Variable range hopping and modulus relaxation in NiFe2O4 ceramics

Highlights: • Three modulus relaxations at T 323K, and two at T < 323K are observed for NiFe₂O₄ ceramics. • Elemental ratio of Fe/Ni ~2.08 suggested the presence of electron and hole charge carriers. • Variable range hopping conduction at T < 323K, and thermally activated hopping conduction at T ≥ 323K. Nickel ferrite ceramics prepared from combustion derived powders have been investigated using dielectric, impedance, and modulus spectroscopy over a wide frequency (0.1 Hz–1 MHz) and temperature range (173–473 K). Step-like dielectric response ${\epsilon }^{\text{'}}\left(T,f\right)$ is related to the surface barrier layer capacitor (SBLC) effect. Dielectric relaxation peaks in ${\epsilon }^{″}\left(f\right)$ masked by the huge dielectric loss are revealed by ${\epsilon }_{der}^{″}\left(f\right)$ derived from the ${\epsilon }^{\prime }\left(f\right)$ data following Kramers – Kronig transformations. Multiple relaxations are evidenced from the asymmetric nature of ${\epsilon }_{der}^{″}\left(f\right)$ peaks. Impedance Cole-Cole analysis reveals two relaxation processes corresponding to grain and grain-boundary effects. A maximum in the grain capacitance (C_g) at 293 K is attributed to the collective influence of electron and hole charge carriers on dielectric polarization. The modulus analysis reveals distinctly three relaxations occurring at temperatures T ≥ 323 K, and only two relaxations for T < 323 K. The variation of modulus relaxation frequencies ln(f_max) with (1/T) suggests different conduction mechanisms for temperature above and below 323 K. The two relaxations observed in the entire temperature range originate due to hopping of holes and electrons at equivalent B-sites of NiFe₂O₄, while the third relaxation seen only at temperatures T > 323 K is attributed to defect hopping. The present analysis based on modulus relaxation, dc conductivity, and power law exponent $s\left(T\right)$ indicates two different conduction mechanisms for Nickel ferrite NiFe₂O₄ with variable range hopping dominating at lower temperatures, and thermally activated hopping at higher temperatures.