Spin-wave eigenmodes of Landau domain patterns

We present micromagnetic simulations of the spin-wave spectra in Landau domain patterns. Ultrashort field pulses of various spatial symmetries are used to excite distinct spin-wave eigenmodes. The frequencies as well as the symmetry of the mode patterns depend on the symmetry of the exciting torque. The latter is determined by the symmetries of the field pulse and the magnetic ground state. Landau domain patterns have collective excitations, i.e., the spin-wave modes cannot be considered for each domain individually. We find transversal modes as well as longitudinal modes as observed experimentally. From the mode profiles an effective dispersion relation is deduced which resembles the dispersion relation for infinitely extended thin films.

Figure 1

(Color online) Spin-wave spectra for three out-of-plane excitations. All excitations were done on a $1\times 1\mu {\mathrm{m}}^2$ $16\text{−}\mathrm{nm}$-thick permalloy square with a sharp pulse of a FWHM of $2.5\mathrm{ps}$. (a) Relaxation without pulse from a nonequilibrium initial state, and excitation (b) with a $20\mathrm{mT}$ pulse having a $\pi ∕2$ rotational symmetry (see inset), (c) with a $20\mathrm{mT}$ pulse having a $\pi$ rotational symmetry (inset), and (d) with a $20\mathrm{mT}$ spatially uniform pulse (inset) on a fully relaxed ground state. The peaks in the spectra are identified for the transversal orders $m=1$, 2, and 3 (see also Figs. 2, 3).

Figure 2

(Color online) Spatial images of the squared amplitude (top row) and the phases (bottom row) of five exemplary longitudinal eigenmodes of the $1\times 1\mu {\mathrm{m}}^2$ square. The numbers $n_1$ indicate the order of the longitudinal modes for the first transversal mode $m=1$ (see Fig. 1).

Figure 3

(Color online) Spatial images of the squared amplitudes (top row) and the phases (bottom row) for a $1\times 1\mu {\mathrm{m}}^2$ square. (a) $m=2$ transversal modes excited by a uniform field pulse [see Fig. 1d]. Dashes indicate modes with maximum excitation in a domain wall. $m=\left(\mathrm{b}\right)$ 2 and (c) 3 transversal modes excited by the field pulses shown in the insets of Fig. 1c.

Figure 4

(Color online) Dispersion relations of the longitudinal eigenmodes for different squares and transversal orders $m=1–3$. Closed symbols denote $m=1$ modes. (Green) diamonds are for the $4\times 4\mu {\mathrm{m}}^2$, (blue) triangles for the $2\times 2\mu {\mathrm{m}}^2$, (red) solid circles for the $1\times 1\mu {\mathrm{m}}^2$, and (black) squares for the $750\times 750{\mathrm{nm}}^2$ pattern. The higher transversal modes [open (red) circles] $m=2$ and 3 are for the $1\times 1\mu {\mathrm{m}}^2$ square. Lines represent analytical predictions for an infinite thin film. The inset shows the local internal field with the colors ranging from fields with values below zero (blue) up to $6.7\mathrm{kA}∕\mathrm{m}$ (red).