Vacancy-assisted hopping transport and microscopic origin of hierarchical spin in spinel Mn_0.5Mg_0.5Cr₂O₄

In this study, we investigate the magnetic ground state, the microscopic origin of glassy dynamics, oxygen vacancies-assisted carrier hopping conductivity, and dielectric properties in the spinel Mn_0.5Mg_0.5Cr₂O₄(MMC) using experimental and density functional theory (DFT) + U calculations. First-principles DFT + U identifies the collinear ground state with Mn↑Cr1↑Cr2↓ spin configuration. The PDOS displays O(2p)-TM(3d) covalency and Heisenberg mapping, resulting in the antiferromagnetic exchange coupling between nearest neighbors, with magnitudes following the orderJ_Cr-Cr>J_Mn-Cr>J_Mn-Mn. DFT + U calculations confirm the coexistence of direct Cr-Cr exchange and indirect oxygen-mediated Mn-O-Cr superexchange in the MMC sample. Spin-resolved PDOS of the oxygen-deficient sample reveals vacancy induced shallow Cr-3d states near the conduction band minimum. These states act as localized polaron trapping sites and generate defect dipoles that can reorient under an applied AC electric field. This vacancy bridged small polaron mechanism governs both dielectric relaxation and charge transport in MMC. The dielectric behavior is consistent with the Maxwell-Wagner interfacial polarization model, which involves conducting grains and resistive grain boundaries, superimposed on an intrinsic, non-Debye relaxation. The AC conductivity is well described by the non-overlapping small-polaron tunneling model, and the DC conductivity obeys the Mott-Davis small-polaron model. MagnetizationM(T) exhibits ferrimagnetic ordering around 25 K, followed by clear thermomagnetic irreversibility below 27 K and a sharp cusp at approximately 15 K. The field-cooled warming trace is non-monotonic, indicating a rugged energy landscape and a broad distribution of barriers. Zero-field-cooled memory measurements reveal a temperature-selective dip that becomes deeper with increasing waiting time. The remanent magnetization and the distinct memory responses observed under positive and negative temperature cycles confirm hierarchical, metastable spin dynamics below the spin-freezing temperature.