Spin-wave localization in tangentially magnetized films

We present an analytical description of localized spin-wave modes that form in a parabolic field minimum in a thin ferromagnetic film. Mode profiles proportional to Hermite functions are eigenfuctions of the applied field and exchange parts of the equations of motion, and also provide a basis for numerical approximation of magnetostatic interactions. We find that the spin-wave modes are roughly equally spaced in frequency and have roughly equal coupling to a uniform driving field. The calculated mode frequencies and corresponding profiles of localized spin-wave modes are in good agreement with micromagnetic modeling and previously published experimental results on multiple resonances from a series of localized modes detected by ferromagnetic resonance force microscopy.

Figure 1

Schematic representation of localized spin-wave mode profiles formed in a film of thickness $a$, magnetized in plane. The field is in the $z$ direction while the magnitude of the field has a parabolic spatial dependence.

Figure 2

Geometry of the model, simulating a ferromagnetic resonance force microscopy experiment. The parameters used in calculations are as follows: tip radius, $R=500$ nm; distance between the tip and the surface of the film, $d=100$ nm; thickness of the film, $a=20$ nm.

Figure 3

The spatial dependence of the normalized dipolar field from the micromagnetic tip. Two-dimensional potential well, calculated by the formula $H_i/H_0=f(y,z)-1$, fitted by two-dimensional harmonic oscillator potential $-C_0+C_z{z}^2+C_y{y}^2$, where $C_0=0.41,C_z=0.543/R^2,C_y=0.0942/R^2$.

Figure 4

Dispersion of spin-wave modes in parabolic field wells calculated by micromagnetic modeling as functions of applied field $H_0$. Material parameters are as follows: saturation magnetization of the film, $M_0=700$ kA/m; gyromagnetic ratio, $\gamma /2\pi =29.6$ GHz/T; exchange stiffness, $A={10}^{-11}$ J/m; and film thickness $a=20$ nm. The parameters of the parabolic well $C_0,C_z,C_y$ approximate a dipole field (see text and Fig. 3). (a) Truncated parabolic well field profile. The dark band corresponds to resonance of the sheet film outside the well, while the lowest solid black line is the uniform-film precession frequency at the minimum field in the well, provided as a reference. (b) Resonances in an “infinite” parabolic well extending over the film plane. The inset shows spectra at 0.1 T for panels (a) and (b), respectively.

Figure 5

Mode profiles of the localized excitations calculated with different approximations. The field and equilibrium magnetization lie along the $z$ axis. Profiles are presented in order of increasing frequency. Top row: Profiles calculated by micromagnetic modeling at ${\mu }_0{H}_0=0.1$ T, corresponding to the shoulder and the first three maxima in the inset of Fig. 4. Middle row: Mode profiles for the diagonal approximation, equivalent to Hermite function basis states ${\phi }_{0,m}$ for $m=\{2,0,4,6\}$. With magnetostatic interactions, the (0,2) mode has a lower frequency than the (0,0) mode. Bottom row: Eigenmode profiles when hybridization via magnetostatic interactions is taken into account, showing strong similarities to the micromagnetic results.

Figure 6

Calculated dispersion of precession modes using (a) Hermite function mode profiles and a diagonal approximation and (b) when hybridization by magnetostatic interactions is taken into account. Similarly to Fig. 4, the lower black dotted lines correspond to the uniform precession frequency in the minimum field.

Figure 7

Mode profiles of the 12 lowest modes, calculated on the basis of an array of functions ${\phi }_{mn}(\eta ,\xi )$ with even indices $\left(m,n\right)$, where both $m$ and $n$ vary from 0 to 28. Corresponding frequencies (intensities) are given in the upper right (lower left) corners of every image. Magnetic parameters and parabolic well were defined in the previous section. Applied field ${\mu }_0{H}_0=0.1$ T.

Figure 8

Intensities vs frequencies for hybridized modes calculated on the basis of an array of functions ${\phi }_{mn}(\eta ,\xi )$ with even indices $\left(m,n\right)$, where both $m$ and $n$ vary from 0 to 28. Intensities describe coupling to a uniform microwave field, as in ferromagnetic resonance experiments. Corresponding profiles for the lowest 12 modes were presented in Fig. 7.