The role of rare-earth Ce-doped in tailoring the structural, optical and magnetic properties of Mg–Zn ferrite nanoparticles having the chemical formula \({\text{M}\text{g}}_{0.5}{\text{Z}\text{n}}_{0.5}{\text{C}\text{e}}_{x}{\text{F}\text{e}}_{2-x}{\text{O}}_{4}\) (where x = 0.00, 0.03, 0.06, and 0.10) has been investigated. X-ray diffraction (XRD) patterns have ensured the single-phase cubic spinel structure for x \(\le\) 0.06, while a small peak of CeO2 associates with the cubic spinel structure for x = 0.10. The theoretical lattice parameters \(({a}_{th})\) of the synthesized nanoparticles have shown a strong agreement with the experimental lattice parameters \(({a}_{\text{e}\text{x}\text{p}})\) for x \(\le\) 0.06. However, for x = 0.10, the value of \({a}_{\text{e}\text{x}\text{p}}\) is reduced, which could be ascribed to the formation of secondary CeO2 phase for higher Ce content. Crystallite size, dislocation density, micro strain, hopping lengths, tolerance factor, bond angles and bond lengths have been estimated in this study. Transmission electron microscopy (TEM) has displayed the nanocrystalline nature of all the samples and the average particle size \(\left({D}_{\text{a}\text{v}\text{g}}\right)\) decreases from 23.17 \(\pm\) 0.54 to 18.35 \(\pm\) 0.68 nm as the content of Ce increases from x = 0.00 to x = 0.06. The optical band gap energy ( \({E}_{g}\) ) of the synthesized ferrite nanoparticles have been obtained using UV–visible spectroscopy, showing a redshift in absorption edge, and the value of \({E}_{g}\) decreases from 2.09 (for x = 0.00) to 1.89 eV (for x = 0.06). The magnetic hysteresis curve shows that all the synthesized samples exhibited a ferromagnetic nature. The value of coercivity (Hc) is found to be 21.85 Oe for x = 0.06, which suggests that the magnetic softness property has been improved due to the reduction of the internal stress and particle size. A lower value of remanent magnetization (Mr) for x = 0.06 has been observed and it is found to be 1.05 emu/g. In addition, a smaller value of squareness ratio, r (0.029) is also obtained along with a lower anisotropy constant. The relatively low anisotropy constant (ka) also supports the soft magnetic nature of the samples. An enhancement of the dielectric constant has been clearly observed for x = 0.03, which could be attributed to the densification of the sample. An impedance spectroscopy measurement shows that the relaxation peaks of Ce-doped Mg–Zn nano-ferrites shifted to higher frequencies. In addition, the variation of ( \(Z^{\prime \prime } /Z_{\max }^{\prime \prime }\) ) with the applied frequency curves has also been confirmed that the influence of Ce has an impact on the relaxation process in Mg–Zn ferrite nanoparticle, which leads to the reduction of the relaxation time. It is inferred that the influence of Ce doping significantly tailors the multifunctional behavior of Mg–Zn ferrite nanoparticles, making them promising for high frequency operation and optoelectronic applications. In this study, the lower value of Hc is obtained for x = 0.06, which clearly confirms the soft ferrites. This property can be utilized in an inductor material as a potential application in high quality of coil. The dielectric properties of the synthesized nanomaterials have also shown the possible application in high frequency devices.
