*3.7. Dielectric Loss Tangent*

Dissipation of energy is measured with respect to alternating external field which is recorded in terms of dielectric loss. The variation in dielectric loss as a function of frequency for varying Zn concentration in hematite is shown in Figure 11.

*Crystals* **2020**, *10*, x FOR PEER REVIEW 14 of 19

**Figure 11.** Variation of dielectric loss with frequency of pure Fe2O3, Zn 2%, Zn 4% and Zn 6% nanoparticles. **Figure 11.** Variation of dielectric loss with frequency of pure Fe2O<sup>3</sup> , Zn 2%, Zn 4% and Zn 6% nanoparticles.

A report by *Iwauchi et al.* [43] showed that conduction hopping and dielectric behavior are

strongly correlated. The dielectric loss is high at lower frequencies, due to the grain boundary behaving as an insulating interface, as the charge carriers undergo space charge polarization [44]. Diamagnetic dopant and organized growth of domains have a major impact on decrease of loss tangent at small frequencies. The loss factor decreases at higher frequencies, due to the mismatch of electrons with applied field frequency, as discussed above. Loss is dependent on various factors, such as ferric and ferrous content, the stoichiometry of material and heterogeneous domain wall geometry. It can be observed that loss tangent has a relaxation peak for α-ZnxFe2−xO3, which is consistent with earlier reports [45,46]. According to Rezlesque model [47], a peak in dielectric loss is expected when the hopping frequency of electrons between Fe2+ and Fe3+ states is in resonance with the external applied electric field's frequency. The maxima in dielectric loss can be expressed as a relation ωτ = 1, where ω is the angular frequency of field and τ is relaxation time for hopping mechanism. The increase in peak height, as well as shifting in peak position with doping of Zn shows the variation in hopping probability of electrons between Fe2+ and Fe3+ states, and this is influenced by the number of Fe3+ ions in the octahedral site [48]. *3.8. AC Conductivity*  To study the hopping mechanism, ac conductivity (σac) versus logf of Zn doped hematite at room temperature is plotted (Figure 12a). At lower frequencies, conductivity seems to be constant and A report by Iwauchi et al. [43] showed that conduction hopping and dielectric behavior are strongly correlated. The dielectric loss is high at lower frequencies, due to the grain boundary behaving as an insulating interface, as the charge carriers undergo space charge polarization [44]. Diamagnetic dopant and organized growth of domains have a major impact on decrease of loss tangent at small frequencies. The loss factor decreases at higher frequencies, due to the mismatch of electrons with applied field frequency, as discussed above. Loss is dependent on various factors, such as ferric and ferrous content, the stoichiometry of material and heterogeneous domain wall geometry. It can be observed that loss tangent has a relaxation peak for α-ZnxFe2−xO3, which is consistent with earlier reports [45,46]. According to Rezlesque model [47], a peak in dielectric loss is expected when the hopping frequency of electrons between Fe2<sup>+</sup> and Fe3<sup>+</sup> states is in resonance with the external applied electric field's frequency. The maxima in dielectric loss can be expressed as a relation ωτ = 1, where ω is the angular frequency of field and τ is relaxation time for hopping mechanism. The increase in peak height, as well as shifting in peak position with doping of Zn shows the variation in hopping probability of electrons between Fe2<sup>+</sup> and Fe3<sup>+</sup> states, and this is influenced by the number of Fe3<sup>+</sup> ions in the octahedral site [48].

#### using log σac versus logf as shown in Figure 12b. In large polaron model, ac conductivity decreases, while in the small polaron model, ac conductivity increases with rise in frequency. In the present *3.8. AC Conductivity*

study, conductivity shows almost linear behavior with increases in frequency that indicates conduction hopping is followed by small polaron mechanism, as evident from Figure 12b. Conductivity is more affected by grain boundaries at lower frequencies, while grains have more impact on conduction at high frequencies [49]. The increase in frequency enhances the hopping of charge carriers between ferric and ferrous ions that leads to increase in conductivity. Low conductivity is observed at lower frequencies which is due to the blocking effects at grain boundaries. To study the hopping mechanism, ac conductivity (σac) versus logf of Zn doped hematite at room temperature is plotted (Figure 12a). At lower frequencies, conductivity seems to be constant and increases with rise in frequency. The type of polarons involved in hopping mechanism was estimated using log σac versus logf as shown in Figure 12b. In large polaron model, ac conductivity decreases, while in the small polaron model, ac conductivity increases with rise in frequency. In the present study, conductivity shows almost linear behavior with increases in frequency that indicates conduction hopping is followed by small polaron mechanism, as evident from Figure 12b. Conductivity is more affected by grain boundaries at lower frequencies, while grains have more impact on conduction at high frequencies [49]. The increase in frequency enhances the hopping of charge carriers between ferric and ferrous ions that leads to increase in conductivity. Low conductivity is observed at lower frequencies which is due to the blocking effects at grain boundaries.

increases with rise in frequency. The type of polarons involved in hopping mechanism was estimated

The relation between frequency and ac conductivity can be depicted as [50],

$$
\sigma\_{ac} = 2\pi f \varepsilon\_0 \varepsilon' \tan \delta \tag{15}
$$

where, f is frequency in Hz. The conductivity decrement with Zn dopant concentration could be described by the microstructures of the material, the probability of hopping and hopping duration of the electrons. This may arises due to the reduction of Fe3<sup>+</sup> ions in octahedral site and creation of Fe3<sup>+</sup> vacancies by substitution of Zn2<sup>+</sup> ions. *Crystals* **2020**, *10*, x FOR PEER REVIEW 15 of 19

**Figure 12.** (**a**) Frequency dependency AC conductivity of pure Fe2O3, Zn 2%, Zn 4% and Zn 6% nanoparticles and (**b**) Linear plot of log σac versus logf. **Figure 12.** (**a**) Frequency dependency AC conductivity of pure Fe2O<sup>3</sup> , Zn 2%, Zn 4% and Zn 6% nanoparticles and (**b**) Linear plot of log σac versus logf.

#### The relation between frequency and ac conductivity can be depicted as [50], *3.9. Modulus Properties*

*3.9. Modulus Properties* 

ᇱߝߝ݂ߨ2 = ߪ (15) ߜܽ݊ݐ where, f is frequency in Hz. The conductivity decrement with Zn dopant concentration could be described by the microstructures of the material, the probability of hopping and hopping duration of the electrons**.** This may arises due to the reduction of Fe3+ ions in octahedral site and creation of Fe3+ The electric modulus plays an important role in the study of conduction and relaxation behavior of materials, and also, in detecting the impedance sources like grains, grain boundary conduction effect, electrode polarization and electrical conductivity. The real (M') and imaginary (M") components of electric modulus can be obtained using dielectric constant (ε 0 and ε") [51]:

$$M^\* = \frac{\varepsilon'}{\varepsilon'^2 + \varepsilon''^2} + i \frac{\varepsilon'}{\varepsilon'^2 + \varepsilon''^2} = M' + iM''. \tag{16}$$

The electric modulus plays an important role in the study of conduction and relaxation behavior of materials, and also, in detecting the impedance sources like grains, grain boundary conduction The real component (M') of electric modulus represents the energy given to the system, and the imaginary component (M") represents the dissipated energy during the conduction process.

effect, electrode polarization and electrical conductivity. The real (M') and imaginary (M") components of electric modulus can be obtained using dielectric constant (ε′ and ε″) [51]: ఌᇱ∗ = ܯ ఌᇱమାఌ"మ + ݅ ఌᇱ ఌᇱమାఌ"మ = ܯ<sup>ᇱ</sup> + ݅ܯ".) 16 ( The real component (M') of electric modulus represents the energy given to the system, and the imaginary component (M") represents the dissipated energy during the conduction process. The frequency dependence of electric modulus (M', M") at room temperature is shown in Figure 13. It is observed that at lower frequencies M' is very small nearly to be zero and a continuous dispersion with frequency increases having a tendency to saturate at a maximum value for all the samples at higher frequencies due to the relaxation process. These observations implies the lack of restoring force for flow of charge carriers under the action of steady electric field. The small value of M' at low frequency supports the long range mobility of charge carriers. While, in higher frequencies, M' increases rapidly with frequency, indicating that the conduction mechanism, which may be due to the short range mobility of charge carriers.

**Figure 13.** Frequency dependence (**a**) real component (M') and (**b**) Imaginary component (M") of electric modulus of pure Fe2O3, Zn 2%, Zn 4% and Zn 6% nanoparticles. The imaginary part of electrical modulus M" shows an increasing trend compared to frequency with relaxation peaks for all samples. The frequency region below peak frequency represents the frequency range by which ions drift to long distance, i.e., performing successful hopping from one site to neighboring site. Whereas, the high frequency region above the peak shows that the carriers are confined to their potential wells and can make localized motion inside the well. The occurrence of peak in electrical modulus M" indicates the transition from long range to short range mobility of charge carriers with rise in frequency. The behavior of the modulus spectrum is indicative of hopping type mechanism for electrical conduction in the system. The broadening of the peaks is the consequence of the distributions of relaxation time arise from the non-Debye type of the material. Further, it is observed that the peaks shift towards the higher frequency side, with Zn doping and the obtained relaxation peaks having resonance peaks, where the oscillating dipoles frequency matches the applied field frequency.

**Figure 12.** (**a**) Frequency dependency AC conductivity of pure Fe2O3, Zn 2%, Zn 4% and Zn 6%

ᇱߝߝ݂ߨ2 = ߪ

where, f is frequency in Hz. The conductivity decrement with Zn dopant concentration could be described by the microstructures of the material, the probability of hopping and hopping duration of the electrons**.** This may arises due to the reduction of Fe3+ ions in octahedral site and creation of Fe3+

The electric modulus plays an important role in the study of conduction and relaxation behavior of materials, and also, in detecting the impedance sources like grains, grain boundary conduction effect, electrode polarization and electrical conductivity. The real (M') and imaginary (M")

The real component (M') of electric modulus represents the energy given to the system, and the

(15) ߜܽ݊ݐ

ఌᇱమାఌ"మ = ܯ<sup>ᇱ</sup> + ݅ܯ".) 16 (

The relation between frequency and ac conductivity can be depicted as [50],

components of electric modulus can be obtained using dielectric constant (ε′ and ε″) [51]:

ఌᇱమାఌ"మ + ݅ ఌᇱ

imaginary component (M") represents the dissipated energy during the conduction process.

ఌᇱ∗ = ܯ

nanoparticles and (**b**) Linear plot of log σac versus logf.

vacancies by substitution of Zn2+ ions.

*3.9. Modulus Properties* 

**Figure 13.** Frequency dependence (**a**) real component (M') and (**b**) Imaginary component (M") of electric modulus of pure Fe2O3, Zn 2%, Zn 4% and Zn 6% nanoparticles. **Figure 13.** Frequency dependence (**a**) real component (M') and (**b**) Imaginary component (M") of electric modulus of pure Fe2O<sup>3</sup> , Zn 2%, Zn 4% and Zn 6% nanoparticles.
