A B-site disordered double perovskite $\mathrm{NdSrCoFe}{\mathrm{O}}_6$ was successfully synthesized by the conventional sol-gel method. Detailed experimental analyses revealed that $\mathrm{NdSrCoFe}{\mathrm{O}}_6$ crystallizes in the orthorhombic $Pnma$ space group, in which ${\mathrm{Co}}^{2+/3+}$ and ${\mathrm{Fe}}^{3+/4+}$ ions are randomly distributed at the BB′ sites, and ${\mathrm{Sr}}^{2+}$ and ${\mathrm{Nd}}^{3+}$ ions are respectively ordered at the A and A′ sites in an alternating arrangement along the $c$ direction. $\mathrm{NdSrCoFe}{\mathrm{O}}_6$ has a semimetallic-to-semiconducting transition nature, and a paramagnetic-ferromagnetic (FM) second-order phase transition originating from the complex hybridization between Co $3d$ and O $2p$ states is also found to occur at $T_{\mathrm{C}}\approx 150$ K. Then the spin coupling between ${\mathrm{Fe}}^{4+}\leftrightarrow {\mathrm{Co}}^{3+}$ and ${\mathrm{Fe}}^{3+}\leftrightarrow {\mathrm{Co}}^{2+}$ randomly distributed on the B and B′ sites leads to a FM cluster spin-glass behavior with characteristic parameters of $k=0.01, T_{\mathrm{SG}}=82.7$ K, $zv=1.89$, and ${\tau }_0=0.46\times {10}^{-4}$ s. Additionally, Griffiths-like phase behavior was observed in the region $T_{\mathrm{C}}<T<T_{\mathrm{GP}}$, with $T_{\mathrm{GP}}=245$ K, consistent with the power law exponent of $\lambda =0.74$. The maximum isothermal magnetic entropy change $-\mathrm{\Delta }{S}_M^{\max }\approx 1.84\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ and relative cooling power $\approx 43.8\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$ under a field of 40 kOe also indicate a magnetocaloric coupling wherein fitted critical exponents $\beta =1.384, \gamma =0.621$, and $\delta =1.421$ are far from any conventional universality class. Density functional theory calculations demonstrated spin short- and long-range ordering competitions for Fe/Co at BB′ sites, which arise predominantly from the stronger negatively charged ligand interaction with Co $3d$ orbitals and the weakest Fe $3d$ orbitals. This unconventional behavior is expected to be the main reason for the experimentally observed magnetic exchange distance decreasing with $J\left(r\right)\approx r^{-4.7}$.

Herein, we carefully investigated the Fe³⁺ doping effects on the structure and electron distribution of Cr₂O₃ nanoparticles using X-ray diffraction analysis (XRD), maximum entropy method (MEM), and density functional theory (DFT) calculations. We showed that increasing the Fe doping induces an enlargement in the axial ratio of c/a, which is associated with an anisotropic expansion of the unit cell. We found that as Fe³⁺ replaces Cr in the Cr₂O₃ lattice, it caused a higher interaction between the metal 3d states and the oxygen 2p states, which led to a slight increase in the Cr/Fe–O1 bond length followed by an opposite effect for the Cr/Fe–O2 bonds. Our results also suggest that the excitations characterize a well-localized bandgap region from occupied Cr d to unoccupied Fe d states. The Cr₂O₃ and Fe-doped Cr₂O₃ nanoparticles behave as Mott–Hubbard insulators due to their band gap being in the d−d gap, and Cr 3d orbitals dominate the conduction band. These findings suggest that the magnitude and the character of the electronic density near the O atom bonds in Cr₂O₃ nanoparticles are modulated by the Cr–Cr distances until its stabilization at the induced quasi-equilibrium of the Cr₂O₃ lattice when the Fe³⁺ doping values reaches the saturation level range.