Intrinsic Nonlinear Ferromagnetic Relaxation in Thin Metallic Films

We propose an intrinsic mechanism for ferromagnetic relaxation in thin films that can dominate competing linear mechanisms even for rapidly relaxing metals. In particular, we use an analytic theory of four-magnon scattering to demonstrate rapid decay for technologically important systems involving high moment materials subject to large rotations. A micromagnetic simulation is used to verify the results.

Figure 1

Time evolution of (a) number of $k=0$ magnons $N_0$, number of all $k\ne 0$ magnons $\sum _{\mathbf{k}\ne 0}{N}_{\mathbf{k}}$, and total number of magnons $N_{\mathrm{t}\mathrm{o}\mathrm{t}}\equiv \sum _{\mathbf{k}}{N}_{\mathbf{k}}$; (b) magnon numbers $N_{\mathbf{k}}$ with different $\mathbf{k}\ne 0$; (c) envelope of the angle $\varphi$ between the average magnetization and the applied field. Iron film ($M_s=1700\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{/}\mathrm{c}{\mathrm{m}}^{\mathrm{3}}$, $\gamma =1.76\times {10}^7(\mathrm{O}\mathrm{e}\mathrm{·}\mathrm{s}{)}^{-1}$, $A=2\times {10}^{-6}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{/}\mathrm{c}{\mathrm{m}}^{\mathrm{3}}$, $K_1=4.8\times {10}^5\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{/}\mathrm{c}{\mathrm{m}}^{\mathrm{3}}$); $D_z=500\AA$, $H_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{l}}=1000\mathrm{O}\mathrm{e}$, ${\varphi }_0=25°$, $T=300\mathrm{K}$.

Figure 2

Theoretical ${\Gamma }_{\mathbf{k}}^{\mathrm{t}\mathrm{h}}$, calculated using Eqs. (2, 4, 5, 6), and simulation ${\Gamma }_{\mathbf{k}}^{\mathrm{s}\mathrm{i}\mathrm{m}}\equiv \mathrm{l}\mathrm{n}[N_{\mathbf{k}}(t_2)/N_{\mathbf{k}}(t_1)]/t$, $t_2=0.2\mathrm{n}\mathrm{s}$, $t_1=0.1\mathrm{n}\mathrm{s}$. Calculation parameters are the same as in Fig. 1.

Figure 3

Damping time ${\tau }_d$ calculated from simulations for ${\varphi }_0=45°$ switching angle as a function of film thickness $D_z$.

Figure 4

Switching pattern in the conditions of experiment [4] with (a) $\alpha =0.037$ and no magnons excitation allowed; (b) $\alpha =0.005$ and magnon-magnon scattering included. Pattern is shown along easy direction $x$, for $y=0$ in terms of paper [4]. $x=0$ corresponds to the origin in [4]. Initially, $M_x/M_s=1$; after the field pulse, regions with $M_x/M_s=-1$ have been switched. Note that the rapid fluctuations would likely be experimentally unresolvable.