Spin wave emission in field-driven domain wall motion

A domain wall (DW) in a nanowire can propagate under a longitudinal magnetic field by emitting spin waves (SWs). We numerically investigated the properties of SWs emitted by the DW motion, such as frequency and wave number, and their relation with the DW motion. For a wire with a low transverse anisotropy and in a field above a critical value, a DW emits SWs to both sides (bow and stern), while it oscillates and propagates at a low average speed. For a wire with a high transverse anisotropy and in a weak field, the DW emits mostly stern waves, while the DW distorts itself and the DW center propagates forward like a drill at a relative high speed.

Figure 1

(a) Schematic diagram of a 1D head-to-head DW (top-left inset) and the snapshot of $\mathbf{s}$ components at $t=200$ for $\alpha =0,J=53.7,D_z=0.317,D_z/D_x=10$, and $H=0.5$. Lower-left inset: The $\mathbf{s}$ profile of DW. Symbols are numerical results, the vertical dashed line indicates the DW center position at $z=0.9$, and solid lines are the Walker profile (4) with $\varphi =Ht$ and $Z=0.9$. Right inset: The simulated motion evolution of the DW center (black curve), the time dependence of the DW center by the collective coordinate model (red curve), and their difference (blue curve). (b) The field dependence of average DW speed $v$ for different $D_z$ and $D_x$. Vertical dashed and solid lines correspond to critical fields $H_{c1}$ and $H_{c2}$, respectively. Inset: Simulated DW center for fields below and above $H_{c1}$ (indicated by arrows in the main figure).

Figure 2

(a) SW frequency $\omega$ as a function of applied field $H$. $\omega$ of the left domain is denoted by squares and that of the right domain by circles. Filled (open) symbols are the results for $D_z=0.317$ and $D_x=0.0317$ $(D_z=0.37$ and $D_x=0.0158)$. (b) SW wave number $k$ as a function of $H$. Upper (lower) part is for the left (right) domain. Solid curves are analytical results of Eq. (7) with ${\omega }_{+}=3H$ (upper) and ${\omega }_{-}=H$ (lower). Vertical lines indicate critical field $H_{c2}$. (c) Symbols are $D_x$ dependence of average DW speed for $D_z=0.317$ and various $H$. Solid curves are parabolic fits to the data.

Figure 3

Snapshot of $\mathbf{s}$ components at $t=200$ for $J=53.7,D_z=0.317,D_z/D_x=10,H=0.5$, and $\alpha =0.001$. Left inset: $\mathbf{s}$ profile of DW. Symbols are numerical results. Vertical dashed line indicates the DW center position at $z=2$, and solid curves are the Walker profile (4) with $\varphi =Ht$ and $Z=2$. Right inset: $Z\left(t\right)$ vs $t$ from simulations (black curve) and from solution of the collective coordinate model [Eq. (5)] (red curve). The difference (blue curve) shows the extra DW propagation due to SW emission.

Figure 4

(a) Snapshot of three spin components at $t=200$ for $J=2,D_x=2,\alpha =0,H=0.02$, and $D_z=0.2$ (left panel) or $D_z=0.02$ (right panel). Inset: Time dependence of the DW center position for $D_z=0.2$ (black line) or $D_z=0.02$ (red line). (b) DW speed $v$ (left axis) and SW wave number $k$ in left domain (right axis) as a function of transverse anisotropy $D_x$. The dashed lines indicate the smallest $D_x$ for drilling motion. Inset: $D_z$ dependence of $D_z+D_{xc}$ for $H=0.02$ (red circles) and $H=0.01$ (black squares). The unit is ${\mu }_0{M}_s^2{a}^3$. (c) $k$ (upper panel) and $v$ (lower panel) as a function of applied field $H$ for $D_z=0.2$. The dashed lines indicate the largest speed from Eq. (5).

Figure 5

Snapshot of spatial distributions of spin components at $t=200$ with $J=2,D_z=0.2,D_x=2,\alpha =0.001$, and $H=0.05$. Inset (1): The time dependence of the DW center position. Inset (2): Enlarged figure near the DW center.