Quantum size effects on magnetism in Eu₂MnHfO₆ nanoparticles: Role of surface-to-volume ratio, magnetic blocking, and magnetic phase transitions

In this paper, we investigate the quantum size effects on Eu₂MnHfO₆ (EMHO) nanoparticles (NPs) synthesized via sol–gel and annealed at 500, 800, and 1000 °C. XRD confirmed a cubic Fd3̅m structure across all samples, with NP sizes increasing from 7.4 nm to 23.8 nm. Smaller NPs displayed features of quantum dots (QD) and weakened Mn/Hf–O–Mn/Hf bonding due to broken surface coordination, as seen in FTIR spectra. Magnetic measurements revealed size-dependent behaviour: smaller NPs exhibited surface-driven phenomena like blocking temperature (T_B) and surface spin effects (T_SE), while larger NPs (23.8 nm) showed bulk-like super-exchange interactions with distinct Néel (T_N) and Curie (T_C) temperatures. Local magnetic inhomogeneities indicating a Griffiths phase (GP) were found in the smallest and largest particles, overlapping with T_C. These novel results underscore how surface-to-volume ratios influence magnetic transitions, supporting a modified Curie–Weiss law that distinguishes surface and volume effects with coordination variations in MNPs.