Electrical Properties of Nanocrystalline and Bulk Nickel Ferrite

Phys. Scr. 97 (2022) 095812 https://doi.org/10.1088/1402-4896/ac87dc
PAPER
Electrical transport properties of nanocrystalline and bulk nickel
ferrite using complex impedance spectroscopy: a comparative study
Sanjeet Kumar Paswan1
, Lagen Kumar Pradhan2
, Pawan Kumar3
, Suman Kumari4
, Manoranjan Kar4
and
Lawrence Kumar1
1
Department of Nanoscience and Technology, Central University of Jharkhand, Ranchi-835205, India
2
Department of Physics, Deogarh College, Sambalpur University Deogarh-768110, India
3
Department of Physics, Mahatma Gandhi Central University, Motihari-845401, India
4
Department of Physics, Indian Institute of Technology Patna, Bihta, Patna, 801106, India
E-mail: lawrencecuj@gmail.com
Keywords: dielectric relaxation, electric modulus, activation energy, stretched exponential parameter, polaron, nyquist plot
Abstract
In this work, the comparative study on the electrical transport properties of nanocrystalline nickel
ferrite (NiFe2O4) and its bulk counterpart has been carried out in detail by using complex impedance
spectroscopy in a wide range of frequencies (100 Hz–1 MHz) and temperatures (40 °C–320 °C). The
dispersive nature of the dielectric constant and loss factor is explained by the Maxwell-Wagner model
and Koop’s phenomenological theory. The value of the dielectric constant for nanocrystalline nickel
ferrite is found to be more as compared to its bulk counterpart. The frequency variation dielectric
permittivity is well fitted with the modified Debye formula, which suggests the presence of multiple
relaxation processes. The temperature dependent ac conductivity follows Jonscher’s universal power
law and reveals the presence of multiple transport mechanisms from small polaron hopping (SPH) to
correlated barrier hopping (CBH) mechanism near 200 °C. The estimated values of Mott parameters
are found to be satisfactory. Thermally activated relaxation phenomena have been confirmed by
scaling curves of imaginary impedance ( 
Z ) and modulus ( 
M ). The comparison between the 
Z and

M spectra indicates that both long-range and short-range movement of charge carriers contribute to
dielectric relaxation with short-range charge carriers predominating at low temperatures while long-
range charge carriers are dominating at high temperatures. Analysis of the semicircular arcs of Nyquist
plot indicates the presence of grain boundary contribution to the electrical conduction process for the
nanocrystalline sample at high temperatures. The non-Debye type of relaxation has been examined by
stretching exponential factor (β) which has been estimated by fitting the modified KWW
(Kohlrausch-Williams-Watts) equation to the imaginary electric modulus curve. The value of β is
found to be strongly temperature dependent and its value for the nanocrystalline sample is less than
that of the bulk system which is explained on the basis of dipole-dipole interaction.
1. Introduction
During the last few decades, spinel ferrites have been extensively studied because of their excellent electrical and
magnetic properties for wide a range of applications [1]. They are considered good dielectric materials and have
enough potential to demonstrate a wide range of applications from microwave frequency to radio frequency [2].
AB2O4 is the general chemical formula of spinel ferrites. Among the spinel ferrite family, nickel ferrite (NiFe2O4)
has long been studied due to its exceptional magnetic and electrical properties. It is regarded as a soft magnetic
material and crystallizes into a cubic spinel structure with
-
Fd m
3 space group. It has low magnetic coercivity, low
eddy current, low dielectric loss, and high electrical resistivity which make it an excellent material for use in
various electronic, electrical, and telecommunication applications [3]. The studies on dielectric, impedance, and
electrical transport characteristics of nickel ferrite are important not only from an application point of view but
RECEIVED
6 June 2022
REVISED
27 July 2022
ACCEPTED FOR PUBLICATION
8 August 2022
PUBLISHED
19 August 2022
© 2022 IOP Publishing Ltd

also considerable from a fundamental perspective and academic interest. The electrical properties of nickel
ferrites are very susceptive to synthesis methods, sintering temperature, sintering time, composition, and grain
size [4]. There are several reports available on variations in dielectric properties of nickel ferrite synthesized by
different methods such as conventional solid-state reaction route, micro-wave synthesis, micro-emulsion, sol-
gel, etc. [5]. Besides this, nickel ferrites synthesized by similar methods are also reported to exhibit different
values of dielectric constant [6, 7]. The electrical characteristics of nickel ferrites can be examined in detail using
complex impedance spectroscopy. Investigation of dielectric and electrical transport behaviour of nickel ferrite
through complex impedance spectroscopy in a wide range of frequencies and temperatures enables the
evaluation of the contributions from grain and grain boundary to the overall electrical properties of the material
[7]. The frequency and the temperature-dependent dielectric study provide valuable information about the
behaviour of localized charge carriers which is very helpful to understand the dielectric polarization mechanism
and dielectric relaxation in spinel ferrite material [8–10]. The electrical analogous circuits containing resistors
and capacitors are used as models during complex impedance analysis to extract the details of the electrical
performance of materials [11]. A good understanding of its electrical behaviour provides the insight to explore
this material in the fabrication of modern electronic devices. The electrical conduction in spinel ferrite occurs
through the polaron hopping mechanism which is referred to as the movement of the charge carrier along with
the elastic distortion field. The hopping of the charge carrier takes place between cations of the same atom
possessing multiple valence states situated at crystallographically octahedral lattice sites (B-sites) [12]. The
mechanism of conduction in NiFe2O4 is controlled by the hopping of electron and hole via - -
+ - +
Fe O Fe
3 2 2
and - -
+ - +
Ni O Ni
2 2 3 path respectively. In addition, grains and grains boundaries also affect the conduction
mechanism significantly where grains have a low resistance while the grain boundaries have high resistance [13].
The electrical conductivity and dielectric behaviour of nickel ferrite in the bulk form have been extensively
reported in the literature [3, 14]. In recent years many authors have observed interesting dielectric and electric
properties of nickel ferrite in the nanocrystalline form due to a remarkable increase in the volume proportion of
grain boundary (high resistive) to that of grain (low resistive) [7, 15, 16]. Nevertheless, the study of the dielectric
and electrical transport behaviour of nanocrystalline NiFe2O4 is still ongoing and is the subject of continuous
investigation due to its key role in designing microwave devices which is a highly desirable area in modern
research and technology. Although a large number of studies on magnetic properties of nanocrystalline nickel
ferrite and its bulk counterpart have been reported in the literature, the electrical properties to the best of our
knowledge have not been studied extensively up to the same extent. In this work, an extensive study on the
electrical transport properties of nanocrystalline nickel ferrite with its bulk counterpart prepared by the citrate
sol-gel method have been carried out. The detailed comparative analysis of the electrical transport properties has
been carried out with the help of complex impedance spectroscopy in a wide range of frequencies and
temperatures. In the present study, an attempt has been made to establish the correlation between polarization
and electrical conduction mechanism with in-depth understanding. The obtained experimental results have
been analyzed using different dielectric and conductivity models.
2. Experimental section
The NiFe2O4 materials were synthesized by the standard citrate sol-gel method. The detail of the method of
preparation of the present NiFe2O4 system by the citrate sol-gel method has been reported in our previous
publication [17, 18]. The obtained powders were calcined at 700 °C and 900 °C for 3 h to get nanocrystalline and
bulk samples, respectively. Furthermore, the calcined powders were pressed into a cylindrical pellet of 10 mm
diameter and 1.2 mm thickness using a hydraulic press by employing 1 ton of pressure. The pellets were sintered
again at 700 °C and 900 °C for 3 h. For electrical measurements, the highly conducting silver paste was applied
on both flat circular surfaces of the pellets and dried in an oven at 120 °C for 30 min for proper binding of the
silver paste on the surface to ensure good ohmic contacts. It works as an electrode for electrical measurements.
The wires were connected with both the silver-painted circular surfaces of the pellets in parallel plate capacitor
geometry with NiFe2O4 material as the dielectric. It is electrically equivalent to capacitance in parallel with
resistance. The electrical measurement of the sintered pellets had been measured by the impedance analyzer
(N4L 1735 impedance analyzer) in a wide range of temperature (40 °C–340 °C) and frequency (100 Hz–1 MHz).
The N4L Impedance analyzer and the temperature-controlled furnace were interfaced with the computer using
the Lab-View program. The input of 2 V peak-to-peak sinusoidal signal was employed in the circuit. The density
of sintered pellet was measured by the Archimedes method employing distilled water using a digital wet balance
(Sartorius CPA225D semi microbalance).
2
Phys. Scr. 97 (2022) 095812 S K Paswan et al

3. Results and discussion
The details of the characterization of the present samples by x-ray diffraction, FESEM, EDS, XPS, Raman
spectroscopy, FTIR spectroscopy, TEM, UV absorption spectroscopy, VSM, and induction heating system have
been presented in our previous publication [17]. Briefly, Rietveld’s refined x-ray diffraction pattern revealed the
single-phase cubic spinel structure of the synthesized samples. The average crystallite size estimated by the
Rietveld analysis for 700 °C and 900 °C annealed samples was found to be around 44 nm and 88 nm,
respectively. The estimated cation distribution by the Rietveld analysis indicated mixed type spinel structure for
the 700 °C annealed sample while the 900 °C annealed sample, was very close to that of inverse type. The XPS
analysis confirmed the presence of Ni and Fe elements in both 2+ and 3+ oxidation states (Ni2+
& Ni3+
, Fe2+
,
and Fe3+
). The estimated size of the nanograin using the FESEM micrograph for the 700 °C annealed sample was
found to be around 59 nm while for the 900 °C annealed sample the grain size was 139 nm. The FESEM
micrograph revealed the size of the particle in the nanometeric range for the 700 °C annealed sample while for
the 900 °C annealed sample, the micron grain size was observed as grains coalesced to form larger grain upon
annealing. The EDS measurement indicated that the synthesized samples were stoichiometric with a 2% error.
The characteristic peak of only Fe, Ni, and O element was evident in the energy-dispersive x-ray spectroscopy
(EDS) spectrum [17].
Density plays a significant role in controlling the physical properties of polycrystalline materials. According
to Archimede’s principle, the density of sintered sample could be expressed as [19]
=
-
´
Density of sample
Weight of sample air
Weight of sample air Weight of sample water
Density of water
( )
( ) ( )
Here the density of water is 1 g cc−1
at room temperature.
The porosity of the sample could be calculated using the following relation [14]
r r
r
=
-
P 1
x
x
( )
Here P represents the porosity, rx corresponds to x-ray density, and r denotes the measured density using the
Archimedes principle. Using the Archimedes method, the measured density (r) for samples sintered at 700 °C
and 900 °C are found to be 4.448 g cc−1
and 4.989 g cc−1
, respectively. The value of x-ray density for 700 °C and
900 °C annealed samples has been reported in our previous publication [17]. The estimated porosity for 700 °C
and 900 °C sintered samples is found to be around 16% and 8%. As expected, the current estimated values of
density and porosity support the fact that increasing the sintering temperature leads to densification because the
thermal energy decreases the pore volume through the driving grain boundary [14].
3.1. Dielectric characteristics
3.1.1. Variation of dielectric constant with frequency at different temperature
The frequency dependence of the real part of the dielectric constant measured at various temperatures ranging
from 40 °C (313 K) to 320 °C (593 K) over the frequency range of 100 Hz–1 MHz for NiFe2O4 sintered at 700 °C
and 900 °Care illustrated in figures 1(a) and (b), respectively.
The observed trend of the plot suggests the existence of more than one type of polarization in the present
samples. In dielectric materials, mainly four types of polarization such as dipolar, ionic, electronic and interfacial
or space charge polarization are responsible for dielectric properties. The dipolar (or orientation) and space
charge polarization play a significant role at low frequency and both these polarizations strongly depend on the
temperature. The ionic and electronic polarizations play dominating role at high frequency and exhibit the
temperature independence. Both the plots show strong dielectric dispersion in the lower frequency range
measured at different temperatures which are mainly due to the interfacial polarization [9, 20–22]. It is evident
from the above plots that, the dielectric constant decreases rapidly with a frequency below ∼103
Hz, while in the
range of 103
Hz to 105
Hz, the dielectric constant decreases slowly with frequency. Finally, the dielectric constant
for both the sintered samples is almost independent of frequency and approaches a constant value above 105
Hz.
The above observation for the samples under study is the typical dielectric behaviour of spinel ferrite material
which is well supported by the earlier reports [23, 24]. Furthermore, this kind of behaviour could be explained by
the phenomenological Koop’s theory which considers the dielectric structure of the spinel ferrite system as a
Maxwell-Wagner (MW) type inhomogeneous medium. According to the Maxwell-Wagner model, the dielectric
structure of polycrystalline spinel ferrite is assumed to be composed of conducting grains separated by highly
resistive grain boundaries. At low frequencies, the insulating grain boundaries are more effective in contributing
to dielectric values than conducting grains. The polarization mechanism in polycrystalline spinel ferrite is
attributed to the hopping of electrons and holes between cations of the same element in different oxidation states
3

over the octahedral site [25]. According to the Rezlescue model, the hopping of electrons between Fe2+
and Fe3+
ions and the hopping of holes between Ni3+
and Ni2+
ions over the octahedral sites (B-sites) are mainly
responsible for dielectric polarization in the nickel ferrite system [15]. When the electric field is applied, the flow
of the electrons takes place in the direction of the field within the grain through hopping. By the process of
hopping, the electric charge carriers (electron and hole) reach the insulating grain boundaries. As the grain
boundaries are a highly resistive part, the electric charge carriers accumulate at the grain boundaries and induce
the large number of space charge polarization at these places which leads to a high dielectric constant at low
frequencies. As the frequency of the alternating electric field increases, the conductive grains become more
active, and the hopping of electric charge carrier’s increases resulting in a decrease in dielectric constant. Beyond
a certain limit of the applied field, the hopping (electronic exchange) of the electric charge carrier (between
Fe3+
-O-Fe2+
and Ni2+
-O-Ni3+
) cannot follow the frequency of the applied field and the electric charge carriers
reverse the direction of their motion. The alteration of their direction lags behind that of the field and
subsequently reduces the electric polarization and hence dielectric constant [25, 26]. It is evident from
figures 1(a) and (b) that in the low-frequency range; the dielectric constant is increased with the temperature.
The dielectric constant increase with temperature at low frequency indicates that space charge polarization is the
main contributor to the dielectric constant as electronic and ionic polarization is known for their temperature
independency. On the other hand, space charge polarization increases with temperature whereas dipolar
polarization decreases with temperature [9].
The possible explanation of the observed effect is that the number of charge carriers gets increase with the
increase of temperature and, the increased numbers of charge carriers build up the enhanced space charge
polarization leading to an increased dielectric constant [26–28]. A comparison of the real dielectric values at a
different frequency and different temperatures for 700 °C and 900 °C sintered samples is presented in table 1. It
is observed that at each particular frequency, the values of dielectric constant measured at different temperature
for 700 °C sintered samples is more than that of 900 °C sintered sample. Two possible explanations for the
observed effect could be provided as follow: (1) as reported in our recent publication [17], the sample NiFe2O4
sintered at 900 °C is in bulk form while the sample sintered at 700 °C is nanocrystalline form. In the case of
nanocrystalline materials, a greater number of smaller grains and grain boundaries are present as compared with
their bulk counterparts. The smaller grains are expected to contribute to high surface polarization owing to the
large surface area of smaller individual grains which results in an enhanced dielectric constant for a 700 °C
sintered sample [8]. In general, nanostructured materials consist of a large number of interfaces. At these
interfaces, one could not rule out the presence of defects that might lead to the change in space charge
distribution. The oscillation of space charge is expected under the influence of applied ac electric field and this
space charge might be trapped by interfacial defects present at the surface of the nanoparticles. This could result
in the formation of additional space charge polarization which leads to enhancing of the dielectric constant
[29, 30]. In addition, one cannot rule out the presence of oxygen vacancies on the surfaces of nanoferrite
materials. The oxygen vacancies might be considered equivalent to positive charges. This might lead to
additional dipole moments giving rise to an enhanced dielectric constant as compared to its bulk counterpart
[30]. (2) The existing literature reveals that one could expect a higher dielectric constant in the spinel ferrite
system if the concentration of Fe2+
cations over the octahedral sites is high [31]. The Fe2+
ions cause the transfer
of charge of type Fe2+
↔Fe3+
ions resulting in the local displacement of electrons in the direction of an electric
Figure 1. The variation of the real part of dielectric constant with frequency for NiFe2O4 sintered at (a) 700 °C and (b) 900 °C.
4

field, which leads to dielectric polarization [21, 32]. In our recent publication, using the study of the x-ray
diffraction pattern employing the Rietveld refinement method, the estimated cation distribution over
tetrahedral and octahedral sites for 700 °C and 900 °C annealed samples are reported to be
(Ni0.16Fe0.84)A[Ni0.84Fe1.16]B and (Ni0.06Fe0.94)A[Ni0.94Fe1.07]B respectively [17]. The estimated cation
distribution revealed the lower concentration of Fe cations over octahedral sites for 900 °C annealed samples.
The presence of both Fe2+
and Fe3+
ions at octahedral sites for the present samples has been established by the
XPS study which has also been reported in our recent publication [17]. Hence, the lower value of the dielectric
constant for the 900 °C annealed sample might be attributed to the reduction of the Fe2+
↔Fe3+
pair at
octahedral sites (B-sites) owing to the lower concentration of iron cations at B sites[32]. The high value of the
dielectric constant for the 700 °C sintered sample makes it suitable for microwave application. A similar type of
behaviour exhibited by the nanocrystalline spinel ferrite system and its bulk counterpart has been reported in the
literature [31, 33]. However, in the literature, a decrease in the dielectric constant for nanocrystalline spinel
ferrite system as compared to its bulk counterpart has also been reported by Saafan et al [9] and Mansour et al
[28].
The sample (NiFe2O4) under study consists of nickel, iron, and oxygen ions. One could expect the
contribution of these ions to the dielectric relaxation process. Hence, the dielectric relaxation behaviour of the
present sample could be studied by employing modified Debye’s relaxation equation assuming that more than
one ion contributes to the dielectric relaxation process [24]. Therefore, the modified Debye’s relaxation function
would be the appropriate model to describe the dielectric relaxation process and the dispersion in the dielectric
constant could be expressed as [24, 34].
e e
e e
wt
¢ = ¢ +
¢ - ¢
+ a
¥
¥
-
1
2
0
2 1
( )
( )
( )
( )
where e¢
¥ represents the dielectric constant at the highest frequency ( at 1 MHz), e¢
0 denotes the dielectric
constant at the lowest frequency ( at 100 Hz), w stands for the angular frequency of the field, and represented as
w p
= f
2 , f is the linear frequency of the applied electric field, t is the Debye average relaxation time and a is the
spreading factor of the actual relaxation times about the mean value. In the present study, the modified Debye
equation has been fitted well to the experimental data at all temperatures which validates the assumption of
more than one ion contributing to the dielectric relaxation process [24]. Typical fitting of the modified Debye
equation to the experimental data measured at 160 °C is depicted in figure 2 for the sample sintered at 700 °C
and 900 °C respectively. The spreading factor (α) and relaxation time (τ) for the sample sintered at 700 °C and
900 °C are enlisted in table 2. The spreading factor (α) varies between 0 and 1 which is an acceptable value
[34–36]. The spreading factor (α) and relaxation time (τ) decrease with the increase in temperature which is the
typical behaviour of high dielectric materials. The reduction of spreading factor and relaxation time with
temperature in the present study is consistent with the literature reported on the spinel ferrite system [37].
Table 1. The values of the real part of dielectric constant e¢ measured at a different
frequencies and temperatures for NiFe2O4 sintered at 700° C and 900 °C.
NiFe2O4 sintered at 700 °C
Temperature e¢ (102
Hz) e¢ (103
Hz) e¢ (104
Hz) e¢ (105
Hz)
40 °C 551 182 63 21
80 °C 1812 570 174 61
120 °C 3737 1286 387 122
160 °C 4023 2275 736 215
220 °C 4310 3808 2955 689
280 °C 11 738 4690 2013 984
300 °C 24 115 7226 2752 687
320 °C 38 568 10 530 4795 838
40 °C 470 131 40 14
80 °C 1624 431 116 34
120 °C 2880 894 272 68
160 °C 3844 2044 665 186
220 °C 4647 4005 2708 521
280 °C 8074 4288 2466 1351
300 °C 14 425 5816 2399 968
320 °C 27 930 10 501 4574 1180
5

The obtained values of spreading factor (α) and relaxation time (τ) for the sample under study are consistent
with the reported literature values [36]. The relaxation of polarization of charges might be due to the existence of
inertia in the charge movement. The length of the hopping between cations over the B-sites, bond length, and
interionic distance considerably affects the mean relaxation time and spreading factor in the spinel ferrite
system [36].
It could be seen that for 700 °C sintered sample, the mean relaxation time and spreading factor are more
than that of the 900 °C sintered sample. As reported in our previous publication [17], the length of hopping
between the cation within the octahedral site and bond length for a 700 °C annealed sample is more than that of
the 900 °C annealed sample which increases in relaxation time and spreading factor for this 700 ° sintered
sample.
3.1.2. Variation of dielectric constant with the temperature at different frequencies
The change in the real part of the dielectric constant with the temperature at selected frequencies for 700 °C and
900 °C sintered samples are shown in figures 3 and 4, respectively. In the temperature range of ∼120 °C–130 °C
and ∼250 °C–350 °C, both the sintered samples show the dielectric relaxations in the frequency range of
102
Hz–105
Hz. The observed broad hump in the frequency range of 102
Hz–105
Hz and temperature range of
120 °C–130 °C might be attributed to the dipolar polarization and dominating role of space charge polarization
due to the presence of crystal defects [32]. A clear peak is observed around the temperature range of 319 °C–
328 °C within the frequency range ∼102
Hz–105
Hz which might be ascribed to the ionic and electronic
polarization of the system [32]. The intensity of the broad hump and the intensity of peak maxima for both the
sintered samples decrease upon increasing the frequency from 102
Hz to 105
Hz. Similar observations have been
reported in the literature for nanocrystalline nickel ferrite systems and bulk cobalt ferrite compounds [34, 35]. It
is to be noted that the peak shift of the maximum dielectric constant takes place towards to higher temperature
upon increasing the frequency from 102
Hz to 105
Hz. Upon increasing the frequency, the charge carriers are
Figure 2. The fitting of the modified Debye equation to the real part of dielectric data measured at 160 °C for the sample sintered at (a)
700 °C and (b) 900 °C.
Table 2. The spreading factor (α) and relaxation time (τ) for the NiFe2O4
sintered at the 700 °C and 900 °C at the selected temperatures.
NiFe2O4 sintered
at 700 °C
NiFe2O4 sintered
at 900 °C
Temperature α τ(s) α τ(s)
40 °C 0.6596 5.6×10−4
0.5702 5.1×10−4
80 °C 0.6076 5.1×10−4
0.5558 4.9×10−4
120 °C 0.5713 3.7×10−4
0.5446 3.4×10−4
160 °C 0.5241 6×10−5
0.4921 1.5×10−4
220 °C 0.4881 6.3×10−6
0.4576 1.2×10−5
280 °C 0.4224 5.2×10−6
0.3954 2.8×10−6
300 °C 0.3517 3.9×10−6
0.3219 1.7×10−6
320 °C 0.2916 2.4×10−6
0.2516 0.9×10−6
6

unable to align with the fast changing alternating field. As a result, there is a decrease in polarization. To restore
the polarization, the system requires higher energy. The desired energy could be provided by increasing
temperature. Upon increasing the frequency, the higher temperature is required for restoration of polarization.
As a consequence, the dielectric maxima shift towards the high temperature side upon increasing the
frequency [34].
It is interestingly observed that the temperature dependence of the dielectric constant at different
frequencies is different. The dielectric constant increases more rapidly at 100 Hz as compared to other
frequencies. The four types of polarizations namely, interfacial, dipolar, ionic, and electronic contribute to the
dielectric constant. The high dielectric constant at 100 Hz is due to interfacial and dipolar polarization and
shows a strong dependence on frequency and temperature [24]. Above 105
Hz frequency, only electronic and
ionic polarization contributes to the dielectric constant and remains independent of temperature. Therefore, the
effect of temperature on high-frequency dielectric constants is not significant and results in low dispersion of
dielectric constant. It can explain the observed dependence of dielectric constant on temperature and
frequencies [24]. It could be seen clearly that in the temperature range of 40 °C–120 °C the dielectric constant is
increasing slowly with temperature in the frequency range of 102
–105
Hz. Whereas in the temperature range
250 °C–320 °C, the increase of dielectric constant with temperature is faster in the frequency range of
102
–106
Hz. It appears that in the temperature range 250 °C–320 °C the charge carriers are receiving enough
thermal energy to overcome the thermal activation energy barrier. As a result, the mobility of the charge carrier
is expected to be increased giving rise to an enhanced rate of hopping of the charge carrier and the formation of
more dipoles. By receiving the thermal energy upon temperature increase, more and more dipole moments are
expected to be oriented in the field direction which might lead to increased dielectric polarization and, hence the
rapid increase in dielectric constant [34]. In the temperature range of 40 °C–120 °C, the mobility of the charge
carrier is expected to be low due to less supplied thermal energy. Hence, one could expect the low rate of hopping
of charge carrier and slow variation of dielectric constant with temperature. It can be seen from the dielectric
versus temperature plot (figures 3 and 4) that the values of the dielectric constant of the 700 °C sintered sample
are more than that of the 900 °C sintered sample. It might be attributed to an additional contribution from
surface polarization to total dielectric polarization due to the nanocrystalline nature of the 700 °C sintered
Figure 3. The variation of real dielectric constant with the temperature at the selected frequency for NiFe2O4 sintered at 700 °C.
7

sample [8]. A more detailed study regarding the effect of microstructure on the electrical properties has been
discussed based on impedance formalism which is presented later below.
3.2. Dielectric loss (tan δ) behaviour
3.2.1. Variation of dielectric loss with frequency at different temperature
In general, the dielectric loss (tan δ) arises in ferrite materials when there is a lag in the dielectric polarization
concerning the applied alternating electric field which is caused by the presence of imperfections and impurities
in the ferrite crystal lattice [11, 38]. The variation of dielectric loss tangent (tanδ) as a function of frequency at
various temperatures for the sample sintered at 700 °C and 900 °C is illustrated in figures 5(a) and (b)
respectively. Both plot shows a decrease in dielectric loss (tan δ) with an increase in frequency at every
temperature. The behaviour of tanδ with frequency at a certain temperature also follows a similar trend as that of
variation of the real part of dielectric constant with frequency at a certain temperature (as illustrated in figure 1).
The tanδ decreases rapidly below ∼103
Hz, while the rate of decrease of tanδ in the range of 103
Hz to 105
Hz
is slow.
Above 105
Hz, the tanδ is almost independent of frequency. The above observation for the sample under
study is well supported by the earlier reports [39]. It could be explained similarly as in the case of dielectric
constant. In brief, the dielectric loss (tanδ) in the spinel ferrite system mainly originates due to the hopping of
localized electric charge carriers and the creation of defect-induced dipoles. In the low frequency region, mainly
the hopping of localized electric charge carriers contributes to high dielectric loss. In the low frequency range,
resistive grain boundaries are more effective. Therefore, higher energy is required for the exchange of charge
carriers through the cation-anion-cation (c-a-c interactions) interaction at the octahedral sites which could
result in high dielectric loss. In the high frequency range, conductive grains are dominant. Therefore, in the high
frequency range, a small amount of energy is required for the exchange of charge carriers over the octahedral
sites which could result in the small value of energy loss [9]. In summary, the observed decrease in tanδ with
frequency in the low frequency range reveals that beyond a certain critical frequency, the hopping rate of charge
carriers in the present NiFe2O4 system lags behind the alternating electric field [2]. Whereas a decrease of
dielectric loss with frequency in high-frequency range suggests the decrease in relaxation of dipole under the
Figure 4. The variation of real dielectric constant with the temperature at the selected frequency for NiFe2O4 sintered at 900 °C.
8

influence of the external applied alternating electric field [9]. Furthermore, the tan δ versus frequency plot shows
that at every particular frequency, dielectric loss tangent values increase with rising in temperature which might
be attributed to the increased conduction of thermally activated charge carriers [40]. A similar observation has
been reported in the literature for spinel ferrite nanoparticle systems [40]. Based on figures 5(a) and (b), the
dielectric loss tangent values at selected frequencies and different temperatures for both the sintered samples are
listed in table 3.
It is observed that for the 700 °C sintered sample, at 40 °C and in the frequency range of 102
Hz–106
Hz, the
loss tangent values are found to be in the range of 0.81–0.006. As discussed in the previous section, the dielectric
constant of 700 °C sintered sample at 40 °C is found to be very high. The dielectric loss value for the 700 °C
sintered samples in the entire frequency range and at 40 °C are found to be less than 1. The very high dielectric
constant and low dielectric loss at room temperature suggest that the sample under study could be used in high
frequency/microwave application because low dielectric loss is essential to intensify the skin depth so that the
microwaves penetrate the bulk of ferrite materials [41, 42]. The loss tangent values of both the sintered samples
at 40 °C and in the frequency range 102
Hz–106
Hz are comparable and it is found to be less than 1. It could be
seen from table 3 that a rise in temperature leads to higher loss tangent values for the 700 °C sintered sample as
compared to that of the 900 °C sintered sample. It might be attributed to oxygen ion vacancies [2]. The low
tangent loss values for 900 °C sintered samples might be attributed to reduced porosity, increased density,
improved connectivity between the grain, and the presence of large grain [14].
Figure 5. The variation of dielectric loss factor (tanδ) with frequency at different temperatures for NiFe2O4 sintered at (a) 700 °C and
(b) 900 °C.
Table 3. Dielectric loss tangent (tanδ) of sintered NiFe2O4 at
selected frequencies and temperatures.
Frequency
Temperature
40 °C 80 °C 120 °C 160 °C
102
Hz 0.81 7.18 20.58 101.12
103
Hz 0.42 5.31 9.36 12.04
104
Hz 0.19 2.66 5.11 2.77
105
Hz 0.07 1.07 1.74 0.86
106
Hz 0.006 0.23 0.62 0.71
102
Hz 0.75 1.69 4.11 20.12
103
Hz 0.35 1.04 1.91 2.82
104
Hz 0.13 0.33 0.75 0.52
105
Hz 0.04 0.11 0.19 0.21
106
Hz 0.003 0.02 0.05 0.16
9

3.2.2. Variation of dielectric loss with the temperature at a different frequency
The variation of loss tangent (tanδ) as a function of temperature at various frequencies of 102
Hz, 103
Hz, 104
Hz,
105
Hz, and 106
Hz for the sample sintered at 700 °C and 900 °C is depicted in ﬁgures 6 and 7 respectively. It
shows the increasing trend of tanδ with the temperature at all frequencies. Careful examination of the plot of tan
δ versus temperature at 100 Hz for both the sintered samples illustrates that the loss tangent values are almost
independent of temperature in the temperature range 40 °C–150 °C. At frequencies 103
Hz, 104
Hz, 105
Hz, and
106
Hz, the rate of increase of tan δ with temperature in the temperature range 40 °C–150 °C is very slow for both
the sintered samples. It could be seen clearly that in the frequency range 103
Hz–105
Hz, the tan δ plot for both
the sintered samples shows a broad hump at a temperature of around 126 °C. It might be attributed to defects
present in the sample [43]. Similar experimental observation in the low temperature range has been reported in
the literature for nickel ferrite nanoparticles synthesized by the co-precipitation method [43]. Beyond 150 °C
and at the all frequency (102
Hz–106
Hz), the increase of tan δ with temperature is very sharp and rapid for both
the sintered samples. This sharp increase in the value of tanδ at the higher temperature is explained on the basis
that the increase in temperature gives rise to lattice vibration which creates a phonon and the interactions of
these phonons with the thermally activated charge carriers give rise to scattering which results in the sharp
increase in tan δ [43, 44]. The presence of crystal defects including oxygen vacancies and dominancy of
conductivity at high temperatures might also be responsible for high dielectric loss [45]. Charge carriers become
more mobile with temperature, increasing dielectric loss and polarization. In addition, the high value of the
dielectric loss at the high temperature might be attributed to charge accumulation at grain boundaries [46]. With
further increase in temperature, the tan δ versus temperature plot shows a decline after a certain maximum value
at 102
Hz and 103
Hz for 700 °C sintered samples whereas the same observation for 900 °C sintered samples has
been found at 102
Hz. This behaviour is typical of relaxation losses [35, 47].
3.3. AC conductivity
Spinel ferrites are reported to have low electrical conductivity. In order to understand the conduction
mechanism for the present samples, the ac conductivity has been calculated from dielectric data using the
Figure 6. The variation of dielectric loss with the temperature at the selected frequency for NiFe2O4 sintered at 700 °C.
10

empirical relation expressed as [21]
s p e e d
= ¢
f
2 tan 3
ac o ( )
Where f is the angular frequency, e¢ is the dielectric constant and the tanδ is the dielectric loss tangent.
The variation of ac conductivity as a function of frequency at various temperatures for the sample sintered at
700 °C and 900 °C is illustrated in ﬁgures 8(a) and (b) respectively.
The literature survey provides the insight that frequency dependent ac conductivity plot for the spinel ferrite
family is characterized by a very weak frequency dependent plateau at lower frequency region and frequency
dispersion at high frequency region at a given temperature[21]. As expected, both plot shown in ﬁgures 8(a) and
(b) illustrates that in the frequency range of 102
to 104
Hz, the ac conductivity is observed to be weakly frequency
dependent for all temperatures whereas the ac conductivity is found to increase strongly with frequency in the
frequency range of 104
–106
Hz for all temperatures. The frequency dispersion in the range of 104
to 106
Hz
corresponds to ac conductivity, whereas the weakly frequency dependent plateau in the range of 102
to 104
Hz
relates to the dc conductivity of the sample [21]. Hence the total conductivity of the samples under study could
be considered a summation of both dc and ac components: s s s
= + .
dc ac DC conductivity is primarily due to
the excitation of electrons in localized states within conduction bands. The ac conductivity is basically hopping
conduction where d electrons hop between octahedral cations to transfer charge [48].
The explanation of the observed frequency dependent ac conductivity curve for insulating spinel nickel
ferrite system is presented as follow: spinel nickel ferrite structurally forms cubic close-packed oxygen lattice
with cations Ni and Fe distributed randomly over in-equivalent crystallographic tetrahedral (A-sites) and
octahedral (B-sites) sites. The cations Ni and Fe are treated isolated from each other because these cations are
surrounded by closed packed oxygen anions. Owing to this there is very little possibility of direct overlap
between the wave functions of these cations present on adjacent sites. As a result, the charge carriers like
electrons and holes associated with Fe and Ni cations are not free to move through the crystal lattice. Hence, the
charge carriers associated with these cations are regarded as localized charge carriers in crystal lattice [16].
However, in the presence of lattice vibrations, sometimes the cations are expected to come close enough. So,
Figure 7. The variation of dielectric loss with temperature at selected frequency for NiFe2O4 sintered at 900 °C.
11

under the application of applied ac electric field, there might be the probability of jumping of the electron from
one cation to another. Inside a crystal lattice, the lattices ions vibrate at finite temperatures and generate a
phonon. The coupling between a hopping electron and phonons at finite temperature distorts the elastic field of
the crystal lattice. The hopping of electrons distorts the electrical charge configuration in the lattice structure.
These hopping electrons along with their corresponding distortion field are known as polaron [13]. The
probability of hopping electrons in crystal lattice depends upon the separation between the cations involved. In
the spinel ferrite system, the two metal cations on the B sites are close together than the two metal cations on the
B and A sites. Therefore, the probability of hopping electrons from A-sites associated with cations to the B sites
associated with cations is very small as compared with that of B-B sites jumping. The extensive literature survey
reveals that electrical conduction in the NiFe2O4 system takes place primarily by the mechanism of polaron
hopping which infers that movement of the electron with its corresponding distortion causes an electrical
conductivity [12]. The polaron hopping mechanism states that under the application of external electric field
there is hopping of polarons (electrons) between ions with different valence states of the same element which are
distributed randomly over crystallographically equivalent two adjacent octahedral sites (B-sites) in spinel lattice.
By the application of an ac electric field, the hopping of electrons between Fe2+
and Fe3+
cations present at the
octahedral (B-sites) sites are mainly responsible for conductivity. In addition to this, there is also a hopping of
hole between Ni3+
and Ni2+
cations present at the octahedral (B-sites) sites which contribute to conductivity
[49, 50]. The available literature reports two types of polaron namely small polaron and large polaron. In the
small polaron model, the ac conductivity increases with frequency whereas the decrease of ac conductivity with
increasing frequency takes place for the large polaron model. In the present study, the ac conductivity is found to
be increased with increasing frequency suggesting the contribution of small polaron to conductivity [51].
Two types of hopping mechanisms are associated with charge carriers inside the crystal lattice. The first one
is nearest neighbour hopping which is also known as successful hopping and another is variable range hopping
which is also regarded as unsuccessful hopping. The total conductivity of the spinel ferrite system is a summation
of the dc and ac parts. The dc conductivity occurs in the low frequency region which could be explained by using
nearest neighbour (successful) hopping mechanism. According to this mechanism, when an ion jumps to a new
site from an initially relaxed local configuration, it would not remain in equilibrium with its neighbour ions. To
stabilize the new position of this jump ion, the neighbour ions have to move from their initial position so that
jump ions become relaxed. It results in a path of long-range translational movement of ions thereby
contributing to dc conductivity. The AC conductivity occurs in the high frequency region and it could be
explained using variable range (unsuccessful) hopping. According to the unsuccessful mechanism, when ions
jump to the new position from the initially relaxed configuration, the ions are unable to stabilize themselves at
the new site. In order to partially relax the configuration after the jump, the ions are forced to jump back to their
initial position causing volatile forward-backward movement. It results in a localized movement which gives rise
to ac conductivity [21, 26, 27].
The observed increase in conductivity with the applied electric field frequency infers that charge carriers can
hop between different localized states more frequently when field frequency increases. Apart from this, the
trapped charge carriers may also be released from the different centers of trapping, participating in conduction
processes along with hopped charge carriers and thus increasing conductivity [52]. Furthermore, the
explanation of the increase in ac conductivity with increasing frequency could also be provided based on
Maxwell-Wagner theory and the Koop model which has been discussed in the dielectric section. According to it,
Figure 8. Frequency dependent AC conductivity plot at different temperatures for NiFe2O4 sintered at (a) 700 °C (b) 900 °C.
12

the insulating grain boundaries are more effective at low frequency thereby limiting the hopping frequency of
charge carriers. Hence, ac conductivity appears to be stagnant in the low frequency region. On the other hand,
increasing the frequency of the electric field increases the hopping frequency of electrons between Fe2+
and Fe3+
ions within the conductive grains. As a result, ac conductivity increases with the frequency of the applied field
[11]. Spinel ferrites have low mobility. The increased ac conductivity does not indicate an increase in the number
of charge carriers within the system, rather it indicates an increase in the mobility of charge carriers (i.e., the rate
of charge carrier hopping) [32].
Over a wide range of frequencies, the crystalline ceramic’s ac conductivity behaviour complies with
Jonscher’s power law, which is given as follows [21, 22].
s s w
= +
T A 4
dc
s
( ) ( )
The s T
dc ( ) is dc conductivity which is independent of frequency whereas dependent on temperature and it is
affected by the charge carriers’drift [50]. The dc conduction s T
dc ( ) arises because of the thermal actuated
transition of electrons from the valence band to the conduction band [13] and due to the non-equilibrium
occupancy of the trap charges [7]. The second term w
A s is associated with pure ac conductivity and it is both
temperature and frequency dependent and related to the dielectric relaxation caused by localized charge carriers
[21, 50]. Here A is the temperature-dependent parameter having a unit of conductivity and it determines the
strength of the polarizability [53]. ‘s’ is the frequency exponent whose value lies between 0 and 1. The value of ‘s’
provides information about the interaction between the charge carriers with surroundings [53]. The constants A
and n are both temperature and composition dependent [10]. If ‘s’ is equal to zero, the electrical conduction is
frequency-independent or dc conduction i.e. charge carriers are free to drift through the system, and for ‘s’ > 0,
the conduction is frequency dependent [10]. The experimental data measured at different temperatures have
been fitted to Jonscher’s power law to extract the value of s ,
dc A and s. Typical fitting of equation (4) to the
experimental data measured at 160 °C is shown in figure 9 for 700 °C. The fitting for the 900 °C sintered sample
is shown in the inset. There is good agreement between the experimental data and fitting.
The parameter ‘s’ has been extracted for both the sintered samples at different temperatures. Figure 10
depicts the variation of exponent ‘s’ with the temperature for both 700 °C and 900 °C sintered samples.
The variation of exponent ‘s’ with temperature could be used to find the mechanism of conduction involved
in the present material [50]. Different theoretical models have been proposed by several authors for the spinel
ferrite systems to correlate the conduction mechanism of ac conductivity with s(T) behaviour. If exponent ‘s’ is
independent of temperature, one could expect the quantum mechanical tunnelling. If exponent ‘s’ increases
along with temperature, a small polaron hopping (SPH) conduction mechanism is predominant. If the
exponent ‘s’ decreases along with temperature, the correlated barrier hopping (CBH) conduction mechanism is
usually associated with conductivity. If exponent ‘s’ decreases with the temperature reaching a minimum value
followed by the increase of s with a further increase of the temperature, then the overlapping large polaron
tunnelling (OLPT) model prevails [49, 50, 54]. It is observed that for both the sintered samples the exponent ‘s’
initially increases with increasing temperature, thereafter, exhibiting a maximum followed by the decrease of ‘s’
with further increase of temperature. This trend in exponent ‘s’ indicates the possibility of a small polaron
hopping (SPH) conduction mechanism up to 200 °C(473 K) followed by a correlated barrier hopping (CBH)
conduction mechanism for temperature above 200 °C (473 K) for both 700 °C and 900 °C sintered samples.
Both the sintered samples depict the transition from SPH to CBH around the temperature of 200 °C (473 K).
Figure 9. Fitting to Jonscher’s power law for NiFe2O4 sintered at 700 °C and 900 °C (inset).
13

In summary, obtained data suggest that in the temperature range 313 K–473 K polaron might contribute to
conduction while in the temperature range 473 K–598 K hopping of charge carriers may predominate [48]. The
observed nature of variation of exponent ‘s’ agrees well with the earlier research work reported in the literature
on the spinel ferrite system [31–33, 48, 49, 55]. The available literature reports that if the value of s is equal to zero
then involved charge carriers are free to migrate inside the crystal lattice (ceramic). The obtained values of s for
the present sintered samples vary from 0.1 to 0.9 which indicates that the motion of the charge carriers is not free
to drift and it leads not only to conduction but also to a source of considerable polarization [21, 24].
In general dc conductivity is predominant at elevated temperatures and low frequencies. In the low-
frequency range, the dc conductivity is attributed to the thermally activated hopping mechanism where there is
long-range translational motion of ions. In the low-frequency range, the variation of dc conductivity with
temperature for the spinel ferrite system could be analyzed using the Arrhenius equation [16, 56] which is
governed by the empirical relation
s s
=
-E
kT
exp 5
dc
a
0 ⎡
⎣
⎤
⎦
( )
Where Ea represents the activation energy, s is the conductivity at temperature T, s0 is a temperature
independent constant and k is the Boltzman constant. The variation s
ln dc as a function of 1000/T for both the
sintered samples is presented in figure 11.
Both sintered samples exhibit an increase in dc conductivity with temperature, indicating that the samples
are semiconducting [16]. The hopping conduction mechanism explains the increase in dc conductivity with
temperature as the result of an increase in the thermally activated drift mobility of charge carriers. The
probability of hopping is determined by the activation energy, which is associated with the electrical energy
barrier encountered by the charge carrier [57]. To estimate the activation energy for the thermally stimulated
hopping process, the dc conductivity data were fitted to the Arrhenius relation. The estimated values of
activation energy for 700 °C and 900 °C sintered samples are 0.266 eV and 0.315 eV, respectively. The estimated
value of activation energy is consistent with the value reported in the literature [11, 16, 53]. It is observed that the
estimated activation energy value for 900 °C sintered sample is slightly more than that of 700 °C sintered
samples. This might be due to changes in the mobility of charge carriers associated with the modification in the
configuration of grain boundary [31]. This also suggests that for 900 °C sintered samples, more energy is
required for electron exchange between Fe2+
and Fe3+
ions. Figure 11 shows that the 900 °C sintered sample has
lower dc conductivity than that of the 700 °C sintered sample. Observations suggest that higher activation
energy is correlated with lower electrical conductivity [16].
Numerous studies on temperature dependent dc conductivity have been performed in low temperature
region (T„293) using Mott variable range hopping (VRH) equation for spinel ferrite system
[24, 53, 56, 58, 59]. However some reports are available in recent literature where temperature dependent dc
conductivity for spinel ferrite system has been analyzed employing Mott variable range hopping equation (VRH)
in the high temperature region (T 300 K) [60–63]. In view of the above, we have analyzed the temperature
dependent dc conductivity of present samples using Mott variable range hopping (VRH) equation in the high
temperature range (T 300 K). The Mott VRH equation is expressed as [24, 53, 56, 64].
Figure 10. Variationof ‘s’ with temperature for NiFe2O4 sintered at 700 °C and 900 °C.
14

s s
= - T T
exp 6
dc o 1
1
4
[ ( ) ] ( )
Where so and T1 are constant. Following the VRH equation, s
log dc has been plotted against
-
T
1
4 which is shown
in figure 12 for both the sintered samples.
The value of T1 could be estimated by linear fitting to the data. Here T1 is the characteristic temperature
coefficient which measures the degree of disorder. It is given as follows [24, 53, 56, 64].
p x
=
b
T
k N E
24
7
F
1 3
( )
( )
Where N EF
( ) represent the density of localized states at the Fermi level and x is the decay length of the localized
polaron wave function. The value is x is supposed to be equal to the cation- cation distance for the octahedral
sites in the NiFe2O4 system. The value of x for the present nickel ferrite system is about 2.94 Å which is reported
in our previous publication [17]. The estimated value of T1 is found to be 3.17×108
K and 4.76×108
K, for
700 °C and 900 °C sintered samples respectively.
Figure 11. Plot of dc conductivity as a function of inverse temperature for sample NiFe2O4 sintered at 700 °C and 900 °C. Solid lines
are fit to the data according to the Arrhenius equation.
Figure 12. Temperature dependent dc conductivity of NiFe2O4 sintered at 700 °C and 900 °C. Solid lines are fit to the data according
to the Mott VRH equation.
15

The activation energy W at a particular temperature T is given by [24, 53, 56, 64].
= b
W k T T
0.25 8
1
1
4
3
4 ( )
Using the value of T1, the calculated values of activation energy (W) at 313 K are 0.213 eV and 0.287 eV for
700 °C and 900 °C sintered samples respectively.
The hopping range of polaron (R) is given by [24, 53, 56, 64].
x
p
=
b
R
k N E T
8
9
F
1
4
1
4
[ ( ) ]
( )
The calculated value of R at 313 K is 4.93 Å and 6.42 Å for 700 °C and 900 °C sintered samples respectively. The
estimated values of T N E W R
, , &
F
1 ( ) are consistent with the values reported in the literature for the nickel ferrite
systems [53, 59, 60, 65]. The Mott VRH model set some criteria which have to be fulfilled for its applicability
such as activation energy should be greater than thermal energy (i.e. > b
W k T where b
k is Boltzman constant)
and x >
R 1[58, 62]. The values of activation energy W with the values of thermal energy at the same
temperature have been compared. It follows that the value of W is eight to nine times greater than the value of
thermal energy. The value xR is much larger than unity. Thus, the calculated Mott parameters for the
investigated sample are found to satisfy all the conditions thereby justifying the applicability of the Mott VRH
model.
The temperature dependent ac conductivity at different frequencies for 700 °C and 900 °C sintered samples
is shown in figures 13(a) and (b), respectively.
The ac conductivity at each frequency for both the sintered samples is found to increase with rising
temperature. The observed behaviour suggests the semiconducting nature of studied samples. As the
temperature increases, conductivity increases due to the increased drift mobility of the charge carriers, which in
turn increases charge hopping [22]. Both the plot shows that for each frequency, there is occurrence of different
slope in different temperature region (linear fitting in Region I and Region II). The ac conductivity values in
region II (low temperature region) increase slowly with increasing temperature whereas in the region I (high
temperature region) the conductivity increases relatively sharp with temperature. It indicates the presence of
multiple conduction mechanisms [32, 66]. In the high temperature region, it appears that the band gap between
the valence band and conduction band is narrowing, allowing electrons to hop easily from the valence band to
the conduction band, providing high conductivity values as compared to lower temperatures [10]. A similar
observation has been reported in the literature for the spinel ferrite system [10, 21].
In general, the conductivity of nanocrystalline spinel ferrite materials is less in comparison to their bulk
counterpart. Nanocrystalline spinel ferrite materials have more disorder with smaller grains and a large number
of insulating grain boundaries which act as a potential barrier to the flow of charge carriers [66]. As expected, the
size of the grains increases with the sintering temperature. Hence the conductivity of the material is expected to
increase with sintering temperature or the increase in grain size. However, in the present study, the reverse trend
is observed. The conductivity for the 700 °C sintered sample is slight more than that of the 900 °C sintered
sample across all temperature and at different frequencies. The slightly higher value of conductivity for 700 °C
sintered sample might be attributed to localized states present in the forbidden energy band gap which could
arise due to lattice imperfections. The presence of these localized states might lower the energy barriers to the
Figure 13. Ac conductivity plot as a function of inverse temperature at different frequencies for (a) 700 °C and (b) 900 °C sintered
samples.
16

electron flow thereby showing enhanced conductivity [8]. Similar observation for spinel ferrite system has been
reported in literature [61, 66]. As reported in our previous publication [17], the cation distribution for
nanocrystalline nickel ferrite is in a random manner whereas for the bulk phase it is very close to that of inverse
type. The variation of cationic distribution inside a crystal lattice may give rise to local compositional
inhomogeneity. As a result, it might lead to local polar clusters and could form the polaron in crystal lattice [56].
In summary, cationic variation, grain boundaries, size of the grain, porosity, crystal defects and stoichiometry
are some of the important factors which might be responsible for different conductivity behaviour of 700 °C and
900 °C sintered samples from each other [38]. In the present study the dc conductivity of 700 °C sintered sample
(in nanophase) is slightly higher than that of 900 °C sintered sample (bulk phase) across all the temperatures
which suggests that the creation of the polaron is easier in the nanocrystalline phase [56].
3.4. Complex impedance studies
To extract more information about the mechanism of electrical transport for the sample, impedance
measurement has been performed as function of frequency over a range of temperatures. The variation of the
real part ¢
Z
( ) of impedance with frequency from 100 Hz to 1 MHz and temperature between 40 °C and 280 °C is
illustrated in figures 14(a) and (b) for NiFe2O4 sintered at 700 °C and 900 °C respectively. Both the plots show
that values of the real part of impedance ¢
Z
( ) decrease with rising of both frequency and temperature. It points
toward an increase of the AC conductivity with the rise in frequency and temperature (discussed in the previous
section) [10]. In the temperature range of 40 °C–120 °C, both the plots show that ¢
Z is found to decrease fast in
the lower frequency region (below 104
Hz) whereas in the temperature range of 180 °C–280 °C. The temperature
dependent plateau on the low frequency side is followed by a nearly negative slope on the high frequency side.
The plateau region in ¢
Z refers a relaxation process [67]. Both the real part of impedance ¢
Z
( ) plots show that
with increasing temperature, the nearly constant segment is becoming predominant (i.e. variation of ¢
Z with
frequency is very slow at higher temperature) which suggests a strengthened relaxation behaviour upon
temperature enhancement [68]. In the low frequency range, the magnitude of ¢
Z is high at low temperature and
it decreases by increasing temperature. The higher value of ¢
Z at low temperature and lower frequencies implies
that electric polarization is larger in the sample and indicates the negative temperature coefficient of resistance
(NTCR) type behaviour [69]. Furthermore, this behaviour changes drastically in the high frequency region
where the data merge above the certain frequency irrespective of temperature. This nature of plots suggests that
there might be reduction in barrier properties of the sample with rise in temperature resulting in release of space
charge. Beside this, the merging of ¢
Z plot suggests the enhancement in AC conductivity [10, 70]. In summary,
¢
Z plot of both the sintered samples provides a sign of increasing conduction with temperature and frequency,
which implies that the present samples are expected to behave like a semiconducting material. The observed
behaviour of real impedance for the studied samples is consistent with reported ones in the literature for spinel
ferrite systems [45, 67, 71, 72]. It is observed that in the low frequency range the value of ¢
Z for 700 °C sintered
sample is low compared to that of 900 °C sintered sample. It reveals that 900 °C sintered sample is more resistant
in the low frequency region.
Furthermore, to support the analysis, the variation of imaginary part 
Z
( ) of impedance with frequency at
various temperatures is shown in figures 15(a) and (b) for NiFe2O4 sintered at 700 °C and 900 °C, respectively.
Both the plots show that, the values of imaginary impedance 
Z
( ) monotonically decrease on increasing
Figure 14. Dependence of real part of the impedance on frequency at different temperatures for NiFe2O4 sintered at (a) 700 °C and (b)
900 °C.
17

frequency in the range 40 °C–120 °C. In this temperature range (40 °C–120 °C) there is absence of peak maxima
in 
Z plot. The absence of peak maxima in these temperature ranges suggests that there is lack of current
dissipation in sample [45]. In the temperature range of 180 °C–280 °C, initially imaginary impedance 
Z
( )
increases with frequencies of applied field, thereafter, reaching to a maximum value showing a peak and then
monotonous decrease with further rise of frequency. The position of the peak corresponds to the relaxation
frequency of the sample. The appearance of peak in the temperature ranges of 180 °C–280 °C indicates the
presence of space charge relaxation effect in the samples [10, 73]. The presence of only one peak indicates the
covering of single relaxation phenomena in the explored frequency scale. It is observed that in the temperature
range of 180 °C–280 °C, the magnitude of imaginary part of impedance 
Z
( ) at the peak 
Zmax
( ) decreases with
increase in temperature and the corresponding peak maximum shifts towards higher frequency region. The
observed behaviour might be attributed to the reduction of electron-lattice coupling and the increase in the rate
of small polaron hopping with temperature rise [21, 73]. The shifting of the peak to higher frequency side with
increasing temperature indicates increasing conductivity in samples. Also broadening of the peak is observed in
both the sample with increasing temperature which indicates the presence of multiple relaxation
processes [45, 74].
Furthermore, the relaxation time (t) can be estimated from the peak maxima in the figure 15 and can be
expressed in the term of peak frequency (fp) as [71]
t
p
=
f
1
2
10
p
( )
The relaxation frequency is found to increase with temperature. As a result, the relaxation time would
decrease with temperature. Therefore, the variation of relaxation time t with temperature is expected to follow
the Arrhenius relation which is expressed as [71]
t t
= - b
E k T
exp 11
a
0 ( ) ( )
Here to is the pre-exponential factor. Ea represents the activation energy for the relaxation process, b
k is the
Boltzmann constant and T is the absolute temperature. The equation (11) can be rewritten as
= +
b
f f
E
k T
ln ln 12
p o
a
( )
Figure 16 represents the Arrhenius plot of the relaxation time for impedance spectra. The linear fitting to f
ln p
( )
versus T
1000
Arrhenius plot yields activation energy. The estimated activation energy is 0.231 eV and 0.292 eV for
700 °C and 900 °C sintered samples respectively. Also, these values are found to be near with that obtained from
the DC conductivity plot. The relaxation activation energies obtained from the conductivity plot are found to
differ slightly over the selected temperature range. This indicates that same kinds of charge carriers are governing
both the process [35, 72].
The temperature dependent relaxation dynamics could also be investigated by studying the scaling
behaviour of imaginary impedance spectra at different temperature which is in line with the widely accepted
scaling model [59]. Figures 17(a) and (b) shows the normalized plot of 

Z
Zmax
versus
f
fp
at various temperatures for
NiFe2O4 sintered at 700 °C and 900 °C, respectively.
Figure 15. Dependence of imaginary part of the impedance on frequency at different temperatures for NiFe2O4 sintered at (a) 700 °C
and (b) 900 °C.
18

Within the temperature range 180 °C–260 °C, the nature of normalized plot is similar for both the sintered
sample. In these temperature range (180 °C–260 °C), the normalized plot overlap on a single master curve. It
suggests that in these temperature ranges, the relaxation time distribution for both the sintered samples is
independent of temperature [74]. In other words, the relaxation mechanism is nearly temperature independent.
However, in the temperature range of 280 °C– 300 °C and the lower frequency range, the plots of both the
sintered samples do not overlap on a single master curve. The nature of plots within these temperature ranges
and frequency for both the sintered sample are different from each other. In the lower frequency range and at
280 °C, a broad hump is observed, and, at 300 °C clear visible peak has appeared in the scaling plot for 700 °C
sintered samples. The appearance of two peaks indicates the presence of temperature dependent distribution of
relaxation times at 300 °C. However, there is the complete absence of a peak in scaling plot for the 900 °C
sintered sample. The scaling plot shows only a broad hump within the temperature range of 280 °C–300 °C and
in the lower frequency range. It is to be noted that the broadening of the hump within the temperature range of
280 °C–300 °C for 700 °C sintered sample is more than that of 900 °C sintered sample which suggests that at a
lower frequency space charge effect is more for 700 °C sintered sample [59]. To understand the detailed
conduction mechanism, the complex impedance plane plots for 700 °C and 900 °C sintered samples at different
selected temperatures over a wide range of frequencies (100 Hz to 1 MHz) have been depicted in ﬁgures 18 and
19 respectively. In the literature the complex impedance plane plots ( 
Z versus ¢
Z ) are also known as Cole-Cole
plots or Nyquist plots.
Figure 16. Plot of f
ln p versus T
1000
for Nie2O4 sintered at 700 °C and 900 °C.
Figure 17. Scaling behavior of 
Z at different temperature for NiFe2O4 sintered at (a) 700 °C and (b) 900 °C.
19

The arrow indicates the direction of the increase in frequency (from low to high). Each point in the Cole-
Cole plot corresponds to a different frequency. The Cole-Cole plots of both the sintered samples are
characterized by the appearance of semi-circular arcs in the measured frequency range. These complex
impedance plots could be used to assess the presence of Debye or non-Debye type dielectric relaxation in the
system. It is well known that for Debye type relaxation, the center of the semicircle is located on the real axis ( ¢
Z
axis), whereas for non-Debye type relaxation the center lies below the real axis ( ¢
Z axis). The observed
semicircles at all temperatures illustrated in the Cole-Cole plots (as shown in ﬁgures 18 and 19) are found to be
depressed semicircles with their centers on a line below the real axis ( ¢
Z axis). Such behaviour indicates a
departure from ideal-Debye behaviour. The non-ideal Debye behaviour might be originating from several
factors such as the orientation of interior grain, distribution of the size of the grain, grain boundaries,
distribution of atomic defects, effect of stress-strain, etc. [73]. Owing to the presence of non-Debye type
relaxation in the system it is expected that in the material all the dipoles are not relaxed with the same relaxation
time. The distribution of relaxation time is expected in the material instead of a single relaxation time. In
summary, the shape of Cole-Cole plots suggests that electrical response is composed of more than one relaxation
Figure 18. Complex impedance plane plot of sample sintered at 700 °C measured at different temperature.
20

phenomenon with the different relaxation time period in the measured frequency and temperature range
[20, 74, 75]. The Cole-Cole plots of both the sintered samples show that the arcs of semicircles change with a rise
in temperature. This behaviour indicates the presence of a thermally activated conduction mechanism. The
Cole-Cole plots of both the sintered samples show that at 40 °C, an incomplete semicircular arc has been
obtained. As the temperature increases from 40 °C, the semicircular arc is progressively developed, and it
becomes a whole semicircular arc till 200 °C. Beginning with 220 °C, a slight segment of the arc is appeared
from the low frequency side which is connected to the semicircle. Furthermore, with further increase in
temperature, the second arc is gradually spreading to 240 °C. When the temperature finally reaches 260 °C, it
can be seen that the second arc is nearly formed. In the temperature range 260 °C–300 °C, there is a clear
appearance of two complete semicircles with the larger one at the high frequency side. It is to be noticed from
Cole-Cole plots that diameters of the semicircles show a decreasing trend with elevation in temperature. It
indicates the increase of conductivity (reduction in total resistance) in samples with temperature rise. The
observed characteristics support the typical negative temperature coefficient of resistance (NTCR) type
behaviour of samples which is usually shown by semiconducting materials [76]. In the literature, it is reported
that for polycrystalline ceramics possessing non-Debye type relaxation, it is assumed that the microstructure of
materials is composed of grains that are separated by well-defined grain boundaries where grains act as parallel
conducting plates and grain boundaries act as resistive plates [77]. By the previously reported work, generally the
Cole-Cole plots of polycrystalline ceramics are composed of three overlapping depressed semicircles depending
upon the electrical properties of the investigated material [78]. The first semicircle at a higher frequency
represents the resistance of grain only while the second semicircle falling in the intermediate frequency range
corresponds to resistance from both grain and grain boundary. Within the intermediate frequency range, the
semicircular arc representing the grain boundaries generally lies on the lower frequency side because the
relaxation time of the grain boundaries is much larger than that of grains [7]. The third semicircle at a much
lower frequency (below 100 Hz) originates due to electrode polarization [27, 79]. At a much lower frequency, the
electrode polarization results in the appearance of spike like feature in Cole-Cole plots [80]. From a close view of
these semicircles at 40 °C, it appears that for both the sintered samples the curve tends to be linearly line up
towards the 
Z axis with a steep positive slope. It clearly indicates the high insulating nature (high resistivity) of
Figure 18. (Continued.)
21

samples at this temperature [73, 81]. In the measured frequency range (100 Hz-1MHZ), we have not observed
spike like feature (inclined straight line) in the lower frequency region. Hence the possibility of relaxations due to
the electrode-sample interface in the present sample is ruled out [80]. It is also well supported by electrical
modulus analysis(discussed in the next section). In accordance with the previous reported work, we assumed
that in the measured frequency range the resulting complex impedance plane plots for both the sintered samples
in the temperature range 80 °C–240 °C is composed of two overlapping depressed semicircular arcs which are
attributed to the considerable contribution from both grain interior as well as grain boundary to electrical
conduction. This is the most accepted approach in literature to interpret the semicircle due to the distribution of
relaxation time [11, 21, 45, 69, 77, 82]. The overlapping condition depends on the difference of grain and grain
boundary relaxation time constant [83]. The assumption that the resulting impedance plot consists of two
overlapped depressed semicircles in the above temperature range, has also been well supported by the ﬁtting
carried out with the help of an electrical equivalent circuit (discussed in the next paragraph). The appearance of
two well resolved depressed semicircles in 260 °C–300 °C indicates that in this range of temperature, the effect of
grain boundary in the conduction process is expected to be more remarkable. The contribution in electrical
Figure 19. Complex impedance plane plot of sample sintered at 900 °C measured at different temperature.
22

conduction is expected to be more from the grain boundary volume as compared to the grain interior. It is to be
noticed that in the temperature range 260 °C–300 °C, the small depressed semicircles in the low frequency range
for the 700 °C sintered sample are more resolved and sharp as compared to the 900 °C sintered sample. It
appears that for 700 °C sintered sample, there is enhanced inter-granular activity in assisting the electrical
conduction. The observed result is in good accordance with the ac conductivity study. The ideal dielectric
materials are expected to have only capacitive contributions. But in real dielectric materials hardly there is 100%
capacitance, it tends to have resistance in it [84]. The semicircular pattern in the complex impedance plane plots
represents the electrical process taking place in dielectric materials which could be thought of as resulting from
the cascading effect of the parallel combination of capacitive and resistive elements [76, 84]. These resistive and
capacitive elements are arising due to the presence of grains and grain boundaries. The high frequency semicircle
side representing grain contribution which arises due to a parallel combination of grain resistance (Rg) and grain
capacitance (Cg) while the low frequency semicircle side corresponds to the grain boundaries effect which arises
due to a parallel combination of grain boundary resistance (Rgb) and grain boundary capacitance (Cgb) [69, 81].
The low frequency semicircle side is assigned to grain boundaries because the charge at grain boundaries relaxes
at a lower frequency due to a larger relaxation time than that of grain [11, 82]. The correlation between the
electrical properties of polycrystalline ceramics it’s their microstructure could be established with the help of an
equivalent electrical circuit model [84]. In the temperature range 40 °C–240 °C, the complex impedance plots
for both the sintered samples are modelled with the proposed electrical circuit as shown in ﬁgure 20(a). It
consists of parallel combination of RC and RCQ. While in the temperature range 260 °C–300 °C, the plots for
both the sintered samples are modelled with an electrical circuit consisting of the parallel combination of RCQ
and RCQ as shown in ﬁgure 20(b).
In the proposed electrical circuit Rg and Rgb represents the grain and grain boundary resistance, while Cg
and represents grain and grain boundary capacitance, respectively. In the circuit, Q is known as the constant
phase element (CPE) which accounts for deviation from Debye-like relaxation (the observed depression of the
semicircle). It is related to resistance and capacitance as =
-
C Q R
n
n
n
1 1
( )
where the values of n lies between 0 and
1. At =
n 0, it is considered to be a pure resistor while for =
n 1it is regarded as a pure capacitor. For ideal
Debye type relaxation the value n is 1 while the deviation of the value of n from unity represents non Debye type
Figure 19. (Continued.)
23

relaxation [85]. In terms of R ,
g R ,
gb Cg and Cgb the total impedance is expressed as [86]
* = ¢ - 
Z Z iZ 13
( )
where
w w
¢ =
+
+
+
Z
R
R C
R
R C
1 1
g
g g
gb
gb gb
2 2
( ) ( )
and
w
w
w
w
 =
+
+
+
Z
R C
R C
R C
R C
1 1
g g
g g
gb gb
gb gb
2
2
2
2
( ) ( )
In the circuit, the capacitor represents the polarization with storage of energy and resistor represents a
conductive path [87, 88]. The electrical parameters of the proposed electrical equivalent circuits for both the
sintered samples at various temperatures are obtained by fitting the experimental impedance data using
Z-SimpWin software. Figures 20(a) and (b) illustrate that Cole-Cole plot calculated theoretically (obtained from
the circuit using Z-SimpWin software) are in good agreement with those obtained from experiments suggesting
the accuracy and validity of the experimental data and theoretical proposed model. The above observation
justifies the correctness of choosing the equivalent circuit as depicted in figures 20(a) and (b). The estimated
values of electrical parameters like R ,
g R ,
gb Cg and Cgb at different temperatures (within 2%–3% fitting error) for
both the sintered samples are enlisted in table 4.
The relaxation time of the conducting electrons at the grain (tg) and grain boundary (tgb) can be calculated
by using the following relation [80]
t = R C 14
g g g ( )
t = R C 15
gb gb gb ( )
The estimated values of tg and tgb at all temperatures are given in table 4. Relaxation time provides a general idea
of electrical process dynamics in the material. Electrical relaxation is slowed down when the relaxation time is
high, and vice versa [80]. The activation energy related to grains and grain boundary for the samples under study
could be estimated from temperature dependent electrical resistivity plots for grains (s µ
g R
1
g
) and grains
boundaries (s µ
gb R
1
gb
) by employing the Arrhenius equation as to = b
R R e
E
k T
0
a
where Ea is the activation
energy. The plots are illustrated in figures 21(a) and (b). The activation energy of grain boundaries and grains is
estimated to be 0.471 eV and 0.422 eV for the 700 °C sintered sample. The estimated activation energy of grain
and grain boundaries for 900 °C sintered sample is found to be 0.381 eV and 0.432 eV. The activation energies
for grain boundaries and grains for the 900 °C sintered sample is less than that of the 700 °C sintered samples. It
might be attributed to reduction of porosity and defects in high temperature sintering processes [89].
As expected, the resistance of grain boundary is found to be larger than the resistance of grain at all
temperature. Therefore, insulating character of grain boundaries is expected to be more than that of grains. The
arrangement of atoms near the grain boundary region is highly disordered which could result in enhanced
electron scattering during the conduction process. The atomic disorder in grain boundary is due to the presence
Figure 20. Electrical equivalent circuit representation of complex impedance plots (a) from 40 °C to 240 °C (b) from 260 °C–300 °C.
24

of voids, stretched and broken bonds, defects, and misplaced atoms. It could act as a charge carrier trap center
and form the barrier layer for charge transport [90, 91]. As expected, both Rg and Rgb decreases with increasing
T. At lower temperatures, a highly resistive grain boundary or grain interior makes it difﬁcult for electrons to
hop at lower temperatures and affects charge mobility. As the temperature increases, grain boundary scattering
decreases, and the thermally activated localized charge carriers are allowed to drift more freely and trapped
charge carriers are released. The increase in electrical conductivity observed at higher temperatures can be
attributed to the increased rate of thermally activated charge carrier hopping conduction mechanism [7, 81, 82].
The signiﬁcant decrease of both the resistances Rg and Rgb with the increasing temperature indicates the NTCR
character of the present sintered samples. The present study corroborates a brick-layer model of a polycrystalline
ceramic material by assigning two semicircular arcs to electrical response at grain interiors and grain boundaries
[21, 74]. The observed semiconducting behaviour of samples under study could also be discussed on the line of
Goodenough argument. In the literature, Goodenough proposed that in rock-salt type structured material like
MnO, FeO, NiO and CoO, there could be the simultaneous presence of both cation-anion-cation (c-a-c)
Figure 21. Arrhenius plots of grain and grain boundary for the samples sintered at (a) 700 °C and (b) 900 °C.
Table 4. Values of impedance parameters at various temperatures were calculated from Cole-Cole plot for the samples
sintered at 700 °C and 900 °C.
700 °C sintered sample
Temp Rgb (W) Cgb (F) Rg (W) Cg (F) tgb(sec) tg (sec)
40 °C 7.93×106
4.22×10−8
2.94×106
3.31×10−8
33×10−2
9.73×10−2
80 °C 1.15×106
2.23×10−8
2.91×105
1.56×10−8
2.56×10−2
4.53×10−3
120 °C 9.79×104
1.85×10−8
9.51×104
1.15×10−8
18×10−4
10×10−4
180 °C 7.66×104
7.46×10−9
4.89×104
5.65×10−9
57×10−5
27×10−5
200 °C 3.19×104
5.97×10−9
1.12×104
3.76×10−9
19×10−5
4.21×10−5
220 °C 1.81×104
2.21×10−9
4.15×103
1.08×10−9
4×10−5
4.48×10−6
240 °C 2.67×103
9.33×10−10
1.23×103
7.96×10−10
24×10−7
9.79×10−7
260 °C 6.74×102
6.16×10−10
3.95×102
4.12×10−10
41×10−8
16×10−8
280 °C 3.36×102
4.07×10−10
1.53×102
2.26×10−10
13×10−8
3.45×10−8
300 °C 2.42×102
1.19×10−10
0.91×102
1.02×10−10
2×10−8
0.92×10−8
900 °C sintered sample
40 °C 9.21×106
5.78×10−8
3.34×106
4.38×10−8
53×10−2
14×10−2
80 °C 3.11×106
3.65×10−8
3.87×105
2.42×10−8
11×10−2
9.36×10−3
120 °C 9.98×104
2.45×10−8
9.97×104
2.73×10−8
24×10−4
27×10−4
180 °C 8.86×104
8.12×10−9
7.27×104
6.48×10−9
71×10−5
47×10−5
200 °C 5.12×104
7.32×10−9
2.28×104
4.71×10−9
37×10−5
10×10−5
220 °C 3.35×104
4.13×10−9
6.14×103
2.94×10−9
13×10−5
18×10−6
240 °C 3.96×103
9.95×10−10
2.29×103
8.87×10−10
39×10−7
20×10−7
260 °C 8.76×102
7.76×10−10
5.56×102
5.67×10−10
67×10−8
31×10−8
280 °C 5.23×102
6.15×10−10
3.11×102
3.54×10−10
32×10−8
11×10−8
300 °C 4.02×102
2.81×10−10
1.27×102
2.83×10−10
11×10−8
3×10−8
25

interactions and cation-cations (c-c) interactions. If the number of electrons in d levels is between 5 and 8
(5  
m 8, m is the number of electrons in d levels), then c-a-c interactions are stronger and it dominate over
the weak c-c interactions. When this situation occurs, it might lead to semiconducting or insulating behaviour.
When c-c interactions are dominant over c-a-c interactions, it might lead to metallic behaviour [92]. In view of
the above, it appears that in the studied temperature range for both 700 °C and 900 °C sintered samples, the c-a-
c interactions at octahedral (B sites) sites are stronger as compared to c-c interaction leading to semiconducting
behaviour [7, 31, 50, 82].
3.5. Complex electrical modulus analysis
Furthermore, complex modulus formalism has been employed to gather more information on the dynamical
aspects of electrical transport phenomena in the dielectric material such as hopping rate of charge carrier, space
charge relaxation phenomena, conductivity relaxation mechanism etc. For this purpose, the frequency
dependent real and imaginary part of electrical modulus data is usually interpreted. The complex electric
modulus *
M is represented as *= ¢ + 
M M iM where ¢
M and 
M are the real and imaginary parts of the
electric modulus. The complex electric modulus *
M is related to the complex dielectric constant *
e by the
relation *
*
e
=
M
1
[93, 94]. The variation of the real part of modulus ¢
M as a function of frequency over a range
of temperature for 700 °C and 900 °Csintered samples is shown in figures 22(a) and (b) respectively.
It is evident from both the plots that in the low frequency region, the ¢
M curves for all the temperatures tend
to merge as a single curve. Furthermore, the value of the real part of the modulus is approaching a very low value
(approaching zero) in the low frequency region. It is due to the absence of restoring force responsible for the
mobility of charge carriers in an induced electric field. The observed behaviour ¢
M in the low frequency region
suggests the negligible contribution of electrode polarization in the studied samples. In the high frequency
region, the ¢
M curves at all temperatures show a tendency of saturation at a maximum asymptotic value due to
the relaxation process. In mid frequency ranges, ¢
M curves show strong dispersion. In this range of frequencies,
there is a continuous increase in dispersion with the rise of frequency. The observed behaviour suggests the
presence of short-range mobility of the charge carrier involved in the conduction process [94, 95].
The variation of imaginary part of the modulus 
M as a function of frequency at different temperatures for
700 °C and 900 °C sintered samples are shown in figures 23(a) and (b) respectively. The plot at all temperature
for both the sintered samples shows a clear resolved peak appearing at a particular frequency which may
originate due to relaxation behaviour. It is evident from figures 23(a) and (b) that the 
M curves at all
temperatures show an asymmetric nature with respect to relaxation peak maxima whose positions are frequency
and temperature dependent. The frequency corresponds to the maximum value of electric modulus is known as
relaxation frequency ( fp). The presence of this frequency in the modulus curve suggests the conductivity
relaxation mechanism.
Also, the asymmetric broadening of the 
M curve indicates the spread of relaxation time. Furthermore, it
suggests the presence of non-Debye type relaxation in the material [45, 73]. Throughout the studied temperature
range, the plot 
M could be divided into two parts concerning relaxation peak maxima. The first part deals with
the left side of the peak maxima which is assigned as a low frequency region. Due to the low range of frequencies,
charge carriers can move over long distances, i.e., they can hop from one site to another, a phenomenon known
Figure 22. Variation of real part of modulus ¢
M with frequency at different temperature for (a) 700 °C and (b) 900 °C sintered
samples.
26

as hopping. The second part deals with the right side of the relaxation peak maximum which is termed as high
frequency region. Charge carriers in high-frequency ranges are spatially confined to their wells and make only
localized motions within them. It implies that charge carriers are mobile over a short distance. The confinement
of charge carriers in the potential well might be due to presence of defects or interfacial layers between the grains
[94, 95]. In summary, the left of the relaxation peak maxima is the area that indicates the conduction process,
while the right of the relaxation peak is the relaxation process. It is evident from figure 22 that the positions of

M peak maxima shift towards the high frequency side with an elevation of temperature. The observed
behaviour could be explained on the basis that localized charge carriers become thermally activated with an
elevation of temperature. The increasing temperature makes the hopping of charge carriers faster and leads to a
decrease in relaxation time and hence increases the relaxation frequency. The shift of the peak maxima towards
to high frequency side with an increase in temperature suggests the presence of a thermally activated relaxation
mechanism for electrical conduction [21].
The temperature dependence of relaxation dynamics could also be evaluated by the electrical modulus
scaling analysis. The scaling plots 
M at various temperatures for 700 °C and 900 °C sintered samples are shown
in figures 24(a) and (b). The 
M curves measured at different temperatures merges into a single master curve. It
implies that the sample shows the same relaxation mechanism at all temperatures i.e., the dynamic relaxation
processes are temperature independent [96]. Furthermore, the literature review reveals that the asymmetric
nature of the 
M plot of polycrystalline ceramics indicates the stretched exponential character of relaxation
times. It is defined by the empirical Kohlrausch, Williams and Watts (KWW) function [2, 15, 38, 69].
Figure 23. Variation of imaginary part of modulus (M’) with frequency at different temperature for (a) 700 °C and (b) 900 °C sintered
samples.
Figure 24. Scaling behaviour of modulus for the sample sintered at (a) 700 °C and (b) 900 °C.
27

The complex electric modulus M*
could be expressed in terms of Fourier transform of a relaxation function
f t
( ) [2, 15, 38].
* ò w
f
= - -
¥
¥
M M t
d
dt
dt
1 exp 16
0
⎡
⎣
⎢
⎛
⎝
⎞
⎠
⎤
⎦
⎥
( ) ( )
Here
e
=
¥
¥
M
1
is the inverse of high-frequency real part of the dielectric constant. The function f t
( ) is the
time evolution of the electric field within the dielectric material [2, 15, 38]. According to Kohlrausch, Williams
and Watts (KWW) the decay function is related to the relaxation time as follows [2, 15, 38, 69]
f
t
b
= - < <
b
t
t
exp , 0 1 17
m
⎜ ⎟
⎡
⎣
⎢
⎛
⎝
⎞
⎠
⎤
⎦
⎥
( ) ( )
Here tm represents the conductivity relaxation time and exponent b corresponds to stretched coefficient. b is a
non-exponential parameter representative of a distribution of relaxation time. While fitting the experimental
data, it is always advantageous to reduce the number of adjustable parameters. In view of this, Bergmann
modified equations (16) and (17) to a simple empirical relation using the imaginary part of the electric modulus.
It was termed as modified KWW function which is expressed as [97, 98]
b
b
b
b
 =

- +
+
+
b
M
M
f
f
f
f
1
1
18
max
max
max
⎜ ⎟
⎜ ⎟
⎡
⎣
⎢
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎤
⎦
⎥
( )
( )
The modified KWW function has been used to fit the 
M data in order to extract the stretched coefficient b.
Typical fitting of modified KWW function to 
M data measured at 200 °C for 700 °C and 900 °C sintered
sample is depicted in figures 25(a) and (b).
The continuous line in the plot denotes the fitted value of 
M whereas the symbol represents the
experimental data. The fitted curve and experimental data show good agreement in the figures 25(a) and (b). The
value of b indicates whether the relaxation present in the material is of Debye or non-Debye type. If the value of
b is unity then there is presence of Debye type relaxation where dipole–dipole interaction is negligible (ideal
dielectric). However, if the value of b is much less than unity then there is considerable dipole-dipole interaction
leading to non-Debye type relaxation [2, 15, 38, 99]. The values of b obtained by fitting is turned out to be less
than unity for both the sintered samples. In the present studied system, the estimated stretched exponent
parameter (β) confirms a non-Debye dielectric relaxation process. The variation of b as a function of
temperature has been depicted in figure 26.
The b valueisfoundtobeincreasedwithincreasingtemperature.Similarobservationhasbeenreportedinrecent
literaturebyHcinietalforspinelferritesystem[100].Inthroughoutstudiedtemperaturerange,thevalueof b forthe
700 °Csinteredsampleislessthanthatofthe900 °Csinteredsampleasillustratedinfigure26.Asimilarkindof
observationhasbeenreportedinrecentliteraturebyHcinietal[100].Theobservedbehaviourcouldbeexplainedon
thebasisofdipole-dipoleinteraction.Inourpreviouspublication,itisreportedthat700 °Csinteredsamplehas
nanocrystallinenaturewhilebulknaturehasbeenobservedfor900 °Csinteredsample[17].Incomparisonto900 °C
sinteredsample,itisexpectedtohaveamorevolumeofgrainboundaryforthe700 °Csinteredsampleduetoits
nanocrystallinenature.Theincreaseofgrainboundaryvolumeimpliesthatnumberofdipolesingrainboundaryis
Figure 25. Fitting of imaginary modulus data measured at 200 °C to modified KWW equation for the sample sintered at (a) 700 °C
and (b) 900 °C.
28

alsoexpectedtoincreasesignificantly.Asaresult,withinthegrainboundariestheinteractionamongthedipolesis
expectedtoincreasewhichmakesthedipolerelaxationslowerleadingtoreducedrelaxationfrequency[14,101].
Theobservedtrendof b asdepictedinfigure26clearlysuggeststhatdipole-dipoleinteractionforthe700 °Csintered
sampleismorethanthatofthe900 °Csinteredsample.
Itisadvantageoustoplottheimaginarypartofimpedance ( 
Z )andimaginarypartofelectricmodulus 
M
versusfrequencysimultaneouslybecauseitprovidesaninsightwhetherarelaxationprocessinthedielectricmaterial
isdominatedbylongrangeorshort-rangemovementofchargecarriers.Furthermore,itconfirmsthepresenceof
Debyeornon-Debyetyperelaxationinthesamples.Ifatanytemperaturethepeaksoccuratthesamefrequencyin
combined 
Z and 
M versusfrequencyplot,thentherelaxationprocessisdominatedbylongrangemovementof
chargecarriers.Furthermore,atanytemperaturetheoverlapofpeaksincombined 
Z and 
M versusfrequencyplot
takesplaceforDebyetyperelaxation.Anydeparturefromthissuggestsnon-Debyetyperelaxationbehaviour.Ifthe
peaksincombined 
Z and 
M versusfrequencyplotoccuratdifferentfrequencies,thentherelaxationprocessis
dominatedbyshortrangemovementofchargecarriersi.e.,localizedrelaxationprocessdominates[83,102].Inthe
studiedtemperaturerange,wehaveobservedtheappreciablemismatchbetweenthepeaksincombined 
Z and 
M
versusfrequencyplot.Thetypicalcombinedplotof 
Z and 
M versusfrequencyfor700 °Cand900 °Csintered
samplesmeasuredat180 °Cisshowninfigures27(a)and(b).
Inthestudiedsystem,thenon-coincidenceofpeakscorrespondingto 
Z and 
M clearlysuggeststhepresenceof
non-Debyetyperelaxationbehaviour[10,102].Inthepresentstudiedsystem,ithasbeenobservedthatmismatch
betweenpeakscorrespondingto 
Z and 
M becomessmallerwithanincreaseintemperature.Inthelower
temperaturerange,relativelylargemismatchbetweenpeakscorrespondingto 
Z and 
M hasbeenoccurred.It
Figure 26. Variation of b (stretched parameter) with temperature for the sample sintered at 700 °C and 900 °C.
Figure 27. Frequency variation of both impedance and electrical modulus data measured at 180 °C for the sample sintered at
(a) 700 °C and (b) 900 °C.
29

Electrical Properties of Nanocrystalline and Bulk Nickel Ferrite

Electrical Properties of Nanocrystalline and Bulk Nickel Ferrite

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