Dynamic transformations of the internal structure of a moving domain wall in magnetic nanostripes

The magnetic field (or electric current) driven domain-wall motion in magnetic nanostripes is of considerable interest because it is essential to the performance of information-storage and logic devices. One of the current key problems is to understand the complex behaviors of oscillatory domain-wall motions under applied magnetic fields stronger than the so-called Walker field, beyond which the velocity of domain walls markedly drops. In a certain range just above the Walker field, the motions are not chaotic but rather periodic with different periodicities of dynamic transformations of a moving domain wall between the different types of its internal structure. In addition, three different periodicities of the dynamic transformations are calculated, which consist of different types of domain-wall structures that are transformed from one type to another. The transformation periods vary with the field strength and the nanostripe width. This phenomenon can be described by the dynamic motion of a limited number of magnetic topological solitons such as the vortex and antivortex confined in nanostripes. These results provide a considerably better understanding and details of the domain-wall motions in soft magnetic nanostripes.

Figure 1

(Color online) The rectangular-shaped nanostripes of thickness $h=10\mathrm{nm}$, length $l=6\mu \mathrm{m}$, variable widths $w$ as noted, and the coordinate system used. The local magnetization configurations at equilibrium for $w=140$ and $240\mathrm{nm}$ display typical head-to-head DWs: TW with the V-shaped polarization and VW with the downward core orientation placed at the stripe center. An applied magnetic field, $H$, is indicated by the blue arrow. The streamlines with small arrows on each DW represent the in-plane components of the local magnetization. The spot in the lower panel corresponds to the vortex core.

Figure 2

(Color online) DW displacement versus time curves for the dynamic DW motions along the long axis of nanostripes of $10\mathrm{nm}$ thickness, $6\mu \mathrm{m}$ length, but different widths under the applied static magnetic fields as noted above. Positions of the motions of the individual DW types were plotted by following the vortex or antivortex core positions where the exchange energy had the maximum at each time moment. All the $y$ axes are of the same scale, starting from the middle position of the long axis of the nanostripes and ending at the position $2\mu \mathrm{m}$ away from the center. All the $x$ axes are of the same scale, ranging from $t=0\text{to}35\mathrm{ns}$. The red straight line in each box is fitted to the data points within the range of $0–2\mu \mathrm{m}$, indicating each average velocity $\overline{v}$ for the given fields and widths. Simulations were also carried out for the $w$ and $H$ values marked by the red solid circles, and the results are shown in the bottom panel. The thick dotted red and solid black lines separate the linear (left region) and oscillatory (right region) motion regimes.

Figure 3

(Color online) Average velocities of DW motions versus $H$ for two cases of the nanostripe width $w=140\mathrm{nm}$ (solid symbols) and $w=240\mathrm{nm}$ (open symbols). The square, circle, and triangle symbols for each $w$ case, respectively, indicate region I, the steady motion of a single domain wall, region II, the oscillatory motion of different internal DW structures that are periodically transformed from one to another during the DW motion; and region III, the magnetization reversal via multivortex (multiantivortex) states. The insets show the representative snapshot images of the dynamic internal DW structures in the different characteristic regions for the case of $w=140\mathrm{nm}$.

Figure 4

(Color online) (a) The three different characteristic periods of the oscillatory $D$ vs $t$ curves. (b) Energy variations with time during the periodic DW transformations in the nanostripes. Individual curves indicate the total $E_{\mathrm{tot}}$, Zeeman $E_{\mathrm{Zeem}}$, exchange $E_{\mathrm{ex}}$, and dipolar $E_{\mathrm{dip}}$ energy terms, compared with the sum of $E_{\mathrm{Zeem}}$ and $E_{\mathrm{dip}}$. The vertical red lines separate the individual regions of the DW transformations.

Figure 5

(Color online) Plane-view images of the local in-plane magnetization $(M_x∕M_s)$ components for the individual internal DW structures in the nanostripes that transform from one type to another in a periodic manner, together with the trajectories of the TW linear motions, and the VW and AVW gyrotropic motions within the nanostripes. The trajectories were plotted by following the maximum exchange energy values at given times [marked in each region and each case indicated by the gray colors in Fig. 4a] for the three different cases of $(w,H)=(140\mathrm{nm},25\mathrm{Oe})$, $(240\mathrm{nm},20\mathrm{Oe})$, and $(240\mathrm{nm},30\mathrm{Oe})$. The solid circles and arrows on the TW types indicate the nucleation sites where the cores of VWs or AVWs are created and the directions of their initial movements, respectively. The arrows on the trajectories indicate the directions of the movement of each DW type. The open gray-colored circles indicate the starting positions of the movements of each DW.

Figure 6

(Color online) Half-integer and integer topological charges $\left(q\right)$ of each type of DWs in the nanostripes represented as magnetic solitons. The topological charges within each DW are indicated by the open and solid circles (semicircles), as noted. The sum of the topological charges of each DW is always zero. The solid-line triangles on the TW, VW, and AVW structures indicate schematically the local magnetization configurations of the head-to-head DWs.