Structural and electrical properties of chromium substituted nickel ferrite by conventional ceramic method

Fig. 1 shows the X-ray diffraction patterns of NiCr_xFe_2−xO₄ ferrites sintered at 1350 °C. The peaks can be indexed to (1 1 1), (2 2 0), (3 1 1), (2 2 2), (4 0 0), (4 2 2), (5 1 1) and (4 4 0) planes. The samples can be identified as a single-phase cubic spinel structure with no extra lines corresponding to any other crystallographic phase. The sharp peaks in the XRD patterns confirmed the formation of a homogeneous single phase cubic spinel structure corresponding to the space group Fd3m for all the samples. The lattice parameters were calculated from the XRD data.

Figure 1

The lattice constant “a” of each sample was calculated using the relation: (1)a=d(h2+k2+l2)1/2 $$a\, = \, d{({h^2} + {k^2} + {l^2})^{1/2}}$$

where d = interplanar distance and (h k l) are the Miller indices.

The variation of lattice constant with Cr concentration is shown in Fig. 2. It is observed that as the Cr content increases the lattice parameter decreases. This variation of lattice parameter can be explained by the difference of ionic radii between Cr³⁺ and Fe³⁺. The ionic radius of Cr³⁺ (0.64 Å) is relatively smaller than that of Fe³⁺ (0.67 Å)[2]. As there occurs a partial replacement of Fe³⁺ in the octahedral sites by Cr³⁺, the shrinkage of lattice parameter happens. It can also be explained by the migration of cations. The linear variation of lattice parameters with Cr concentration confirms the applicability of Vegard’s law.

Figure 2

The X-ray density (ρ_x = 8 M/Na³, where M is the molecular weight of the corresponding composition and N is Avogadro’s number) and the bulk density (ρ_b = m/V, where V = a³ is the volume of the cubic unit cell) were calculated using the lattice parameter “a” obtained from the XRD measurements.

Both the bulk density and the X-ray density decrease with the increase in Cr concentration. This is due to the smaller molar mass of the Cr (51.996 g/mol) cation substituted in place of Fe with larger molar mass (55.845 g/mol)[4]. It is observed that the bulk density is lower than the Xray density. This is due to the existence of pores which arise during the preparation and/or the sintering process of the samples. As porosity is inversely related to the density, in general, the values of porosity increase with the increase in Cr addition.

The porosity was calculated from the bulk density and X-ray density using the formula: (2)P=(1−ρb/ρx)×100% $$P\, = \,(1 - {\rho _b}/{\rho _x})\, \times \,100\% $$

The calculated lattice parameter, bulk density, X-ray density and porosity are presented in Table 1.

Resistivity depends on the microstructure of ferrite. With increasing porosity, resistivity also increases, since pores do not take part in conduction[4]. Substitution of Cr³⁺ in place of Fe³⁺ increases the DC electrical resistivity as a result of the formation of stable electric bonds between the Cr³⁺ and Fe²⁺, placing Fe²⁺ charge carriers in localized states. The decrease in Fe³⁺ ion content leads to the decrease of ferrous ions formed. This leads to the migration of some Fe³⁺ ions from B sites to A sites, and as a result, the number of Fe³⁺ and Fe²⁺ ions at the B sites decreases. The increase in Cr³⁺ content leads to replacement of Fe³⁺ ions at B sites. Consequently, the resistivity increases on increasing substitution of Cr³⁺ ions[5].

DC electrical resistivity has been measured in air as a function of temperature between room temperature and 450 °C. Each sample was polished and silver paste was applied on both sides of the polished pellet together with two thin copper wires. Fig. 3 depicts that DC resistivity decrease with increasing temperature which means that the conductivity increases. The decrease in resistivity is attributable to the increase in the drift mobility of electric charge carriers due to enhanced oscillation of crystal lattice. This decrease in resitivity indicates the semiconductor nature of all the samples. In polycrystalline ferrites, the bulk resistivity is the combination of crystalline resistivity and the resistivity of the crystalline boundaries. Boundary resistivity is greater than the crystalline resistivity. Consequently, the boundary resistivity has a great influence on the bulk resistivity[6]. An increasing tendency in DC resistivity is observed with increasing Cr concentration[7].

Figure 3

In ferrites, the resistivity (ρ) at absolute temperature (T) is represented by the relation: (3)ρ=ρ0exp(ΔE/kT) $$\rho \, = \,{\rho _0}\exp (\Delta E/kT)$$

where k is the Boltzmann constant and ΔE is the difference between the activation energies in ferro-magnetic (E_f) and paramagnetic (E_p) regions. The activation energy is required to release an electron from Fe²⁺ to the neighboring ion. These extra electrons can contribute the electrical current by jumping or hopping process. According to the Verwey model, the electrical conduction in ferrite can take place by exchanging electrons among the octahedral ions (hopping mechanism) [8–12]. The normal distance between tetrahedral-tetrahedral or tetrahedral-octahedral is large in comparison to octahedral-octahedral so that the electronic exchange between the cations on these sites is low.

The variation of resistivity against inverse of temperature of the samples is shown in Fig. 4. The slope in the low temperature region is regarded as ferromagnetic and it changes into paramagnetic in high temperature region. Activation energy is attributed to a change in conduction mechanism or phase transition from ferromagnetic to para-magnetic. The curves show that the slope changes with temperature which indicates that the activation energy changes with temperature too. The temperature at which the slope is changed does not match the Curie temperature of the respective sample[10]. From the slope (Fig. 4), ΔE (presented in Table 1) was calculated using the formula [11]: (4)ΔE=slope×4.606×8.62×10−5eV $$\Delta E\, = \, slope \times 4.606 \times 8.62 \times {10^{ - 5}}{\rm{eV}}$$

Figure 4

The activation energy in ferromagnetic region is smaller than in the paramagnetic region. In the ferromagnetic region, the activation energy is caused by electron hopping through a single-charged oxygen vacancy and it is attributed to the magnetic spin disordering due to decrease in the concentration of current carriers [11-14]. In the paramagnetic region, the activation energy is above 0.5 eV, which is much larger than the ionization energy for donor or acceptor or for Fe²⁺−Fe³⁺ transition energy. Therefore, the band type conduction is not possible and conduction is due to hopping of polaron [15]. The general trend of the value of the activation energy (ΔE) is to decrease with the increase of Cr concentration.

The variation of AC resistivity with frequency at room temperature is shown in Fig. 5. For all the samples, it is observed that as frequency increases the resistivity decreases. The dispersion of AC resistivity in low frequency region is large and above 10 MHz it remains constant up to frequency of 120 MHz. This dispersion of resistivity can be explained on the basis of Koop’s phenomenological theory. AC resistivity decreases as the frequency increases since at high frequency polarization of the dipoles decreases [2].

Figure 5

Above a certain frequency, the electron exchange does not follow the applied alternating field. The increase in frequency enhances the hopping frequency of the charge carriers between Fe²⁺ and Fe³⁺ ions. These enhanced electrons increase the conduction and thereby decrease the resistivity [16]. The AC resistivity is found to increase with the increase in Cr concentration in the low frequency region.

The real part of dielectric constant was calculated from the formula: (5)ε′=CLε0A $$\varepsilon \prime \, = \,{{CL} \over {{\varepsilon _0}A}}$$

where C is the capacitance in pF, L is the thickness or height of the pellet, A is the cross-sectional area of the circular surface of the pellet and ε_o = 8.854 × 10⁻¹⁴ F/cm is the free space permittivity. Fig. 6 shows the dependence of ε′ on frequency. From these curves it is evident that as the frequency increases, the dielectric constant decreases. The dielectric constant is dependent on frequency in the lower frequency region and independent in the higher frequency region. At higher frequency these values are almost constant. When the electric field propagates with high frequency, the ionic and orientation sources of polarization of the dipoles decrease and beyond a certain frequency electron exchange does not follow the alternating field, which results in a constant value of dielectric constant. This behavior is attributed to the Maxwell-Wagner type space charge polarization which agrees with the Koop’s phenomenological theory [17]. The decrease of ε′ with Cr content resulted in a decrease of dielectric loss [18]. At lower frequency the high value of ε′ is due to the predominance of the species, such as Fe²⁺ ions, oxygen vacancies, grain boundary defects, interfacial distribution, etc. During the sintering process, electron exchange occurs between Fe²⁺↔ Fe³⁺ which leads the displacement of electron in the direction of applied field that determines the polarization [19].

Figure 6

In polycrystalline ferrites, the dielectric loss tangent is the result of energy dissipation in a dielectric which is due to the lagging of polarization with respect to the applied alternating field. Fig. 7 depicts the dielectric loss behavior of the NiCr_xFe_2−xO₄ system which shows asymmetrical behavior. In this figure, there is a peak in the power loss owing to the transfer of the maximum electrical energy to the oscillating ions when the frequency of the external AC field is equal to the hopping frequency of the charge carriers [20]. The occurrence of the loss peak in the tanδ versus frequency is associated with the correlation between the hopping conduction mechanism and dielectric behavior of spinel ferrites.

Figure 7