1 Introduction

In the last decencies, manganese oxides have been extensively investigated in the scientific research field [1,2,3]. Such materials constitute a promising family of inorganic systems [4, 5]. Due to their unusual physical properties, manganites are used in enormous applications [6,7,8]. For examples, high electrical conductivity characteristic makes them functional for constructing solid oxide fuel cell (SOFC) [9, 10]. Owing to their important magnetocaloric effect and colossal magneto-resistance, manganites were explored in the magnetic cooling system, computer memory system, and magnetic sensors [11,12,13,14]. The physical properties of these systems can be improved by modifying several parameters, such as the nature of the dopant element, A-site deficiency, substitution sites, and preparation route [15,16,17,18,19]. In Ref [20], the studies of the praseodymium manganite (PrMnO3) have demonstrated an interesting magnetic against Neel temperature (TN). It exhibits antiferromagnetic insulator character at TN = 100 K and anisotropic behavior at TN < 100 K [21, 22]. Chukalkin et al. [23] have found that the contribution of lanthanum ion (La3+) leads LaMnO3 to be characterized by a weak ferromagnetic moment and by an inclined antiferromagnetic structure. The difference between the ionic radius of Pr3+ and La3+ allows Pr-based manganites to be the subject of material research [20]. From the magnetic analysis, Pr1−xCaxMnO3 compounds exhibit different behavior: ferromagnetic insulator for x < 0.3 and an antiferromagnetic charge ordering behavior for x = 0.3–0.5 [24]. Indeed, due to their multifunctional properties, Pr1−xCaxMnO3 entice the attention of many groups of research with calcium composition [8, 25,26,27,28]. It is important to mention that such systems have interesting electrical properties that allow them to be characterized by low-dimensional ferroelectricity [29]. Generally, for perovskite oxides, previous studies [30,31,32,33] have intensively investigated the electrical properties of such a material family. In the literature [34,35,36,37], experimental results have established that the physical properties of the manganite system can be influenced by the substitution at A-site. Such a substitution improves their electrical properties. Many works [18, 38, 39] have demonstrated that Pb-doping in manganite enhances the electrical conductivity. It is observed that the insertion of Pb in lanthanum-calcium manganite increases the cell volume, Curie temperature (TC), as well as the electrical transition temperature [40]. In previous work, we have shown that the lowest obtained value of the activation energy (Ea) corresponds to the Pb-doped Pr0.65Ca0.35MnO3 [18]. Besides, dielectric results show that the Pr0.65Ca0.25Pb0.1MnO3 sample has better dielectric properties than Cd and Sr doped ones. Moreover, it has been observed that in La0.7Ca0.3−xPbxMnO3 the electrical conductivity increases, and the metal–insulator transition temperature shifts towards higher temperatures [41]. Finally, of all the four samples Pr0.67A0.33MnO3 (A = Ca, Sr, Pb, and Ba), it has been reported that Pr0.67Pb0.33MnO3 compound is found to exhibit the highest magnetoresistance [42].

In this work, Pr0.65Ca0.25Pb0.1MnO3 ceramic was elaborated by solid-state reaction. Then, electrical properties are investigated.

2 Experimental details

Pr0.65Ca0.25Pb0.1MnO3 compound is prepared by conventional solid-state reaction using highly pure stoichiometry powders of Pr6O11, CaCO3, PbCO3, and MnO2. During the process, the precursors were mixed and calcined at 900 °C for 12 h. Then, the obtained powder was pressed into pellets and sintered at 1000 and 1200 °C for 48 h. To conduct electrical measurements, a thin silver film was deposited through a circular mask of 6 mm of diameter on both sides of the pellet. Consequently, a configuration of a plate capacitor was obtained. Therefore, measurements of both electrical and dielectric properties were performed on an Agilent 4294A analyzer and Janis VPF800 cryostat. In addition, a wide range of frequency [40–108 Hz] and temperature [80–400 K] were used for measurements.

3 Results and discussions

3.1 X-ray diffraction

To determine the crystalline structure of Pr0.65Ca0.25Pb0.1MnO3, room temperature powder X-ray diffraction (XRD) pattern was collected and shown in Fig. 1. The Rietveld refinement of the XRD measurement suggests the sample belongs to the Pnma space group. The structural parameters of the unit cell are a = 5.4521 Å, b = 7.6886 Å and c = 5.4457 Å.

Fig. 1
figure 1

Rietveld refined powder XRD pattern of the Pr0.65Ca0.25Pb0.1MnO3 compound

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3.2 Ac-conductivity analysis

The ac-conductivity (σac) spectra, at several temperature ranges of 80–100 K, 120–160 K, and 180–400 K for Pr0.65Ca0.25Pb0.1MnO3 are shown in Fig. 2a–c, respectively. In the temperature range of 80–100 K (Fig. 2a), two linear slopes were observed. Hence, the σac can be described by Bruce equation [43,44,45]:

$$\sigma_{{{\text{ac}}}} \left( \omega \right) = \sigma_{{{\text{dc}}}} + A_{1} \omega^{S1} + A_{2} \omega^{S2}$$
(1)
Fig. 2
figure 2

Frequency dependence of ac-conductivity (σac) for Pr0.65Ca0.25Pb0.1MnO3 compound at different temperatures ranges [80–100 K (a), 120–160 K (b) and 180–400 K (c)]. d Temperature dependence of the frequency exponent s1 and s2

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where σdc is the dc-conductivity, A1, A2 are constants, s1, and s2 are the frequency exponents. Then, for the temperature range [120–160 K] (Fig. 2b), a large plateau appears at low-frequency corresponding to the σdc. Such behavior is accompanied by a change in slope at high frequency. So, the σac can be expressed by the Jonsher law [43,44,45]:

$$\sigma_{{{\text{ac}}}} \left( \omega \right) = \sigma_{{{\text{dc}}}} + A\omega^{S1}$$
(2)

Beyond 180 K (Fig. 2c), a metallic behavior appears. Therefore, the conductivity response can be analyzed using the Drude model [46]:

$$\sigma_{{{\text{AC}}}} = \frac{{\sigma_{{{\text{DC}}}} }}{{1 + \left( {\omega \tau_{s} } \right)^{2} }}$$
(3)

where τs represents the relaxation time.

For the temperature dependence of the frequency exponent s, different models have been proposed [47, 48]. The variations of the exponent s1 and s2 as a function of temperature are shown in (Fig. 2d). From s1 (T) plot, two different conduction mechanisms were determined: the non-overlapping small polaron tunneling (NSPT) [49, 50], and the correlated barrier hopping (CBH) mechanism [51,52,53]. The reported finding displays that the frequency exponent s1 increases with the temperature until 140 K. This result indicates that the NSPT model is a suitable one to describe the conduction mechanism. In a covalent solid, if the incorporation of charge carriers to a site leads to a large degree of local lattice distortion, then the NSPT mechanism can be appeared [54]. According to this model, the exponent s1 can be described by the following expression [55]:

$$\mathop s\nolimits_{1} = 1 + \frac{{4\mathop k\nolimits_{{\text{B}}} T}}{{W_{{\text{H}}} }}$$
(4)

where WH is the polaron hopping energy.

For T > 140 K, s1 decreases with increasing temperature. Consequently, the correlated barrier hopping (CBH) is the appropriate model to characterize the electrical conduction. According to this model, the conduction is generated by a polaron hopping mechanism trapped in two defect centers, over the Coulomb barrier that isolated it [55]. Then, the frequency exponent is described by the relation [55]:

$$\mathop s\nolimits_{1} = 1 - \frac{{6\mathop k\nolimits_{{\text{B}}} T}}{{W_{{\text{m}}} }}$$
(5)

The variation of the second frequency exponent s2 obtained at low-temperature increases with increasing temperature indicating that the NSPT is the adequate model for describing the conduction mechanism.

The evolution of ac-conductivity as a function of temperature at different frequencies for Pr0.65Ca0.25Pb0.1MnO3 is shown in Fig. 3a. The obtained result can be divided into two regions. In the first region (R-I), it is noticed that the conductivity increases with both temperature and frequency. Such variation can be related to the thermally enhanced drift mobility of charge carriers [56]. In the second region (R-II) and at low frequencies, a saturation region was detected. Then, a semiconductor–metal transition temperature was observed at T = 180 K for 400 kHz. Figure 3b shows the plot of σac.T versus 1000/T at different frequencies. The activation energies are deduced using the following relation [57]:

$$\sigma_{{{\text{AC}}}} .T = \sigma_{{0}} \exp \left( {\frac{{ - E_{{{\text{ac}}}} }}{KT}} \right)$$
(6)
Fig. 3
figure 3

a Variation of ac-conductivity (σac) as a function of temperature at selected frequencies. b Variation of (σac.T) versus (1000/T) for Pr0.65Ca0.25Pb0.1MnO3

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At low frequencies, it is found that the activation energy values decrease with increasing frequency from Ea1 = 55 meV to Ea1 = 44 meV and from Ea2 = 137 meV to Ea2 = 120 meV at high and low temperature ranges, respectively. Such behavior confirms that the hopping conduction is the predominant mechanism [58, 59]. An increase in the applied field frequency leads to an improvement in electronic jumps between the localized states [60]. Hence, the activation energies decrease with increasing frequency. For 400 kHz, an only activation energy value was deduced Ea2 = 83 meV, confirming the appearance of metallic behavior (as previously reported in Fig. 3a).

3.3 Dc-conductivity analysis

The evolution of the dc-conductivity (σdc) as a function of temperature for Pr0.65Ca0.25Pb0.1MnO3 is shown in Fig. 4. As can be observed, the σdc increases with the temperature confirming a semiconductor behavior. Indeed, the increase in temperature provokes the increase of density of free carriers, which leads to the reduction of the encountered barriers and the improvement of electrical conductivity. At a specific temperature of Tsat = 250 K, the conductivity reaches a saturation region. This behavior suggests that the trapping centers are completely emptied. For Ts = 140 K, the change in slope can be related to the change in the conduction process.

Fig. 4
figure 4

Temperature dependence of the electrical conductivity σ(T) of Pr0.65Ca0.25Pb0.1MnO3

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3.4 Complex impedance analysis

The evolution of the real part of impedance (Z′) versus frequency at different temperatures for the investigated compound is shown in Fig. 5a. The found result indicates higher values of Z′ at low frequency. Then, it decreases with increasing temperature and frequency, suggesting an increase of the ac-conductivity [61]. Also, at high frequencies and for selected temperature, the merge of Z′ confirms the presence of space charge polarization [62, 63]. From Z′ spectrum, we can determine the Average Normalized Change (ANC). The temperature dependence of ANC for the investigated compound is illustrated in Fig. 5b. The ANC appears to be constant at low temperatures. Then, it decreases with temperature. Such evolution confirms the existence of different conduction mechanisms [64]. The variation of dANC/dT with temperature is illustrated in Fig. 5c. A change in the slope of dANC/dT was observed at TANC = 140 K, which is considered as a temperature at which the trapped centers seem to have vanished. The obtained temperature value is identical to the value evaluated from σdc(T) curve (TANC = Ts) as shown in Fig. 4.

Fig. 5
figure 5

a Evolution of Z′ as a function of frequency at different temperatures for Pr0.65Ca0.25Pb0.1MnO3. b, c Temperature dependence of average normalized change (ANC) and dANC/dT

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The normalization curve of Z″/Z″max at different temperatures is shown in Fig. 6a. The spectrum is characterized by the appearance of a peak for each temperature. When the temperature increases, the peaks shift towards high frequencies, confirming the presence of a relaxation phenomenon. The observed peak does not lie at the same frequency. Such behavior specifies the non-Debye type of relaxation in our material [65]. Furthermore, for T = 80 K, we notice the presence of the second peak at high frequencies. This result proves the existence of the second relaxation.

Fig. 6
figure 6

a Variation of Z″/Z″max versus frequency at different temperatures. b Frequency dependence of d(Z′/Z′max)/df and Z″/Z″max at T = 100 K for Pr0.65Ca0.25Pb0.1MnO3

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At a fixed temperature, T = 100 K, for example, the variation of d(Z′/Z′max)/df and Z″/Z″max with frequency is shown in Fig. 6b. It is clear that the minimum of Z′ and the maximum of Z″ do not merge, indicating the deviation from the Debye’s model [65].

The Nyquist diagram is an exciting method to distinguish the contribution of the grains and grain boundary in the conduction mechanism and to model the sample by an equivalent electrical circuit. In Fig. 7, we present the Nyquist diagram for Pr0.65Ca0.25Pb0.1MnO3 at T = 80 K. Two semi-circles are observed. At high frequencies, the semi-circle corresponds to the contribution of the grain. Then, at low frequencies, it represents the grain boundaries contribution [66]. The equivalent circuit is shown in the inset of Fig. 7 implying two parallel circuits: [Rg//CPEg] in series with a circuit [Rgb//CPEgb]. Rg and Rgb characterize the transport through grains and grain boundaries, respectively. The CPE impedance is determined by the following expression [67]: \(Z_{{{\text{CPE}}}} = \frac{1}{{Q\left( {jw} \right)^{\alpha } }}\), where Q is the capacitance value of the CPE impedance, and α (0 < α < 1) represents the deviation from Debye’s model.

Fig. 7
figure 7

Nyquist plot at T = 80 K for Pr0.65Ca0.25Pb0.1MnO3 compound

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3.5 Dielectric study

The evolution of capacitance versus frequency at different temperatures for Pr0.65Ca0.25Pb0.1MnO3 ceramic is reported in Fig. 8. At low frequency, the capacitance increases with temperature, but it decreases with frequency. Based on the work of Bahgat and Abou-Zeid [68], the increase of the capacitance with temperature can be attributed to the reduction in bond energies. The same behavior was recently reported by M’nassri et al. [69]. In addition, the rise in temperature is followed by several effects. It reduces the interatomic forces, and it improves the orientational vibrations. On the other hand, an increase in temperature leads, as well known, to an increase in thermal agitation. Hence, the orientational vibrations are troubled. At high frequency, the capacitance sharply decreases. Such evolution may be due to the disappearance of space charges.

Fig. 8
figure 8

Variation of the capacitance versus frequency at different temperatures for Pr0.65Ca0.25Pb0.1MnO3 compound

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4 Conclusion

This paper presented an electrical characterization of Pr0.65Ca0.25Pb0.1MnO3 ceramics elaborated by the solid-state method. The admittance spectroscopy technique was used to conduct electrical measurements. It is found that the investigated ceramic exhibits a semiconductor behavior. In addition, a saturation region seems to appear at high temperatures (Tsat = 250 K). The ac-conductivity analysis indicates the presence of the hopping process. From such an analysis, the temperature dependence of the frequency exponent s suggests that the NSPT and CBH are the appropriate models to characterize the electrical conduction mechanisms. Then, complex impedance analysis proves the contribution of grains and grain boundaries in the conduction process and confirms the presence of a relaxation phenomenon. Also, dielectric properties are found to be strongly dependent on both temperature and frequency.