Velocity Enhancement by Synchronization of Magnetic Domain Walls

Magnetic domain walls are objects whose dynamics is inseparably connected to their structure. In this Letter, we investigate magnetic bilayers, which are engineered such that a coupled pair of domain walls, one in each layer, is stabilized by a cooperation of Dzyaloshinskii-Moriya interaction and flux-closing mechanism. The dipolar field mediating the interaction between the two domain walls links not only their position but also their structure. We show that this link has a direct impact on their magnetic-field-induced dynamics. We demonstrate that in such a system the coupling leads to an increased domain wall velocity with respect to single domain walls. Since the domain wall dynamics is observed in a precessional regime, the dynamics involves the synchronization between the two walls to preserve the flux closure during motion. Properties of these coupled oscillating walls can be tuned by an additional in-plane magnetic field enabling a rich variety of states, from perfect synchronization to complete detuning.

Figure 1

Coupled domain walls dynamics. (a) Magnetic bilayer track containing two up-down chiral Néel walls coupled by a dipolar field. (b) When an out-of plane field $B_z$ is applied, the top and bottom DW magnetization angles (seen separately in the top view) tilt. DWs are described by magnetization angles ${\phi }_b$ and ${\phi }_t$ inside the bottom and top walls and by their relative position $\delta q$. (c) Kerr micrographs in the initial state [coupled DW (blue triangle)] and after application of seven 30-ns-long 150-mT field pulses (bottom), which result in decoupled DWs (green, top layer; red, bottom layer). Dark, light gray, and gray contrasts inside the stripe correspond to $\uparrow \uparrow$, $\uparrow \downarrow$, and $\downarrow \downarrow$ magnetization directions, respectively. (d) Single DW velocity at ${\mu }_0{H}_z=77\mathrm{mT}$ for an up-down DW, as a function of an in-plane field. (e),(f) DW velocity as a function of an out-of-plane field at ${\mu }_0{H}_x=0\mathrm{mT}$ (e) and $+12\mathrm{mT}$ (f). In (d)–(f), the color code is the same; full and empty symbols correspond to the experiments and 2D micromagnetic simulations, respectively. Insets in (e),(f) show the calculated curves by the minimal model [8], using the coupled DWs effective DMI constant (strong coupling approximation).

Figure 2

Simulations illustrating the DW oscillation synchronization. (a) Sketch of the magnetization angle in the center of the bottom (${\phi }_b$) and top (${\phi }_t$) DWs. In the absence of the in-plane field (top), the two oscillations are detuned, while the appropriate in-plane field (bottom) ensures their locking. (b) Time evolution of $m_y$ for a 1D wire containing a pair of up-down DWs at ${\mu }_0{H}_z=125\mathrm{mT}$, for ${\mu }_0{H}_x=0\mathrm{mT}$ (top) and $+12\mathrm{mT}$ (bottom). The coupled DW velocity is $98\mathrm{m}/\mathrm{s}$ [inset of Fig. 1]. Green and red curves represent the top and bottom DWs, respectively. The blue curves denote the relative distance between the walls. (c) Evolution of $m_y$ at ${\mu }_0{H}_z=125\mathrm{mT}$ for a 2D system in the middle of the stripe as indicated by the dashed line in Fig. 3. (d) Power spectral density of $m_y$ calculated from 1D model (b) for a 20-ns time window, as a function of the in-plane field for the bottom (d) and top (e) DWs. Vertical dashed lines show lower and upper bounds of individual (decoupled) DWs. Gray dashed and dotted lines represent the minimal model [8] results for the coupled (using $D_{\text{eff}}$) and single DWs (using $D_b$ and $D_t$), respectively. (f) Geometric mean of the values presented in (d) and (e), which underlines the synchronization regions.

Figure 3

Simulation of DW motion in a 2D disordered medium. (a) Calculated evolution of the average distance between bottom and top DWs at ${\mu }_0{H}_z=125\mathrm{mT}$ for ${\mu }_0{H}_x=+12$ (black), 0 mT (red), and $-5\mathrm{mT}$ (blue). The inset shows the resulting coupled DWs velocities as a function of the in-plane field. (b) Snapshots of the two DWs dynamics at 5 ns, corresponding to (a). The dark gray contrast corresponds to the $\uparrow \uparrow$ state, light gray to the $\uparrow \downarrow$ state, and white contrast to the $\downarrow \downarrow$ state. The $m_y$ component is reflected by the red ($m_y=1$) and blue ($m_y=-1$) color scheme. The size of the moving box centered on the top wall is $1\times 1\mu {\mathrm{m}}^2$.