Hopping conduction mechanism and impedance spectroscopy analyses of La0.70Sr0.25Na0.05Mn0.70Ti0.30O3 ceramic

The perovskite sample La0.7Sr0.25Na0.05Mn0.7Ti0.3O3 (LSNM0.70T0.30) was produced via a solid-state route process. The frequency dependence of electrical conduction plot established that according to the Jonscher law. The electrical conduction process was based on both theoretical conduction models assigned to the non-overlapping small polaron tunneling model at low temperatures and correlated barrier hopping mechanism at high temperatures. Detailed investigation of impedance data revealed a non-Debye-type relaxation occurring in the polycrystalline. In addition, the dielectric response confirmed the dominance of the Maxwell–Wagner model and Koop’s phenomenological theory effect in conduction phenomenon. The values of permittivity is high for LSNM0.70T0.30 were observed. These values make this composition interesting for microelectric applications. In the thermal study, the relaxation processes observed by electrical conductivity, impedance, and modulus are associated with singly and doubly ionized oxygen vacancies for the lower and higher temperature, respectively.


Introduction
Demand for powerful nanotechnology application is generating challenge among scientists and engineers to provide a stable and flexible system. In fact, more compounds have been suggested in this context and some perfection was achieved in various study fields by the perovskite system, for example, magnetotunable photocurrent activity, Spin Hall-Magnetoresistive devices and electrocatalytic activity [1][2][3].
Perovskite manganese oxide with an AMnO 3 general formula may be the next-generation sample for clean energy and large-scale energy storage due to its dual feature of magnetic/electric properties and its diverse applications [4,5]. Its multiferroic nature demonstrates the strong influence of its morphological characteristics on the magneto-transport behavior explained in the framework of Zener's theory [6,7].
However, manganite exhibits an important effect known as colossal magneto-resistance (CMR) closely related to its magnetic transition between a paramagnetic/insulating phase and ferromagnetic/metal phase [8][9][10]. The insulator-metal transition is due to the partial doped of the rare earth (RE) site by alkaline earth (AE) ions, causing the manganese ions to be trained in a combination of Mn 3? / Mn 4? ions. It was thought that spin system and magneto-transport characteristics are related by the double exchange interaction caused by the motion of the e g electrons for Mn 3? / Mn 4? ions [11][12][13][14]. The mixed valence state of manganese (Mn 3? /Mn 4? ) of these compounds displays the prominent role to alternate.
S. El Kossi et al. [15] explore the effect of incorporating a compound to affect the electrical characteristics, i.e., the connection between the electrical and the chemical characteristics. This variation in the conductivity value can be explained by morphological and the random distribution of cations in the lattice by raising concentration of the oxygen vacancy (V o ), resulting from the effects of the charge compensation produced via the incorporation of (Sr, Na) ions into the La-site and Ti ion into site Mn for LaMnO 3 .
Furthermore, the structural properties of manganite are also influenced by the elaboration method which affects the microstructural properties of the compounds and the crystallite size and play an important role in the manipulation of the physical properties of the compounds. In fact, solid-state methods are well established in the synthesis of complex oxides as they allow to obtain phase pure products and to control precisely their stoichiometry. Also, the solid-state method is more used in commercial production, due to its operability of preparation process. However, there is many type of synthesis technique as sol-gel [16] and molten salt [17], but the traditional solid state is the most known due to its simple preparation, low price and high impact to form a crystal compound.
In the present investigation, we report on the conduction process, the electrical characteristics and the behavior of the relaxation process of LSNM 0.70-T 0.30 ceramic synthesized via the solid-state process. In addition, we discuss the conduction mechanism and dielectric relaxation determined by impedance spectroscopy (IS), showing the low acquired dielectric loss.

Experimental procedure
In this work, the sample LSNM 0.70 T 0.30 was produced by the solid-state process reported in the previous research [18]. Reagent-grade oxide powder, La 2 O 3 , SrCO 3 , Na 2 O 3 , TiO 2 and MnO 2 (all Aldrich make 99.99%) are used as the starting material. The sample was calcined at 1000°C with intermediate grindings to remove impurities like carbon present, using temperature controlled programmable muffle furnace. The calcined powder LSNM 0.70 T 0.30 has pressed isostatically into circular pellet form under 8 t/cm 2 (of about 2 mm thickness) using a hydraulic press and at 1350°C for 48 h in air with intermediate remilling and repulsive to obtain the final product. Finally, these pellets were quenched to room temperature. This step was carried out in order to preserve the crystalline structure at the annealing temperature [19]. The synthesis procedure is (stepwise) given in Fig. 1.
The phase purity and crystallinity of LSNM 0.70 T 0.30 were identified by powder X-ray diffraction analysis with an XPERT-PRO diffractometer with a graphite monochromatized Cuka radiation (k Cuka-= 1.54 Å ).The data were recorded over the angular range of 10-100°with a step size of 0.017°and counting time of 18 s per step. The X-ray pattern after the Rietveld refinement is examined in the previous paper [18]. Note that LSNM 0.70 T 0.3 crystallizes in rhombohedral (R-3c) pattern with no visible secondary phase, with lattice parameters a = b = 5.536 (3) Å , c = 13.438 (3) Å and cell volume V = 356.75 (2) Å 3 .
The dielectric data were derived from ceramic disks after depositing gold electrodes on the circular faces by cathodic sputtering via an IS data employ an N4L-NumetriQ type PSM1735. After applying the electrodes, Agilent 4294A analyzer was used to collect conductance and capacitance measurements over a wide range of frequency (40-10 6 Hz). The amplitude of the applied AC signal is 50 mV. A liquid nitrogen cooled cryostat from Janis Corporation is used to provide temperature variation from 80 to 440 K.

Conductivity analysis
The conductivity spectra r(x) of the LSNM 0.70 T 0.30 compound at various temperatures over the wide frequency interval are illustrated Fig. 2a. In the lower frequency range, the r(x) increased with the rise of temperature. The r(x) against frequency data exhibited more model properties such as dispersion, low frequency plateau and high frequency dispersion. The change in the r(x) slopes was found in the lower and higher frequency areas. The apparent low frequency plateau is frequency independent that correlates to the DC conductivity (r dc ) and reveals longdistance mobility of the charge carriers. This low frequency plateau rose with increasing temperature, causing long-distance motion of the charge carriers. The frequency dispersion, produced by short-and long-distance mobility of charge carriers, is due to the input of electrical conductivities in LSNM 0.70 T 0.30 compound, respectively.
In fact, the r dc rose with raising the temperature, indicating a semiconductor nature in the LSNM 0.70-T 0.30 compound.
Generally, the total electrical conductivity r(x) of the LSNM 0.70 T 0.30 sample respects Jonscher's law described as [20]: where Ax s represents the transport behavior of the charge carrier within the polycrystalline compounds and 'n' gives details on the level of collaboration between mobile carriers and lattice. Both parameters 'A' and 'n' depend on the temperature and the nature of the compound.

AC conductivity study
The experimental conductivity measurements were adjusted by the equation (Eq. (1)) to determine the exponents (n) with temperature. Figure 2b depicts an example of adjustment of the data for the temperature 100 K. Figure 2c displays the evolution of the parameter ''n'' which rose with temperature increase (in the range [80 K-200 K]) and then decreased in the range [200 K-440 K], indicating that both conduction processes are present. So, depending on this evolution, non-overlapping small polaron tunneling (NSPT) and correlated barrier hopping (CBH) are the suitable models, respectively. Similar results were found for La 0.67-x Eu x Ba 0.33-Mn 0.85 Fe 0.15 O 3 manganites [21], La 0.9 Sr 0.1 MnO 3 ceramic [22]. For the NSPT model, the parameter ''n'' rose with temperature increase, in a covalent solid. The supplementing of the charge carriers to a site caused local deformation of the system. The mixture of electrons and its local deformation forms a Polaron and the energy of the states diminished via an order of W p (the Polaron Energy).
The term ''Small Polaron'' suggests that these particles are so focused which made their deformation clouds non-overlapping [23]. In the NSPT model, the process relates to the potential barrier (W H ) between both sites. r ac is obtained from the expression [24]: where N(E F ), a -1 and R x are, respectively, the density of states close the Fermi level, the spatial extension of the polaron and the tunneling distance. The exponent ''s'' may be expressed according to [25]: where W H is the potential barrier, s 0 (= 1/x = 1/ 2pf) is conductivity relaxation time and k B is Boltzmann's constant.
At high values of the ratio W H =k B T, Eq. (3) becomes: In the CBH model, the parameter ''n'' decreased with temperature increase. The conduction was carried by the jump of electrons above a Coulomb barrier which divides it [26].
r ac is obtained from the expression: where n is polaron number implicated in jumping behavior, N indicates density of localized states in which the carriers are present, N p characterizes density of localized states in which carriers jump, e 0 indicates the dielectric constant of the free space.
In this type of conduction, the potential barrier (W M ) in the CBH model is described as follows [27]: To estimate large W M /K B T values, the parameter ''n'' was minimized to: The W H and W M were determined from the linear adjustment of the empirical data 1-s against temperature. Their values were W H = 0.087 eV and W M-= 0.12 eV (Fig. 2c).
In addition, as observed, temperature increase led term increase in the binding energy of the charge carriers that can hardly hop from one site to the next with more energy needed [28].

DC conductivity study
r dc of the LSNM 0.70 T 0.30 sample is a concept of thermal process and according to Arrhenius' law described as follows: where r 0 is a pre-exponential parameter and K B is Boltzmann's constant.
The values of E a are determined via the slope of Log (r dc .T) against 1000/T in the inset of Fig. 2d. So, E a values around 0.16 eV found for conduction are very similar to these of the E a of V o in the perovskite type [29]. The transport mechanism can take place by a single or bipolaron hopping behavior over the barrier between the grain and grain boundary sites in the CBH model. The small polaron is produced from Jahn-Teller electron-phonon combination of the Mn 3? ions. On the variation of the temperature, the bipolaron can be created as the number of free carriers rises. Indeed, the rise in temperature causes an elevation in the density of free carriers, which leads to the reduction of the encountered barriers and the enhancement of conductivity [30][31][32].
In addition, the reaction of the chemisorbed oxygen ions species (O -, O 2-) on the nano-surface is accompanied with liberate of electrons to the conduction band which leads to the observed change in conductance increases (Fig. 2a). The same process has been demonstrated in some sample [33].
The scaling process of conductivity spectra of LSNM 0.70 T 0.30 ceramic is illustrated in Fig. 2e. We can note that all the curve superimposed to produce a unique master line. The overlapping of measurements at several temperatures indicates that the process of electrical ion conduction in the LSNM 0.70-T 0.30 sample is not linked to temperature, but to a short distance attributed to the polarization of the electrode, in the low frequency interval and at high temperature.

Complex impedance analysis
The variation of the real part (Z 0 ) of impedance against the frequency for LSNM 0.70 T 0.30 ceramic is displayed in Fig. 3a for a T range [80 K-440 K]. The magnitude of Z 0 is nearly independent of frequency, at a fixed temperature and at low frequencies, which indicates a Negative Temperature Coefficient of the Resistance [34]. In addition, the significant reduction in the magnitude of Z 0 with temperature rise shows the rise in the conductivity. This may be explained to improve mobility and reduced trapped charge density [35]. The evolution of Z 0 against the temperature gradually decreased and then disappeared by further increasing the frequencies. This behavior can validate the existence of polarization of the space charge in the LSNM 0.70 T 0.30 sample [32]. Also, we may see a shift in the plateau of Z 0 , indicating existence of a frequency relaxation behavior in the LSNM 0.70 T 0.30 ceramic [36,37].
Further investigations on the intrinsic property of the compounds were performed. The average Normalized Change (ANC) analysis is described via the following equation [38]: where Z 0 (Z' value at frequency near to zero) can be roughly evaluated via the extrapolation of the low frequency response, DZ 0 = Z 0 high -Z 0 low with Z 0 low / Z 0 high are the Z 0 value at low/ high frequency f low / f high and Df = f high -f low . The temperature dependence of the ANC is illustrated in Fig. 3b. As expected, on rising temperature, we can see clearly that the shape of the flow curves reveals the presence of various conduction mechanisms and density of trapped charges markedly decreased. The amount of releasing trapped charge from such intrinsic defects in materials was examined by the slope of the differential of ANC. Noticeably, a modification of the slope of the differential of the ANC (inset of Fig. 3b) was detected around 250 K for LSNM 0.70 T 0.30 ceramic. The spectrum of impedance (Z 00 ) against the frequency at [80 K-440 K] for LSNM 0.70 T 0.30 ceramic is displayed in Fig. 3c. All data presented a peak at a specific frequency f max . The maximal value is indicated Z 00 max . The relaxation frequency f max changed to the higher frequency with temperature increase, confirming presence of a relaxation behavior in temperature and frequency [26]. The relaxation time is calculated according to the equation s = 1/(2pf max ).
The reciprocal relationship between s and temperature is illustrated in Fig. 3d. The temperature connected of s Z obeys the Arrhenius' function as follows: where k B is Boltzmann constant and s Z 0 is the preexponential factor.
The E Z a values obtained via fitting of the Arrhenius graph (shown in Fig. 3d) are represented as ln(s Z ) as a function of 1000/T. We remark that E Z a is about 0.13 eV. This process is related to V o [39].
The Z 00 / Z 00 max curve against of frequency for the LSNM 0.70 T 0.30 ceramic is displayed in Fig. 3e. It is obvious that temperature evolution caused the collapse of all the lines into one line with minor differences. Thus, the repartition of the relaxation behavior can be regarded as unrelated with temperature. In addition, the Full Width at Half maximum was about 1.14. The findings indicated that the distribution of relaxation at various times is invariant and mechanism caused is a non-Debye type [40,41].
Furthermore, Fig. 3f illustrates, at T = 160 K, curve Z 00 /Z 00 max and d(Z 00 /Z 00 max )/df for the LSNM 0.70 T 0.30 ceramic. As seen, the position of the maximum of the data of Z 00 /Z 00 max did not coincide with the position of the data of d(Z 00 /Z 00 max )/df. This shift or non-coincidence indicates the non-Debye mechanism [42,43].

Electric modulus analysis
Complex modulus M* is calculated as follows [44]: where (M 0 , Z 0 ) and (M 00 , Z 00 ) are the real and imaginary parts of M* and Z*, respectively; C 0 is the vacuum capacitance of the measured cell and the electrodes with an air gap of the LSNM 0.70 T 0.30 ceramic thickness.
The frequency related to M 0 (x) at selected temperatures [80-440 K] of LSNM 0.70 T 0.30 ceramic is presented in Fig. 4a. At low frequency, M 0 (x) data tended to zero.
The appearance of the conduction behavior of short-distance mobility can account for the monotonic dispersion when raising the frequency and show the low, contribution of the electrode phenomenon [45]. Figure 4b illustrates the plot of M 00 (x) of LSNM 0.70 T 0.30 ceramic for selected temperatures. We observed well-resolved peaks in typical frequency (f max ). In addition, the asymmetric character of M 00 (x) is noticed in the region of dispersion and the peak moved to the higher frequency side on raising the temperature. This suggests that the rise in relaxation with temperature increase is caused by the thermal activation of charge carriers.
Indeed, this process of thermally activated process, the relaxation time (s), calculated via relationship (2pf max s ¼ 1), is obtained from Arrhenius expression: where (s 0 and k B ) are, respectively, pre-exponential factor and Boltzmann's constant. E M a was obtained via setting Ln (s) vs 1000/T (  Fig. 4c). As expected, the reported E M a value of the LSNM 0.70 T 0.30 ceramic was approximately the same as that obtained by measuring r(x) and in the region of those obtained for some other samples. This implies that conduction mechanism and the relaxation behavior are connected to same source.
Comparing our results with many report paper proved that dielectric relaxation processes have an important role in the industrial application of compounds [46,47]. Figure 4d demonstrates the scaled coordinates M 00 (x) for the spectra mentioned above, displayed on the modulus line via obtained the graph M 00 (x)/ M 00 max (x) of the LSNM 0.70 T 0.30 sample vs the normalized frequency x/x max . If all scaling curve values collapse into one main graph, then distribution of relaxation times is not influenced by the T [48].
As expected, a dispersive behavior emerged of the LSNM 0.70 T 0.30 , showing that relaxation behavior for the LSNM 0.70 T 0.30 is of non-Debye process and described by presence of a poly-dispersive relaxation process in LSNM 0.70 T 0.30 , which is in consistent with results of the IS.

Dielectric results
To evaluate the dielectric behaviors, complex permittivity e * (x) was utilized. This latter was can be extracted from the Z* as [49]: where e 0 (x), e 00 (x) are the real part and the imaginary part of e * (x), respectively. Figure 5a displays the frequency related to the e 0 (x) at selected temperatures of LSNM 0.70 T 0.30 sample.
At low frequencies, the e 0 (x) that tends not to be influenced by frequency exhibits colossal values that may be ascribed to the effect of the interfacial polarization achieved using Koop's theory [50], known as Maxwell-Wagner-Sillars phenomenon. Indeed, Maxwell-Wagner mechanism may be described by investigating the heterogeneity of the structure with respect to semiconductor grains divided by insulating grain boundaries with diverse electrical conductivities. Thanks to the electron jump between Mn 3? / Mn 4? , the charge carriers stuck up at the grain boundaries and created a polarization. As frequencies were increased, charge carriers reversed their orientation more frequently at grain boundary and, as a result, reduced the polarization [51]. The deformation produced by rise of temperature, at fixed frequency, can account for the reduction of the e 0 (x) in LSNM 0.70 T 0.30 compound. The same process has been demonstrated in some sample semiconductors [52,53]. Figure 5b displays the change of the frequency of e 00 (x) at several temperatures of the sample LSNM 0.70 T 0.30 . As can be observed, at low frequencies, the data showed high values of e 00 (x) confirming presence of all polarization phenomena in the LSNM 0.70 T 0.30 sample.
However, as frequency increased, e 00 (x) decreased as the electric dipoles located in the LSNM 0.70 T 0.30 cannot follow the electric field applied to the AC [54]. It should be noted that the absence of peaks in the data of e 00 (x) suggests that the polarization mechanism in the LSNM 0.70 T 0.30 is driven via a jumping behavior as described previously [55].
In fact, e 00 (x) can be represented via the Giuntini expression provided via [56]: where In these equations, a (T) is a constant related only to temperature, m is a parameter explaining the interaction between electric dipoles, and W C is the maximum potential barrier height. We plotted the evolution of the Ln (e 00 ) v.s Ln (x) at several temperatures (in the inset of Fig. 5c). The resulting linear graphs with slopes in the range of -1 suggest that dielectric losses are controlled by the r dc process [49].
These plots were applied to calculate the values of W c depicted in Fig. 5d. As observed, Wc rose due to the deformation produced via temperature increase.
The similar process was found in Pr 0.  [57].
In order to verify the dominance of r dc on the dielectric properties of the LSNM 0.70 T 0.30 manganite resulting in the elimination of the relaxation behavior in e 00 (x) curve, we estimated the dielectric loss by the formula below, that is usually used at low frequencies [58]: So, at low frequency, the e * (x) is almost entirely assigned to r dc . Therefore, relaxation mechanism in dielectric curves is totally obscured by r dc .

Conclusion
The LSNM 0.70 T 0.30 manganite was prepared via solidstate process. The electrical characteristics are studied by impedance spectroscopy over a wide interval of frequency and various temperatures. The complex impedance investigation reveals the presence of an electrical relaxation mechanism in the LSNM 0.70 T 0.30 sample. The rac conductivity established that based on Jonscher law. rac is described at low temperatures by non-overlapping small polaron tunneling model and at high-temperature correlated barrier hopping model. From DC conductance measurement, the electronic conduction appears to be thermally activated, suggesting the existence of semiconductor behavior. Additionally, the dielectric constant curves were applied to study the relaxation dynamics of charge carriers. In fact, Debye-like relaxation was interpreted using the polarization of spatial charges following Maxwell-Wagner model and Koop's phenomenological theory.