponding ρ _th. This may be due to the presence of some pores in the bulk samples which may be formed and developed during the samples preparation or sintering process (percentage of porosity is given in Table 1). The decrease in density with the increase in BST content may be attributed to the fact that the molecular weight of BST (230.70 g/mol) is lower than that of BFGO (322.96 g/mol). The addition of BST in BFGO leads to a variation in bonding and makes a change in inter-atomic distance and density.

Fig. 2

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	Fig. 2 Variation of ρ th and ρ Ba, and variation of V and < rA> b with BST content of various xBST-(1 - x)BFGO ceramics sintered at 950 ° C for 1 h

Figure 2b shows the variation of unit cell volume (V) and average ionic radius in A-site < r_A> of xBST-(1 - x)BFGO ceramics with BST content. The V and also < r_A> are found to increase with increasing BST content. The < r_A> = x × 0.95R_Ba²⁺ + x × 0.05R_Sr²⁺ + (1 - x) × R_Bi³⁺ , where R_Ba²⁺ = 1.35 Å , R_Sr²⁺ = 1.18 Å and R_Bi³⁺ = 1.08 Å , < r_A> in twelve-coordination (Shannon ionic radius) increases from 1.03 to 1.107 Å as BST increases. While in B-site, < r_B> = x × R_Ti⁴⁺ + (1 - x) × 0.9R_Fe³⁺ + (1 - x) × 0.1R_Gd³⁺ , where R_Ti⁴⁺ = 0.68 Å , R_Fe³⁺ = 0.64 Å and R_Gd³⁺ = 0.94 Å , < r_B> in twelve-coordination increases slightly with regard to < r_A> from 0.670 to 0.675 Å . Thus, the regular enhancement of the ionic radius in the A-site cation induces a larger A-site which causes an expansion in octahedral space of the perovskite structure and ultimately results in an increase in V.

Figure 3 shows the FESEM micrographs (FESEM) of xBST-(1 - x)BFGO ceramics. All the ceramics show dense microstructure. The average value of grain size (D) is determined using linear intercept technique. The values of D are found to be in the range of 2.17-2.60 μ m which suggests that BST has an influence on the modification of grain size of xBST-(1 - x)BFGO ceramics. The addition of BST in BFGO initially increases the grain size for x = 0.10 and then decreases for higher content of BST (for x > 0.10). Maximum value of grain size is found to be 2.60 μ m for 0.1BST-0.9BFGO samples. The considerable increase in grain size for x = 0.10 may be due to the uniform grain growth, as shown in Fig. 3b. With the higher addition of BST, the populations of smaller grains are dominating over the larger grains and decrease the uniformity. Such behavior of grain growth reflects the competition between the driving force of grain boundary movement and the retarding force exerted by pores because grain growth is closely related to the grain boundary mobility. When the driving force of the grain boundary in each grain is homogeneous, the sintered body attains a uniform grain, whereas the non-uniform grain growth occurs due to the existence of inhomogeneous driving force.

Fig. 3

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	Fig. 3 FESEM micrographs of various xBST-(1 - x)BFGO ceramics sintered at 950 ° C for 1 h: ax = 0.00; bx = 0.10; cx = 0.20; dx = 0.25

3.4.1 Magnetic Hysteresis Loop Measurement

Figure 4 shows the magnetic hysteresis loops (M-H curve) of xBST-(1 - x)BFGO ceramics (sintered at 950 ° C) measured at room temperature. Significant enhancement of magnetization is observed due to the addition of BST in BFGO although all the samples are not saturated even at an applied magnetic field of 20 kOe (1592 kA/m). The values of remnant magnetization (M_r) and coercivity (H_c) have been determined. It is noticed that the values of M_r for various xBST-(1 - x)BFGO solid solution increase with the increasing BST content. The M_r is found to be 0.011-0.080 emu/g which indicates the enhancement of M_r due to the addition of BST. The highest value of M_r is found to be 0.080 emu/g for 0.25BST-0.75BFGO solid solution. The values of M_r are relatively very small as compared to other ferromagnetic compounds. Unsaturated M-H behavior and small remnant magnetization is the indication of antiferromagnetic with weak ferromagnetism [24]. The obtained result is better than that of other previously reported BFO-based materials [21, 25]. The calculated values of M_r and H_c are listed in Table 2. It is known that the H_c depends on anisotropy constant, average grain size, domain wall energy and several other factors. We calculated the anisotropy constant (K) using the following relation [26]: , where K is the anisotropy constant and M_s is the saturation magnetization. The values of K with increasing BST concentration is presented in Table 2. In our results, both H_c and K decrease with increasing BST content, and this indicates the enhancement of magnetic properties.

Fig. 4

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	Fig. 4 M-H hysteresis loops of various xBST-(1 - x)BFGO ceramics sintered at 950 ° C for 1 h: ax = 0.00; bx = 0.10; cx = 0.20; dx = 0.25

Table 2

Table 2

Table 2 Remnant magnetization (Mr), coercivity (Hc) and magnetic anisotropy (K) of xBST-(1 - x)BFGO ceramics at room temperature x Mr (emu/g) Hc (kOe) K (kOe emu/g) 0.00 0.014 0.791 0.160 0.10 0.011 1.106 0.175 0.20 0.012 0.012 0.003 0.25 0.080 0.005 0.002

Table 2 Remnant magnetization (Mr), coercivity (Hc) and magnetic anisotropy (K) of xBST-(1 - x)BFGO ceramics at room temperature

BFO is basically antiferromagnetic in magnetic order with negligible magnetic moment due to its spiral spin structure [27]. By the addition of elements in both A- and B-sites, the spiral spin order can be suppressed, and as a result the latent magnetization is released [28]. The enhancement of magnetization in xBST-(1 - x)BFGO may be attributed due to the change of canting angle in Fe-O-Fe by a lattice distortion made by Ti⁴⁺, Ba²⁺, Sr²⁺ and Gd³⁺ ions and results a suppression of the spiral spin structure of BFGO [27, 29]. The addition of BST in BFGO only suppresses the spiral spin structure and could not destroy it completely.

3.4.2 Complex Initial Permeability

Complex initial permeability provides the information which is useful in understanding the mechanism of controlling the dynamic properties and also the adaptability of magnetic materials for high-frequency uses. The complex initial permeability is given by μ _i^* = - , where and are the real and imaginary parts of initial permeability, respectively. The describes the energy that stores in the system, and it represents the component of magnetic induction B in phase with the alternating magnetic field H. The describes the dissipation of energy expressing the component of B out of phase with H. Figure 5a shows the variation of for xBST-(1 - x)BFGO ceramics with frequency measured at room temperature. It is noticed that the value of for all the compositions remains almost constant up to a certain frequency and then decreases which indicates the frequency stability of for the studied compositions. It is also observed that the value of is increasing with BST content. Initial permeability in magnetic materials originates because of the spin rotation and domain wall motion. So, the initial permeability can be written as [30]: = 1+χ _spin + χ _dw, where and denote the magnetic susceptibility of spin and domain wall motion, respectively, and where γ is the domain wall energy. More precisely, can be explained using the Globus-Duplex relation [31]: , where K is the anisotropy constant. Hence, it is obvious that is inversely proportional to K as well as H_c. Figure 5b presents the variation of and H_c of various xBST-(1 - x) BFGO ceramics sintered at 950 ° C as a function of BST content. From this figure, it is found that increases, whereas H_c decreases with the increasing of BST content.

Fig. 5

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	Fig. 5 Variation of with frequency a, and Hc with BST content b and tanδ with frequency c of various xBST-(1 - x)BFGO ceramics sintered at 950 ° C for 1 h

The value of is found to decrease with frequency because of the fact that at higher frequencies, pinning points are found to originate at the surface of samples from the impurities of grains and intragranular pores. This phenomenon in turn obstructs the motion of spin and domain walls. Hence, the contribution of spin and domain wall motion toward to the increment of permeability decreases [31]. On the other hand, it is difficult to explain compositional dependence of for the case of polycrystalline materials because is sensitive to many factors. The value of depends on stoichiometry, composition, average grain size, impurity contents, coercivity, porosity, etc. As mentioned above in BFO, oxygen vacancies were created due to the reduction of Fe³⁺ ions and volatilization of Bi³⁺ ions during sintering process. The oxygen vacancies may affect domain wall motion by the formation of mechanical barrier against the domain wall [32]. As mentioned above, concentration of oxygen vacancy decreases with the increase in BST content, so that the motion of domain wall is facilitated in xBST-(1 - x)BFGO ceramics with the addition of BST, which results in the increase in .

The plot of magnetic loss, tanδ (= / ), as a function of frequency is shown in Fig. 5c. It is observed that tanδ decreases with increasing frequency and remains constant after 1 MHz. The decrease in tanδ with frequency might be attributed to spin inversion losses as suggested by Gama and Rezende [33] or due to the reduction of activity of space charges because space charges cannot follow the rapid variation of the alternating field at high frequency. The values of tanδ for all the compositions are decreased very significantly, and it may be due to the reduction of imperfection after the addition of BST in BFGO.

3.5.1 Dielectric Properties at Room Temperature

Figure 6a shows the variation of dielectric constant (ε ' ) with frequency at room temperature for xBST-(1 - x)BFGO ceramics sintered at 950 ° C for 1 h. It is observed that at low frequency, ε ' has high value and then decreases gradually with the increase in frequency. This type variation of ε ' is explained on the basis of polarization mechanisms. Contribution of different polarization mechanisms varies at different frequencies. The total polarizability may be separated into three parts: dipolar, ionic and electronic. For the case of polycrystalline compounds, there is also an interfacial or space charge polarization. The space charge polarization may originate in the prepared material from the point defects during sintering process. At low frequency, all the different types of polarization are in action, but major contribution to ε ' comes from space charge polarization. The space charges are able to follow the variation of the applied electric field at low frequency. They do not get enough time to build up and cannot follow the variation of field frequency at high frequency. This results in less contribution of space charges at high frequency toward ε ' and decrease in ε '. On the other hand, the dipolar and the ionic contributions are also small at high frequency due to the inertia of the molecules and ions.

Fig. 6

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	Fig. 6 Variation of ε ' a and tanδ b with frequency, and dependence of ε ' c and tanδ d on BST content of various xBST-(1 - x)BFGO ceramics sintered at 950 ° C for 1 h

Figure 6b shows the variation of dielectric loss (tanδ ) with frequency. The value of tanδ is high at low-frequency region, and then, it decreases significantly with the rise in frequency. An increasing trend at high frequency is also noticed for the case of x = 0.20 and x = 0.25 samples which may originate from polarization relaxation [34]. Dependence of ε ' and tanδ of xBST-(1 - x)BFGO ceramics on the concentration of BST at different frequency is shown in Fig. 6c and d. It is observed that the value of ε ' for all compositions is increasing with BST concentration. It is also noticed that BST added samples have lower value of tanδ than that of BFGO and tanδ is decreasing with the increase in BST. The reduction in tanδ is found to be 62.5% than that of BFGO. In BFO, a charge imbalance is created due to the volatilization of Bi³⁺ and the reduction of Fe³⁺ ions during the sintering process [35] and thus creates oxygen vacancies. These oxygen vacancies are responsible for the reduction of dielectric properties in BFO. Addition of BST in Gd-doped BFO compensate Bi³⁺ by the substitution of Ba²⁺ and Sr²⁺ on A-site and Fe³⁺ by the substitution of Ti⁴⁺ or Gd³⁺ ions on B-site as illustrated in:

(3)

Hence, oxygen vacancies are reduced when BST is mixed with BFGO. The decrement of the oxygen vacancies results in improved dielectric properties.

3.5.2 Temperature-Dependent Dielectric Properties

Figure 7a shows the variation ofε /with temperature for various xBST-(1 - x)BFGO ceramics measured at 1 kHz. Two different peaks at around 100 and 370 ° C are noticed for the case of pure BFGO ceramics. The peak at lower temperature disappears due to the addition of BST in BFGO samples. An irregularity in temperature-dependent dielectric measurements may originate from different factors such as structural transitions, magnetoelectric effect which is the intrinsic properties of multiferroic materials or the enhancement of charge carrier’ s movement to the interface for the case of polycrystalline materials [36]. The peak at around 100 ° C may be originated due to inhomogeneous mixing of constituent elements in BFGO which was also mentioned in the previous report [21]. The temperatures for which maximum dielectric constant was obtained (T_m) are calculated for all the compositions. These values are about 360, 345, 320 and 305 ° C for x = 0.00, 0.10, 0.20 and 0.25, respectively, at 1 kHz, and the T_m shows decreasing trend as x content increases. This may be attributed to the introduction of Sr²⁺ ions at A-site in the prepared solid solutions because Sr²⁺ ions have an important role to reduce the phase transition temperature [37]. However, the maximum value of dielectric constant () at T_m is found to increase from 3395 for x = 0.10 to 7158 for x = 0.25 at 1 kHz which is higher than that of BFO (≈ 40) and lower than that of BST (≈ 11, 880) [38]. This suggests that BST has an influence on the improvement of ε ' in xBST-(1 - x)BFGO ceramics.

Fig. 7

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	Fig. 7 Variation of ε ' a and tanδ b with temperature of xBST-(1 - x)BFGO ceramics at 1 kHz frequency sintered at 950 ° C for 1 h

Figure 7b shows the variation of tanδ with temperature for all the compositions. The value of tanδ is found increasing slowly with the increase in temperature, and after a certain temperature (≈ 250 ° C), the loss is found increasing rapidly. The existences of space charges in polycrystalline materials become more active at low frequency and also at higher temperature. The thermally activated space charges are responsible for the increment of tanδ at the higher temperature.

3.6.1 Complex Impedance Analysis

Complex impedance spectroscopy is a powerful technique to characterize many of the electrical properties of polycrystalline materials and their interfaces. It may be used to explain the dynamics of mobile or bound charges in the grain or grain boundaries. The impedance behavior is described by the classical model known as Debye model, and it is written in the form:

(4)

where

(5)

(6)

Figure 8a shows the variation of real part of impedance (Z' ) with frequency at room temperature of various xBST-(1 - x)BFGO ceramics with x = 0.00, 0.10, 0.20 and 0.25. As shown in Fig. 8a, the value of Z' gradually decreases with increasing frequency up to a certain limiting frequency (≈ 5 kHz), and then, it becomes almost frequency-independent. The decrease in Z' indicates that the conduction is increasing with frequency. The higher values of Z' at lower frequencies mean that the polarization in the ceramics is larger because at lower frequency all kinds of polarization are present. It is also observed that the value of Z' for all compositions coincides at higher frequencies. The merger of Z' at higher frequencies indicates possible release of space charge polarization [39]. Variation of Z' with frequency at different temperatures (150-300 ° C) for the composition x = 0.25 is shown in Fig. 8b. The values of Z' decrease in the low-frequency region (up to a certain frequency ≈ 100 kHz) with temperature, and then, they appear to merge at the higher frequency. This indicates that the frequency at which the space charge becomes free also depends upon the temperatures.

Fig. 8

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	Fig. 8 Variation of Z′ with frequency at room temperature for various xBST-(1 - x)BFGO ceramics a, and at different temperatures for 0.25BST-0.75BFGO ceramics b

Figure 9a shows the variation of imaginary part of impedance (Z'' ) with frequency at various temperatures (150-300 ° C) for the composition with x = 0.25. The variation of Z'' with frequency exhibits a peak for all samples. At higher frequency, it also decreases with increasing temperature due to the reduction of loss in the resistive part of the sample. This plot is useful for the determination of relaxation frequency (f_r) at different temperatures. Relaxation occurs in resistive materials due to the presence of immobile species at lower temperature and defects at higher temperature. As shown in Fig. 9a, the relaxation peak is found to shift toward higher frequency with increasing temperature. The values of f_r have been determined from the frequency versus Z'' curves and listed in Table 3. The variation of f_r with temperature follow the Arrhenius relation: f_r = f_oexp(-E_a/K_BT), where f_o is the pre-exponential factor, K_B is the Boltzmann constant, E_a is the activation energy for the hopping mechanism and T is the absolute temperature. Figure 9b presents the Arrhenius plot of f_r for the impedance spectra. The values of E_a have been determined from the slope of the straight line and found to be 0.240 eV for the temperature region 150-300 ° C.

Fig. 9

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	Fig. 9 Variation of Z' ' with frequency a, and Arrhenius plot of frb for 0.25BST-0.75BFGO composition at different temperatures

Table 3

Table 3

Table 3 Values for grain resistance (Rg), grain boundary resistance (Rgb), relaxation frequency (fr), grain capacitance (Cg) and grain boundary capacitance (Cgb) of 0.25BST-0.75BFGO Temperature (° C) Rg (kΩ ) Rgb (kΩ ) fr (kHz) Cg (nF) Cgb (nF) 50 1060 - 1.09 0.14 - 100 1250 - 1.09 0.12 - 150 55 50 6.63 0.44 0.48 200 25 15 17.00 0.37 0.62 225 9.50 4.00 40.27 0.42 0.98 250 3.80 2.20 88.22 0.48 0.82 275 1.90 0.90 178.70 0.47 0.99 300 1.05 0.48 309.41 0.49 1.07

Table 3 Values for grain resistance (Rg), grain boundary resistance (Rgb), relaxation frequency (fr), grain capacitance (Cg) and grain boundary capacitance (Cgb) of 0.25BST-0.75BFGO

Polycrystalline ceramics consist of grains and grain boundaries. Grain and grain boundary contributions to the electrical properties of dielectric materials are normally analyzed by the complex impedance spectroscopy following brick-layer model [40]. For this, the sample is represented by an equivalent circuit which consists of three parallel RC circuits connected in series. These RC elements correspond to grains, grain boundaries and electrode-specimen interface in order of decreasing frequency. Each RC element of the equivalent circuit gives rise to a semicircle. A single semicircle in Z' versus Z'' plot indicates grain effects, and the second semicircle indicates the presence of grain boundary effects. If a third semicircle is present, it indicates the electrode effects. Figure 10a shows the plot of Z' versus Z'' (Cole-Cole plot) for different compositions of xBST-(1 - x)BFGO taken over a wide range of frequencies (20 Hz-100 MHz) at room temperature. All the compositions exhibit a single semicircular arc starting from the origin. The absence of second semicircle in the complex impedance plots indicates that the materials have only grain effect to the conductivity mechanism at room temperature. It is also realized that patterns of the semicircular arc are changing with composition which implies a modification of the resistance/reactance ratio with the changing of BST content in the ceramics. It is obvious from the figure that the diameter of the arc increases with increasing BST which suggests that the value of grain resistance (R_g) is increasing with increasing BST content. Figure 10b-i shows the Cole-Cole plot (Z' versus Z'' plot) of 0.25BST-0.75BFGO ceramic at different temperatures (50-300 ° C). A single semicircular arc is found to observe up to 100 ° C. With increasing temperature, the arc shifts toward the higher value of Z' axis. Above 100 ° C, a complete semicircle is observed on the high-frequency side and a small tail of semicircular arc on the low-frequency side. The second semicircular arc gradually spreads with further increasing temperature. The high-frequency semicircle indicates the grain property of the materials, while the low-frequency semicircular arc presents the effect for grain boundaries. The values of grain resistance (R_g) and grain boundary resistance (R_gb) have been calculated from the intercepts on the Z' axis. The grain capacitance (C_g) and grain boundary capacitance (C_gb) have been also calculated using the relations: C_g = 1/(R_gω _g) = 1/(2π f_rR_g) and C_gb = 1/(R_gbω _gb) = 1/(2π f_rR_gb). The values of these parameters are listed in Table 3.

The electrical conductivity (σ ) is a thermally activated process and important to describe electrical behavior of a material. In the present study, the DC conductivity (σ _dc) is calculated using the relation σ = t/(AR), where R is the resistance of the compound (resistance of grain or grain boundary). The variation of logσ versus 10³/T for both grain and grain boundary is shown in Fig. 10j. The nature of the variation for both R_g and R_gb is linear and obeys the Arrhenius relation: σ _dc = σ _oexp (-E_a/K_BT), where the symbols have their using meanings. The values of E_a have been calculated from the slope of the logσ versus 10³/T curves. For grains, the value of E_a is found to be 0.16 eV in the temperature range 50-150 ° C and 0.33 eV in the temperature range 200-300 ° C. For grain boundaries, the value of E_a is found to be 0.29 eV in the temperature range 150-300 ° C. It is observed from the Fig. 10j that in both the cases (grain and grain boundary), the σ _dc increases with increasing temperature. From Table 3 it is also noticed that both R_g and R_gb decrease with the increase in temperature. It follows typical semiconductor behavior of the material with the characteristics of negative temperature coefficient of resistance (NTCR).

Fig. 10

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	Fig. 10 a Cole-Cole plots of Z′ and Z′ ′ at room temperature for different compositions of xBST-(1 - x)BFGO ceramics; b-i Cole-Cole plots for 0.25BST-0.75BFGO composition at different temperatures; j Arrhenius plots of grain and grain boundary conductivity for 0.25BST-0.75BFGO composition

3.6.2 Complex Modulus Analysis

The complex modulus spectroscopy is a very convenient tool to determine, analyze and interpret the electric transport properties in the materials having the smallest capacitance. The modulus spectra are particularly useful for separating spectral components of the materials having similar resistance but different capacitance. The complex electric modulus is represented by the following expression:

(7)

whereM' = ω C_oZ'' and M'' = ω C_oZ' , and the symbols have their usual meanings.

The variation of real part of modulus (M' ) with frequency at various temperatures (150-300 ° C) for x = 0.25 is shown in Fig. 11a. The value of M' is found to be very low (nearly zero) at low frequencies, and it increases with the increase in frequency. This continuous dispersion on increasing frequency may be contributed to the conduction phenomena due to short-range mobility of charge carriers. It is possibly related to the lack of restoring force leading the mobility of the charge carriers under the action of an induced electric field [41].

Fig. 11

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	Fig. 11 Variation of M′ a and M'' b with frequency at various temperatures, Arrhenius plot of fmaxc, and M′ versus M′ ′ curves at different temperatures d for 0.25BST-0.75BFGO composition sintered at 950 ° C for 1 h

Figure 11b shows the variation of imaginary part of electric modulus (M'' ) for x = 0.25 with frequency at different temperatures (150-300 ° C). It exhibits a peak at the frequency above 10³ Hz. These peaks indicate the transition from long-range to short-range mobility of charge carriers with the increase in frequency. The frequency region below the peak maximum determines the range in which charge carriers are mobile over long distance, whereas the region above peak maxima establishes the charge carriers are confined to potential well. It is also observed that the peaks are shifting toward higher-frequency side with the increase in temperature. This behavior indicates that the dielectric relaxation is thermally activated in this compound in which hopping mechanism of charge carriers dominate intrinsically. This type of effect has been observed in the modulus spectra of previous report [42]. The observed asymmetry in peak broadening indicates the spread of relaxation with different time constants, and hence, the relaxation in the material is of non-Debye type. Value of M'' at lower frequency is found to be low which indicates that the electrode polarization makes a negligible contribution to the material. The value of frequency at maximum M'' (f_max) has been determined from the frequency versus M'' curves. The temperature dependence of f_max curve is shown in Fig. 11c which follows the Arrhenius relation: f = f_oexp(-E_a/K_BT), where the symbols have their using meanings. The value of E_a has been determined from the slope of the Arrhenius plot and found to be 0.246 eV which is very close to the values of E_a obtained from Z'' spectra. The closer value of the activation energies obtained from the impedance spectra and the analysis of modulus spectra suggests that the mobility of charge carrier in the compound is due to the same mechanism.

In order to avoid the uncertainty arising out of the presence of grain/grain boundary effect at different temperatures, the impedance data are re-plotted in the modulus formula. Complex impedance spectra give more emphasis to element with larger resistance, whereas complex electric modulus plots highlight those with smaller capacitance. The modulus plots of M' versus M'' at various temperatures (150-300 ° C) for x = 0.25 are shown in Fig. 11d. The modulus spectra show a change in its shape with the rise in temperature and suggest a probable change in the values of the capacitance as a function of the temperature. It is worth noting that at low frequencies, data exhibit a common origin almost identical with the origin of the graph. The coincidence of the beginning of the semicircles with the origin of the graph is a clear indication that no other relaxation process is present at lower frequencies in the studied material.

Figure 12 shows the frequency response of Z'' and M'' for x = 0.25 at 250 and 300 ° C. It is seen that the peaks of Z'' and M'' for both cases do not coincide which indicates that non-Debye-type relaxation is present in this compound. For a Debye-type relaxation, the peaks of both impedance and modulus plots should coincide. Any departure from this suggests non-Debye-type relaxation. Moreover, the frequency versus Z'' and M'' combined plots are used to distinguish whether a particular relaxation process in the material is due to the long-range movement of the charge carriers or the short-range movement of the charge carriers. The non-coincidence of the peaks corresponding to Z'' and M'' suggests that the relaxation is due to the short-range movement of charge carriers which was also observed in M' versus frequency spectra.

Fig. 12

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	Fig. 12 Frequency responses of Z′ ′ and M′ ′ at 250 ° C a and 300 ° C b for 0.25BST-0.75BFGO composition sintered at 950 ° C for 1 h