Domain-wall skyrmions in chiral magnets

Domain-wall skyrmions are skyrmions trapped inside a domain wall. We investigate domain-wall skyrmions in chiral magnets using a fully analytic approach. Treating the Dzyaloshinskii-Moriya (DM) interaction perturbatively, we construct the low-energy effective theory of a magnetic domain wall in an $O\left(3\right)$ sigma model with the DM interaction and an easy-axis potential term, yielding a sine-Gordon model. We then construct domain-wall skyrmions as sine-Gordon solitons along the domain wall. We also construct domain-wall skyrmions on top of a pair of a domain wall and an anti-domain wall. One characteristic feature of domain-wall skyrmions is that both skyrmions and antiskyrmions are equally stable inside the domain wall, unlike the bulk in which only one of them is stable.

Figure 1

The energy density and magnetization plots for a domain wall with a kink on it. For simplicity we have chosen $\tilde{\kappa }=\frac{1}{5}, \alpha =\theta =X=c=0$, to give a kink centered at zero on the domain wall.

Figure 2

Schematic figures of superpositions of skyrmions and domain walls to demonstrate the relative orientation; these are plotted for $\theta =\alpha =0$. The upper figure is an anti-domain-wall skyrmion superposition while the lower figure is a skyrmion domain-wall superposition. In these schematics the cross signifies a downward-pointing magnetization $n_3=-1$ while the circle with a dot in the middle signals an upward-pointing arrow $n_3=1$, e.g., one that is pointing out of the page. There are two further pictures that could be made where the skyrmion is on the opposite side of the domain wall; these also showcase an attractive interaction between the skyrmion and the domain wall.

Figure 3

Left: Energy density of a spiral configuration with $\tilde{\kappa }=\frac{1}{5}$. Right: A magnetization plot of the $k=\frac{99998}{100000}$ spiral on a domain wall. Comparing the magnetization plot in Fig. 1 we see that the spiral involves multiple kinks along the domain wall.

Figure 4

The energy for the superposition against the domain-wall separation for the case of $m=1$.

Figure 5

The energy density and magnetization plots for the domain-wall–anti-domain-wall superposition of Eq. (28) with $X_1=0, X_2=10, {\phi }_1$ (${\phi }_2$) is a kink (antikink) centered at $x_2=\pm 2$.