Ferrites having the general formula of MFe2O4 (M= Ca2+, Co2+, Ni2+, Mn2+, Mg2+, Zn2+ etc) are the group of hybrid functional materials. They are very superior over other nano materials as they can exhibit very good electrical and magnetic properties simultaneously at high frequency regimes. These materials are very useful in designing electrical, magnetic, microwave and other advanced electronic devices . They also find wide range of technological applications in information storage systems, transformer core, multilayered chip inductors, magneto caloric refrigeration and magnetic diagnostics [1-3] due to their predominant properties such as high electrical resistivity, high saturation magnetization, low power losses and high initial permeability [4,5]. The main advantage of these ferrites is due to their large compositional variability, which can be achieved by the substitution of different cations in the host crystal structure (at tetrahedral and octahedral sub-lattices) and as a result various physical, electrical, structural and magnetic properties can be tuned, which makes them excellent multi functional materials . Also due to the combined effect of magnetic and electrical properties ferrites they are very good candidate for multiferroic materials. Recently kumar et al have reported that ferrites behave like multiferoic materials .
The structural and electrical properties can be improved by the proper choice of divalent cations to replace Ca2+ ions, in the parent lattice of calcium ferrite which causes a structural variation in the host lattice. The distribution of cations among A- sites and B-sites is responsible for mobility of the charge carriers which plays a very important role in the observed electrical and dielectric behaviors (dielectric constant and losses) in ferrites [8, 9]. Calcium ferrite is a mixed spinal ferrite along with moderate magnetization, magneto resistance and very high magneto crystalline anisotropy. The preparation methods, substitutions and the sintering temperature have profound influence on the structural, electrical and magnetic properties of ferrites [10, 11]. Several methods have been employed to synthesize the nano ferrites with controlled microstructure and particle size including co-precipitation , sol–gel , reverse micelles , hydrothermal  and solvothermal . The Impedance spectroscopy is an important tool to study the electrical conduction behavior of the ferrites. The complex impedance studies were carried out as a function of frequency of the applied AC field to understand the dielectric dispersion and relaxation behavior of synthesized nano ferrites.
The reports found on the literature on dielectric feature and impedance spectroscopy of Barium substituted Calcium ferrite is scarce. This material has been extensively studied due to its high resistivity. So the analysis in order to understand the source of resistance, contribution of resistances from grain boundaries and grains, to resolve the relaxation contributions and conduction phenomenon in these materials is more intriguing. As per the best of our knowledge, many researchers have tried to investigate and analyze the properties like structural, electrical and dielectrical properties of ferrites using different experimental techniques but an adequate analysis of impedance spectroscopy and electric modulus of Ba substituted calcium ferrite is still lacking and offers further investigations.
In the present work we attempt to study the influence of Ca: Ba ratio on the structural, electrical and dielectric properties of the barium doped calcium ferrites synthesized by novel solution combustion method. The solution combustion method offers many advantages like it is rather simple, economical, less time consuming and scalable for industrial applications compare to other methods. The variations in the conductivity, dielectric constant and dielectric loss tangent, complex impedance, electric modulus with composition, frequency have been studied and the results are discussed.
MATERIALS AND METHODS
The barium substituted calcium ferrites BaxCa1-xFe2O4 (where x= 0.00 to 0.25 in steps of 0.05) BCAF were prepared by solution combustion method  using stoichiometric ratio of oxidizer and fuel. AR grade (Aldrich) Barium nitrate (Ba (NO)3.6H2O), ferric nitrate (Fe (NO3)3) and Calcium nitrate hexa hydrate (Ca (NO3)2.6H2O are used as oxidizers and ODH used as a fuel . The metal nitrates and ODH were dissolved in double distilled water using a magnetic stirrer to obtain the uniform mixture. This mixture was taken in a Pyrex dish and kept in a pre-heated muffle furnace maintained at 450 ± 10oC. Initially, the solution boils and subsequently frothed to yield fine brown coloured powder. The combustion process completes in 20 min. The obtained samples were then ground properly to get fine powder. The structural studies were carried out using X-ray diffractometer and the average size was calculated using Debye-Scherrer formula, Ѕ = kλ / βcosθ where k = 0.9 is the correction factor to account for the particle shapes, β is the full width at half maximum (FWHM) of the most intense diffraction peak, λ is the wavelength of a Cu-Kα radiation = 1.5406 Å and θ is the Bragg’s angle. The surface morphology of all the samples was studied by SEM (JSM-6490) at a voltage of 5 kV. The average particle size is estimated by TEM micrographs using Image J software were taken by FETEM with electron energy of 200 kV. The dielectric and a. c. conductivity studies were carried out using programmable impedance analyzer (Waynkerr 6500 B) at room temperature in the frequency range of 100Hz-5MHz.
RESULTS AND DISCUSSIONS
The Powder X-ray diffraction patterns of all samples of BCAF are shown in the Fig. 1. PXRD pattern reveals that the synthesized samples were found to be polycrystalline in nature having orthorhombic structure and it is well matched with JCPDS No 32-168. All the diffraction peaks were indexed as (320), (2 01), (131), (311), (141) (401) and (161) which confirms the phase formation of the prepared ferrites with space group Pnam (no. 62). Broad diffraction peaks indicates that the ferrite particles are of nano-sized. The crystallite size of samples were estimated using Debye-Scherrer formula and it is found that, the average crystallite size increases with increase in concentration of Ba2+.
The increase in crystallite size is due to the substitution of elements at different site that changes the unit cell parameters or volume because of the difference in ionic radii. In the present system, radius of Ba2+ (=1.35 Å) is larger than that of Ca2+ (1.00Å), Therefore, substitution of Ba2+ at the A-site may results in an increase in the crystal size as its composition is varied. The variation of crystallite size with different concentration of the Ba is shown in the Table 1.
Morphology analysis by SEM
The scanning electron micrographs (SEM) of the calcium nano ferrite sample x=0.00 are shown in Fig. 2a and Fig. 2b, the barium substituted calcium nano ferrite sample for the concentrations x=0.25 are shown in Fig. 2c and Fig. 2d. The SEM micrographs shows the presence of uniform mono phase microstructure. The micrographs reveal that the synthesized ferrites are highly porous in nature, which is due to the evolution of the gases during the solution combustion process. The porous structure has a large specific area which results in the improvement of active surface suitable for many useful applications in sensor based devices . SEM micrographs also show that sub micrometer sized primary particles are agglomerated into the larger secondary particles with slightly spherical shapes and making a compact network, hence these materials may be used for frequency dependent conductivity applications like EMI shielding, antenna and core of inductors etc . The agglomerated nature is usually observed in many nano crystalline ferrites. The EDAX measurements for calcium nano ferrite for x=0.00 is shown in Fig. 3a and for the concentrations x=0.25 are shown in Fig. 3b. It reveals that the Ca: Ba: Fe ratio in the matrix is close to the nominal compositions with an overall a slight excess of Fe observed. The EDAX results suggest that the precursors have undergone the chemical reaction fully to form the BCAF ferrite materials.
Particle size analysis by TEM
The TEM micrographs of CaFe2O4 are shown in Fig. 4. The images show slightly spherical particles, with less regularity in size. The particles are agglomerated due to their slow growth during the solution combustion process . The particles size were measured using “ImageJ” software. The particle size obtained from TEM was about 30 nm and the results are in very good agreement with the average crystallite size calculated by Debye-Scherer formula using XRD data. The HRTEM image has a lattice spacing of 3.5 nm which confirms the crystalline nature of the sample.
The study of electrical properties of ferrites is very useful, in order to unfold their suitability for the devices of frequency dependant applications. These properties depend upon materials composition, method of preparation, sintering temperature and cation positions in the unit cell. Dielectric parameters like dielectric constant, complex dielectric constant, dielectric loss and ac conductivity of BaxCa1-xFe2O4 (where x= 0.00 to 0.25 in steps of 0.05) ferrites were investigated in the frequency range 100Hz–5MHz at room temperature.
The conduction mechanism in ferrite sample is mainly due to charge libration and hopping of electrons between the ions of same element which are formed during sintering, which exist in more than one valence state ( Fe2+ and Fe3+) and distributed randomly at B-site as explained by verwey et.al . The electronic exchange between Fe2+ and Fe3+ ions results in the local displacement of charges in the direction of the applied field, which is responsible for polarization in ferrites . The rate of hopping depends on the concentration of Fe2+ and Fe3+ ion pairs at B-site. Since the increase in frequency enhances electron hopping frequency of the charge carriers between Fe2+ and Fe3+ the conductivity increases. The dependence of conductivity on frequency can also be explained on the basis of poloron hopping model reported by austin-mott  according to which σAC shows increasing trend with increase in frequency for small range poloron hopping
The equation for total electrical conductivity is given by
σ(f,T)= σdc(T) + σac(f,T) (1)
σdc is the d.c conductivity depends only on temp, the second term σac is the A.C conductivity which depends both on temperature and frequency attributed to the electron hopping mechanism at the B-site. The frequency depandance of σac is expressed by the formula
σac (f)=Afn (2)
where A and n are constants.
The variation of σac with frequency is shown in the Fig. 5.It is also observed from the figure that the value of σac shows almost frequency independent nature up to frequency of 60 kHz, as per the frequency independent conductivity observed in the case of band conduction. Whereas beyond this frequency 60 kHz (hopping frequency) the value of σac increases with frequency for all the samples of BCAF. The transition frequency from σdc to σac is termed as hopping frequency, indicating the transition from band conduction to poloron hopping conduction [25,26].
Also it can be seen from the figure that the conductivity increases with increase in concentration of Ba2+ at the expense of Ca2+. This increase in conductivity is due to increase in electron hopping process which arises due to the reduction in the field strength of surrounding ionic sites (Ca2+=0.35 and Ba2+=0.27) at Fe ions site.
The frequency dependent complex dielectric function is generally described using the expression:
ε*(ω)=ε′(ω) - iε′′(ω) (3)
ε′ is the real part of the dielectric constant, ε′′ is the imaginary part of the dielectric constant.
ε′ = C d / (εo A) (4)
Where C, d, A and εo represent the capacitance (in farad), thickness (in meters), cross-sectional area of the flat surface and constant of permittivity for free space (εo = 8.86 × 10-12 F/m).
Fig. 6 describes the variation of (ε′). The decrease in real part of dielectric constant (ε′) with increase in frequency is explained on the basis of relative frequencies of the applied electric field and electron hopping. If the frequency of the electron hopping between the Fe3+ and Fe2+ ions at octahedral site (B-site) is higher when compared to the frequency of the applied AC field, the hopping of electrons can able to interact with the field reversals of the applied electric field very easily, resulting in a higher value of dielectric constant (ε′). This tendency holds good for low frequencies. Whereas, at higher frequencies since the hopping electrons are required to move across a resistive body of the sample, they can no longer follow the field reversals of frequency of the applied field, which results in phase lagging and lowering of the dielectric constant (ε′). Consequently, the electron exchange between Fe3+ and Fe2+ is perturbed at high frequencies, which explains the lower value of dielectric constant (ε′) at high frequencies, which is in good agreement with the Koops’ model . According to Koop’s model, the ferrite consists of two layers of Maxwell-Wagner type , known as the grains (a more conducting layer) and the grain boundaries (a poor conducting layer). The number of grain boundaries increases with decrease in grain size, which contributes to the dielectric constant at lower frequencies while the grains have low dielectric constants and are effective at high frequencies. Further the species contributing to the polarizability lags behind the applied field.
From the figure it is also observed that dielectric constant is maximum for the sample for the composition x=0.00 compared to the other samples, this increased dielectric dispersion can be explained on the basis of concentration of ferrous ions at the octahedral site. For x=0.00 the concentration of ferrous ions is higher compared to the other compositions of BCAF ferrites at the octahedral site. Due to this fact it is possible for these ions to get polarized maximum. Whereas for the other compositions the Ba ions increases at the octahedral site causing the decrease in the number of ferrous ions responsible for polarization , resulting in the low value of dielectric constant.
Dielectric loss(Tan δ)
The dielectric loss factor tanδ estimates the loss of energy within the material which occurs due to the phase difference at a particular frequency and is given by the relation:
Fig. 7 shows the variation of dielectric loss with the frequency. The value of Tan δ depend upon no of factors such as stoichiometry, Fe2+ content and structural homogeneity which in turn depend upon the composition and sintering temperature of the samples. It can be seen from the figure that the value of tanδ for all the samples of BCAF has been found to decrease with increase in frequency. This nature of dielectric loss can be explained on the basis of koops phenomenological theory of dielectrics . As per this theory at lower frequencies the value of resistivity is high due to grain boundaries hence more energy is required exchange of electron between Fe2+ and Fe3+ ions resulting in high energy loss leading to large value of tanδ. Where as at high frequencies small energy is enough for the electron transfer due to low resistance resulting in low energy loss. Many partial peaks can be observed from the figure for all the compositions of BCAF, the peaks are formed whenever the frequency of the hopping charge carriers coincides with the frequency of the applied field. This behavior of ε′ and tan δ observed for BCAF ferrite samples are in good agreement with those reported by S.M. Ramay et al, M. Hashim et al and T. Ramesh et al [32-34]
Fig. 8 shows the complex impedance spectra for all the compositions of BCAF at room temperature. Impedance spectroscopy is an important method to understand the electrical properties of ferrites. According to Maxwell-wagner model a ferrite material is assumed as piled up layers of conducting plates (grains) separated by resistive plates (grain boundaries). The impedance response of such materials may consists of contributions from the grains, grain bounderies and the material electrodes in the order of decreasing frequency. The impedance of grains, grain boundaries and impedance electrodes results in a separate semicircles in complex impedance plane as they posses different relaxation times.
According to the cole-cole type of distribution, complex impedance is composed of two overlapping semicircles.
First one at low frequency corresponds to grain boundary and the second one at high frequency is due to the grain, based on the distribution of relaxation times . If the time constant of the different process differs by a factor of less than hundered then impedance responses due to grain boundary and the grain may overlap.
It can be noticed from the figure that the complex impedance spectra for all the compositions of BCAF shows only one semi circular arc corresponding to the grain boundary, which indicates that grain boundary resistance is very high for all the samples. The high value of the grain boundary resistance may be due to the increased disorder atomic arrangement, increased surface to volume ratio, porosity and decrease of Fe3+ ions. Among the different compositions of BCAF, the grain boundary resistance is small for x= 0.00 comparatively. Further from the impedance plots the data do not represent the full semi circle rather they are partial semicircles. Also it is observed that the centre of semicircles lies below the abscissa (Z′) axis which suggests the dielectric relaxation is of non-Debye type in the all BCAF samples. It is also noticed that the radius of the semicircle is different for different samples because of variation in the relaxation time [36,37].
Electric Modulus analysis
From the analysis of complex electric modulus the electrical responses of the material and electrical relaxation of ionic solids can be understood. Typical features M* as a function of frequency may be broad, asymmetric peak in the imaginary part and a sigmoidal step in the real part. In order to reconfirm the relaxation behavior in BCAF sample the real and imaginary parts of the electric modulus are calculated using the relations
Frequency dependence of the M′ for the BCAF samples are illustrated in the Fig. 9. The M′ shows continuous dispersion with increase in frequency. Dispersion at low frequency suggest that the conduction phenomenon due to short range mobility of charge carriers. It is also found the dispersion shifts towards the high frequency region for the Barium content X=0.15, which suggests the long range mobility of the charge carriers .
The variation of M′′ as a function of frequency of different Ba concentration is illustrated in Fig. 10. The Fig shows the well resolved peak in the pattern. The peak in the low frequency region determines the range in which the charge carriers can move over long distances. The peak towards the high frequency region determines the range in which the charge carrirs are spatially confined to their potential wells and being mobile over short distances .
Thus the peak frequency indicates the transition from long range to short range mobility with increase in frequency. The peak frequency shifts towards the left for the higher values composition. The characteristic frequency at which M′′ is maximum correspond to relaxation frequency form which relaxation time can be evaluated using the formula. τM′′ = [1/(2πfM′′)]. The relaxation time and frequency are provided in Table 1. The given plot is asymmetric nature with respect to peak maxima and the peaks are considerably broader on either side of the maxima, this type of nature is being the consequence of the distributions relaxation time due to non-debye nature of the material. It is observed that τ is of the order milliseconds , it is noteworthy that τ is typically of the order of 100 ns in semiconductors  and they are on the order of picoseconds in bulk materials due to the high density of scattering centers [40,41]. Such a long dielectric spin –relaxation time in the BCAF nano composites can be very useful in nano scale spintronic devices.
The present work summarizes the sol–gel synthesis of Ba2+ doped BaxCa1-xFe2O4 (where x=0.0, 0.05, 0.1, 0.15.0.2& 0.25) by solution combustion method using ODH as a fuel. The PXRD patterns confirm the formation of single phase. The crystallite size was estimated through the Scherer’s formula is found to be in the range 27-44 nm. AC conductivity increases linearly with the applied field. The real and imaginary part of permittivity have been found to decreases with increase in frequency of the applied field as well as with the dopant concentration due to electronic hopping between Fe3+ and Fe2+. Complex impedance analysis for the ferrites under study reveals that grain boundary resistance is dominant for all the Ba-substituted calcium ferrite. Also, the impedance and modulus analysis confirms the presence of non-Debye type of relaxation in Ba substituted Calcium ferrite. The synthesized material possesses lower value of dielectric loss at high frequency which are suitable frequency dependant automotive applications and long dielectric spin times have also been determined from the electric modulus spectra that make the BCAF nano composite as a promising candidate for nano scale spintronic devices.
The authors would thank Principal and management of Sai Vidya institute of technology Bangalore and A.C.S College of Engineering, Bangalore. We are thankful Dr. B.M. Nagbhushan. H.O.D, Department of chemistry, MSRIT, Bangalore, we are also thankful to Dr. Murugendrappa, Department of Physics, BMSCE for providing the lab facilities to carry out electrical properties.
CONFLICT OF INTEREST
The authors declare that there are no conflicts of interest regarding the publication of this manuscript.