Unified theory of magnetization temperature dependence in ferrimagnets

TL;DR

The paper tackles the challenge of predicting the temperature dependence of spontaneous magnetization in multi-sublattice ferrimagnets, using YIG as a concrete test case. It extends the Weiss–Heisenberg mean-field framework and the spin-wave approach to a unified SW–MFA description, introducing a minimal model with a single tuning parameter $\delta$ to reconcile low-temperature spin-wave suppression with high-temperature mean-field behavior. The key result is a quantitatively accurate description of $M(T)$ across the entire range up to the Curie temperature, with a near-mean-field $M(T) \propto (T_{\rm C} - T)^{1/2}$ scaling over most of the range and $T_{\rm C}$ matched to the experimental value by choosing $\delta = 0.428$. This approach provides a practical framework for predicting magnetization in complex ferrimagnets and can be extended to other multi-sublattice systems and garnet-like structures, potentially informing magnonic device design and fundamental magnetic theory.

Abstract

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 zero to the Curie temperature in simple cubic ferromagnets, EuO and EuS. We extend this approach to compute and analyze $M(T)$ for multi-sublattice collinear ferrimagnets such as Yttrium Iron Garnet $Y_3Fe_5 O_{12}$. We analyzed and generalized for multi-sublattice collinear ferrimagnets two well-known approximations describing $M(T)$. The first approach is the Bloch-3/2 law, which describes the suppression of $M(T)$ due to spin-wave excitation, and is valid in the low-temperature limit $T << T_c$. The second one is Weiss's mean-field approximation, which provides a reasonable description of $M(T)$ near $T_c$. Using a single tuning parameter, we combine these two approaches to describe $M(T)$ for any $0<T<T_c$. 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 dependences $M(T)$ follow the mean-field prediction $\sqrt{T_c - T }$ for almost all temperatures.

Figures (5)

• Figure 1: (a) The crystallographic structure of YIG features a body-centered cubic (BCC) unit cell with the O$_{\rm h}$ symmetry group, occupying half of a cube. This structure includes four formula units (Y$_3$Fe$_5$O$_{12}$)$\times 4$, amounting to a total of 80 atoms. Among these, there are 20 magnetic iron ions (Fe$^{3+}$) divided into two groups: 8 Fe$^{3+}$ ions occupying the (a) octahedral sites, represented as a blue ball and a blue-shaded area, and the remaining 12 Fe$^{3+}$ ions located in (d) tetrahedral sites, represented as a green ball and yellow-shaded area. The dodecahedra (c) site occupied by Yttrium ions is shown as a black ball and green-shaded area, the O$^{2-}$ ions are drawn in red. (b) The exchange pathways used in the effective Heisenberg Hamiltonian are labeled according to the sites they are connecting: $J_{\rm ad}$ as (ad), two different in symmetry (aa) interactions, are denoted as $J_{\rm aa1}$ and $J_{\rm aa2}$, etc.
• Figure 2: (a) Temperature dependence of the normalized mean spin $\overline S^{z}(T)/ \overline S^{z}(0)$, equivalent to the normalized magnetization $M(T)/M(0)$, according to Eq. \ref{['Snorm']}. Various experimental and theoretical results are labeled by circled numbers ⓝ of colors matching the color of the lines and described below. The close-up of the high-$T$ range marked by a rectangle with a pink background is shown in the inset. (b) The close-up of the low-$T$ range, marked by a rectangle with a light-blue background in the main panel (a). Various experimental datasets and theoretical results are labeled as follows. Experimental data: blue dots ① -- 1964-Anderson results Anderson1964, orange dots ② -- 1974 data Hansen and others Hansen1974, black solid line ③ -- our current results of a 1 mm YIG sphere magnetized in the ⟨111⟩ direction were measured via VSM. Violet dashed line ④ -- all-temperature fit of experimental data ② $\overline{S}^{z}=A (T_{_{\rm C}}-T)^\beta$ with $A$ = 1.045 and $\beta \simeq 0.50\pm 0.005$; for details see Appendix \ref{['A6']}. Blue solid line ⑤ -- numerical solution of MFA Eqs. \ref{['4']} and dashed blue line ⑥ -- near-$T_{_{\rm C}}$ fit for MFA solution ⑤, $\propto \sqrt{T_{_{\rm C}}-T}$; Green solid line ⑦ -- numerical analysis of Eq. \ref{['S-Td']} for spin-wave (SW) approximation (SWA) to $\overline{S}^z(T)/ \overline{S}^{z}(0)$, exact at low temperature; Red solid line ⑧, -- the unified approach Eq. \ref{['26']}, with the magnon frequencies $\omega_j(k,T)$ from \ref{['T-dep1']} and $\delta=0.428$.
• Figure 3: The plots of an analytical result of Ref. Cherepanov1993 for all 20 branches of the magnon energy spectra in YIG for $\bm k \| [110]$ (left part of the $\bm k$ axis and for $\bm k\| [100]$ (right part of the $\bm k$ axis). Three vertical axes represent various ways to quantify the magnon energy and frequency. The modes corresponding to (a) -interactions are shown as dashed blue lines; the (d)-modes are plotted as dashed green lines; the (ad)-modes are shown as solid blue lines, and the (da)-modes are shown as solid green lines. Horizontal black dashed line indicates the room temperature $T=300$ K, and the horizontal red dashed line marks the YIG Curie temperature $T_{_{\rm C}}\approx 560\,$K. These spectra are quite similar to recent results by Princep et al. Princep2017aPrincep2017j as listed in Tab. \ref{['t:2']}.
• Figure 4: The numerical solution for the normalized mean spin $\overline S^{z}(T)/ \overline S^{z}(0)$. The results of numerical solutions are denoted as follows: Solid blue line ⑤ (the same as in Fig. \ref{['F:2']})-- the normalized total spin, defined by Eq. \ref{['Sz']}, Green dashed and solid lines ⑨ and ①0 represent the normalized spins of sublattice (a) $-\overline S^{z}_\mathrm{a}(T)/\overline S^{z}_\mathrm{a}( 0)$, and sublattice (d) $\overline S^{z}_\mathrm{d}(T)/\overline S^{z}_\mathrm{d}( 0)$ respectively, defined by Eqs. \ref{['4']} based on exchange integrals from Cherepanov1993. Vertical dashed lines mark the Curier temperatures $T_{_{\rm C}}^{\rm Exp}$ (black) and $T_{_{\rm C}}^{\rm MF}$ (red). Inset: high-resolution close-up of the low-$T$ range marked in the main panel as a rectangle with light-blue background.
• Figure 5: Experimental and numerical results for $\overline{S}^{z}/\overline{S}^{z}(0)$ vs the temperature $T$. Experimental results ①, ②, and ③ are the same as in Fig. \ref{['F:2']}. Numerical solutions of Eq. \ref{['S-Td']} are represented by solid lines ⑦, ①1 and ①2. For the green line ⑦ (also shown in Fig. \ref{['F:2']}) we used Eq. \ref{['S-Td']} with temperature-independent frequencies $\omega_j(k)$. For the yellow line ①1 we applied in Eq. \ref{['S-Td']} temperature-dependent frequencies $\omega_j(k,T)$, Eq. \ref{['T-dep']}, and for the magenta line ①2 $\omega_j(k,T)$--Eq. \ref{['T-dep1']} with $\delta=0.5$. For both, we used the experimental results for magnetization $\overline{S}^{z}$ from the experiment ②. Inset: high-resolution close-up of the low-$T$ range marked in the main panel as a rectangle with a light-blue background.