Domain Wall Tilting in the Presence of the Dzyaloshinskii-Moriya Interaction in Out-of-Plane Magnetized Magnetic Nanotracks

We show that the Dzyaloshinskii-Moriya interaction (DMI) can lead to a tilting of the domain wall (DW) surface in perpendicularly magnetized magnetic nanotracks when DW dynamics are driven by an easy axis magnetic field or a spin polarized current. The DW tilting affects the DW dynamics for large DMI, and the tilting relaxation time can be very large as it scales with the square of the track width. The results are well explained by an extended collective coordinate model where DMI and DW tilting are included. We propose a simple way to estimate the DMI in magnetic multilayers by measuring the dependence of the DW tilt angle on a transverse static magnetic field. These results shed light on the current induced DW tilting observed recently in $\mathrm{Co}/\mathrm{Ni}$ multilayers with structural inversion asymmetry.

Figure 1

(a) Schematic of the tilted DW. (b) Micromagnetic configuration of a 100-nm-wide track with $D=2\mathrm{mJ}/{\mathrm{m}}^2$ and a transverse magnetic field ${\mu }_0{H}_y=100\mathrm{mT}$. (c) DW tilt angle as a function of ${\mu }_0{H}_y$ for several values of $D$ and (d) as a function of $D$ for ${\mu }_0{H}_y=100\mathrm{mT}$. Dots are the results of micromagnetic simulations, whereas the continuous lines are the results of the collective coordinates model described later.

Figure 2

Dynamics of the DW driven by an external magnetic field $H_z$ for a 100-nm-wide nanotrack. (a) Magnetization pattern of the DW for different values of $H_z$ with $D=2\mathrm{mJ}/{\mathrm{m}}^2$. Tilt angle (b) and velocity (d) of the DW as a function of $H_z$ for different values of $D$. The inset in (d) shows the DW velocity $v_n$ in the direction perpendicular to the DW surface ($v_n=v\cos \chi$). (c) Time dependence of the tilt angle for ${\mu }_0{H}_z=100\mathrm{mT}$ applied at $t=0$ and different track widths $w$ for $D=2\mathrm{mJ}/{\mathrm{m}}^2$. In (b)–(d), the results of the micromagnetic simulation (resp. CCM) are plotted in colored dots (resp. continuous lines). The dashed (resp. continuous) black line in (d) (resp. (d), inset) corresponds to the prediction of the standard (q, $\psi$) model for $D=2\mathrm{mJ}/{\mathrm{m}}^2$.

Figure 3

Dynamics of the DW driven by the spin-orbit torque [${\mu }_0{H}_{\mathrm{SO}}=0.1\mathrm{T}/({10}^{12}\mathrm{A}/{\mathrm{m}}^2)$] for a 100-nm-wide nanotrack. The results of the micromagnetic simulation (CCM) are plotted in colored dots (continuous lines). (a) Magnetization pattern of the DW for $D=2\mathrm{mJ}/{\mathrm{m}}^2$ and different values of $J$. The white arrow indicates the current direction. (b),(d) Tilt angle (b) and velocity (d) of the DW as a function of $J$ for different values of $D$. The inset in (d) shows the DW velocity in the direction perpendicular to the DW surface $v_n$ for $D=2\mathrm{mJ}/{\mathrm{m}}^2$. The continuous black line is the result of the standard ($q$, $\psi$) model. (c) Time dependence of the DW tilt angle for different track widths $w$ for a current of density $0.25\times {10}^{12}\mathrm{A}/{\mathrm{m}}^2$ applied at $t=0$ ($D=2\mathrm{mJ}/{\mathrm{m}}^2$).