1 Introduction

According to their behavior, the ferroelectric materials with perovskite-like structure may be divided into different classes depending on whether they are classical or relaxor ferroelectrics [1]. Relaxor ferroelectrics are characterized by their broad dielectric transition, known as diffuse phase transition with frequency dispersion [2, 3]. Many investigations have been devoted to the study of the relaxation parameters in order to achieve their use in satisfactory capacitors and actuators [4]. However, most typical relaxors are lead-based perovskite compounds, which are environmentally unfriendly because of the toxicity of lead [5, 6]. Up now, current research is oriented to environment friendly applications using lead-free-materials [7–9]. Ceramics based on barium titanate (BaTiO3) are frequently used to manufacture multilayer ceramic capacitor and thermistors owing to their high dielectric constant [10]. Homovalent or heterovalent substitutions of barium or titanium ions were shown to give rise to various behaviors and properties [11, 12]. In addition, the aim for application was to prepare lead-free ceramics with such properties close to room temperature. In fact, up to now, lead free relaxor presented value of T m often widely lower than room temperature [13].

The aim of the present work is to obtain a relaxor with T m near the room temperature by adding Pr3+ at the Ba2+ site and Sn4+ at the Ti4+ site simultaneously. The details of relaxor and ferroelectric properties of the Ba1−3x Pr2x Ti0.95Sn0.05O3 (BPTS-x) ceramics were investigated.

2 Experimental procedure

In analogy to our previous work [14], the conventional solid-state reaction technique was used for the synthesis of BPTS-x. The starting materials were reagent BaCO3, Pr6O11, TiO2, and SnO2 taken in stoichiometric ratio. The surface of the sintered specimens were analyzed through scanning electron micrograph (SEM) using a Philips XL30 Scanning Electron Microscope. Room temperature powder X-ray diffraction patterns were recorded on Siemens D-5000 diffractometer using CuK α radiation (λ=1.54059 Å) in the angle 20∘≤2θ≤120∘ with 18 s couting time for each step of 0.016∘ in order to determine the structure for all prepared ceramic composition. For the dielectric characterization, the sintered disc (of 1-mm thickness and 8-mm diameter ) was polished, electroded with fine silver paint, and then connected to the LCR meter.

3 Results and discussion

3.1 Structural analysis

The phase identification of polycrystalline \(\mathrm{Ba}_{1 - 3x}^{2 +} \mathrm{Pr}_{2x}^{3 +} \mathrm{Ti}_{0.95}^{4 +} \mathrm{Sn}_{0.05}^{4 +} \mathrm{O}_{3}^{2 -}\) (x=0.01, 0.03, and 0.05) pellets was performed by X-ray diffraction with Cu Kα radiation. The X-ray diffraction pattern for a typical x=0.01 sample is shown in Fig. 1. The diffraction results indicate that all the samples are in single phase without any impurity and the samples have a rhombohedral lattice structure with space group \(R\bar{3}c\) in which Ba/Pr atoms are at 6a(0,0,1/4) positions, Ti/Sn at 6b(0,0,1/2), and O at 18e(x,0,1/4) position. The structural parameters are obtained by refining the experimental data by Rietveld FULLPROF program (version 1.9c-May 2001-LLB-JRC) [15] and are listed in Table 1 along with the respective residuals viz. R wp (weighted pattern), R p (pattern), and R exp (expected pattern) and goodness of fit χ 2. The lattice parameters and the unit cell volume V obtained decrease nearly monotonically with increasing Pr-concentration because of the increase of smaller cation Pr3+ for Ba2+.

Fig. 1
figure 1

Rietveld plots for Ba0.97Pr0.02 Ti0.95 Sn0.05O3 compounds synthesized by solid-state reaction; I obs (solid), I cal (circles), Bragg positions (bars), and I obs−I cal (solid, lower)

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Table 1 Some physical compounds of Ba1−3x Pr2x Ti0.95Sn0.05O3
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Notice that the stability of the Ba1−3x Pr2x Ti0.95Sn0.05O3 compounds having perovskite-type structure can be discussed in term of the tolerance factor value defined by Goldschmidt as \(t = \frac{(r_{\mathrm{A}} + r_{\mathrm{O}})}{\sqrt{2} (r_{\mathrm{B}} + r_{\mathrm{O}})}\) [16], where r A, r B, and r O are ionic radii of A, B, and oxygen, respectively, in the perovskite ABO3 structure.

The distorted perovskite-type structure is stable over the range 0.75≤t≤1.06. Based on Shannon’s ionic radii [17], , , , , and . The calculated t values (see Table 1) were included in the distorted perovskite stability range.

The SEM micrograph shows homogenous microstructure and uniform grain size distribution with average particle size. In addition, we can clearly observe that the grain size is different in all the samples and estimated to be mostly within 295–326 nm (see Table 1).

The particle sizes obtained by SEM are much larger than those calculated by X-ray diffraction (Table 1), which indicates that each particle observed by SEM basically consists of several crystalline grains. The experimental density is determined from the weight and geometrical dimensions of the cylindrical pellets and then compared to theoretical density determined from X-ray measurements. The compactness is thus calculated as the ration C=d exp/d theor. Table 1 shows the values of d exp, d theor, and C.

3.2 Dielectric properties

Temperature dependence of dielectric constant for BPTS-x ceramics measured at 1 kHz are shown in Fig. 2. The dielectric constant show a typical behavior of a ferroelectric material: it increases with temperature rise to its maximum value (\(\varepsilon'_{\mathrm {m}}\)) at the Curie temperature T C (or at the maximum temperature T m). For x=0.01, 0.03, and 0.05, the values of T C and ε′m are listed in Table 2.

Fig. 2
figure 2

Dielectric constant spectra of the entire system ceramics

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Table 2 The parameters C, T dev, T m, ε′m, T 0, and ΔT m for Ba1−3x Pr2x Ti0.95Sn0.05O3 at 1 kHz
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It is observed that the compounds undergo a transition phase from ferroelectric to paraelectric for all the samples. As the x value increases from 0.01 to 0.05, ε max decreases, and T C shifts to lower temperature, resulting from the replacement of the Ba2+ site by Pr3+. However, a similar observation has been made by Hongwei Chen [18] for BaZr x Ti1−x O3.

In most ferroelectrics, the relationship between temperature and dielectric constant above the Curie temperature can be expressed fairly accurately with simple equation called the Curie–Weiss law [19, 20]:

$$ \frac{1}{\varepsilon'} = \frac{T - T_{0}}{C}, $$
(1)

where T 0 is the Curie–Weiss temperature, and C is the Curie–Weiss constant. The inverse dielectric constant (1/ε′) as a function of temperature at 1 kHz for all ceramic samples is plotted in Fig. 3. The experimental data were fitted by Eq. (1) using a fitting program, and the fitting results for C and T 0 are summarized in Table 2. For the classical ferroelectrics, the order of the ferroelectric–paraelectric phase transition can be determined from the value of T 0. In case T 0≠T m, the phase transition is of the first order. If T 0=T m, this transition is of second-order type [21]. In our study, it was apparent from Fig. 3 that the values of T 0=T m for all the samples, which indicates the typical second-order ferroelectric phase transition.

Fig. 3
figure 3

The plot of inverse dielectric constant as a function of temperature at 1 kHz for BPTS-x (x=0.01, 0.03, and 0.05) ceramics

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Deviation from the Curie–Weiss law can be defined by ΔT m as follows [19, 22]:

$$ \Delta T_{\mathrm{m}} =T_{\mathrm{dev}}-T_{\mathrm{m}}, $$
(2)

where T dev denotes the temperature above which the dielectrice constant follows the Curie–Weiss law.

T dev and ΔT m are also listed in Table 2. The difference in T m, ΔT m, is commonly used to measure the extent of frequency dispersion and thereby the degree of relaxor behavior, with a larger ΔT m indicating stronger relaxor behavior [23, 24]. As listed in Table 2, the values of T dev, T m, and ΔT m decrease gradually with the augment of Pr-doping level, which indicates that paraseodyme can decrease the diffuseness of the phase transition of BTS ceramics.

In order to further characterize the dielectric dispersion and diffuseness of the phase transition, a modified empirical expression proposed by Uchino and Nomura [25] was employed in the present work:

$$ \frac{1}{\varepsilon'} - \frac{1}{\varepsilon'_{\mathrm{m}}} = \frac{(T - T_{\mathrm{m}})^{\gamma}}{ C'}, $$
(3)

where C′ and γ are assumed to be constant. The value γ evolves from γ=1 for Curie–Weiss behavior to γ=2 for ideal relaxor ferroelectric behavior. Based on the temperature plots of ε′ at 1 kHz, the graphs of log(1/ε′−1/ε′m) versus log (T−T m) for the Ba1−3x Pr2x Ti0.95 Sn0.05O3 ceramics were plotted, giving the results shown in Fig. 4. A linear relationship is observed for Pr-doped BTS ceramics. By least-squared fitting the experimental data to the equation, the γ was determined. As the paraseodyme content increases from 0.01 to 0.05, the diffuseness constant γ decreases from 1.70 to 1.15. It is evident that the diffuseness of the phase transition of Pr-doped BTS ceramics decreases with the increase of Pr content. This is because the ferroelectric ceramics with small grain sizes exhibit more remarkable diffuse phase transition [26].

Fig. 4
figure 4

Plots of log(1/ε′−1/ε′m) vs log(T−T m) for ceramic BPTS-x (x=0.01, 0.03, and 0.05) at 1 kHz (symbol: experimental data: solid line: fitting to Eq. (3))

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As is seen from a number of recently reported relaxors [27, 28], the slope of ε′ at T>T m can be scaled with the empirical Lorenz-type relation:

$$ \frac{\varepsilon_{\mathrm{A}}}{\varepsilon'} = 1 + \frac{(T - T_{\mathrm{A}})^{2}}{2\delta^{2}}, $$
(4)

where δ is a measure of the degree of diffuseness of the permittivity peak, T A (<T m) and ε A (>ε m) are the fitting parameters defining the temperature and magnitude of the Lorenz peak. This equation gives a more agreeable description of the experimental data than the previously used relation \(\frac{\varepsilon_{\mathrm{m}}}{\varepsilon'} - 1 = \frac{(T - T_{\mathrm{m}})^{\gamma}}{2\delta ^{2}}\) [29]. It describes the static conventional relaxor susceptibility, which provides the dominant contribution to the permittivity peak at temperatures above the temperature of the dielectric maximum. All the samples show good agreements to this formula above T m. A curve fitted by Lorenz-type relation for Ba1−3x Pr2x Ti0.95 Sn0.05O3 ceramic is shown in Fig. 5. The values of δ decrease with increasing Pr content, i.e., from δ=53.70 for x=0.01 to δ=42.21 for x=0.05. This indicates that the degree of relaxor behavior decreases with increasing Pr content, which is consistent with the above-mentioned observation and previous results [30, 31] on other systems.

Fig. 5
figure 5

Dielectric constants of BPTS-x (x=0.01, 0.03, and 0.05) as a function of temperature at 1 kHz

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3.3 Frequency dependent conductivity studies

It well known that conductivity in ferroelectrics appreciably affects the properties in the sense that there will be a competition between the growth of the ferroelectric phase and the free charge carriers. Conductivity studies help to illustrate the effects of conductivity not only on the domain structure and its motion, but also on the nature of charge carriers.

Variation of ac conductivity with frequency can be understood from Fig. 6.

Fig. 6
figure 6

Variation of conductivity (σ) of BPTS-x (x=0.01, 0.03, and 0.05) with frequency at room temperature. The inset shows the nonlinear fitting of conductivity (σ)

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The conductivity plots of BPTS-x possess the following characteristics:

  1. (i)

    Plateau region at lower frequencies corresponds to σ dc.

  2. (ii)

    Dispersion at high frequencies. The frequency at which the dispersion takes place is known as hopping frequency [32].

  3. (iii)

    With respect to the increase in Pr content in the compound, ac conductivity was found to be increasing.

The ac conductivity behavior obeys the Jonscher’s power law governed by the relation [33]

$$\sigma= \sigma_{\mathrm{dc}} + A\omega^{s}, $$

where σ dc is the dc conductivity in a particular range of temperature, A is a temperature-dependent parameter, and s is the temperature-dependent exponent in the range of 0≤s≤1. The exponent s represents the degree of interaction between mobile with the lattices around them, and the prefactor exponent A determines the strength of polarizability [34]. The detailed analysis of σ ac of BPTS-x suggests that the universal power law is obeyed. It is confirmed by the typical fit of the above equation to the experimental data (inset in Fig. 6).

The value of parameter s calculated from the slope σ vs log(f) curves is found to be less than unity and decreases with increase in temperature. The trend of change in s with temperature is shown in Fig. 7. The temperature dependency of s of disordered systems has been explained by two physical processes such as correlated barrier hopping (CBH) [35] and quantum mechanical tunneling (QMT) like electron tunneling (ET) [36], small polaron tunneling (SPT) [35], and large polaron tunneling (LPT) [37]. From the trend of change in s with temperature, it is presumed that the correlated barrier hopping is suitable. According to this model, the conduction occurs via polaron hopping process where two polarons simultaneously hop over the potential barrier between two charged defect states and the barrier height is correlated with the inter-site separation via a Coulombic interaction. Therefore, the conduction in the system could be considered due to the short-range translational type hopping of charge carriers [38–40]. This indicates that the conduction process is thermally activated. The exponent s for this model is given by

$$ s = 1 - \frac{6k_{\mathrm{B}}T}{ [ W_{\mathrm{m}} + k_{\mathrm{B}}T\ln(\omega\tau_{0}) ]}. $$
(5)

The first approximation of this equation gives the simple expression for s as [41]

$$ 1 - s = \frac{6k_{\mathrm{B}}T}{W_{\mathrm{m}}}, $$
(6)

where W m is the binding energy, which is defined as the energy required to remove an electron completely from one site to another site. The characteristic decrease in slope with the rise in temperature is due to the decrease in binding energy [42] as illustrated in Fig. 7. Using the values of the binding energy minimum hopping distance R min is calculated [43]:

$$ R_{\min} = \frac{2e^{2}}{\pi\varepsilon_{0}\varepsilon W_{\mathrm{m}}}, $$
(7)

where e is the electronic charge, ε 0 is the permittivity of free space, and ε is the dielectric constant.

Fig. 7
figure 7

Variation of index s and binding energy (W m) with temperature for BPTS-x. The inset shows the variation of binding energy hopping (R min) with temperature at 1 kHz

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The inset in Fig. 7 shows the variation of R min with temperature at 1 kHz. A similar order of decrease in R min was also reported by K. Prasad et al. for the (Na0.5Bi0.5)ZrO3 system [44].

4 Conclusion

Pr-substituted BaTi0.95Sn0.05O3 ceramics were synthesized via the solid-state reaction method. Perovskite single-phase BPTS-x compounds for 0.01≤x≤0.05 were obtained. The Rietveld refinement shows that the composition has a rhombohedric perovskite structure with space group \(R\bar{3}c\). Dielectric study of the composition shows typical characteristics of relaxor ferroelectrics, with broad and dispersive permittivity maxima, based on empirical parameters (ΔT m, δ, and γ). The ac conductivity was found to obey the universal power law. The pair approximation type correlated barrier hopping (CBH) model accurately describes the mechanism of charge transport in BPTS-x system. Such compositions could be excellent materials both for highly stable ferroelectric memory and for high-temperature piezoelectric applications.