Interaction between propagating spin waves and domain walls on a ferromagnetic nanowire

We numerically investigate the interaction between propagating spin waves and a transverse domain wall in a nanowire by using micromagnetic simulations. In order to understand the mechanisms that lead to domain wall motions, we calculate domain wall velocity in a defect-free nanowire and the depinning fields for a pinned domain wall that is depinned in and against the direction of the spin-wave propagation. We find that the physical origin of the spin-wave-induced domain wall motion strongly depends on the propagating spin-wave frequency. At certain spin-wave frequencies, transverse domain wall vibrations lead to transverse wall displacements by the spin waves, while at other frequencies, large spin-wave reflection drives domain wall motion. By analyzing the depinning field calculations, the different underlying physical mechanisms are distinguished.

Figure 1

The initial spin configuration of the trapped head-to-head TW nucleated at the position of a square notch in the center of the nanowire. The square notch size is$5\times 5$ nm${}^2$. In the bottom we zoom into the central part of the wire to show the internal structure of the TW pinned at the notch. In order to generate SWs, we apply a localized sinusoidal field at a position denoted as the SW source. The SW source is located at 500 nm from the center of the nanowire. To depin the TW, we apply an external field $B_{\mathrm{ext}}$ along the $\pm x$ direction.

Figure 2

(a) TW displacement as a function of simulation time ($\Delta t$ $=$ 20 ns, $f_H$ $=$ 5.25, 6.25, and 11.25 GHz). (b) The TW velocity ($v_{\mathrm{TW}}$) as a function of SW frequency ($f_H$ $=$ 3 GHz 30 GHz, $\Delta f_H$ $=$ 0.25 GHz, and $\Delta t$ $=$ 20 ns). (c) The SW amplitude ($=$ FFT amplitudes of $M_z$) as a function of the position from the SW source ($x=0$) without a TW ($f_H$ $=$ 5.25, 6.25, and 11.25 GHz).

Figure 3

Spin-wave amplitudes with and without a TW for two spin-wave frequencies (a) $f_H$ $=$ 6.25 GHz and (b) $f_H$ $=$ 11.25 GHz. These frequencies correspond to the positive peaks in Fig. 2b.

Figure 4

(a) The SW amplitudes as a function of time ($t$ $=$ 5.0–5.5 ns with $f_H$ $=$ 6.0 GHz, $B_0$ $=$ 200 mT, and $B_{\mathrm{ext}}$ $=$ 0.5 mT) at the TW position ($x$ $=$ 500 nm and $x$ $=$ 700 nm) (black line, the SW amplitude without a TW; red line, the SW amplitude with a TW). (b) The FFT spectra of the SW amplitude for the case of (a) at the TW position ($x$ $=$ 500 nm). (c) The SW amplitudes as a function of time at $x$ $=$ 500 nm ($t$ $=$ 5.0–10.0 ns with $f_H$ $=$ 11.0 GHz, $B_0$ $=$ 200 mT, and $B_{\mathrm{ext}}$ $=$ 0.8 mT). (d) The FFT spectrum of the SW amplitude for the case of (c) at the TW position. Inset shows the snapshots of the $y$-component magnetization for two different simulation times.

Figure 5

(a and c) $\Delta B_{\mathrm{dep}}$ calculations with the external fields along the $\pm x$ direction ($B_0$ $=$ 150 and 200 mT). (a) $\Delta B_{\mathrm{dep}}$ calculations with the external fields along the $+x$ direction [$\Delta B_{\mathrm{dep}}$ $\equiv$ 1.7 mT $-B_{\mathrm{dep}}\left(f_H\right)$]. (b) The normalized $M_x/M_S$ curves with various external fields (up, $f_H$ $=$ 6.0 GHz; down, $f_H$ $=$ 11.0 GHz). $B_0$ $=$ 150 mT is used (blue line, $f_H$ $=$ 6.0 GHz and $B_{\mathrm{dep}}$ $=$ 1.0 mT; red line, $f_H$ $=$ 11.0 GHz and $B_{\mathrm{dep}}$ $=$ 1.3 mT). (c) $\Delta B_{\mathrm{dep}}$ calculations with the external fields along the $-x$ direction [$\Delta B_{\mathrm{dep}}\equiv -$ 1.7 mT $-B_{\mathrm{dep}}\left(f_H\right)$]. (d) The normalized $M_x/M_S$ curves with various external fields (up, $f_H$ $=$ 8.5 GHz; down, $f_H$ $=$ 11.0 GHz). $B_0$ $=$ 200 mT is used.