Walker solution for Dzyaloshinskii domain wall in ultrathin ferromagnetic films

We analyze the electric current and magnetic field driven domain-wall motion in perpendicularly magnetized ultrathin ferromagnetic films in the presence of interfacial Dzyaloshinskii-Moriya interaction and both out-of-plane and in-plane uniaxial anisotropies. We obtain exact analytical Walker-type solutions in the form of one-dimensional domain walls moving with constant velocity due to both spin-transfer torques and out-of-plane magnetic field. These solutions are embedded into a larger family of propagating solutions found numerically. Within the considered model, we find the dependencies of the domain-wall velocity on the material parameters and demonstrate that adding in-plane anisotropy may produce domain walls moving with velocities in excess of 500 m/s in realistic materials under moderate fields and currents.

Figure 1

A snapshot of tiltless DW driven by current $\mathbf{j}$ and out-of-plane magnetic field $\mathbf{h}$ in a ferromagnetic nanostrip with DMI and two anisotropies (larger out of plane and smaller in plane along the strip axis) from a simulation of Eq. (1).

Figure 2

The dimensionless DW velocity $V$ at zero current as a function of the propagation angle ${\varphi }_0$ obtained from the solution of Eq. (1) for $k_z=1.674$, $k_x=0.167$, $\kappa =0.366$, $\alpha =0.5$, and $h_z=-0.0183$ corresponding to the parameters in the text. The simulated data are indicated by the blue dots connected with a dashed blue line. The red solid line shows the dependence given by Eq. (10) with $\mathbf{j}=0$. The green dots indicate the velocity for the Walker solution from Eq. (17), corresponding to the special values of ${\varphi }_0$ obtained from Eq. (16) and indicated by dotted lines.