Nontrivial temperature dependence of ferromagnetic resonance frequency for spin reorientation transitions

We find unusual temperature dependence of the ferromagnetic resonance (FMR) frequency $f_{\mathrm{R}}$ for the spin-reorientation (SR) transition, in which the easy axis changes depending on temperature, observed in the Nd permanent magnet, ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$: $f_{\mathrm{R}}\sim 0$ below the SR transition temperature ($T_{\mathrm{R}}$), drastic increase of $f_{\mathrm{R}}$ around $T_{\mathrm{R}}$, and decrease from a peak at higher temperatures. It is nontrivial that the SR transition causes the unusual behavior of the FMR frequency in a wide temperature region. We show the mechanism of the temperature dependence by theoretical and computational analyses. We derive a general relation between $f_{\mathrm{R}}$ and magnetizations to help the understanding of the mechanism, and clarify that the fluctuation of the transverse magnetization is a key ingredient for the resonance in all temperature regions.

Figure 1

(a) Temperature dependences of $m^2, m_z^2$, and $m_{xy}^2$ for the Nd magnet model (1). (b) $\theta$ dependence of the per-site anisotropy energy of Eq. (2) for the Nd magnet model.

Figure 2

Temperature dependences of $f_{\mathrm{R}}$ (blue diamonds) and $f_{\mathrm{SW}}$ (red circles) for the Nd magnet model.

Figure 3

For $D_1=1$ and $D_2=0$ in the minimal model (6), temperature dependences of $m^2, m_z^2$, and $m_{xy}^2$ are given in (a) and those of $f_{\mathrm{R}}$ (blue circles) and $f_{\mathrm{SW}}$ (red circles) are plotted in (b). For $D_1=1$ and $D_2=-0.7$, the former and latter dependences are depicted in (c) and (d), respectively. $k_{\mathrm{B}}=1$ was set. The time step for Eq. (5) $dt=0.005$ was used for the evaluation of $f_{\mathrm{R}}$.

Figure 4

Temperature dependences of $T{m}_z$ (blue circles) and $1/m_{xy}^2$ (red diamonds) for the minimal model (6) with (a) $D_2=0$ and (b) $D_2=-0.7$, and for (c) the Nd magnet model (1).