Dynamics of a vortex domain wall in a magnetic nanostrip: Application of the collective-coordinate approach

The motion of a vortex domain wall in a ferromagnetic strip of submicron width under the influence of an external magnetic field exhibits three distinct dynamical regimes. In a viscous regime at low fields the wall moves rigidly with a velocity proportional to the field. Above a critical field the viscous motion breaks down, giving way to oscillations accompanied by a slow drift of the wall. At still higher fields the drift velocity starts rising with the field again but with a much lower mobility $dv/dH$ than in the viscous regime. To describe the dynamics of the wall, we use the method of collective coordinates that focuses on soft modes of the system. By retaining two soft modes, parametrized by the coordinates of the vortex core, we obtain a simple description of the wall dynamics at low and intermediate applied fields that applies to both the viscous and oscillatory regimes below and above the breakdown. The calculated dynamics agrees well with micromagnetic simulations at low and intermediate values of the driving field. In higher fields, additional modes become soft and the two-mode approximation is no longer sufficient. We explain some of the significant features of vortex-domain-wall motion in high fields through the inclusion of additional modes associated with the half antivortices on the strip edge.

Figure 1

(Color online) A head-to-head vortex domain wall in a long strip of Permalloy 200 nm wide. The thickness in the out-of-plane direction is 20 nm. Simulation using OOMMF (Ref. 9). The winding numbers of the three topological defects are labeled. The shaded region indicates out-of-plane magnetization within the vortex core.

Figure 2

A very simple (four-Néel-wall) model of the vortex domain wall (Ref. 29). The vortex core is denoted by the filled circle. Four Néel walls separate regions of uniform magnetization. Shaded areas indicate a buildup of magnetic charge. The panels show states with a fixed longitudinal coordinate of the vortex, $X=\text{const}$. The transverse coordinates are $Y=w/2$, $w/4$, 0, $-w/4$, and $-w/2$.

Figure 3

(Color online) A sketch of the domain-wall drift velocity $V\left(H\right)$. Lower panels show the vortex motion trajectory (solid lines) and equipotential lines (dotted) at different field magnitudes. (A) below the Walker breakdown; (B) just above the Walker breakdown; (C) small dissipation regime; (D) small dissipation regime with dominating Zeeman force.

Figure 4

(Color online) The drift velocity $V_d$ of the domain wall as a function of the applied field $H$ for a Permalloy strip of width $w=200\text{nm}$ and thickness $t=20\text{nm}$. Dashed vertical lines mark the critical fields $H_{c-}$ and $H_{c+}$.

Figure 5

(Color online) The time constant ${\tau }_1$ is determined by fitting the numerically calculated time dependence of the $Y$ coordinate to a decaying exponential; see Eq. (27).

Figure 6

(Color online) Numerical simulations above the critical field show repeated oscillations in the width of the wall, measured as the distance between the half antivortices. The patterns for various field values are offset vertically by 150 nm for clarity. Each wall width oscillates near $w=200\text{nm}$. The vortex crossing time $T$ and the decay time of wall-width oscillations ${\tau }_2$ are shown by arrows.

Figure 7

(Color online) Top row: The left-hand side of Eq. (37) for the two-coordinate model plotted for the first three passes of the vortex across the strip under an applied field of 32 Oe. Bottom row: When a finite mass $m=1.1\times {10}^{-23}\text{kg}$ is included for the edge defects, as in Eq. (47), the total $x$ momentum of the modes is conserved.

Figure 8

(Color online) Image of a wall moving to the right with an applied field of 4 Oe. Shaded regions indicate out-of-plane magnetization.

Figure 9

(Color online) Top: Numerically determined $x$ positions of the vortex and the two half antivortices in a 20-nm-thick, 200-nm-wide Permalloy strip as functions of time for $H=32\text{Oe}$. Bottom: The longitudinal positions of the vortex and the two half antivortices in the model with massive edge defects. The vortex merges with an edge defect each time it hits the edge. Note that the oscillation of that edge defect is suppressed compared to the other.

Figure 10

(Color online) Oscillations in the width of the wall in a strip with thickness $t=20\text{nm}$ and width $w=200\text{nm}$ in a field of $H=32\text{Oe}$. The upper curve shows the prediction of the model using the same parameters as in Fig. 9. It is offset vertically by 100 nm from the numerically determined result for clarity.

Figure 11

Three-Néel-wall model of the vortex domain wall proposed in Ref. 11. Solid lines denote straight Néel-wall regions and dashed lines denote regions in which the Néel walls are parabolic.

Figure 12

(Color online) Value of the scaling function $f$ for various values of the aspect ratio $w/t$.