Size and interface effects on ferromagnetic and antiferromagnetic transition temperatures

A simple and unified model, which is based on the energy-equilibrium criterion between the spin-spin exchange interactions and the thermal vibration energy of atoms at the transition temperature and a size-dependent Debye temperature function ${ϴ}_{\mathrm{D}}\left(D\right)$, has been established for the size, dimension, and interface effects on the Curie temperature $T_c\left(D\right)$ and the Néel temperature $T_{\mathrm{N}}\left(D\right)$ of low-dimensionally ferromagnetic and antiferromagnetic nanocrystals, where $D$ denotes the diameter of nanoparticles and nanorods or the thickness of thin films. In terms of this model, $T_c\left(D\right)$ and $T_{\mathrm{N}}\left(D\right)$ functions are predicted to increase or decrease with dropping $D$, depending on the interaction strength at the film/substrate interface when the interface exists. The predicted results are consistent with available experimental measurements for Fe, Co, Ni, Gd, Ho, ${\mathrm{Co}}_1{\mathrm{Ni}}_1$, ${\mathrm{Co}}_1{\mathrm{Ni}}_3$, ${\mathrm{Co}}_1{\mathrm{Ni}}_9$, ${\mathrm{Fe}}_3{\mathrm{O}}_4$, $\mathrm{Mn}{\mathrm{Fe}}_2{\mathrm{O}}_4$, CoO, NiO, and CuO low-dimensional nanocrystals.