Systematic motion of magnetic domain walls in notched nanowires under ultrashort current pulses

The precise manipulation of transverse magnetic domain walls in finite/infinite nanowires with artificial defects under the influence of very short spin-polarized current pulses is investigated. We show that for a classical $3d$ ferromagnet material like nickel, the exact positioning of the domain walls at room temperature is possible only for pulses with very short rise and fall time that move the domain wall reliably to nearest neighboring pinning position. The influence of the shape of the current pulse and of the transient effects on the phase diagram current-pulse length are discussed. We show that large transient effects appear even when $\alpha =\beta$, below a critical value, due to the domain wall distortion caused by the current pulse shape and the presence of the notches. The transient effects can oppose or amplify the spin-transfer torque (STT), depending on the ratio $\beta /\alpha$. This enlarges the physical comprehension of the domain wall (DW) motion under STT and opens the route to the DW displacement in both directions with unipolar currents.

Figure 1

(a) Simulated structure: planar nanowire with ten symmetric double notches. The equilibrium position of a pinned DW is shown in the finite case. (b) Definition of current pulse with its temporal characteristics. Two successive current pulses with opposite polarity are shown. The normalized potential pinning energy for finite and infinite cases as determined by micromagnetic simulations (symbols) are shown in (c) and (d), respectively. The line is a fit as described in the text.

Figure 2

Phase diagram for a DW at $T=0$ K in a finite (a) and infinite (b) nanostrip in the parameter space stable-time–current amplitude with $t_r=t_f=5\mathrm{ps}$ and $t_z=10$ ns. The total time of the periodic pulses is 350 ns. The micromagnetic results (scattered symbols) are compared to the 1D model (colored regions). The diagrams show only a few bands due to the finite size of the wire or due to the number of notches used. The upper right region is due to the finite size of the nanowire or of the simulated window (infinite case).

Figure 3

Influence of the pulse shape on the phase diagram for a DW at $T=0$ K in an infinite nanostrip. The parameter space is the total time $\left(t_r+t_s+t_f\right)$ vs current amplitude. (a) Symmetric pulse with $t_r=t_f=5$ ps and $t_z=10$ ns; asymmetric pulse with (b) $t_r=5$ ps, $t_f=300$ ps, and $t_z=10$ ns and (c) $t_r=300$ ps, $t_f=5$ ps, and $t_z=10$ ns. The upper right region is due to the finite size of the simulated window.

Figure 4

Probability of DW motion in different bands for the finite strip (a) and the infinite strip (b)–(d) at $T=293$ K. (a), (b) A symmetric pulse is applied with the characteristics $t_r=t_f=5$ ps, after an initial and final $t_z$ of 10 ns. (c) An asymmetric pulse is applied with pulse shape $t_r=5$ ps, $t_f=300$ ps, after an initial and final $t_z$ of 10 ns. (d) An asymmetric pulse is applied with pulse shape $t_r=300$ ps, $t_f=5$ ps, after an initial and final $t_z$ of 10 ns. The damping parameter is taken as $\alpha =0.02$ and $\beta =2\alpha$. The band pockets, which appear in the panels (b) to (d) on the left, correspond to the band $-1$. The dotted lines are guide to the eyes and represent the band's edges.

Figure 5

Influence of the damping parameter $\alpha$ and nonadiabatic parameter $\beta$ on the phase diagram for an infinite strip at $T=0$ K: (a) $\alpha =0.05$ and $\beta =0$, (b) $\alpha =0.02$ and $\beta =0$, (c) $\alpha =\beta =0.05$, and (d) $\alpha =\beta =0.02$. A sequence of symmetric pulses with the same characteristics as in Fig. 2 are applied. The scattered symbols represent the micromagnetic results, while the colored regions are calculations with the 1D model. Trajectories in the phase space $\left(X,\psi \right)$ corresponding to the $-1$ band in panels (b) [(d)]: (e) $t_s=250$ ps, $j_e=3.4\mathrm{A}/\mu {\mathrm{m}}^2,\alpha =0.02,\beta =0$ and (f) $t_s=300$ ps, $j_e=3.6\mathrm{A}/\mu {\mathrm{m}}^2,\alpha =\beta =0.02$. The scattered symbols and the full line represent the micromagnetic and the 1D results, respectively. The dotted part of the full line indicates the applied pulse duration. (g) DW angle variation for $\alpha =0.02,\beta =0$ for several points in the panel (b). (h) Comparison of the DW angular variation for $\alpha =\beta =0.02$ and $t_s=300$ ps, $j_e=3.6\mathrm{A}/\mu {\mathrm{m}}^2$ for the infinite strip with notches [band $-1$ in panel (d)] and strip without notches.

Figure 6

Influence of the pulse raise time $t_r$ on the phase diagram for a DW at $T=0$ K in an infinite nanostrip for different parameters $\alpha$ and $\beta$. The parameter space is the raise time vs current amplitude. In all cases, $t_s=t_f=5$ ps and $t_z=10$ ns. (a) $\alpha =0.02$ and $\beta =0.04$, (b) $\alpha =\beta =0.02$, and (c) $\alpha =0.05$ and $\beta =0.1$. In (a) the micromagnetic results (scattered symbols) are compared to the 1D model (colored regions), while in (b) and (c) only 1D model results are shown. At high currents, the micromagnetic results give several unintended states [the empty scattered symbol region to the right of (a)].