Study of Urbach energy and Kramers-Kronig on Mn and Zn doped NiFe₂O₄ ferrite nanopowder for the determination of structural and optical characteristics

M_xNi_1-xFe₂O₄ spinel ferrite (M = Mn, Zn, and x = 0, 0.05) has been successfully synthesized by co-precipitation technique with hydrazine hydrate reduction agent (instead of NaOH) and Ethylene glycol surfactant. The XRD spectra of the samples illustrated high crystallinity. The structural characterization of pure and doped fcc NiFe₂O₄ were calculated by Scherrer, Modified Scherrer, Williamson-Hall, and SSP methods. In comparison of several methods, the Scherrer method is unreasonable method and W-H method has an acceptable range and can calculate both < L > and strain without restriction. The specific surface area in Zn-doped increased, demonstrate increment of adsorption properties in Ni ferrite structure. TEM images revealed the shape of grains is spherical, cubic, and irregular, with a grain size in the range of 35-65 nm. Hysteresis loops illustrated the magnetic behavior of samples. From the reflectance data, the band gap energies were estimated at 1.984, 1.954, and 1.973 eV for un-doped, Mn, and Zn-doped NiFe₂O₄ respectively (red shift). The almost low value of Urbach energy for pure, Mn, and Zn -doped NiFe₂O₄ indicates low structural disorder, which can approve the high crystallinity of samples. Direct band gap energy (E_g), refractive index, and extinction coefficient were estimated by the Kramers-Kronig method with linear optical evaluations. The E_g by K-K method is in good agreement with the E_g by Kubelka-Munk function.