Magnetic domain-wall dynamics in wide permalloy strips

Domain walls in soft permalloy strips may exhibit various equilibrium micromagnetic structures depending on the width and thickness of the strip, ranging from the well-known transverse and vortex walls in narrow and thin strips to double and triple vortex walls recently reported in wider strips [V. Estévez and L. Laurson, Phys. Rev. B 91, 054407 (2015)]. Here, we analyze the field driven dynamics of such domain walls in permalloy strips of widths from 240 nm up to $6 \mu \mathrm{m}$, using the known equilibrium domain wall structures as initial configurations. Our micromagnetic simulations show that the domain wall dynamics in wide strips is very complex, and depends strongly on the geometry of the system, as well as on the magnitude of the driving field. We discuss in detail the rich variety of the dynamical behaviors found, including dynamic transitions between different domain wall structures, periodic dynamics of a vortex core close to the strip edge, transitions towards simpler domain wall structures of the multi-vortex domain walls controlled by vortex polarity, and the fact that for some combinations of the strip geometry and the driving field the system cannot support a compact domain wall.

Figure 1

(a) Geometry of the permalloy strip. (b) A top view of the magnetization in the initial state. Magnetization points along the long axis of the strip within the two domains (as indicated by the arrows) forming a head-to-head configuration. The domain wall separating these domains has a geometry-depedendent equilibrium structure. Here, we consider systems where the latter is either a vortex wall (VW), double vortex wall (DVW), or a triple vortex wall (TVW). To study the DW dynamics, an external magnetic field $B_{\mathrm{ext}}$ is applied along the long axis of the strip.

Figure 2

$v\left(B_{\mathrm{ext}}\right)$ curves for different strip geometries, where the VW is the equilibrium DW structure. (a) For two strips, both having the same width of $w=240$ nm and in-plane cell size 5 nm, and different thicknesses ${\mathrm{\Delta }}_z=10$, 20 nm. (b) The same for a strip with $w=768$ nm, ${\mathrm{\Delta }}_z=15$ nm and in-plane cell size 3 nm. For 3 mT $\le B_{\mathrm{ext}}\le$ 3.1 mT the DW is unstable; the DW width grows without bound.

Figure 3

Vortex wall dynamics in a strip with $w=240$ nm and ${\mathrm{\Delta }}_z=10$ nm for $B_{\mathrm{ext}}=1.5$ mT (i.e., above the Walker field $B_{\text{W}}=1.2$ mT). The DW dynamics repeats a cycle where the vortex core first moves out of the strip through the top strip edge, forming an ATW, followed by the injection of a vortex core from the top edge with the opposite polarity, which subsequently moves across the strip to the bottom edge, reversing the ATW magnetization. Then a vortex, again with a polarity opposite to that of the previous one, is nucleated from the bottom edge, and moves to the top edge, again reversing the ATW magnetization. Then the same process repeats.

Figure 4

Vortex wall dynamics in a strip with $w=240$ nm and ${\mathrm{\Delta }}_z=10$ nm for $B_{\mathrm{ext}}=4$ mT (above the Walker breakdown, $B_{\text{W}}=1.2$ mT). Now the dynamics alternates between ATWs of different magnetization, with its reversal mediated by either a vortex or an antivortex core moving across the strip width.

Figure 5

The trajectory $y_{\text{c}}$ of the vortex core in the vicinity of the edge of the system for $w=768$ nm, ${\mathrm{\Delta }}_z=15$ nm, and $B_{\mathrm{ext}}=1.5$ mT, illustrating the attraction-repulsion effect. The red line at $y_c=$ 0 corresponds to the strip edge. The inset shows an example of the DW configuration with the vortex core close to the edge of the strip.

Figure 6

(a) Walker breakdown-type of dynamics for $w=768$ nm, ${\mathrm{\Delta }}_z=15$ nm, and $B_{\mathrm{ext}}=2.5$ mT. The vortex core exhibits asymmetric lateral oscillations, leaving the strip at the top edge, but reversing the direction of motion and polarity of the core well inside the strip when closer to the bottom edge. (b) The instability occurring for $B_{\mathrm{ext}}=3$ mT, where the DW width grows without limit as the leading HAV edge defect moves faster than the combination of the trailing HAV and the vortex core.

Figure 7

VW dynamics for $B_{\mathrm{ext}}=3.3$ mT in a strip with $w=768$ nm and ${\mathrm{\Delta }}_z=15$ nm. The trailing HAV edge defect repeatedly emits a vortex-antivortex pair, which then annihilates, and the process is repeated.

Figure 8

$v\left(B_{\mathrm{ext}}\right)$ curves for a VW and a DVW initial state in a strip with $w=1536$ nm and ${\mathrm{\Delta }}_z=15$ nm, where both structures are equilibrium DWs. In-plane cell size equal to 3 nm. The gray areas indicate unstable regions (no compact DWs).

Figure 9

DW dynamics starting from a VW initial state for $w=1536$ nm, ${\mathrm{\Delta }}_z=15$ nm and $B_{\mathrm{ext}}=2.1$ mT. The DW dynamics shows the nucleation of an antivortex into the strip from the top edge, followed by the injection of a vortex. The antivortex moves towards the vortex that was previously in the strip, leading to an annihilation process between them. After that, the DW structure reduces its width, resulting a VW. Then the process is repeated.

Figure 10

DW dynamics starting from a VW initial state for $w=1536$ nm, ${\mathrm{\Delta }}_z=15$ nm and $B_{\mathrm{ext}}=3.8$ mT. Very complex, possibly chaotic dynamics is observed.

Figure 11

VW velocity $v$ as a function of the width of the strip $w$ when the thickness and field are fixed to ${\mathrm{\Delta }}_z=15$ nm and $B_{\mathrm{ext}}=0.3$ mT, respectively. $w$ is in logarithmic scale. (Inset) VW velocity $v$ as a function of ${\mathrm{\Delta }}_z$ for $w=768$ nm and $B_{\mathrm{ext}}=0.3$ mT.

Figure 12

The equilibrium DW structures in terms of their constituent elementary topological defects, i.e., vortices with a winding number +1 (full circles) and edge defects with winding number $-1/2$ (empty circles). (a) Vortex wall, (b) double vortex wall, and (c) triple vortex wall.

Figure 13

DVW dynamics in a strip with $w=1536$ nm and ${\mathrm{\Delta }}_z=15$ nm for $B_{\mathrm{ext}}=0.3$ mT and different polarities of the vortices. (a) Parallel polarity; dashed lines help to see the transient relative motion of the vortex cores, before the steady state motion with a nonevolving DVW structure is reached. (b) Antiparallel polarity; DVW is not stable, and becomes a VW.

Figure 14

Top view sketch of the different polarity configurations for a TVW structure.

Figure 15

TVW dynamics in a strip with $w=5120$ nm and ${\mathrm{\Delta }}_z=25$ nm for $B_{\mathrm{ext}}=0.05$ mT and two different configurations of the polarity. (a) TVW structure is stable against this field. (b) The initial TVW structure transforms into a DVW, which then exhibits steady dynamics within the simulation time scale.

Figure 16

TVW dynamics in a strip with $w=5120$ nm and ${\mathrm{\Delta }}_z=25$ nm for $B_{\mathrm{ext}}=0.1$ mT and two different configurations of the polarity. (a) The initial TVW structure transforms first into a DVW, and then into a structure exhibiting the topology of a VW. (b) The initial TVW structure transforms into a DVW, which then exhibits steady dynamics within the simulation time scale.

Figure 17

Snapshots from a strip with $w=5120$ nm and ${\mathrm{\Delta }}_z=5$ nm with $B_{\mathrm{ext}}=0.9$ mT. For these conditions the system cannot support a compact DW, resulting in an instability where a complex magnetization pattern fills the strip.