Localized Ferromagnetic Resonance in Inhomogeneous Thin Films

The effect of sample inhomogeneity on the ferromagnetic resonance linewidth is determined by diagonalization of a spin wave Hamiltonian for ferromagnetic thin films with inhomogeneities spanning a wide range of characteristic length scales. A model inhomogeneity is used that consists of size $D$ grains and an anisotropy field $H_p$ that varies randomly from grain to grain in a film with thickness $d$ and magnetization $M_s$. The resulting linewidth agrees well with the two-magnon model for small inhomogeneity, $H_{p}D\ll \pi M_{s}d$. For large inhomogeneity, $H_{p}D\gg \pi M_{s}d$, the precession becomes localized and the spectrum approaches that of local precession on independent grains.

Figure 1

Precession frequencies of 3001 spin waves for a 3 nm thick rectangle of uniform Permalloy, $10\mu \mathrm{m}\times 2.5\mu \mathrm{m}$. A 114 mT field is applied in plane to give a uniform precession frequency of $\omega (k=0)/(2\pi )=10\mathrm{G}\mathrm{H}\mathrm{z}$.

Figure 2

Calculated FMR response for a 3 nm thick film of Permalloy with 20 nm grains and a 10 mT rms perturbation field. (a) FMR intensity and simulated FMR signal, (b) half of the grain structure, and (c) the corresponding half of the eigenmode indicated by the arrow in (a) showing the collective response of this mode to a uniform driving field.

Figure 3

Calculated FMR response for a 3 nm thick film of Permalloy with $1.4\mu \mathrm{m}$ grains and a 10 mT rms perturbation field. (a) FMR intensities and signals calculated from eigenmode intensity and from the local resonance approximation, (b) half of the grain structure, and (c) the corresponding half of the eigenmode indicated by the arrow in (a) showing local precession appearing predominantly in two of the grains.

Figure 4

Peak-to-peak (filled symbols) and standard deviation (open symbols) linewidths as a function of grain size. Results are shown for three values of rms perturbation field, $H_p^{\mathrm{r}\mathrm{m}\mathrm{s}}$. The characteristic value of $H_{p,c}$ for the transition from collective to local behavior is shown. The left and right axes are related by Eq. (2).