1 Introduction
During the last decades, rare earth orthoferrites have received special attention thanks to their vast range of technological applications [1]. In fact, they can be used as catalysts [2], fuel cells [3], oxygen permeation membranes and gas sensors [4, 5]. Lanthanum ferrite oxide, LaFeO3, has been considered as one of the most important of the orthoferrite materials which exhibits a high stability over a wide range of temperatures [6].
Physical properties of LaFeO3 are principally controlled by several factors. Among them doping is an effective way to improve their physical properties and to develop higher performances which are required for desired applications.
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Our literature review shows that various reports are available on the study of the useful properties of LaFeO3 by replacing La-site with alkaline metal such as Sr and Ba.
Recently, Cyza et al. [7] reported that the doping of LaFeO3 with Sr leads to a decrease of grains size and on the surface which positively affects the physical properties such as sensitivity for the gases. On their part Yao et al. [8] synthesized La1−xSrx FeO3 by sol–gel method. They found that as Sr content increases, the structural phase varies from orthorhombic to rhombohedral and cubic phase. They also investigated on the formaldehyde gas-sensing properties of Sr doped LaFeO3. The experimental results show that the optimum operating temperatures of La1−xSrxFeO3 varies with Sr content. On the other hand Wang et al. [9] have done a detailed study of the Influence of Sr substitution on thermoelectric properties of La1−xSrxFeO3 ceramics. They found that the increase of Sr content reduces significantly both the electrical resistivity and the Seebeck coefficient, but slightly increases the high-temperature thermal conductivity. Adiabatic polaron hopping mechanism is found responsible for the conductivity in La1−xSrxFeO3, and the conductive carriers are thermally activated. Reactivity of La1−xSrxFeO3 perovskite oxides for chemical-looping reforming of methane was evaluated by He et al. [10]. They found that the substitution of La by Sr in LaFeO3 would improve the amount of adsorbed oxygen of the perovskites, and inhibit methane decomposition in the methane reaction. Dielectric properties of La1−xSrxFeO3 were also studied by Chern et al [11]. They observed that the activation energy decreases when the Sr-doping increases, with a strong dependence of temperature and frequency and high real dielectric constant.
On the other hand, the barium-doped lanthanum ferrite has also been the focus of intense studies. Sun et al. [12] reported a research on the structural and electrical properties of nanocrystalline La1−xBaxFeO3 for gas detection application. The results show that appropriate Ba-doping can restrain the growth of the grain size and improve the response of the LaFeO3 based sensor to ethanol gas. Furthermore, the best response to ethanol gas was observed with Ba content equal to 0.25 mol. Based on solid-state reaction route, Masud et al. [13] prepared the La0.5Ba0.5FeO3 system. They reported in detail the dielectric behavior of this system. Also, they observed that La0.5Ba0.5FeO3 exhibit high dielectric permittivity. A complex impedance and modulus spectroscopic study shows two relaxation processes corresponding to grain and grain boundary contributions in this system. They demonstrated that magnetic field-dependent non-ferroelectric high dielectric behavior of La0.5Ba0.5FeO3 might be important for its applications in devices. Zhang et al. [14] also investigated the physical properties of the same system. They observed interesting magnetic phase transition at low temperature under external pressure.
In our previous work [15], we have studied in detail the effects of strontium doping on the structural, morphological, optical and electrical properties of the LaFeO3 doped barium, prepared according to the sol–gel method. The results obtained show that these properties have low strongly affected by the strontium content.
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In present work, we are trying to investigate the effect of Sr-doping on the structural and dielectric properties of La0.75Ba0.25−xSrxFeO3 nanocrystallines using impedance spectroscopy technique.
2 Experimental details
La (NO3)3·6H2O, Sr (NO3)2, Ba (NO3)2 and Fe (NO3)3·9H2O of analytic purity were used as raw materials to prepare the La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) ferrites by sol–gel method [16]. Firstly, all nitrates were separately dissolved in distilled water. Then citric acid and ethylene glycol were added to the solution under continuous stirring to get a homogeneous solution. This solution was heated until the gel formation. The obtained gel was dried and calcined at 300 °C for 12 h. At the end of the procedure, the resultant powders were pressed into pellets and heated at 900 °C for 24 h.
An X’Pert PRO MPD X-ray diffractometer with monochromator Cu-Kα radiation (λCu = 1. 5405 Å) was used to carry out the phase identification for the obtained samples. Complex impedance spectroscopy was used to study the electrical properties of our materials. This technique describes the electrical processes that occur in a system by applying an ac signal as input pressure [17]. It is used to analyze the phenomena of electrical conduction of materials and to separate the different contributions that could constitute this system (grain and grain boundary). The electrical measurements were carried out on pellet disks of about 8 mm in diameter and 1.3 mm in thickness in the temperature range from 250 to 310 K, in the frequency range from 100 Hz to 1 MHz, using an Agilent 4294 impedance analyzer.
3 Results and discussion
3.1 XRD studies
Figure 1a shows the XRD patterns of La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) samples. A perovskite structure consisting of an orthorhombic phase was revealed by XRD patterns, in accordance with what previously reported [15]. The patterns show a slight shift in the most intense peak toward higher 2θ values with Sr content increasing, this indicates that the substitution of Sr has smaller ionic radii than the Ba. The shift of diffraction peaks toward higher angle reveals that the cell parameters of samples continuously decrease with the increase of Sr content. A typical Rietveld refinement analysis of x = 0.15 sample, using the FullProf software [18], is shown in Fig. 1b and their crystal structure is presented in Fig. 1c. The results of the refinement structural parameters of our samples are reported in Table 1. As clearly shown in this table, the bond lengths and bond angles show slight deviation when the doping increases.
Fig. 1
XRD patterns for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.1 and 0.15) compounds (a) rietveld refinement plot for x = 0.15 sample (b) and crystal structure (c)
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Table 1
Refined structural parameters for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds
x = 0.05 | x = 0.10 | x = 0.15 | |
|---|---|---|---|
a (Å) | 5.551 (8) | 5.544 (6) | 5.539 (7) |
b (Å) | 7.859 (4) | 7.849 (4) | 7.842 (5) |
c (Å) | 5.563 (8) | 5.556 (6) | 5.552 (7) |
FeO (1) | 1.9874 (8) | 1.9746 (8) | 1.9710 (80) |
Fe–O (2) | 1.44 (8) | 2.395 (7) | 2.374 (18) |
Fe–O (2) | 2.24 (7) | 1.882 (3) | 1.91 (2) |
〈Fe–O (2)〉 | 1.889 | 2.083 | 2.085 |
〈La(Ba, Sr)–O〉 | 2.24 | 2.375 | 2.32 |
Fe–O(1)–Fe | 162.69 (7) | 167.24 (4) | 168.26 (4) |
Fe–O(2)–Fe | 129 (3) | 132.8 (5) | 132.1 (8) |
〈Fe–O– Fe〉
| 143.345 | 150.02 | 150.18 |
3.2 Impedance studies
Impedance spectroscopy is a well-known as a powerful technique for investigation of the electrical properties of materials. Here, the real part of impedance (Z′) as a function of frequency at several temperatures is shown in Fig. 2. The behavior of Z′ observed in our samples shows three distinct regions in which changes are clearly visible with change in temperature and frequency. The first one (R-I) is at low frequency in which the values of Z′ are frequencies independent and decrease with temperatures increasing. In the second one (R-II), the magnitude of Z′ is found to decrease gradually with the rise of both frequency and temperature, due to the high mobility of charge carriers that indicates an increase in AC conductivity with increasing both temperature and frequency. At higher frequencies (R-III), the Z′ merges irrespective of temperature. This result may possibly be related to the release of space charge as a result of reduction of the barrier properties of the materials [16, 19].
Fig. 2
The frequency dependence of the real (Z′) part of the complex electrical impedance at several measurement temperatures
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The plots of the frequency dependence of imaginary part of impedance Z″ for all compounds is shown in Fig. 3. One can see that the Z″ values attains a peak maxima at a particular frequency (fmax) known as the relaxation frequency [20]. It is worth noting that the peaks are shifted to higher frequencies with increasing temperature. Such behavior indicates the presence of relaxation in our materials [21].
Fig. 3
The frequency dependence of the imaginary (Z″) part of the complex electrical impedance at several temperatures
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To provide the characteristic activation energy of the relaxation process, the plot of Ln(fmax) (relaxation frequencies corresponding to Z″max) versus 1000/T is shown in Fig. 4. The temperature dependence of the relaxation frequency follows the Arrhenius law:
Fig. 4
The Arrhenius plots shows depedence fmax versus 1000/T. Inset figure is the activation energy Ea values
$${f_{\text{max} }}={f_0}\exp \left( { - {{{E_a}} \mathord{\left/ {\vphantom {{{E_a}} {{k_B}T}}} \right. \kern-0pt} {{k_B}T}}} \right)$$
(1)
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Figure 5 shows Nyquist plots of La0.75Ba0.25−xSrxFeO3 compounds. The impedance spectra are characterized by the appearance of semicircles where the diameter decreases with increasing temperature, referring to pronounced increase of DC conduction [22]. The impedance data are fitted using Z-view software [23] and the best fit are obtained when employing an equivalent circuit, as shown in Fig. 6, consisting of a series combination of grains boundary (Rgb–CPEgb) and grains (Rg–Cg).
Fig. 5
Nyquist plot of impedance data at several temperatures
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Fig. 6
The equivalent circuit for impedance
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The constant phase element (CPE) impedance is given by the following relation:
$${\text{Z}_{\text{CPE}}}=\frac{1}{{\text{Q}{{\left( {\text{j}\upomega } \right)}^\upalpha }}}$$
(2)
where Q is a proportional factor, \(\omega\) is the angular frequency, and α is the parameter which estimates the deviation from the ideal capacitive behavior. α is zero for the pure resistive behavior and is the unit for the capacitive.
The values of all fitted parameters are presented in Table 2. We can see from this table that the grain boundary resistance values decrease with the increase of temperature, indicating a semi-conducting behavior for all samples [21]. Furthermore, it has been found that the values of Rgb are larger than those of Rg. This is assigned to the fact that the atomic arrangement near the grain boundary region is disordered, resulting in an increase of the electron scattering.
Table 2
Parameters of equivalent circuits deduced from the complex diagram at several temperatures for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds
T (K) | 250 | 260 | 270 | 280 | 290 | 300 | 310 | |
|---|---|---|---|---|---|---|---|---|
x = 0.05 | Rgb (\(\times \;{10^5}\; \Omega\)) | 71.54 | 43.03 | 14.78 | 11.17 | 7.38 | 4.48 | 3.93 |
CPEgb (\(\times\; {10^{ - 12}}{\text{F}}\)) | 4.66 | 4.70 | 6.565 | 7.62 | 8.27 | 4.17 | 0.041 | |
\(\alpha\)
| 0.9709 | 0.973 | 0.965 | 0.969 | 0.9703 | 1.039 | 1.092 | |
Rg (\(\times \;{10^5}\;\Omega\)) | 4.266 | 4.348 | 4.349 | 4.002 | 3.849 | 4.172 | 1.793 | |
Cg (\(\times\; {10^{ - 13}}{\text{F}}\)) | 1.548 | 1.586 | 1.605 | 1.625 | 1.69 | 1.768 | 1.322 | |
x = 0.10 | Rgb (\(\times\; {10^5}\; \Omega\)) | 5.598 | 2.183 | 2.145 | 1.179 | 0.9443 | 0.6134 | 0.4088 |
CPEgb (\(\times \;{10^{ - 11}}{\text{F}}\)) | 2.779 | 3.767 | 3.188 | 4.763 | 5.151 | 5.326 | 5.132 | |
\(\alpha\)
| 0.9564 | 0.9263 | 0.9244 | 0.9068 | 0.9045 | 0.9055 | 0.91 | |
Rg (\(\times \;{10^3}\; \Omega\)) | 4.923 | 6.789 | 5.709 | 4.231 | 2.778 | 1.949 | 1.404 | |
Cg(\(\times {10^{ - 8}}{\text{F}}\)) | 9.451 | 9.131 | 9.05 | 8.161 | 7.61 | 7.136 | 6.7 | |
x = 0.15 | Rgb (\(\times {10^5}{{{\Omega}}}\)) | 7.101 | 4.014 | 2.451 | 1.439 | 0.9604 | 0.627 | 0.3389 |
CPEgb (\(\times {10^{ - 11}}{\text{F}}\)) | 2.579 | 30.913 | 3.814 | 2.9 | 4.118 | 4.709 | 10.57 | |
\(\alpha\)
| 0.9325 | 0. 915 | 0.9013 | 0.9216 | 0.9021 | 0.898 | 0.8604 | |
Rg (\(\times {10^4}{{{\Omega}}}\)) | 5.498 | 3.444 | 1.042 | 1.005 | 0.3994 | 0.346 | 0.1133 | |
Cg (\(\times {10^{ - 10}}{\text{F}}\)) | 0.4704 | 0.616 | 2.218 | 4.897 | 2.358 | 1.18 | 2.182 |
The logarithmic variation of the deduced Rgb and Rg with the inverse of temperature is shown in Fig. 7a and b. The activation energy can be obtained using the Arrhenius law:
Fig. 7
The Arrhenius plots shows dependence resistances (a) grain boundary \(({R}_{gb})\) and (b) grain \(({R}_{g})\) versus 1000/T
$${R_{g,gb}}={R_0}\exp \left( { - {{{E_a}} \mathord{\left/ {\vphantom {{{E_a}} {{k_B}T}}} \right. \kern-0pt} {{k_B}T}}} \right)$$
(3)
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It is worth noticing from Fig. 7 that the activation energy values for grain boundaries are found to be slightly higher than that of grains indicating higher resistive behavior than that of grains.
3.3 Conductivity studies
In order to understand the conduction behavior and to determine the parameters that may control the conduction processes in the La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) samples, we plotted the variation of AC conductivity with frequency at operating temperatures from 250 to 310 K. As shown in Fig. 8, the conductivity increases with increasing frequency and temperature in all the samples. We notice that at low frequencies, the conductivity spectra are independent of the frequency for each temperature. For high frequencies values, these spectra are found to increase with frequency. According to Jonscher’s power-law [24], the frequency dependence of the conductivity is expressed as:
Fig. 8
Frequency dependency of conductivity for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds
$$\upsigma \left( \upomega \right)={\upsigma _{{\text{dc}}}}+{\text{A}}{\upomega ^{\text{S}}}$$
(4)
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In this expression \({\sigma }_{\text{d}\text{c}}\) is the direct current conductivity, A is a constant depending on temperature and the exponent “S” represents the degree of interaction between mobile ions and its surrounding lattices.
Experimental conductivity spectra are adjusted using the equation (Eq. 4). The fitting results are summarized in Table 3. The DC conductivity as a function of the inverse of the absolute temperature is shown in Fig. 9. This variation can be described by the Arrhenius equation as [25]:
Table 3
The Jonscher’s power-law fitting results of ac conductivity for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds
T(K) | 250 | 260 | 270 | 280 | 290 | 300 | 310 | |
|---|---|---|---|---|---|---|---|---|
x = 0.05 | σdc (10−5 S m−1) | 3.347 | 5.526 | 9.137 | 14.042 | 24.100 | 31.571 | 47.432 |
A (× 10−12) | 2.803 | 2.911 | 2.949 | 7.516 | 8.604 | 5.269 | 7.269 | |
S | 1.094 | 0.987 | 0.901 | 0.866 | 0.823 | 0.838 | 0.902 | |
x = 0.10 | σdc (10−5 S m−1) | 4.285 | 7.031 | 11.321 | 16.670 | 26.630 | 41.087 | 61.427 |
A (× 10−10) | 8.695 | 8.513 | 7.350 | 5.473 | 2.716 | 1.109 | 0.646 | |
S | 0.745 | 0.776 | 0.799 | 0.833 | 0.888 | 0.948 | 0.987 | |
x = 0.15 | σdc (10−5 S m− 1) | 3.938 | 6.931 | 11.879 | 19.334 | 30.835 | 46.002 | 67.394 |
A (× 10−10) | 8.501 | 11.481 | 10.669 | 9.069 | 7.933 | 7.469 | 6.667 | |
S | 0.758 | 0.761 | 0.770 | 0.788 | 0.811 | 0.843 | 0.87 |
Fig. 9
Variation of the \({\text{Ln}}\left({\sigma }_{dc}\right)\) as a function 1000/T for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds. Inset figure is the activation energy Ea values
$${\upsigma _{\text{dc}}}={\upsigma _0}\exp \left( { - {{{\text{E}_\text{a}}}/{{\text{k}_\text{B}}\text{T}}}} \right)$$
(5)
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The variation of the exponent “S” as a function of the temperature for all the compounds is shown in Fig. 10. As we can notice for x = 0.05, that this parameter decreases with increase in temperature and attains a minimum value then further increases suggesting the conduction to be mediated by overlapping large polaron tunneling (OLPT) process [26]. While, for x = 0.10 and x = 0.15, “S” increases with temperature increasing, indicating that the non-overlapping small polaron tunneling (NSPT) model is applicable for the both samples [27].
Fig. 10
Temperature dependence of the exponent S for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds
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The OLPT model (x = 0.05):
The overlapping large polaron tunneling (OLPT) model proposed by Long [28] where the exponent “S” depends on both temperature and frequency. According to this model, the large polaron wells at two sites overlap there by reducing the polaron hopping energy as follows [29, 30]:where \({ r}_{p}\) is the large polaron radius, R is the polaron intersite separation and \({W}_{HO}\) is given by:where εp is the effective dielectric constant. Assuming \({W}_{HO}\) is constant for all sites, whereas the intersite separation R is a random variable.
$${W_H}=~{W_{HO}}~\left( {1 - {r_p}/R} \right)~~$$
(6)
$${W_{HO}}={e^2}/4{\varepsilon _{\text{p}}}{{\text{r}}_{\text{p}}}$$
(7)
The AC conductivity in overlapping large polaron tunneling conduction mechanism is [31]:
$${\sigma _{AC}}\left( \omega \right)=\frac{{{\pi ^4}}}{{12}}{e^2}{\left( {{k_B}T} \right)^2}{\left[ {N\left( {{E_F}} \right)} \right]^2}\frac{{\omega R_{\omega }^{4}}}{{2\alpha {k_B}T+\left( {{W_{HO}}{r_P}/R_{\omega }^{2}} \right)}}$$
(8)
In this expression, \({\text{k}}_{\text{B}}\) is the Boltzmann constant, N is the density of states at the Fermi level, \(\alpha\) the spatial extent of the polaron, and \({R}_{\omega }\) is the tunneling distance.
\({R}_{\omega }\) can be calculated by the quadratic equation:
$${\text{R}^{\prime}}_{\upomega }^{2}+\left[ {{{\text{W}}_{{\text{HO}}}}\upbeta +{\text{Ln}}\left( {\upomega {\uptau _0}} \right)} \right]{\text{R}^{\prime}} - {{\text{W}}_{{\text{HO}}}}{{\text{r}^{\prime}}_{\text{P}}}\upbeta =0$$
(9)
where
$${{\text{R}^{\prime}}_{{\varvec{\omega}}}}=2{{\varvec{\alpha}}}{{\text{R}}_{{\varvec{\omega}}}},\quad {{\varvec{\beta}}}=\frac{1}{{{{\text{k}}_{\text{B}}}{\text{T}}}},\quad {{\text{r}^{\prime}}_{\text{p}}}=2{{\varvec{\alpha}}}{{\text{r}}_{\text{p}}}$$
The frequency exponent S in OLPT model can be evaluated as [26]:
$${\text{S}}=1 - \frac{{8{\upalpha}{{\text{R}}_{\upomega}}+\frac{{6{{\text{W}}_{{\text{HO}}}}{{\text{r}}_{\text{p}}}}}{{{{\text{R}}_{\upomega}}{{\text{k}}_{\text{B}}}{\text{T}}}}}}{{{{\left( {2{\upalpha}{{\text{R}}_{\upomega}}+\frac{{{{\text{W}}_{{\text{HO}}}}{{\text{r}}_{\text{p}}}}}{{{{\text{R}}_{\upomega}}{{\text{k}}_{\text{B}}}{\text{T}}}}} \right)}^2}}}$$
(10)
Parameters N(EF), α, WHO and rp for La0.75Ba0.20Sr0.05FeO3 obtained from fitting of experimental data are given in Table 4. It can be observed that N(EF) and rp increase with frequency. In fact, the increase in the frequency stimulates the mobility of free charge which may cause the increment of the radius of the polaron and thus justifying the elevation of the AC conductivity with frequency.
Table 4
Parameters used to OLPT model fitting for La0.75Ba0.25−xSrxFeO3 (x = 0.05) compound at various frequencies
Frequencies (kHz) | N(EF) (1034 eV−1 m−1) | \(\alpha\) (Å−1) | \({W}_{HO}\) (eV) | \({r}_{p}\) (Å−1) |
|---|---|---|---|---|
500 | 13.144 | 4.412 | 0.308 | 2.996 |
700 | 9.539 | 3.246 | 0.366 | 2.404 |
1000 | 9.107 | 3.308 | 0.199 | 1.980 |
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The NSPT model (x = 0.10 and x = 0.15):
According to the NSPT model, the exponent S is given by [32]:
$$S=1+\frac{{4{{\text{k}}_{\text{B}}}{\text{T}}}}{{{{\text{W}}_{\text{m}}} - {{\text{k}}_{\text{B}}}{\text{TLn}}\left( {\upomega {\uptau _0}} \right)}}~~~~$$
(11)
A first approximation of this equation, it reduces to the simple expression for the exponent S as:
$$S=1+\frac{{4{{\text{k}}_{\text{B}}}{\text{T}}}}{{{{\text{W}}_{\text{m}}}}}$$
(12)
The values of Wm (the bonding energy of the carrier in its localized sites) are determined from the Eq. 12 using the slope of the S (T) curves. The values are obtained in the order of 0.108 eV and 0.156 eV for x = 0.10 and 0.15 respectively. According to this model, the AC conductivity can be described by the following expression [33]:
$${\upsigma _{{\text{AC}}}}=\frac{{{{\left( {\uppi {\text{e}}} \right)}^2}{{\text{k}}_{\text{B}}}{\text{T}}{\upalpha ^{ - 1}}\upomega {{\left[ {{\text{N}}\left( {{{\text{E}}_{\text{F}}}} \right)} \right]}^2}{\text{R}}_{{\text{W}}}^{2}}}{{12}}{\text{~}}$$
(13)
Whither \({ \text{R}}_{\text{w}}\) is the tunneling distance is given by:
$${{\text{R}}_{\text{w}}}=\frac{1}{{2{\upalpha}}}\left[ {{\text{Ln}}\left( {\frac{1}{{{\upomega}{{\uptau}_0}}}} \right) - \frac{{{{\text{W}}_{\text{H}}}}}{{{{\text{k}}_{\text{B}}}{\text{T}}}}} \right]$$
(14)
In Eqs. (13) and (14), the term \({{\upalpha }}^{-1}\) is the spatial extension of the polaron, \(\text{N}\left({\text{E}}_{\text{F}}\right)\) represents the density of states near the Fermi level and WH is the polaron hopping energy.
The \(\text{N}\left({\text{E}}_{\text{F}}\right)\), \(\alpha\) and WH were adjusted to fit the calculated curves of Ln(\({\sigma }_{ac}\)) versus T on the experimental curves. The obtained parameters are summarized in Table 5. As seen clearly that the both values of \(\text{N}\left({\text{E}}_{\text{F}}\right)\) and α increases with frequency, which is in accord with the literature [34]. Furthermore, the \(\text{N}\left({\text{E}}_{\text{F}}\right)\) values are reasonably high which suggests that the jump between site pairs dominates the charge transport mechanism in La0.75Ba0.25−xSrxFeO3 samples [35].
Table 5
Parameters used to NSPT model fitting for La0.75Ba0.25−xSrxFeO3 (x = 0.10 and 0.15) compounds at various frequencies
500 kHz | 750 kHz | 1 MHz | |
|---|---|---|---|
x = 0.10 | |||
N(EF) (1038 eV−1 m−1) | 1.8743 | 9.833 | 73.540 |
α (Å−1) | 1.12 | 1.52 | 1.79 |
WH (eV) | 0.287 | 0.221 | 0.171 |
x = 0.15 | |||
N(EF) (1039eV−1 m− 1) | 0.855 | 4.309 | 13.346 |
α (Å−1) | 1.49 | 1.66 | 1.82 |
WH (eV) | 0.353 | 0.247 | 0.115 |
3.4 Dielectric studies
The complex dielectric function is expressed as:
$${\varepsilon ^*}=\varepsilon ^{\prime}+{\text{j}}\varepsilon ^{\prime\prime}$$
(15)
where ε′ and ε″ are the real and imaginary parts of the complex permittivity. At selected temperature, we have plotted in Fig. 11 the frequency dependence of the imaginary part ε″ of the complex permittivity for x = 0.05, 0.10 and 0.15. From this plot it is clear that ε″ decreases rapidly with increase in frequency in the lower frequency region and become very stable at high frequency region. In the low frequency region, the values of the imaginary part of dielectric constant are very high. Indeed, the exchange of electrons between the Fe2+ and Fe3+ species serves to increase the orientation of dipoles and to the accumulation of charges at the interface between the grain and grain boundaries, which birth to all types of polarization such as electrodes, atomic and ionic [36]. On the other hand, at high frequency region, the exchange between the Fe2+ to Fe3+ species do not follow the alternating applied electric field; this leads the dielectric constant to remain constant across the measured temperature range [37].
Fig. 11
Variation of ε″ with frequency at several temperatures for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds
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The behavior of ε″ as a function of both frequency and temperature can be analyzed according to Giuntini et al [38]:
$$\varepsilon ^{\prime\prime}\left( \omega \right)=\left( {{\varepsilon _0} - {\varepsilon _\infty }} \right)2{\pi ^2}N{\left( {ne/{\varepsilon _0}} \right)^3}KT\tau _{0}^{m}W_{M}^{{ - 4}}{\omega ^m}={\text{B}}{{\upomega}^{\text{m}}}$$
(16)
In this equations ε0 is the static dielectric constant, ε∞ the dielectric constant at “infinitely high” frequencies, N is the concentration of the localized sites, n is the number of charge carriers and WM is the maximum barrier.
Figure 12 illustrated the variation of Ln(ε″) versus Ln(\(\omega\)) at different temperatures for all samples. These curves exhibit a series of straight lines with different slopes. From these slopes, the power “m” can be evaluated for different temperatures.
Fig. 12
Variation of ln(ε″) with ln(ω) at several temperature for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds
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The variation of the power m (T) is plotted in the inset of Fig. 12. As clearly indicated in the insert, for x = 0.05, the values of m decreases with increasing temperature and reaches a minimum value then increases slightly with increasing temperature which suggests the OLPT model [39]. An increase in m with the increase in temperature for both compounds x = 0.10 and 0.15 indicates the NSPT model [40]. It is obvious that both exponent “S” and exponent “m” have the same behavior trend. In fact, this result is satisfying the empirical law and confirms the correlation between ac conduction and the dielectric properties of semiconductor materials [41, 42].
3.5 Electrical modulus studies
The complex electric modulus formalism is necessary for the insight of the dielectric properties of the prepared samples. In Fig. 13 the plot of imaginary part of the electric modulus M″ at different temperatures is shown. It’s worthy to note that the M″ curve exhibits a maximum at a characteristic frequency, which is known as the relaxation frequency (fr). This maximum was shifted toward higher frequency side with increasing temperature. This behavior suggests that the dielectric relaxation is activated thermally and that a hopping process of charge carriers dominates intrinsically in grains [43].
Fig. 13
Frequency dependence of imaginary part of the electric modulus M″ at several temperatures for La0.75Ba0.25−xSrxFeO3 (x = 0.05, 0.10 and 0.15) compounds
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Figure 14 displays the variation of the relaxation frequency with inverse of the temperature. It is noted that this variation exhibits an activated behavior follows the Arrhenius relation [44]:
Fig. 14
Variation of the relaxation frequency with inverse of the temperature. Inset: the activation energy Ea values
$${f_r}={f_0}\exp \left( { - {{{E_a}} \mathord{\left/ {\vphantom {{{E_a}} {{k_B}T}}} \right. \kern-0pt} {{k_B}T}}} \right)$$
(17)
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The activation energy values obtained from the fitting of curves are 0.303, 0.361 and 0.404 eV for x = 0.05, 0.1 and 0.15, respectively. It is observed that these values increase with the Sr-doping and that are in good agreement with those determined from the dc conduction.
4 Conclusion
In summary, we have synthesized the La0.75Ba0.25−xSrxFeO3 perovskite with x = 0.05, 0.10 and 0.15 by sol–gel method. The impedance spectra are characterized by the appearance of semicircular arcs, well modeled in terms of electrical equivalent circuit confirmed the contribution of the grain and grain boundary. The Jonscher’s power law is well fitted to the ac conductivity data. From fitting parameter, the temperature dependence of the frequency exponent “S” showed that for x = 0.05 the conduction mechanism can be attributed to the overlapping large polaron tunneling (OLPT) model and to non-overlapping small polaron tunneling model (NSPT) for the x = 0.10 and 0.15. Activation energies are extracted from several analyses: complex impedance and modulus spectra and DC conductivity spectra dependence temperature using the Arrhenius behaviors. It was found that the activation energies increase when the Sr2+ is increased. From the permittivity analysis, according to Giuntini’s equation, the behavior of ε″ as a function of both frequency and temperature is analyzed and the temperature dependence of “m” is correlated with the AC conductivity.
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