Unified theory of magnetization temperature dependence in ferrimagnets

Recent advancements in spintronics and fundamental physical research have brought increased attention to the rare-earth-based magnetically ordered materials. One of the important properties of these materials is the temperature dependence of the spontaneous magnetization $M(T)$. Recently, a successful framework was proposed for the theoretical description of $M(T)$ across the entire temperature range from $T=0\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ to the Curie temperature ${T}_{{}_{\mathrm{C}}}$ in simple cubic ferromagnetics, EuO and EuS. We extend this approach to compute and analyze $M(T)$ for a more complex magnetic material: multi-sublattice collinear ferrimagnets such as yttrium iron garnet (YIG) ${\mathrm{Y}}_{3}{\mathrm{Fe}}_{5}{\mathrm{O}}_{12}$. YIG is a unique material that has a high Curie temperature well above room temperature. It features a cubic crystallographic structure characterized by low anisotropy and minimal spin-orbit interaction. These properties result in an exceptionally low spin-wave damping, which is important for fundamental studies as well as engineering applications. The primary challenge in describing ferrimagnets theoretically arises from its intricate magnetic structure, characterized by multiple magnetic sublattices. We analyzed and generalized for multi-sublattice collinear ferrimagnets two well-known approximations describing $M(T)$. The first approach is the Bloch $\frac{3}{2}$-law, which describes the suppression of $M(T)$ due to spin-wave excitation, and is valid in the low-temperature limit $T\ensuremath{\ll}{T}_{{}_{\mathrm{C}}}$. The second one is Weiss's mean-field approximation, which provides a reasonable description of $M(T)$ near ${T}_{{}_{\mathrm{C}}}$. Using a single tuning parameter, we combine these two approaches to describe $M(T)$ for any $0\ensuremath{\le}T\ensuremath{\le}{T}_{{}_{\mathrm{C}}}$. The theoretical result for $M(T)$ aligns well with our measurements and the previously available experimental data across the entire temperature range. We also demonstrate that experimental and theoretical dependencies $M(T)$ follow the mean-field prediction $\sqrt{{{T}_{{}_{\mathrm{C}}}\ensuremath{-}T}^{\phantom{\ensuremath{\int}}}}$ for almost all temperatures.