Magnetic soliton rectifier via phase synchronization

Most of the existing research on the dynamics of magnetic solitons such as a domain wall (DW) has focused on the effect of DC forces, where the induced velocity is determined by the force strength. Here we show that AC forces such as an oscillating magnetic field or current are also able to move a DW straight via synchronization between the DW angle and the phase of the AC force. The resulting DW velocity is solely proportional to the driving frequency of the AC force, but the strength of the AC field just affects the frequency range for criteria for the phase-locking behavior. The AC-force-driven DW motion is shown to exhibit a phase locking-unlocking transition, a critical phenomenon akin to the Walker breakdown of a DC-bias-driven DW motion. Our work shows that a DW can be driven straight by synchronizing its angle to AC forces and thereby demonstrates a proof-of-concept of magnetic soliton rectifiers (i.e., DC motion induced by AC forces), shedding a light on the hitherto overlooked utility of internal degree of freedom for driving magnetic textures.

Figure 1

Domain wall (DW) subjected to a DC and an AC fields. (a) Schematic for a DW motion in a ferromagnet with perpendicular magnetic anisotropy subjected to a DC field $H_{\mathrm{z}}$ (along the $z$ axis) and an AC field $H_{\mathrm{y}}cos\left(\omega t\right)$ (along the $x$ axis). Blue area, magnetization-up state (circled dot); red area, magnetization-down state (crossed circle). (b) Schematic for an angular motion of the magnetization of the DW and an AC field rotating at the frequencies $f_{\mathrm{M}}$ and $f$, respectively. (c)–(e) Time dependence of the magnetizations ($M_x,M_y,M_z$) along the ferromagnetic strip for the DC field $H_z=-3$ mT and the AC field $H_y=40$ mT with the AC-field frequency $f=200$ MHz.

Figure 2

DW angular and linear motion in the phase-locked and the phase-unlocked regimes. Panels (a) and (b) show the DW velocity $V$ as a function of the frequency $f$ of the AC field in the presence of the DC field, $H_{\mathrm{z}}=\pm 3$ mT and $H_{\mathrm{y}}=40$ mT. The symbols represent the results of the micromagnetic simulations and the purple solid line indicates the analytic result based on Eq. (4). The phase-locking regime (red area) and phase-unlocking regime (blue area) are shown in (b). Panels (c)–(e) represent the time dependence of the magnetization of $M_y$ in the phase-locked regime (for $f=100$ and 200 MHz) and unlocked regime (for $f=500$ MHz), respectively.

Figure 3

DW velocity induced by the resonance between the DW angle and the AC field. Panels (a) and (b) show the DW velocity $V$ as a function of the frequency $f$ of the AC field in the presence of the DC field, $H_{\mathrm{z}}=\pm 1$ and 3 mT, respectively. The symbols and the lines are from micromagnetic simulations and analytical solutions [Eq. (6)], respectively. (c) The upper critical frequencies $f_{\mathrm{c}}^u\left(\equiv {\omega }_{\mathrm{c}}^u/2\pi \right)$ for the phase locking-to-unlocking transition as a function of the AC-field magnitude. The symbols and the lines are from micromagnetic simulations and analytical solutions [Eq. (8)], respectively.

Figure 4

DW velocity induced by the resonance between the DW angle and the AC current. (a) Schematics for the current-induced DW motion, where a ferromagnet is subjected to STT by a DC current density $J_{\mathrm{DC}}$ through the magnetic layer and SOT by an AC current density $J_{\mathrm{AC}}cos\left(\omega t\right)$ through a heavy-metal layer. (b) The DW velocity $V$ as a function of the frequency $f$ of the AC current. The symbols and the lines are from micromagnetic simulations and analytical solutions [Eq. (A4)], respectively. (c) The upper and the lower critical frequencies $f_{\mathrm{c}}^{u,l}$ for the phase locking-unlocking transition. The symbols and the lines are from micromagnetic simulations and analytical solutions [Eq. (8)], respectively.