Direct and Alternate Current Conductivity and Magnetoconductivity of Nanocrystalline Cadmium-Zinc Ferrite below Room Temperature

Nanocrystalline cadmium-zinc ferrite samples were prepared by ball milling method and its electrical transport property were investigated within a temperature range 77 K ≤ T ≤ 300 K in presence of a magnetic field up to 1 T and in a frequency range 20 Hz to 1 MHz. The investigated samples follow a simple hopping type charge transport. The dc magnetoconductivity has been explained in terms of orbital magnetoconductivity theory. The alternating current conductivity follows the universal dielectric response σ(f)  Tf. The values of ‘s’ have a decreasing trend with temperature. The temperature exponent ‘n’ depends on frequency. The dielectric permittivity of the samples depends on the grain resistance and interfacial grain boundary resistance. The ac magnetoconductivity is positive which can be explained in terms of impedance of the sample.


Introduction
The potential application and unusual properties of nanocrystalline materials has made them an object of interest for many researches.The unique properties of these classes of materials come due to the presence of its atoms at the grain boundaries or interfacial boundaries in comparison to coarse grained polycrystalline counterparts.Nanocrystalline spinel ferrite is a group of technologically important nanomaterials that has a potential application in magnetic, electronic and microwave fields [1][2][3][4][5].Due to their relatively insulating behaviour, they are used as high frequency magnetic materials [6].In various fields like magnetic recording medium, information storage, colour imaging, bio-processing magnetic refrigeration and magneto optical devices [7][8][9], the use of nanoferrites has made them a very important material for industrial application where as the reduction of particle size to nanometre scale level, different new mechanism like super paramagnetism, quantum mechanical tunnelling, spin canting etc. makes them interesting among scientists to study their transport properties [10,11].
The general formula of ferrite are represented as M Fe O , where M is a divalent metallic ion.The spinel structure has a unit cell that consists of a cubic close-packed array of 32 oxygen ions with 64 tetrahedral sites (T sites) and 32 octahedral sites (O sites); but only eight of the T sites and 16 of the O sites are filled.A large number of investigations on structural and magnetic properties like magnetisation measurement, Mossbauer spectroscopy, neutron scattering etc. have been done on the spinel oxide nanoparticles over the last few years [10][11][12][13][14][15][16][17][18][19][20][21][22][23].The dielectric behaviour on high energy ball milling ultrafine Zinc ferrite above room temperature was reported by Shenoy et al. [17].They suggested that the defects caused by the milling, produce traps in the surface layer contributes to dielectric permittivity via spin polarised electron tunnelling between grains.Ravinder et al. [24][25][26] studied the electrical conductivity of cadmium substituted manganese ferrite and cadmium substituted copper ferrites and nickel ferrites above room temperatures.The electrical conduction in these ferrites was explained on the basis of the hopping mechanism.They show a transition near the Curie temperature in the conductivity versus temperature curve and also suggested the activation energy in the ferromagnetic region is in general less than that in the paramagnetic region.But a systematic analysis on the electron transport mechanism of cadmium substituted zinc ferrite below room temperature is still lacking.
Thus, the detailed electrical transport properties like ac and dc conductivity, ac and dc magnetoconductivity and dielectric properties of nanocrystalline Cd-Zn ferities reported within a temperature range 77 to 300 K and a frequency range 20 Hz to 1 MHz in presence as well as absence of a magnetic field up to 1 T.

Experimental
Accurately weighed starting powders of CdO (M/S Merck, 98% purity), ZnO (M/S Merck, 99% purity) and α-Fe 2 O 3 (M/S Glaxo, 99% purity) taken in 0.5:0.5:1mol% were hand-ground by an agate mortar pestle in a doubly distilled acetone medium for more than 5 h.The dried homogeneous powder mixture was then termed as unmilled (0 h) stoichiometric homogeneous powder mixture.A part of this mixture was ball milled for 3 h, 8 h, 20 h and 25 h duration at room temperature in air in a planetary ball mill (Model P5, M/S Fritsch, GmbH, Germany), keeping the disk rotation speed = 300 rpm and that of the vials~450 rpm respectively.Milling was done in hardened chrome steel vial of volume 80 ml using 30 hardened chrome steel ball of 10 mm dia, at ball to powder mass ratio 40:1.
The X-ray powder diffraction profiles of unmilled and all ball milled samples were recorded (step size = 0.02 ˚ (2), counting time = 5 s, angular range = 15 ˚ -80 ˚ 2) using Ni-filtered CuK α radiation from a highly stabilized and automated Philips generator (PW1830) operated at 40 kV and 20 mA.The generator is coupled with a Philips X-ray powder diffractometer consisting of a PW 3710 mpd controller, PW1050/37 goniometer and a proportional counter.The Rietveld's analysis based on structure and microstructure refinement method [27][28][29][30][31] has been employed which is considered to be the best method for microstructure characterization and quantitative estimations of multiphase nanocrystalline material containing several number of overlapping reflections of ferrite phase for both the unmilled and ball milled powder sample.
The electrical conductivity of the samples was measured by a standard four probe method by using 8 1 / 2 -digit Agilent 3458 multimeter and 6514 Keithley Electrometer.The ac measurement was carried out with a 4284A "Agilent Impedance" analyzer up to the frequency 1 MHz at different temperatures.Liquid nitrogen cryostat was used to study the temperature dependent conductivity by the ITC 502S Oxford temperature controller.To measure the ac response, samples were prepared as 1cm dia pellets by pressing the powder under a hydraulic pressure of 500 MPa.Fine copper wires were used as the connecting wire and silver paint was used as coating materials.The experimental density of the pressed pellets has been calculated from the relation ρ exp = m/r 2 h, where, m, r and h are mass, radius and thickness of the pellet respectively.The measured density lies in the range 5.12 g/cc to 6.38 g/cc.The percentage of error in determining the density is very small (0.21%).On the other hand density has been calculated from X-ray data by using the relation ρ theo = 8M/N A V cell , where, M is the molar mass of the sample, N A is the Avogadro's number and V cell is the unit cell volume.It is observed that the d exp is 80% to 84% of d theo for the investigated samples.The capacitance (C P ) and the dissipation factor (D) were measured at various frequencies and temperatures.The real part of ac conductivity, real and imaginary part of dielectric permittivity have been calculated using the relations σ / (f) = 2fε 0 ε // (f), ε / (f) = C P d/ε 0 A and ε // (f) = ε / (f)D respectively, where ε 0 = 8.854  10 -12 F/m, A and d are the area and thickness of the sample respectively.C P is the capacitance measured in Farad; f is the frequency in Hz.The magnetoconductivity was measured in the same manner varying the transverse magnetic field B  1 T by using an electromagnet.

Results and Discussion
The XRD powder patterns recorded from unmilled and ball milled powder mixture of CdO, ZnO and α-Fe 2 O 3 are shown in Figure 1.The powder pattern of unmilled mixture contains only the individual reflections of ZnO, CdO and α-Fe 2 O 3 phases.The intensity ratios of individual reflections are in accordance with the stoichiometric composition of the mixture.It is evident from the figure that the particle size of the starting materials reduces very fast as their peaks become broadened rapidly in the course of milling.The ferrite reflections were appeared clearly in 3 h milled sample and the content of ferrite phase increases continuously with milling time as noticed up to 25 h of milling.The CdO phase was not used up completely in the process even after 25 h of milling, whereas the other two starting phases vanishes completely in the course of milling.This indicates to the fact that the formed ferrite phase is a non-stoichiometric in composition.There must be a number of vacancies in the tetrahedral sites of the spinel ferrite lattice due to these unreacted Cd 2+ ions.
Figure 2 shows the Rietveld's fitting outputs of unmilled and all ball milled powder patterns.Peak positions   phase.It indicates that there are tetrahedral vacancies in normal spinel structure of (Zn,Cd) ferrite which are supposed to fill with Cd 2+ ions.
Figure 3 shows the variation of relative phase abundances of different phases with increasing milling time.The content (mole fraction) of ZnO phase decreases very rapidly and after 3 h of milling it becomes almost nil, whereas the variation of CdO content shows that initially the phase was utilized rapidly in ferrite phase but at the higher milling time, the rate of inclusion becomes very slow and till 25 h milling the phase was not completely incorporated in the ferrite matrix and ~0.04 mole fraction of the phase remained unreacted.The α-Fe 2 O 3 phase content decreases in a moderate rate and after 8 h of milling it almost vanishes.The ferrite phase content increases sharply up to 8 h of milling and then approaches towards a saturation to ~0.96 mole fraction (Table 1).It may be noticed that the ferrite content increases considerably until the α-Fe 2 O 3 phase was used up completely and after that a slight increment up to 25 h milling is due to a very slow diffusion of Cd 2+ ions into the ferrite matrix.All these variations in contents indicate that Zn 2+ ions have occupied the tetrahedral positions quite rapidly but the Cd 2+ ions took longer time and even after 25 h of milling some tetrahedral positions remained vacant due to insolubility of CdO in ferrite matrix.It is therefore obvious that the prepared ferrite phase is a Zn-rich non-stoichiometric (Zn,Cd)Fe 2 O 4 normal spinel with tetrahedral vacancies.
The nature of variation of lattice parameter of the cubic (Zn,Cd)Fe 2 O 4 phase with increasing milling time is shown in Figure 4.It can be seen from the plot that the lattice parameter of ferrite phase formed after 1h of milling reduces rapidly within 3 h of milling from the value 0.859 nm to ~0.849 nm (Table 1) and then remained almost invariant up to 25 h of milling.In the course of milling, ZnO phase utilized completely in tetrahedral vacancies but CdO phase was not utilized completely and some tetrahedral sites remained unoccupied even after 25 h of milling.
The scanning electron micrograph of the samples CZF3h and CZF20h are shown in Figures 5(a) and 5(b).The pictures show that the samples are not closely packed and consist of several grains.The grains are well resolved and have almost circular (spherical) shape.It is clearly seen in the micrographs that the grains are at nanoscale.The average grain size determined from SEM was noted as 35 nm -40 nm.
The dc conductivity of the Cd-Zn ferrite samples have been measured in the temperature range 77 K ≤ T ≤ 300 K.It is observed that the incorporation of Cadmium atom within zinc ferrite reduces the dc conductivity in compare to our previous study [32].The room temperature con-  ductivity σ(300 K) and conductivity ratio σ r [= σ(300 K)/ σ(77 K)] of the investigated samples increases with the increasing of milling time.The value of conductivity ratio changes between 1.02  10 2 to 2.48  10 3 which is smaller than Zn-ferrite (σ r = 1.12  10 3 to 1.21  10 6 ) [32].The contacts between the nanoparticles are the reason behind this larger conductivity ratio.The conductivity variation of all the samples has the characteristic of a semiconductor i.e. their conductivity increases with in- creasing temperature.The variation of conductivity with temperature is indicated in Figure 6.Lu et al. [18] reported the similar type of variation of conductivity in case of nanocrystalline Zinc ferrite samples.Figure 6 shows the linear variation of ln[σ(T)] with 1/T indicating a simple hopping type charge transport in all the investigated samples [33].The values of activation energy of different samples have been obtained from the slopes of different straight lines.The values of E a with milling time are shown in the inset of Figure 6.The values of E a increase with increasing milling time and hence with decreasing particle size of the samples.The increasing milling time may decrease the size of the metal core which in turn increases the activation energy.
The influence of magnetic field on the dc conductivity of the samples has been observed under a magnetic field of strength < 1 T. The magnetoconductivity ratio of the milled samples increases with increasing milling time which may be explained in terms of a simple phenomenological model named as orbital magnetoconductivity theory (Forward interference model) [34,35].Again, for the unmilled sample, there is a decrease in conductivity where t 1 = 5/2016.The variation of magnetoconductivity ratio with magnetic field intensity is shown in Figure 7.
The different points in Figure 7 are the experimental data for different samples and the solid lines are the theoretical best fit obtained from Equation (1) for milled samples and from Equation (2) for unmilled sample.For milled samples the value of C sat and B sat can be obtained as a fitting parameter of Equation ( 1) and the values are ratio with increasing magnetic field strength which may be explained by wave function shrinkage model [36].
The orbital magnetoconductivity theory predicts the forward interference among the random paths in the hopping process between the two sites spaced at a distance equal to optimum hopping distance resulting in positive magnetoconduction and can be expressed as where C sat is a temperature independent parameter and B sat is the magnetic field where the magnetoconductivity is saturated [= 0.7(h/e)(8/3) 3/2 (1/L 2 loc )(T/T mott ) 3/8 ].Where L loc is the localization length and T mott is the Mott characteristic temperature.In wave function shrinkage model, the average hopping length reduces due to the contraction of wave function of electrons under the influence of a magnetic field.As a result, the conductivity decreases with increasing magnetic field.Under a small magnetic field, the magnetoconductivity ratio can be expressed as [36]     The ac conductivity of Cd-Zn ferrite samples are investigated in the frequency range 20 Hz to 1 MHz and in the temperature range 77 K ≤ T ≤ 300 K.A general feature of the amorphous semiconductors or disordered systems is that, in addition to the dc conductivity contribution σ dc , the real part of complex ac conductivity σ / (f) is found to follow the so called universal dielectric response behavior, which can be expressed as [37][38][39]     where σ dc is the dc conductivity, α is the temperature dependent constant and the frequency exponent s ≤ 1.
The value of σ ac (f)(the frequency dependent of conductivity) has been determined upon subtraction of the dc contribution from the total frequency dependent conductivity σ / (f). Figure 8 shows the linear variation of ln[σ ac (f)] with ln(f) at different temperatures for the sample 25 h.All the other samples behave in a similar manner.The value of 's' has been calculated from the slope of these linear variation.Figure 9 shows the variation of 's' with temperature.A weak variation of 's' with temperature is observed up to 200 K but at higher temperature (T > 200 K), the value of 's' decrease with increasing temperature.In general, the conduction process of disordered systems is governed by two physical processes such as correlated barrier hopping (CBH) [39] and quantum mechanical tunneling (electron tunneling [40], small polaron tunneling [39] and large polaron tunneling [38].For different conduction process, the nature of temperature dependency of 's' are different.So the exact nature of charge transport may be obtained experimentally from the temperature dependence of 's'.According to the electron tunneling theory 's' is independent of temperature, whereas for small polaron tunneling 's' increases with increasing accordance with the CBH model.According to this model, the charge carrier hops between the sites over the potential barrier separating them and the frequency exponent 's' can be written as temperature.But in case of correlated barrier hopping model 's' only de- where W H is the effective barrier height and  0 is the characteristic relaxation time.For large value of W H /k B T, the value of 's' can be considered as independent of frequency as there is a very small variation of 's' with frequency [41].Again for the investigated samples, the linear variation of ln[σ ac (f)] with ln(f) indicates that 's' is frequency independent.Therefore the experimental data has been fitted with Equation ( 4) with W H and  0 as the fitting parameters.In Therefore the trend of variation 's' with temperature indicates that the ac conductivity of the investigated samples can be explained by the CBH model.The temperature dependence of ac conductivity is shown in Figure 10 for the sample CZF25h for different some frequencies.A weak variation is observed at lower temperature (T < 200 K) in comparison to high temperature (T > 200 K).At a particular frequency the real part of complex conductivity increases with temperature and is found to follow a power law σ / (f)  T n , which is shown as the solid lines in Figure 10.The values of n have been calculated from the power law fitting and found to be strongly frequency dependent for all samples.For different frequency ranging from 1 kHz to 1 MHz, the values of 'n' vary between 15.4 to 10.6 for CZF25h.According to the CBH model [37][38][39][40] the ac conductivity σ / (f) is expressed as σ / (f)  T 2 R ω 6  T n with n = [2 + (1s)ln(1/ω 0 )] for broad band limit and σ / (f)  R ω 6  T n with n = (1 -s)ln(1/ω 0 ) for narrow band limit, where R ω = e 2 /{εε o [W H -k B Tln(1/ω 0 )]}.The theoretical values of 'n', have been calculated taking s = 0.39 at 300 K and 0.88 at 77 K for different frequencies and the value of  0 = 3.26  10 -14 s.The variation of the calculated values of 'n' are in the range 15.61 to 11.40 at 300 K and 4.68 to 3.84 at 77 K for broad band limit and 13.61 to 9.40 at 300 K and 2.68 to 1.84 at 77 K for narrow band limit with frequency variation from 1 kHz to 1 MHz.The ex- perimental values are close to the theoretical values of broad band limit at the higher temperature range but at lower temperature, there is a discrepancy between theoretical and experimental result.
In general, interfacial polarization is exhibited by the ferrites due to structural inhomogenities and existence of free charges [42].The hopping electrons at low frequencies may be trapped by the inhomogeneties.At a particular frequency, the increase in ε / (f) with temperature is due to the drop in the resistance of the ferrite with increasing temperature.Electron hopping is promoted by the low resistance and hence resulting a larger polarizability or larger ε / (f).The variation of real part of dielectric permittivity ε / (f) with frequency for different samples are shown in Figure 11 at T = 300 K.The dielectric permittivity increases sharply with decreasing frequency for all the samples and such behaviour can be attributed to the presence of large degree of dispersion due to charge transfer within the interfacial diffusion layer present between the electrodes.The magnitude of dielectric dispersion depends on the temperature.At lower temperature the relaxation process becomes easier due to the freezing of the electric dipoles and thus there is decay in polarization with respect to the applied electric field.So a sharp increase in ε / (f) is observed at lower frequency region.Therefore the inhomogeneous nature of the samples containing different permittivity and conducting regions, governs the frequency behaviour of ε / (f) were the charge carriers are blocked by the poorly conducting regions.
The effective dielectric permittivity of this type of inhomogeneous systems is explained in terms of Maxwell Wegner capacitor model [43,44] as which the complex impedance can be modelled by an equivalent circuit con-  sisting of resistance and capacitance due to grain and interfacial grain boundary contribution and can be expressed as where sub-indexes 'g' and 'gb' refer to the grain and interfacial grain boundary respectively, R is the resistance, C is the capacitance, ω is 2πf and C 0 is the free space capacitance.The real part of the complex impedance for different samples have been calculated from the experimental data for real (ε / ) and imaginary (ε // ) part of the dielectric permittivity.Figure 12 shows the variation of the real part of the complex impedance of different samples with frequency at room temperature and Figure 13 shows the same variation for 20 h and 25 h samples in presence of a magnetic field.The points in both figures are the experimental data and the solid lines are the theoretical best fit obtained from Equation (6).It is observed from both figures that the experimental data are reasonably well fitted with the theory.The grain and grain boundary resistance and capacitance have been extracted from the analysis at room temperature whose   The variation of real part of the ac conductivity for different samples at room temperature and f = 1 MHz under the influence of magnetic field is shown in Figure 14.With increasing magnetic field, there is an increase in conductivity for different milled samples, but for unmilled sample opposite behaviour has been observed.At present, no theoretical model is found in literature which can explain directly the behaviour of ac conductivity in presence of magnetic field.The SEM micrograph reveals that the investigated samples are heterogeneous in nature with spherical grains.Thus the materials consist of grain and interfacial grain boundary regions.For such heterogeneous samples, it has been already discussed that the dielectric property and impedance depend on the grain and grain boundary resistance and capacitance.The real part of ac conductivity is related to the dielectric response by the relation σ / (B,f) = ωε 0 ε // (B,f) .As the value of ε // (B,f) is dependent on the grain and grain boundary resistance and capacitances, the ac conductivity can be written as [45]  where The change in the value of any of these resistance by the magnetic field will affect the value of the conductivity.From the analysis of the real part of complex impedance in presence of constant magnetic field at 0.76 T, it has been found that the value of grain and grain boundary resistance increases by the application of magnetic field for unmilled sample, whereas it decreases for milled samples.Hence the total contribution due to grain and grain boundary resistance (R = R g + R gb ) decrease with increasing magnetic field for milled samples.Thus the influence on ac conductivity by magnetic field is due to the change in grain and grain boundary resistance by the applied magnetic field.But due to the inability of the analytical expression, the measured data can not be compared with the theory.Thus a more explicit theoretical and experimental study is required to reveal the true mechanism of magnetic field dependent ac conductivity.

Conclusions
The different Cd-Zn ferrite samples had been prepared by the high energy ball milling method.The samples were characterized by XRD which confirms the formation of normal spinel structure with tetrahedral vacancies with particle size 7 nm.SEM picture reveals that the different samples consist of grains of almost spherical shape.The dc conductivity of different Cd-Zn-ferrite follows a simple hopping time of charge conduction mechanism.The magnetic field dependent conductivity of the differ- ent investigated samples increases with increasing magnetic field for milled samples whereas decreases with increasing magnetic field for unmilled sample and those can be explained in terms of orbital magneto conductivity theory and wave function shrinkage model respectively.The real part ac conductivity follows the power law σ / (f ) α f s .The temperature dependence of universal dielectric response parameter 's' was found to follow correlated barrier hooping change transfer mechanism.At a particular frequency the conductivity of the investigated samples follows σ / (f ) α T n , where the values of n strongly depend on frequency.The frequency dependent real part of complex permittivity shows large degree dispersion at low frequency , but rapid polarization at high frequencies which can be interpreted by Maxwell Wegner capacitor model.The grain resistance and capacitance was found to be smaller than the grain boundary resistance and capacitance and the total resistance due to grain and grain boundary decrease due to the application of a magnetic field.The ac resistivity of milled samples was found to show positive (increasing with increasing magnetic field) variation in presence of magnetic field which may be due to the variation of grain and grain boundary resistances by the application of a magnetic field.
Direct and Alternate Current Conductivity and Magnetoconductivity of Nanocrystalline Cadmium-Zinc Ferrite 227 below Room Temperature

Figure 1 .
Figure 1.X-ray diffraction patterns of unmilled (0 h) and ball-milled CdO + ZnO + α-Fe 2 O 3 powder mixture for different durations of ball-milling.ofall reflections of all four phases are marked and shown at the bottom of the plot.Residual of fitting (I 0 -I c ) between observed (I 0 ) and calculated (I c ) intensities of each fitting is plotted under respective patterns.In the present analysis, all recorded XRD patterns of ball milled samples were fitted well only with normal spinel ferrite

Figure 2 .
Figure 2. Observed () and calculated (-) XRD patterns of unmilled (0 h) and different ball-milled samples revealed from Rietveld's powder structure refinement analysis.Residues of fittings are shown under the respective patters.Peak positions of the phases are shown at the base line.

Figure 3 .
Figure 3. Variations of mole fraction of different phases in ball-milled CdO + ZnO + α-Fe 2 O 3 powder mixture with increasing milling time.

Figure 6 .Figure 7 .
Figure 6.Variation of dc conductivity of the different samples with temperature.Inset shows the variation of activation energy of different samples with milling time.

Figure 8 .
Figure 8. Variation of ac conductivity of CZF25h sample with frequency at different constant temperature.

Figure 9 .
Figure 9. Variation of frequency exponent 's' with temperature for different samples.The different solid lines are fitted with Equation (4).

Figure 9 ,
the points are the experimental data and solid lines are the theoretical best fit values obtained from Equation (4).The best fitted values of the parameters W H and  o (at a fixed frequency of f = 1 MHz) are in the range 0.82 to 1.34 eV and 1.36  10 -14 to 7.63  10 -13 s, respectively for different samples.W H has a greater value than the activation energy measured from grain and grain boundary contribution and the values of characteristic relaxation time  0 has a similar value as expected for typical inverse phonon frequency.

Figure 10 .
Figure 10.Variation of ac conductivity of CZF25h sample with temperature at different constant frequency.The lines are fitted with the equation σ / (f)  T n .

Figure 11 .
Figure 11.Variation of real part of dielectric permittivity of different samples with frequency at T = 300 K.

Figure 12 .
Figure 12.Variation of real part of impedance of different samples with frequency at room temperature.The solid lines are fitted with Equation (6).

Figure 13 .
Figure 13.Variation of real part of impedance of CZF20h and CZF25h samples with frequency at room temperature in presence of magnetic field of 0.76 T. The solid lines are fitted with Equation (6).

Table 1 . Microstructure parameters of unmilled and ball-milled CdO, ZnO and α-Fe 2 O 3 powder mixture as revealed from Rietveld's analysis of XRD data.
*Error limits.
value lie within the range 19.8 kΩ to 0.32 MΩ for R g , 0.33 MΩ to 2.79 MΩ for R gb , 0.16 to 1.94 nF for C g and 0.20 to 1.62 nF for C gb for different samples Q without magnetic field and 22.03 kΩ to 0.24 MΩ for R g , 0.35 MΩ to 2.05 MΩ for R gb , 0.18 to 1.99 nF for C g and 0.21 to 1.69 nF for C gb for different samples in presence of a magnetic field.As the resistance due to interfacial grain boundary is much larger than the grain resistance, it may conclude that the grain boundary contribution dominates over the grain contribution.